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VOLUME62
I ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
C*-Algebras Volume 5: Selected Topics
Comeliu Constantinescu Departement Mathematik, ETH Zurich CH-8092 ZLirich Switzerland
I
.
2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
ELSEVIER SCIENCE B.V. Sara .Burgerharlstraat 25 P.O. .Box 211 , 1000 AB Amsterdam, The Netherlands
• 200 I Elsevier Science B. V. AU rights reserved. This worlc is protected under copyright by Elsevier Science, and the followina tctms and eonditions apply 10 its liSe Photocopying Single photocopies of single chopws may be made for personal use as allowed by national copyright laws. Pcnnisslon of tbe Publisher and payment of a (.., is r<:quircd for all other photocopying, including multiple or systematic oopying, copying for advertising or prornotlonaJ pur:poscs, resale:, and aU f"""" of doc:umcnt delivay. Special rates OR available for ediiC3tional instirutions that wish to malce photocopies for non·profit educational classroom usc. Pcrmisoions may be sought directly fiom Elsevier Scicnc:c Olobal Rights llcpoJ1mcm, PO Box 800, Oxford OX5 I OX, UK; phone: (+44) I 865 843lt30, fax: (+44) 1865 853333, e-mail:
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First edition 200 I
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ISBN (ibis volume): IS.BN (5 volume set):
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Preface Functional analysis plays an important role in the program of studies at the Swiss Federal Institute of Technology. At present, courses entitled Functional Analysis I and II are taken during the fifth and sixth semester, respectively. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88, I was fortunate enough to have an eager enthusiastic group of students that I had already encountered previously in other lecture courses. These students wanted to learn more in the area and asked me to design a continuation of the courses. Accordingly, I proceeded during the academic year, following, with a series of special lectures, Functional Analysis III and IV, for which I again distributed typewritten lecture notes. At the end I found that there had accumulated a mass of textual material, and I asked myself if I should not publish it in the form of a book. Unfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contained material that should have been included in the first two portions. And so I came to the conclusion that I should write everything a n e w - and if at a l l - then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost imperceptibly and at the end it had more than doubled. Also, the English language turned out to be a stumbling block for me; I would like to take this opportunity to thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aquilino, who wrote, first a WordMARC TM, and then a I ~ F ~ TM version with great competence, angelic patience, and utter devotion, in spite of illness. My thanks also go to the Swiss Federal Institute of Technology t h a t generously provided the infrastructure for this extensive enterprise and to my colleagues who showed their understanding for it.
Corneliu Constantinescu
XVll
Table of Contents of Volume 5
xix
Introduction 6
Selected Chapters of c·- Algebras . . . . . . . . . . . . . . . . . 6.1 £''- Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 1.1 6.1.2 6.1.3 6.1.4. 6.1.5
6.2
Characteristic Families of Eigenvalues . . . . . . . . . C haracteristic Sequences .. Properties of the .C"-spaces Hilbert-&:hmidt Operators . The Trace . . . . . . . . . .
6.1.6 Duals of .C"-spaces . . . . . 6.1.7 Exterior Multiplication and cP- Spaces 6.1.8 The Canonical Projection of the Tridual of JC 6.1.9 Integral Operators on Hilbert Spaces .. Selfadjoint Linear Differential Equations . . . 6.2.1 Selfadjoint Boundary Value Problems . . 6.2.2 The Regular Sturm- Liouville Theory .. 6.2.3 Selfadjoint Linear Differential Equations on T 6.2.4
3 3 3 10 21
46 56 72
79 102 116 124 125 139 150
Associated Parabolic and Hyperbolic Evolution Equations 153
6.2.5 Selfadjoint Linear Partial Differential Equations . . . . . 184 6.3
6.2.6 Associated Parabolic and Hyperbolic Evolution Equations 192 Von Neumann Algebras . . . . 202 6.3.1 The Strong Topology . . . . . . . . . 203 218 6.3.2 Bidual of a c ·-algebra . . . . . . . . 6.3.3 Extension of the FUnctional Calculus 263 6.3.4 Yon Neumann-Algebras . . 283 6.3.5 6.3.6 6.3.7 6.3.8 6.3.9
The Commutants . . . . . . . . . . . Irreducible Representations . . . . . Commutative von Neumann Algebras . Representations of w·- Algebr as Finite-dimensional c •- aJgebras . . . .
293 299 320 325
334
xvm
Table of Contents
6.3.10 A generalization . . . . .
7
C'- algebrM Generated by Groups . 7.1 Projective Representations of Groups 7.1.1 Schur functions .. .. .. . 7.1.2 Projective Representations . 7.1.3 Supplementary Results . 7.1.4 Examples . . .. .. .. . . 7.2 Clifford Algebras . .. . . . . . . . 7.2.1 General Clifford Algebras . . 7.2.2 Clp,q . 7.2.3 Cf(N)
355 369 369 369
404 431
466
492 492
518 538
Name Index
559
Subject Index.
563
Symbol lndcx
592
Copyngh!cd malcria
xix
Introduction This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of "Definiton- T h e o r e m - P r o o f - E x a m p l e - Exercises". The proofs are rather thorough and there are many examples. The first part of the book (the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations - applications of the theory of Banach algebras. The second part of the book (the last four chapters, resp. the last three volumes) is devoted to the theory of Hilbert spaces, once again in the general sense of the term. It begins with a chapter (Chapter 4, resp. Volume 3) on the theory of C*-algebras and W*-algebras which are essentially the focus of the book. Chapter 5 (resp. Volume 4) treats Hilbert spaces for which we had no need earlier. It contains the representation theorems, i.e. the theorems on isometries between abstract C*-algebras and the concrete C*-algebras of operators on Hilbert spaces. Chapter 6 (which together with Chapter 7 makes Volume 5) presents the theory of s of operators, its application to the self-adjoint linear (ordinary and partial) differential equations, and the von Neumann algebras. Finally, Chapter 7 presents examples of C*-algebras defined with the aid of groups, in particular the Clifford algebras. Many important domains of C*-algebras are ignored in the present book. It should be emphasized that the whole theory is constructed in parallel for the real and for the complex numbers, i.e. the C*-algebras are real or complex. In addition to the above (vertical) structure of the book, there is also a second (horizontal) division. It consists of a main strand, eight branches, and additional material. The results belonging to the main strand are marked with (0). Logically speaking, a reader could restrict himself/herself to these and ignore the rest. Results on the eight subsidiary branches are marked with (1), (2), (3), (4), (5), (6), (7), and (8). The key is
xx
1. Infinite Matrices 2. Banach Categories 3. Nuclear Maps 4. Locally Compact Groups 5. Differential Equations 6. Laurent Series 7. Clifford Algebras 8. Hilbert C*-Modules These are (logically) independent of each other, but all depend on the main strand. Finally, the results which belong to the additional material have no marking and - from a logical perspective - may be ignored. So the reader can shorten for himself/herself this very long book using the above marks. Also, since the proofs are given with almost all references, it is possible to get into the book at any level and not to read it linearly. We assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point-set topology, and integration theory. The book addresses itself mainly to mathematicians, or to physicists interested in C*-algebras. I would like to apologize for any omissions in citations occasioned by the fact that my acquaintance with the history of functional analysis is, unfortunately, very restricted. For this history we recommend the following texts. BIRKHOFF, G. and KREYSZIG, E., The Establishment of Functional Analysis, Historia Mathematica 11 (1984), 258-321. 2. BOURBAKI, N., Elements of the History of Mathematics, (21. Topological Vector Spaces), Springer-Verlag (1994). 3. DIEUDONNI~, J., History of Functional Analysis, North-Holland (1981). 4. DIEUDONN]~, J., A Panorama of Pure Mathematics (Chapter C III: Spectral Theory of Operators), Academic Press (1982). HEUSER, H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die werdende Functionalanalysis), Teubner (1986), 3. Auflage (1992). KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Symposia in Pure Mathematics 38 I (1982), 1-18. MONNA, A.F., Functional Analysis in Historical Perspective, John Whiley& Sons (1973).
xxi
8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359-382. There is no shortage of excellent books on C*-algebras. Nevertheless, we hope that this book will be also of some utility to the mathematics community.
VOLUMES: SELECTED TOPICS
6. Selected Chapters of C*-Algebras
6.1 t : P - S p a c e s The theory of compact operators may be taken substantially further in the case of Hilbert spaces. The s
(p e {0} U [1, co[) consist of classes of such
operators. These spaces bear a particular relationship to the Banach spaces ep , and their theory is a special case of what is sometimes called "non-commutative integration theory". The spaces s s and /:2 are particularly important. They comprise the sets of compact, nuclear, and Hilbert-Schmidt operators, respectively. /:2 is itself a Hilbert space and may be identified with the Hilbert space of square summable kernels in the case of measure spaces. This allows us to view such operators as integral operators. /:1 can be identified with the dual of s
and the dual of s
can be identified with the Banach space /:
of all operators. In particular, the algebra of operators on a Hilbert space is a W*-algebra. This provides information on the dual of E. We present some examples of differential equations as an application of this theory.
E, F, G, H denote Hilbert spaces throughout this section.
6.1.1 C h a r a c t e r i s t i c F a m i l i e s of E i g e n v a l u e s D e f i n i t i o n 6.1.1.1
( 0 ) Let u be a compact operator on E . A characte-
ristic f a m i l y of eigenvalues o f u is a family (o~,.),~i in IK\{O} such that for every ~ E IK\{O} Card{L C I I ~ = c~} agrees with the algebraic multiplicity of c~ with respect to u. If (a~),ei is a characteristic family of eigenvalues of u, then
6. Selected Chapters of C*-Algebras
{a~l~ E I} = ap(u)\{0} and
(OQ)rEI E Co(I) (Theorem 3.1.5.1 a),b),c)). P r o p o s i t i o n 6.1.1.2
( 0 ) Let u be a self-normal compact operator on E .
Take (a,)~el in ]K\{0}. Then the following are equivalent:
a)
( ~ ) t e , is a characteristic family of eigenvalues of u.
b)
There is an ortonormal family (x~),el in E such that
tEI
a =~ b. Given a E IK, define I~ := {~ 9 I l a , = a} and let 7r~ denote the orthogonal projection of E onto Ker ( a l - u). Take a E ]K . By Proposition 5.3.3.21, Ker ( a l - u) is the algebraic eigenspace of u corresponding to a , so that Dim K e r ( a l - u) = Card{~ E I l a ,
= a} = Card I~.
Let (x~)~eIo be an orthonormal basis of K e r ( a l - u). Since Ker ( a l - u) A_ Ker (~1 - u) for distinct a, r
IK (Corollary 5.3.4.5), the family (x~)~ei is orthonormal. By
Proposition 5.3.4.7 a) (and Theorem 3.1.5.1 b)), tt-- ~ O/71"a . aEIK
Hence u vanishes on {x~ ] t E I} • and ux~ = a~x~ for every t E I . By Theorem 5.5.5.4 (a & jl) =:~ j3,
tEI
b :=v a. Take a E IK\{0}. Since
6.1 s
ZtX t :
OQX t
for every ~ 9 I , we have that Ker (c~l - u) = {x~lc~~ = ce}•177 by Proposition 5.5.5.7 b). By Corollary 5.2.3.9, {x, I c~, = c~}•177is the closed vector subspace of E generated by {x, I c~, - c ~ } . Thus DimKer(c~l - u) = Card{~ 9 I l a ~ = a } . By Proposition 5.3.3.21, Ker (c~1 - u) is the algebraic eigenspace of u corresponding to a , so that Card{c 9 I I c~, = c~} is the algebraic multiplicity of c~ with respect to u. I P r o p o s i t i o n 6.1.1.3
( 0 )
Let u be a self-normal compact operator on
E , (a,),ez a characteristic family of eigenvalues of u , f C C(a(u)) such that f (O) = 0 whenever 0 9 a ( u ) , and put
J := {~ e / I f ( c ) r 0}. Then (f(~,)),ej is a characteristic family of eigenvalues of f ( u ) .
By Corollary 5.5.6.5, f ( u ) is compact. By Proposition 6.1.1.2 a :=> b, there is an orthonormal family (x~),~z in E such that
By Theorem 5.5.6.1 a =~ f,
LEI
Hence, by Proposition 6.1.1.2 b ~ a , eigenvalues of f ( u ) . C o r o l l a r y 6.1.1.4
(f(c~,)),cz is a characteristic family of I
Let u be a self-normal compact operator on a Hilbert
space, (ce~),~i a characteristic family of eigenvalues of u, and take n C IN. Then ( ~ ) ~ 1
is a characteristic family of eigenvalues of u n .
C o r o l l a r y 6.1.1.5
( 0 )
I
Let u be a selfadjoint compact operator on a
Hilbert space and (a~)~ei a characteristic family of eigenvalues of u . Then
(sup{c~, 0})~ei (resp. (sup{-c~, 0})~eI) is a characteristic family of eigenvalues of u + (resp. u - ) .
6. Selected Chapters of C*-Algebras
This follows from Proposition 6.1.1.3 and Theorem 4.2.2.9 b) (and Theorem 5.3.1.13). I C o r o l l a r y 6.1.1.6
Let u be a self-normal compact operator on a Hilbert
space and (c~)~I a characteristic family of eigenvalues of u. Then (]c~])~e, is a characteristic family of eigenvalues of lul.
This follows from Proposition 6.1.1.3 and Proposition 4.2.5.4 (and Theorem
5.3.1.13)). C o r o l l a r y 6.1.1.7
m Let u be a normal compact operator on a complex Hilbert
space and (a,)~e, a characteristic family of eigenvalues of u. Then (-5~)~, (rasp. (re a~),e,, rasp. (im a,),e,) is a characteristic family of u* (rasp. re u, rasp. im u).
The assertion follows from Proposition 6.1.1.3 and Corollary 4.1.3.8. P r o p o s i t i o n 6.1.1.8
I
Let Jr be a set of self-normal compact operators on E .
Then the following are equivalent:
a) jc is commutative. b)
There is an orthonormal basis A of E and an f E IK yxm with
xEA
.for every u E j z . If these conditions are fulfilled, then
xEA
for all u, v E jc.
a =v b. Given (u, a) E .T" x IK, let 7ru~ denote the orthogonal projection of E onto Ker ( a l -
u). By Proposition 5.3.4.7 g),
for every u E jc, and so {~1(~,~)
E 7 x ~:} c 7 ~ 9
6.1 s
7
Take u , v E 9v and c~ E ]K. Since ~cc is commutative (Proposition 2.1.1.17
e)), V7ru ~ --
i.e. K e r ( a l -
7ruoLV
u ) r e d u c e s v (Corollary 5.2.3.12 b)).
Given g E IK7 , define Fg := M Ker (g(u)l - u) uE9 c
and let Ag be an orthonormal basis of F 9 . Since
F~• for distinct g, h r
7 (Corollary 5.3.4.5), it follows that A:=
U
Ag
gEIK ~
is an orthonormal set of E . Assume there is a u E ~ and an c~ c ]K\{0} such that F " - A • O Ker (c~1 - u) ~: {0}. Order the set {FA(gKer(c~l-u~))
~{0}
((u~,c~))~ e, f a m i l y i n
~xIK}
by inclusion. Since F is finite-dimensional (Theorem 3.1.5.1 a)), the above set has a least element
Take v E ~ . Since for every g c IK ~- , Fg reduces v (Corollary 5.3.2.10), F and G also reduce v. Hence there is a ~ E IK with G N Ker (/~1
-
v) # {0}.
Since G is a least element we get G C Ker ( ~ 1 - v). Thus there is a g c ]K J= with
6. Selected Chapters of C*-Algebras
G c Ker ( g ( v ) l for every v
v)
6 ~ . We deduce t h a t G C F a = A , • C All ,
which is a contradiction. Hence for every u 6 .T" and c~ E ] K \ { 0 } , A • M Ker (c~l - u) = {0}. Hence
NKeru-F~
A•
lEA
•177
u6~-
A•
{0},
i.e. A is an o r t h o n o r m a l basis of E . Take u E ~ ' . Given c~ 6 ]K, put B~ := A f'l Ker (c~l - u ) . B~ is an o r t h o n o r m a l basis of Ker (c~l - u ) , and so
xEBa
if c~ # 0 (Proposition 5.5.1.7). For each x E A , let f ( u , x ) denote the c~ 6 IK with x 6 B~. T h e n
= sea(u)
ck6IK
Z
..uo=
aElK\{0}
Z aeIK\{0}
xeBa
= xEA
( T h e o r e m 5.5.6.1 a :=v c). b :=v a and the final assertion are easy to see. Corollary
6.1.1.9
I
Let u,v be self-normal compact operators on E , and
(c~)~ei, (/3;~)~eL characteristic families of eigenvalues of u and v, respectively.
II UV = VU -- O,
then the family obtained by joining the above two families is a characteristic family of eigenvalues of u + v.
6.1 f_.P-Spaces
9
By Proposition 6.1.1.8 a =~ b, there is an orthonormal basis A of E and f, g E IK A with
xEA
xEA
fg=O. Now u + ~ - ~(f(~)
+ g(x))(-Iz)x,
xEA
and the assertion follows from Proposition 6.1.1.2 b ==> a.
I
6. Selected Chapters of C*-Algebras
10
6.1.2 C h a r a c t e r i s t i c S e q u e n c e s
Definition 6.1.2.1 ( 0 )
(von Neumann, Schatten,
1948)
Take
u E f_.(E, F ) . Given n E IN, define ~n := {G vector subspace of ElDim G < n} O,(u) := inf
{llulaZllla
e ~,}
(the n - t h characteristic number of u ), and O(u) := (On(U))ner~ (the characteristic sequence of u) N(u) := {n E IN[0n(u) -7(=0}.
(On(U))nEg(u) is called the characteristic family of u. For p e [1, c~], define EP(E, F ) : = {u E s s Given u 6 s
:= s
F)lO(u) E t~},
E).
F ) , define
Ilullp :-IIO(u)ll~
(the p-norm of u).
Define further /:~
F) := K:(E, F ) ,
/:~
:= K:(E),
and
Ilull0 := Ilull
for every u E Z:~
F). An initial segment of IN is a subset N of IN such that given m E ]N
and n E N , m < n =~ m E N . 0(u) is decreasing and N(u) is an initial segment of IN. We have f_..~176 and
01(u) = Ilull = Ilull~ for every u E /:(E, F ) . We shall show that the p-norm is a norm (Theorem 6.1.3.12 a)) and that for p = 1 the symbols /21 and [[-[[1 introduced above agree with the ones introduced in Definition 1.6.1.1.
6.1 s
( P r o p o s i t i o n 6.1.2.2 b)
11
1
if
n < DimE
L0
if
n > DimE.
a) n e lN==~ O.(1E) = {
F o r p 9 [1, c~[, 1E 9 s
E finite-dimensional =::v II1,~11,,= (DimE)~.
~
a) Assume that n < D i m E and take G 9 ~n. Then G • =/- {0}, so that
II1EIG'II
:
O n ( E ) : 1.
1,
If n > D i m E , then E 9 ~n, E •
{0},
IIIEIE•
O,
O,~(E) = O.
b) follows from a). P r o p o s i t i o n 6.1.2.3
m
( 0 ) If (x,y) 9 E x F and p 9 [1,cx~] then f
e,-,((-Ix)y)-- ~ Ilxll Ilyll
I,
0
<-I~)y 9 s
if
n-
if
n 9 INk{l},
1
II<'l~>yllp--Ilxll Ilyll.
We have
01((-Ix)y)--II(.Ix>ull-
Ilxll Ilyll.
Since the restriction of <.lz>y to {x} • vanishes identically, we see that
o.(<.1~>~) - 0 for every n E IN\{1}. P r o p o s i t i o n 6.1.2.4 u e s
m
( 0 ) Take p,q C [1, c~] with p < q, and
Then u 9 Lq(E, F) and
This follows immediately from
(Proposition 1.1.2.6 b),e)),
m
6. Selected Chapters of C*-Algebras
12
If u,v 6 s
P r o p o s i t i o n 6.1.2.5
then
e~+~_~(~ + v) < e~(u) + e~(~) for all n,p E IN. Take G1 E ~n and G2 E qp and let G be the vector subspace of E generated by G1 + G2. Then G E ~n+p-1 9 Thus
o~+,_,(u + v) <_ II(u + v)la•
<_ Ilulamll + Ilvla•
_< Ilula~ll + Ilvla~ll.
Since G1 and G1 are arbitrary, 0~+,_~ (u + v) < 0~ ( u ) + 0~(v). P r o p o s i t i o n 6.1.2.6
m
( 0 ) Take n E IN and put
~" := {v E s
l D i m l m v < n}.
Then
]:or every u E s
F).
Take v E jc. Since the map (Kerr) l
x:
>Imv,
>vx
is la:jective, we get successively, Dim ( K e r r ) •
D i m I m v < n,
(Kerr) • E ~ ,
e,~(u) _< Ilul(Kerv)•
= IlulKervll- I1(~- v)lKervll _< Ilu- vii e.(,,) _< d,(,).
Now take G E ~n and put V
:~
UOTr
G .
Then v E 3e, so that d~(~)
_< I1~ - vii -
II~ o
( 1 E -- ~ G ) I I - -
Ilu ~
~11
--
II~la'll 9
Since G is arbitrary, d,(u) _< e~(u), e~(~) = d , ( ~ ) .
I
6.1 f_.P-Spaces
Corollary 6.1.2.7
Given u e L ( E , F ) , v E s
~m+n_l(V Take Uo E / : ( E , F ) , vo e s
13
G ) , and m, n C IN,
U) ~ ~m(U)~n(V) .
o
G) with
DimImuo<m,
DimImvo
Then (v - v0) o (~ - ~o) = v o ~ - (v o uo + vo o ~ - vo o ~o)
and Im (v o u0 + Vo o u - vo o uo) C v (Im uo) + Im vo,
Dim Im (v o uo + vo o u - Vo o uo) _< Dim v(Im uo) + Dim Im vo < m + n - 1. We deduce t h a t
em§
(V 0 U) ~ II(v -- VO) 0 ( ~ -
~0)11 ~ I1~ -- UOII IIv -- ~011
(Proposition 6.1.2.6). Since uo and vo are arbitrary, e~§
(v
o
~) < e ~ ( u ) e ~ (v)
m
(Proposition 6.1.2.6). Corollary 6.1.2.8
If u, v E s
F) and n C IN, then
le~(~) - e~(~)l ~ I1~ - ~11. If w C s
F ) and DimImw < n,
then, by Proposition 6.1.2.6,
e~(~) ~ Ilu- wll-
I1(~- ~)+ (~ - w)ll ~ I1~- vii + IIv- wll.
By Proposition 6.1.2.6 again,
e.(~) ~ Ilu - vii + e. (v) so that
6. Selected Chapters of C*-Algebras
14
Corollary 6.1.2.9 valent: a)
u is compact.
b)
O(u) E co.
( 0 ) Tak~ ~ c Z~(E, F). Th~n th~ foOo~ng ~
a :=~ b. Choose c > 0. By Corollary 5.2.3.4, there is a v E s
~q~-
F) with
Ilu- vii < ~. Put n : - 1 + D i m I m v E IN. Then
en(u) ~ Ilu- vii < ~, by Proposition 6.1.2.6. Hence O(u) E Co. b =~ a. Take e > 0. There is an n E IN with On(u) < e. By Proposition 6.1.2.6, there is a v E s
F ) with Dim Imv < n ,
Ilu- vii < c.
Thus u E s
F ) = K:(E, F )
by Corollary 5.2.3.4. C o r o l l a r y 6.1.2.10
I ( 0 ) For every u E s s
C s
g(u) is finite and so
C ]C(E, F)
for every p E [1, oc[. By Proposition 6.1.2.6, N(u) is finite. Thus
~s(E, F) c L~(E, F). The inclusion
L~(E, F) c ~(E, F) follows from Corollary 6.1.2.9 b :=v a.
I
6.1 f3-Spaces
15
P r o p o s i t i o n 6.1.2.11 ( 0 ) Let N be an initial segment of IN, (Xn)neN and (Yn)~eN orthonormal families in E and F , respectively, and take a
(O~)neN C co(N), such that (lOnl)neN decreases. Put
u= ~
en(l~>y~
nEN
(Corollary 5.5.5.5). Then N(u) C N and
for every n C N . Take n E N . Let G be the vector subspace of E generated by (Xm)me~n and take H C ~n. Then (TrHXm)mEIN~ is linearly dependent. Thus there is a family (am)mean in IK, such that not every c~m vanishes and n
~
Olm7rHX m --" 0 .
rn--1
We deduce that 0 # x := ~
o~mXm C G N H •
m=l
IlutH•
a~
Iluxll2: ~
lem(XlXm)l 2
mCN
mElNn
IlulH'll ~ levi,
Now take n E N . Let G be the vector subspace of E generated by
(Xm)me~_l and take x C G •
II~xll ~ = ~
Then
lem
mEN m>n
by Bessel's Inequality, and so
mCN m>n
I(~lx~)l ~ < le~l~llxll ~
6. Selected Chapters of C*-Algebras
16
0~(u) ~ IlulGlll < Io,~1,
e~(u) = levi. Take n E IN\N. Since DimImu < n, it follows from Proposition 6.1.2.6 that
e~(u)=O. Corollary
6.1.2.12 ( 0 )
Let (x~)~e, and (Y~)~c, be orthonormal families
in E and F , respectively. Take f E co(l) define
u := ~
f(~)(.Ix~}yt
tEI
(Corollary 5.5.5.5), and take p E [1,oc]. Then u belongs to s
iff
f E l:'~(I), and in this case
Ilull, = Ilfll,. Corollary 6.1.2.13
( 0 ) Let u be a compact operator on E and N an
initial segment of IN. If (Oln)neY iS a characteristic family of eigenvalues of u such that (ic~nl)neg is decreasing, then the following are equivalent:
a) u is self-normal. b) there is an orthonormal family (Xn)neg in E with
nEN
If these conditions are fulfilled, then:
c) N D N(u) and On(u)= ]~,] f o r e w ~ y n e N . d) If p E [1, c~[U {0}, then u E s
iff (an)new E gP(N), and in this
case
Ilullp = II(~n)n~NIIp"
6.1 LP-Spaces
17
a :=> b follows from Proposition 6.1.1.2 a =~ b. b =~ a. Let (x~)~ei be an orthonormal basis of E with N C I . Then = [ c~x~ UXn
(
0
if nEN if n E I\N.
By Proposition 5.5.5.7 e), u is self-normal. c) follows from Proposition 6.1.2.11. d) follows from c). C o r o l l a r y 6.1.2.14 ( 0 ) Let u be a selfadjoint operator on E . p C [1, oc[. Then the following are equivalent:
m Take
~) ~ e ~ , ( E ) .
b) ~§ ~- e ~:"(E). If these conditions are fulfilled, then
a) or b)implies that u is compact (Corollary 6.1.2.10). Let (c~)~i be a characteristic family of eigenvalues of u. By Corollary 6.1.1.5, (sup{a~, 0})~ci and (sup{-c~, 0})~ei are characteristic families of eigenvalues of u + and u - , respectively, and the assertions now follow from Corollary 6.1.2.13 d). i C o r o l l a r y 6.1.2.15 Let u be a self-normal operator on E , and take n C IN and p e [1, oc]. Then u n e f 3 ( E ) iff u e s and in this case
By Corollary 5.3.3.11, u is compact iff u n is compact. Hence, by Corollary 6.1.2.10 (and Proposition 4.2.4.2 b)), we may assume that u is compact. The assertion now follows from Corollary 6.1.1.4 and Corollary 6.1.2.13 d). i C o r o l l a r y 6.1.2.16
( 0 ) Let u be a positive compact operator on E and
f a continuous increasing real function on a(u) with f(O) = 0 whenever 0 C a ( u ) . Then (On(U))neg(~) is a characteristic family of eigenvalues of u and O(f(u)) = (f(On(U)))neiN. In particular,
e(~ ~) = ( ~ ( ~ ) ~ ) ~ for every a C ]0, oc[.
18
6. Selected Chapters of C*-Algebras
By Corollary 6.1.2.13 a ~ b & c, (0n(U)),EU(~) is a characteristic family of eigenvalues of u and there is an orthonormal family (Xn)neN(~) in E such that
U"- Z On(U)('lXn}Xn" hEN(u) By Theorem 5.5.6.1 e =~ f, f(u)=
Z f(On(U))('lXn}Xn. heN(u)
Hence, by Corollary 6.1.2.13 b =~ c,
O(f(U)) = ( f (On(u)))nON 9
Proposition 6.1.2.17 each n E IN put
( 0 )
I
Let u be apositive operator on E and for
sup (uxlx). Fee.. xe(F.U)#
c~, := inf
Then the following are equivalent: a)
u is compact.
b)
(llux~ll)~,
c)
((uxlx))xEA E co(A) for any orthonormal basis A of E .
d)
((uxnlxn))nE~ E co for any orthonormal sequence (xn)ne~ in E .
e co(I)
for any orthonormal family (x,),E, in E .
If these conditions are fulfilled, then On(u) "-O~n for every n E IN and there is an orthonormal family (Xn)neg(~) in E such that
nEN(u) a ==> b follows from Corollary 5.5.1.9. b =~ c =r d is trivial. d =~ e. Assume that
6.1 s
a:= By recurs•
19
lim a . > O .
n---~O0
we construct an orthonormal sequence
(xa)ne~ in E such t h a t
OL
for every n E IN. Take n E IN, and suppose the sequence has been constructed up to n -
1. Let F denote the vector subspace of E generated by
{xmlm E
IN,-1}. Then F E ~n, so t h a t Ot
(uxlx) > an > - - .
sup xE(F•
2
Hence there is an x , E F j- such t h a t Ol
IIx.II- 1,
> -{.
This completes the inductive construction. The existence of the orthonormal sequence
(Xn)ner~ in E contradicts d).
e : a a. Let F be a vector subspace of E . Then sup zE(F •
(uxlx}=
sup {ulxlu 89 xe(F i )#
sup xE(F •
Ilu~xll=--11~89177 ~
Hence an =
inf Ilu 89
2=
FErn
On(u~) ~.
1
By Corollary 6.1.2.9 b ==~ a, u~ is compact. Hence u is compact. Now we prove the final assertion. By Corollary 6.1.2.13 a =a b & c, there is an orthonormal family (xn)ner~(u) in E such t h a t
nEN(u)
Take a vector subspace F of E . Then sup zE(F•
(uxlx > <
Crn= inf
sup
sup
Iluxllll~ll
< II~IF~II,
xE(F•
and so
FErn xE(F•
(uxlx)<
inf IlulY•
-- FErn
To prove the reverse inequality, we may assume t h a t n E
N(u). Let G be the
vector subspace of E generated by (xm)me~= and take F E q=. There is a y E G A F • with I l y l l - 1. Now
20
6. Selected Chapters of C*-Algebras
n
sup (~xlx) > xE(F'I-) #
(~yly)-Y'~ e~(~)(ylx.~)(xmly)--m=l
n
= ~
em(~)l(x.~ly)l ~ > en(u)
ram-1
I(ylx~)l ~ = e,~(~), 'rn=l
and so an=
inf
sup
FErn xE(FJ.)#
(~l~)>__O.(u).
m
C o r o l l a r y 6.1.2.18 ( 0 ) Let u be a positive operator on E and take p, q E [1, oc[. Then u E ~P(E) iff u~ E f~q(E), and in this case
I1~11~ = I1~ I1~. We may assume that u is compact (Proposition 5.3.3.10 b), Corollary 6.1.2.10). By Corollary 6.1.2.16,
=
(~)~).~,
and the assertion now follows (Corollary 6.1.2.13 a ::v d).
II
6.1 s
21
6.1.3 P r o p e r t i e s of the /:P-spaces T h e o r e m 6.1.3.1
( 0 )
Take u c /C(E,F)
and put v := (u* o u ) 1 9 Take
w C f~(E, F) such that
Keru=Kerw,
u=wov,
(Proposition 5.5.5.10).
a)
There are orthonormal families (xn)n~N(u) and (Y~)neN(u) in E and F , respectively, such that
u= ~
e~(~)(-Ix~)v~.
new(u) b)
e(~) = O(~*) =
O(v).
nEN(u)
d) n E N ( u ) =:=>yn = w x n . v is compact (Proposition 5.3.3.10 b)). By Proposition 6.1.2.17, there is an orthonormal family (Xn)nEN(v) such that
v= Z
e~(v)(.Ix~)~
n~N(v)
Given n E N ( v ) , put Yn :-- W X n .
By Proposition 5.5.5.10 f), (Yn)neN(v) is an orthonormal family in F and
nEN(v)
It follows that
nEN(v)
(Proposition 5.3.2.13 b) ), and so e(~) = e(~*) = e(~)
by Proposition 6.1.2.11.
I
6. Selected Chapters of C*-Algebras
22
Corollary 6.1.3.2
( 0 ) Take p E {0} U [1, oo] and u E f_.(E,F). Then
the following are equivalent:
a) u E f_2(E,F). b) u* E / Y ( F , E ) . c)
(u* o ~)~ e L ~ ( E ) .
If these conditions are fulfilled, then
Ilullp - II~,*llp - II (u* o u) 89lip. By Corollary 6.1.2.10, Proposition 5.3.3.10 b), and Theorem 5.3.1.4, we may assume that u is compact, and so the assertion follows from Theorem 6.1.3.1 b). I Corollary 6.1.3.3 an a E ]0, c~[ with
( 0 ) Take u E f_,(E,F) and v E ]C(E,G). If there is
(~* o ~)~ < (v* o v) o ,
then:
a) On(u) < On(v) for every n E IN. b) If p E {0} U [1,c~] and v E s
then u E s
G) and Ilullp
Ilvll, 9
a) Given x E E,
(Corollary 5.3.3.7), and so
On(U)--0n((U* OU)89 =0n((U* ou)~ 2Aft~ On((V* oV)a) 29-ff--0n(V) (Corollary 6.1.2.16, Theorem 6.1.3.1 b), Proposition 6.1.2.17). b) follows from a). Definition 6.1.3.4
( 0 ) Take u E I C ( E , F ) . A Schatten decomposition
of u is an expression of the form
nEN such that:
I
6.1 s
23
1) N is an initial segment of IN.
2) (0n)new is a decreasing family in JR+\{0}. 3)
(Xn)neY and (Yn)ncg are orthonormal families in E and F , respectively.
4) u =
E e~(-I~)y~. nEN
By Proposition 6.1.2.11, N = N ( u ) and
for every n 9 N . By Theorem 6.1.3.1 a), every compact operator admits a Schatten decomposition. C o r o l l a r y 6.1.3.5 ( 0 ) If u 9 K:(E,F) and v 9 s then there are orthonormal bases (x~)~ei, (Y~)~L of E and F , respectively, such that n 9 I ML :~
(vuxnixn) = (uvYniYn),
9 I \ L ====v<~x~lx~>)~ e L \ I
~
o,
( ~ v y ~ l y ~ ) = o.
In particular, ((vux~ix~))~e, is summable iff ((uvy~ Y~))~eL is summable and in this case
~EI
AEL
Let
e,~(.l=,~)yn nEN
be a Schatten decomposition of u and let (x~)~ei, (Y~)~eL be orthonormal bases of E and F , respectively, such that N=INL.
By Theorem 5.5.5.4 b ::v e & k,
e I\N ~ e L\N ~
(~x~lx~) - 0 ,
(uvy~iy~)-- (vy~iu*y~)- O.
I
~4
6. Selected Chapters of C*-Algebras
( 3 ) Th~ definitions of L:I(E,F) and I1"11~ giv~ in Definition 1.6.1.1 and Definition 6.1.2.1 coincide in the case of Hilbert spaces.
P r o p o s i t i o n 6.1.3.6
We use /21(E,F)old, II'llxold and /21(E,F)new, II.[l~.ew to denote the corresponding symbols introduced in Definitions 1.6.1.1 and 6.1.2.1, respectively. Take u E ~.(E,F). By Corollary 6.1.2.10 and Corollary 3.1.1.5, we may assume that u is compact. Let
)-~'~ o,~(. Ix,~) y,~ nEN
be a Schatten decomposition of u (Theorem 6.1.3.1 a) ). First suppose that u E / : l ( E , F ) n e w . Then u E/:l(E,F)old and II?-llllold
~~
IlOnXnll IlYnll
nEN
: ~ On - - I l u l l l n e w
9
nEN
Now suppose that u E L;l(E,F)old and let ((a~,bt))~el be afamilyin E x F , such that Ilatll Ilbtll < o c ,
u = ~(-lat)bt.
tEI
tel
Then
On = (uxniYn) = ~-~(xnlat)(btlYn) <_ ~ tel
On <_ ~ nEN
~
](xniat)(b~lyn)],
~El
I(Xnia~)(btiYn)l <_ ~
nEN tel
IlatlI ]lbtil < oc
tel
by Proposition 5.5.1.19. It follows that u E/21(E,F)lnew and
II~ll~new --< Ilulliold Lemma
m
( 0 ) If n E IN and (C~k)kE~n , (~k)ke~n are families in
6.1.3.7
IR , then n
n-1
k
n
k--1
k-1
i--1
k=l
Now n-1
k
k=l
i-1
k-1
6.1 s
25
n-1
k
n-1
k
n
k=l
i--1
k=l
i--1
k=l
k--1
i:1
k=2
i=1
k=l
I k=l
Lemma
6.1.3.8
i=1
k=l
k=l
( 0 ) Take n C IN, and let ( a i ) i e ~ , (~i)ie~n be decreasing
families in IR+ such that k
k
i=1
k=l
for every k E IN~. Then n
i--1
i--1
for every p E [1, c~[. For every x E ]R n we need in the proof the family ('xi)ielN,~ obtained by writing the coordinates of x as a decreasing family. Formally this can be described in the following way. Given i c INn and x C ]R ~ , define PAi " - {A C IN~lCard (INn\A) < i},
xi := inf suPlxkI,
~ ' - - (xi)ie~,.
AE~i kEA
Furthermore, put
-(~)~o,
Z--(9~)~o,
K - - {~ e ~ 1 ~ = ~3}, and let L denote the absolutely convex hull of K in IRn . Since K is finite, L is compact. Let x' be a linear form on IR n . P u t 7
-
(x'(ek))k~o
6. Selected Chapters of C*-Algebras
26
Then, by Lemma 6.1.3.7, n-1
Ix'(~)l <_
k
I'y~l~ - ~ ( ~ k k=l
-- ~k§
~
k=l
n-1
k
k=l
i=1
i=1
n
n
i=1
n-1
I'Y,I --<
I'Y~I+ '~n
i=1
k=l
k
= ~(~
- ~§
k=l
E ~, +
~o
i=1
_<~ ( ~ - ~§
~
k=l
~, -< i=1
~
~, +
i--1
~, = i=1
~. k=l
There is an x E K with 9 '(~)
= ~ ~kZk. k--1
Thus Ix'(c~)l _< sup x'(x) <_ sup x'(x). xEL
xEK
By Corollary 1.3.1.7, a E L. Hence there is a family (6x)xEK in ]R such that
xEK
xEK
We deduce that
xEK
xEK
~~, < ~,~. i=1
L e m m a 6.1.3.9
i=1
( 0 ) Let (Ol,X)(~,~)eZ• be a family in ]K such that ~ E I ==> E AEL
[oz.x[ < 1,
I
6.1 f3-Spaces
A 9 L==~ ~
27
[a~ 1<_1.
Take p 9 [1, oc], x 9 gP(L), and for each ~ 9 I put ataxa.
Y~ := L AEL
Then
Y := (Y~)I,EI 9 ~.P(I),
Ily[[p ~ [[X[Ip .
It is obviously sufficient to consider the case when I and L are finite. We m a y also assume t h a t each a ~
and x~ is a positive real n u m b e r . Finally, we
m a y a s s u m e t h a t I = L = INn for some n 9 IN and t h a t
(Xi)ielNn, (Yi)ie~n
are decreasing. T h e n for k 9 INn, k
k
~
k
(~a~j)xj
j=l
k
= ~-~xj + ~
i=1
j=l
j=l
k
( ~ o z ~ j - 1 ) x j <_ i=1
k
k
k
k
k
k
j=l
j=l
i=1
j=l
j=l
i--1
_~~x3 § ~ (~o~,- ~)x~- ~ j § ~ (F~o~3)~- ~ , so t h a t
k ~Yi=
k ~--~ ( ~ O l i j X j ) = ~-~ (~-~Ctij)Xj -i=l j = l j=l i=1
i=1
k
k
k
j--1
i--1
j--k+l
k~
~(k
~Zxj§247 j=l
j=l
i=1
j=k+l
=~x~§ j=l
j=l
i-=l
( j=l
~oij)xk = i=1
~,~)~_~= i=1
i=1 j=l
j=l
By L e m m a 6.1.3.8,
Ilyll~ < Ilxll~.
m
28
6. Selected Chapters of C*-Algebras
( 0 ) Take u E s and p E [1, oo[. If $2 denotes the class of families ((a~,b~))~EI such that (a~)~ei and (b~)~E1 are orthonormal families in E and F, respectively, then
P r o p o s i t i o n 6.1.3.10
E
[(ua~Ib~)[P]((a~'b~))tEIE fl}.
0~(u)P = sup { E
nEIN
tel
First we prove the inequality _<. By Theorem 5.5.5.1 d =~ a, we may assume u to be compact. Let
heN
be a Schatten decomposition of u. Then
for every n E N , and so, by Proposition 6.1.2.11, o~(u), = ~ hEN
_< sup { E
I(~x~ly.)l ~ <
hEN
](uatlb'}l'l ((a~'b~))~e'E $2}.
eEI
Now we prove the reverse inequality. Take ((a~, b~))~e~ E s that u E s F). Let
We may assume
hen
be a Schatten decomposition of u (Corollary 6.1.2.10). Then
(ua~lb~) - E O~(a*lxn)(ynlb~) heN
for every ~ E I. By Proposition 5.5.1.19,
E I(a~lxn)(ynlb~}l <- IIaLII IIb~ll -< 1 hen
for every ~ E I and ~EI
Thus
E I(ua~lb~}lP<- E O~ = E On(u)" ~E I
heN
nEIN
(Proposition 6.1.2.11, Lemma 6.1.3.9). Hence sup { E ~E I
I(ua~ b~)lPl ((a~'b~))~E' E f2} <_ E On(u)P" nElN
I
6.1 s
Corollary 6.1.3.11
( 0 )
29
If u is a self-normal compact opertor on E ,
then
E
0n(u)P = sup { E
nEIN
I(uxix}iP] A orthonormal basis of E } ,
xEA
for every p E [1, c~[.
By Theorem 5.5.6.1 a ==> e, there is an orthonormal basis A of E and an f E co(A) with
xEA
Then
f(x)
= (~lx)
for every x E A. By Corollary 6.1.2.12,
nEIN
xE A
xE A
The assertion now follows from Proposition 6.1.3.10.
Theorem 6.1.3.12
( 0 )
I
Let p ~_ [1,(x~[.
a) EP(E,F) is a vector subspace of 1C(E, F) and the map EP(E,F)
>]R+,
u,
>llull~
is a norm. We consider /:P(E, F) with this norm. b) E2(E, F) is complete.
c) If (x~)eeI, (Y~)eE/ are orthonormal families in E and F , respectively, and (C~)~E, E gP(I), then the family (c~(.Ixe)y~)~ e, is summable in s F). d) s
is a dense vector subspace of s
a) Take u, v E s and let (x,)e~,, (Ye),e, be orthonormal families in E and F , respectively. Then
( E i~(~§ v)x~l~l~) ~ : ( ~ I~x~l~l § Ivx~l~l') ~ ~_ eel
eel
30
6. Selected Chapters of C*-Algebras
§ (Z eel
ll lt § II ll.
LEI
(Proposition 6.1.3.10). Hence u + v E Z:P(E, F) and
Ilu + vll~ ~ I~11, + I1~11, (Proposition 6.1.3.10). It is easy to see that au E / : ( E , F) and that
II~ullp = I~1 Ilullp for every a E]K. The assertion now follows with the help of Proposition 6.1.2.4. b) Let (u~)~E~ be a Cauchy sequence in /:P(E, F ) . By Proposition 6.1.2.4 (and Corollary 6.1.2.10, Theorem 3.1.1.3), (un)~e~r converges in K:(E, F) to a u. Let c > 0. There is an no E IN such that
Ilum- u~ll~ < for all m , n E IN\INno. Let (x,)~e,, (Y,),E, be orthonormal families in E and F , respectively. Then I((~m - u~)x~ly31 ~ <_ I 1 ~ - ~11~ < ~ LEI
for all m, n E IN\IN~o (Proposition 6.1.3.10). Hence
I((~ - u~)x~ly~)l ~ <- c~, ~EI
u - u~ E/:P(E, F ) ,
Ilu- ~11~ <
for every n E IN\INno (Proposition 6.1.3.10). By a), u E/:P(E, F) and (un)nE~ converges to u in E P ( E , F ) . Thus f 3 ( E , F ) is complete. c) Take e > 0. Choose J E ~ l ( I ) , such that I~l~<~ ~ ~EI \ J
Then
~~(-I~)y~ tEK
_<
I~l ~
<~
p
for every K E ~ 3 f ( I \ J ) (Proposition 6.1.3.10). By b) and Proposition 1.1.6.6, the family (a,(.iz~)y~)~e, is summable in /2P(E,F). d) follows from Corollaries 6.1.2.10, 6.1.2.12, and Theorem 6.1.3.1 a).
I
6.1 s
C o r o l l a r y 6.1.3.13 ( 0 ) p e {0} U [1, c~], then: ~)
If u E s
31
v E f~(F,G), w E s
and
~ e ~ ~ e~(w o v o ~) < I l w l l e n ( v ) l l ~ l l 9 ,f
WOVOU
b)
v e Z:"(F,G) ~
c)
[
c s
H) ,
I1~ o v o ~11~ < II~Jl Ilvll,ll~ll 9
The map s
~, s
v:
>w o v o u
is continuous and linear with norm at most Ilwll Ilull.
d)
L:P(E) is an involutive Banach algebra. If p e [1, oc[ then this Banach algebra is quasiunital iff E is one-dimensional.
a) The inequality
follows immediately from the definition. By Theorem 6.1.3.1 b), we deduce that 0~(v o ~) = e ~ ( ( ~ o u)*) -- e ~ ( u , o v*) < il~*lle~(~*) = e~(~)ll~ll 9
Hence en(~ o v o u)<
Ilwllen(v o ~)<
II~llen(~)ll~ll 9
b) follows from a) and Proposition 3.1.1.11. c) follows from b) and Theorem 6.1.3.12 a). d) By Theorem 6.1.3.12 b) and Corollary 6.1.3.2, s is an involutive Banach space. By b) and Proposition 6.1.2.4, s is an involutive Banach algebra. Let ~ be an approximate unit of E . Assume that E is not one-dimensional. Then there are x, y E E such that
Ilxll- Ilyll
= 1,
x _1_ y .
Then, by b), II((-J~)x + ( . l y ) u ) o
for every u E s
~11~ __ IJ~ll. _<
# . We get by Proposition 6.1.2.11,
2~ - I I ( . I x ) x + ('IY)YlIp = lim It(('lx)x + ('ly)y)~
which is the expected contradiction.
-< 1 E
32
6. Selected Chapters of C*-Algebras
C o r o l l a r y 6.1.3.14
( 0 )
Take p 9 {0} tO [1, oo]. Then f_2(E) is an invo-
lutive ideal of f_.(E) and an involutive unital E(E)-algebra.
The assertion follows from Theorem 6.1.3.12 a),b) and Corollary 6.1.3.13 b),d).
9
C o r o l l a r y 6.1.3.15
(Fatou's Lemma) Take p 9 [1, co[, u 9 f_.(E, F) , and let
(Un)~er~ be a bounded sequence in s u 9 Ep(E,F)
converging pointwise to u. Then
and
Ilull~ _ lim inf Ilu~llp. n--). oo Let (x~)~e,, (Y~)~E, be orthonormal families in E and F , respectively. By Proposition 6.1.3.10,
~_~I(ux, ly~)lp ~ tel
l(unx~ly~)l p ~ lim ioonfIlunll~, <
lim__inf ~
co,
tel
u 9 f_F(E, F ) , and
llullp < lim inf llun]Ip,
m
n--+ o o
Remark. The hypothesis that (un)ne~ is bounded in E P ( E , F ) cannot be relinquished, as the following example shows: Take p := 1, E "- F "= g2. Put
u "= ~
1
~(.le,)ek,
k=IN
and un "=
~(-lek)ek, k=l
for every n 9 IN. C o r o l l a r y 6.1.3.16 Let IK =(P,. Take p e {0}U[1, oo[ and u e E ( E ) . Then iff reu, im u 9 ZF(E) and in this case
u 9 s
sup{llreull p , Ilimullp} _< Ilullp _< llreu[lp + Ilimullp. If u E EP(E), then, by Corollary 6.1.3.2, u* e EP(E) and
Ilu*llp = I1~ Ip. By Theorem 6.1.3.12 a), re u, imu E EP(E) and
6.1 s
II~e~ll, < I1~11,,
33
Ilim~ll, -< I1~11-
then, by Theorem 6.1.3.12 a), u E f3(E) and
If re u, im u E s
I1~11, _< Ilre ull, + Ilim ull,. Corollary 6.1.3.17
sup{ E
Let u E s
I
and p E [1, oe[. If IK =(~ and
l(uxlx)lp A ~176176
of E } < oo
then u E s (E) . Since <(re
u)xlx )
:
re
(uxlx)
((im ~)xl~> : im (uxlx) for every
x E A,
it follows
sup{xEAE I< (~e ~)~1~> I'
A orthonormal basis of E } < oe,
s u p { E[((imu)x[x)[pxeA A orthonormal basis of E } < oe. By Corollary 6.1.3.11, reu, imu E s 6.1.3.16.
Hence u E s
by Corollary I
Corollary 6.1.3.18 ( 0 ) Let u be a compact self-normal operator on E . Then there are an orthonormal basis A of E and an f E IK A such that
ux : f (x)x for every x E A. If A and f are as above, then: a)
Given p E [1, oe[U{0},
u
Es
iff f E gP(A) and in this case II~ll~ = IIfll,
and
x6A
6. Selected Chapters of C*-Algebras
3r
b)
ap(u) = f (A) .
xEA
d)
If c~ E IK\{0}, then -1
Im (c~l -- u) = f (a) •
and -1
1
1
f(x)(y]x) --I
yeA\ ! (,~) for every y E Im (al - u). By Corollary 6.1.2.13 a =~ b, there are an orthonormal basis A of E and an f E]K A such that u~ : f(~)~
for every x E A. a) By Theorem 5.5.5.4 a & jl =~ j2 & j3, f E co(A) and
xEA
By Corollary 6.1.2.12, u E s
iff f E t~P(A) and in this case Ilull, = IlIII..
Moreover, by Theorem 6.1.3.12 c), if f E t~P(A) then
u = ~
f(x)(.lx)x
(in s
xEA
b) follows from Proposition 5.5.5.7 c). c) follows from a). d) follows from a) and Theorem 5.5.6.1 e =~ h. P r o p o s i t i o n 6.1.3.19 If E # {0}, F # {0}, and p E [1, oc[, then s is separable iff E and F are separable.
I
F)
6.1 f_.P-Spaces
35
First suppose that E and F are separable. Let A and B be countable dense subsets of E and F , respectively. Take (x,y) C E x F . There are sequences (x~)~er~, (y~)ner~ in A and B , respectively, which converge to x and y, respectively. Take n E IN. Then II(-Ix~>y~
- (-Ix>yll~
~ II<-Ix~ -
x>y~llp -4-II(l~>(y~ -
y)ll~ =
= IIx~ - xll Ily~ll + Ilxll Ily~ - yll
(Proposition 6.1.2.3), so lirnc(-Ixn)y,~ = (.[x>y. Let C be a countable dense subset of IK and let 2" be the set of linear combinations of the elements of {(<'lx>yl(x, y) c A x B} with coefficients from C . By the above considerations
s
F ) C .~,
where 9r denotes the closure of $" in s
s
F) is dense in s
F). Since ~ is countable and since F) (Theorem 6.1.3.12 d)), it follows t h a t s F)
is separable. Now suppose that F is not separable. Since s it follows that s
F) is not separable. Hence s
F) is isomorphic to F , F) is not separable.
Finally, suppose that E is not separable. By Corollary 6.1.3.2, the map
s
>s
u,
> u*
is an isometry of real Banach spaces. By the above considerations, s not separable.
II
P r o p o s i t i o n 6.1.3.20
(Lebesgue's Dominated Convergence Theorem) Take
u C s p E [1, oc[, v E s s (E, F) such that
and let (u~)~er~ be a sequence in
0m(u~) ___0m(~) for all m,n E IN and lim Ilun - ull = O. n---~ o o
Then u c s
F) is
F) and lim Ilun - u[Iv -- O.
6. Selected Chapters of C*-Algebras
36
C h o o s e m E IN. T h e n
0m(u. - u) <_ I1~. - ull for every n E IN, so that lim Om(Un -- U) = O.
n---+oo
B y P r o p o s i t i o n 6.1.2.5,
O,n(~) < e~ (~. -- ~) + Om(~.) < I1=" -- ~11 + era(v),
Ore(u) < Ore(v),
u Es
0~m_~(u.- ~) _< Om(U.)+ Om(U)<_ 2Om@), 0 ~ ( ~ . -- ~) _< Om(~.) + 0,.+I(U) <_ 20m(V) 9 Take ~ > 0 . There is an mo E ]IN with cx;
cp
0re(v)" < -g. TI~=Tn0
There is, furthermore, an no C IN such that 2too
Z
cp
0m(~. -- u)~ < 7 '
m=l
for every n E IN, with n _> no. It follows that
oo IIUn
- - "/ill ~ - - ~
em( an --
/z)P - -
m=l 2too
= ~
oo
oo
em(U n I u)P _~_~
m---1
02(mo+m)(?.tn _ u)P _~ ~
m=l cp
e,(mo+m)-I
(Un -- ?-t)p <
m=l cx;
oo
< -~ -~-2 ~
Omo+m(V)+ 2 ~ Omo+m(V) < m=l m=l cP
cP
cP
<7+-s163 for every n C IN, with n > no, i.e.
lim Ilu~ - u[I p = O.
n---),cx:)
I
6.1 s
Theorem
37
( 0 ) Let p,q be conjugate exponents. Take
6.1.3.21
u E E.P(E,F) and v E f_.q(F,G). Then v o u E Et(E,G) and (HOlder inequality).
By Corollary 6.1.3.13 b),
p, q E [1, c~[.
we may assume that
E Ozm<'lam)X~ '
E
mEM
Let
fin <'lyn)bn
mEN
by Schatten decompositions of u and v, respectively. Put 1
= ((~ o ~)* o (v o ~ ) ) ~ ,
By Proposition 5.5.5.10, there is a t E s
G) such that
Ker(vou)-Kert,
vou-tow.
Let A be an orthonormal basis of E and take B E ~ s ( A ) . Then W
--
t*
oVOU
(Proposition 5.5.5.10 e)), so ( ~ x l x > = <(t* o v o ~ ) ~ 1 ~ ) = ( ~ x l v * t z > =
= <~-'~ (~,~(xlam)xm ] E - ~ { t x l b ~ ) y ~ } = mEM
nEN
= E E a'~fl~(x[a'~}(t*b~ix)(xmlyn) mEM nEN
for every x E A (Proposition 5.3.2.13 b)). It follows that
E (wxix} = E E a'~'~(xmiY~)@s•177177177 xEB
mEM nEN
(Proposition 5.5.1.7). We have
E ](x'~lYn)@B•177177 nEN
= ~
I(xmly~><~lt~.~a~)t <_ Ilx~ll I I t ~ . ~ a m l l < 1
nEN
for every m E M and
38
6. Selected Chapters of C*-Algebras
E
[(x'~IYn)(Tcs"t*b'~iTrsz•
=
mEM
= E
I(xmIy,)(Trs•177
iam)I <- Iiy,][ iITrB•177
1
mEM
for every n E N (Propositions 5.2.3.1 a ~ b, 5.5.1.19, 5.5.5.10 c)). Given rn E M , put ~/~ :=
Y~~.(xmiY.)(~rB'lt'bnITrs•
9
hEN
By Lemma 6.1.3.9, ('~m)mEM E tq(M) and II('~m)mEMiiq ~ ]I(~n)nEINliq = Iiv[Iq 9
By the classical HSlder inequality, E
I<WXIZ)] <~ E
xEB
IC~rn'~rnI -~ ]I(OLrn)rnEMiipl[(~/rn)rnEMI]q <-- IillIiP[ vl]q"
mEM
Since B is arbitrary,
xEA
By Corollary 6.1.3.11, w E/~I(E) and
I1~11, < Ilull~llvllq. It follows that v o u = t o w E f-.I(E,G),
I•v 0 UI] 1 = IIt 0 wl] 1 <~ ]lt[[ IIwII1 <~ I]ltllP]iYiI q (Corollary 6.1.3.13 b), Proposition 5.5.5.10 c) ). P r o p o s i t i o n 6.1.3.22
( 0 ) Take u,v E f_,(E, F) with uov*=O
a)
(Keru) • C Kerv,
b)
[[u § v i i - sup(llull, Ilvll).
and v* o u = O .
(Keru*) • C Kerr*,
Imu 2_ I m v .
6.1 s
c)
39
If u, v are compact, then there are orthonormal families (xL),~i in E and (YL)~el in F , respectively, and functions f , g 9 ]K I such that u-
~
f(~)(.Ix~)y~,
v-
tel
d)
~j-~g(t)(.Ix,)y, ,
f g = o.
tel
If p 9 [1, c~[ and u, v 9 s
F) , then
I1~ + vll~ = Ilull~ + t1~11~. a) We have Im v* C Keru
Im u C Kerv*
SO
(Kerv) •
CKeru,
(Keru*) •
(Imv) •
(Proposition 5.3.2.4) and (Keru) •
(Kerv) •
Imu/Imv.
b) We may assume that fl~l < I1~11. Put G "- Ker u.
By a), I1~11 = IIw~zll = I1(~ + ~ ) ~ x l l
< I1~ + vii I ~ x l l _< i1~ + vii IIx I
for every x E E , so that
By a), II(u + v)xll 2 = Ilux + vxll 2 = Ilu~clx + w ~ x l l 2
-
-
= I l u ~ l x l l 2 + Ilwcxll 2 < Ilull211~cxll 2 + llvll211~xll 2 <
4o
6. Selected Chapters of C*-Algebras
for every x 9 E , and so
+
< I1 11.
c) Let
nEN(v)
hEN(u)
be Schatten decomposition of u and v. Then
{an[n 9 N(u)} C I m u ,
{bnln 9 N(v)} C Imv
and
{x~ln 9 N(u)} C (Kern) -L,
{y~ln 9 N(v)} C (Kerv) -L
(Theorem 5.5.5.4 a =v g). By a),
{an]n 9 N(u)} _L {bnin 9 N(v)},
{xnln e
N(u)} _L {ynln
9
N(v)},
and now the assertion follws. d) follows from c) and Corollary 6.1.2.12.
m
( 0 ) If E and F are infinite-dimensional then for every u E ~ ( E , F ) # there is a pair (x,y) E E x F with x # O, y # 0 such
P r o p o s i t i o n 6.1.3.23
that
flu • (.[x)yl] <_ 1. In particular, K:(E,F) # has no extreme points and K:(E,F) is not a dual space. Let
nEN
be a Schatten decomposition of u. First suppose that N is finite. Then there are
6.1 s
41
with
ilxll = Ilyli = 1. By Proposition 6.1.3.22 b),
ilu + (.Ix)yll < 1. Now suppose that N is infinite. Then there is an n C N with 0n < 1. Put x := (1
-On)Xn,
Y :: Yn.
By Corollary 6.1.2.12, Ifu + (.Ix)yll _< 1. u is not an extreme point of )U(E, F ) # , since
+ (.Ix)y e ~ ( E , F ) # and 1 ~=~(~ +
1
(-I~)y) + ~ ( ~ - (-ix)y)
By Corollary 1.3.1.12, K:(E, F ) is not a dual space. Lemma 6.1.3.24
m
Let n C IN and let u be a nilpotent operator on IK n . Then
there is an orthonormal basis ( x m ) m e ~
of ]K n such that uxm
belongs to the
vector subspace of IK n generated by (Xp)pelNm_ 1 for every m E INn.
We prove the assertion by complete induction, u* is nilpotent, and so it is not invertible. Hence there is an xn E ]K n with
iiz~ii = 1,
~*Xn
=
O.
Thus
for every y C ]K n , i.e. Im u C {xn} • . By the hypothesis of the induction, there is an orthonormal basis (Xm)me~n_l of (Xn} • such that, for every m C I N n - l , UXm belongs to the vector subspace of {xn} • generated by (Xp)pe~n_ 1 The family (Xm)me~n property,
has the required m
6. Selected Chapters o] C*-Algebras
42
For every u E ~ ( E ) , there is an orthonormal basis A
P r o p o s i t i o n 6.1.3.25
of the closed vector subspace of E generated by 1 U Eo(~- -&~) ~<\{0}
(Definition 3.1.3.18) such that ((ux[x))~eA is a characteristic family of eigenvalues of u. Take a E IK\{0}. By Theorem 3.1.3.17 b),d), E a ( 1 -
~u) is finite--
dimensional and invariant with respect to u, and the map Ea(1-~u)
>E,(1-~u),
x,
>(1-
u)x
is nilpotent. By Lemma 6.1.3.24, there is an orthonormal basis (x~)ne~q(~) of Ea(1 - ~u) such that z a n Ea(1 -
~lu)
~UXan belongs to the vector subspace of
generated by (x~m)me~_~ for any n E INq(~) .
By Proposition 3.1.4.3, the set
is linearly independent. By Theorem 3.1.5.1 c), B is countable. We write B as a family (Yn)nEY such that N is an initial segment of IN, and that for every n E N ,
a Y n - uyn belongs to the vector subspace of E generated
by (Ym)me~n-1 9 Let (Xn)ne y be the family constructed from (Yn)nEN by the Gram-Schmidt orthonormalization. (x~)neN is an orthonormal basis of the closed vector subspace F of E generated by
U Eo(1-in) Take c~ E a(u)\{0} and choose n E Y such that yn is an algebraic eigenvector of u corresponding to c~. Then ~x~ - uxn belongs to the vector subspace of E generated by (xm)me~_l 9 It follows that -
<~nlx~>
:
<~lx~>
- ~x~lxn>
:
0
= ~
Hence Card {n ~
Nl(uxnlx~) :
~}
agrees with the algebraic multiplicity q(c~) of c~ with respect to u and
((UX,~lXn))n~N is a characteristic family of eigenvalues of u.
I
6.1 s
43
C o r o l l a r y 6.1.3.26 (H. Weyl) If u is a compact operator on E and (a,)~ei is a characteristic family of eigenvalues of u, then
~EI
nEN(u)
By Proposition 6.1.3.25, there is an orthonormal family (x,),e, in E such that (ux~lx~) = a~
for every ~ E I . It follows that
~EI
LEI
nEIN
(Proposition 6.1.3.10).
I
P r o p o s i t i o n 6.1.3.27 map s
Take p E {0} U [1, oo[ and u E f_.P(F,G). Then the
E) # x E.(G, H) #
>f_.P(U, H) ,
(v, w) ,
> w o u o v*
(Theorem 6.1.3.12 a), Corollary 6.1.3.13 b)) is continuous. (Recall that s denotes the set s
Case 1 3 (x, y) E F x G, Take (v0, w0) E s
IIw
E)F#
E) # endowed with the topology of pointwise convergence.)
E) x s
o ~ o v* -
for every (v, w) E s 6.1.2.3), so that
H ) . Then
~ o o u o v:,l p - - I I ( . I w ) w y
_< II(-I(v -
--II(v
u = (Ix)y
vo)x)wyllp
§
II(-Ivox)(~
(-Ivo~/~oyll,,
- wo)yll,,
-
- vo)xll II~,yll + II~oxll II(w - ~o)yll E) x s
H) (Proposition 5.3.2.13 c),d), Proposition
lim w o u o v* (v,w)~+(vo,wo) Case 2 u E
-
f-.,S(F,
G).
--
*
W 0 o U o ?30 .
6. Selected Chapters of C*-Algebras
44
This follows from Case 1 and Corollary 5.5.1.11. Case 3 u 9
G)
Take r > 0. By Corollary 5.2.3.4 and Theorem 6.1.3.12 d), there is a u0 9 s (F, G) such that I 1 ~ 0 - ull~ < ~
We have IIw o u0 o v* - w o u ~ ~*ll~ -
IIw o (u0 - u) o ~*llp ~_
< I1~11 I1~0- ~11~11,11 < for every (v, w) 9 s F) # x L(G,H) # (Corollary 6.1.3.13 b)). By Case 2, the map associated to u0 is continuous, so our map, being the uniform limit of continuous maps, is continuous, m
Proposition 6.1.3.28 Let A be an orthonormal basis of E and take p 9 {0} U [1, oc[. /f 1K = r then the vector subspace of ~EP(E) generated
by { (.Ix + iky)(x + iky)lx, y 9 A, k 9 IN4} (Proposition 6.1.2.3) is dense in s Let 9r be the vector subspace of s
generated by
{(.Ix + iky)(x + iky)lx, y e A, k 9 IN4} and ~ its closure in s Step1
(-y)xE~
for all x, y E A
Since the map E•
~(E),
(~,~)'
~<'1~)~
is sesquilinear (Proposition 5.3.2.13 a)), 4
1 ('lY)X -- ~ E ik('lx + iky}(x + iaY) e .T k--1
6.1 s
45
by the polarization identity (Proposition 2.3.3.2 d)).
Step2
( . I x ) ~ E ~ for every (x,~) E A x E
Take e > 0. There is a B E ~s(A) such that
yEB
Thus
yEB
yEB
yEB
(Proposition 6.1.2.3). Since e is arbitrary, it follows from Step 1 that
{1~)~ e :X. Step3
( . l ~ ) r i E ~ for all ~ , r / E E
Take e > 0. There is a B E ~s(A) such that
xEB
Thus
xEB
xEB
xEB
(Proposition 6.1.2.3). Since c is arbitrary, Step 2 implies that (-I~)rl E )r. Step 4
The assertion
By Step 3 and Corollary 5.5.1.11,
Ls(E) c 7 . By Theorem 6.1.3.12 d) and Corollary 5.2.3.4, J= = Z ' ( E ) .
i
6. Selected Chapters of C*-Algebras
46
6.1.4 H i l b e r t - S c h m i d t O p e r a t o r s
De~itio~
6.1.4.1
( 0 ) rh~ o p ~ t o ~ oS L~ o~ ~U~d ~ b ~ t - S ~ h ~ d ~
operators (Carleman, 1921)
P r o p o s i t i o n 6.1.4.2 F , respectively, then
( 0 ) If A and S are orthonormal bases of E and
xEA
for every u E s
(x,y)EA•
yEB
F ) . If the above sums are finite, then u is compact.
By Parseval's equation,
Z
Jluxll2 - ~
xEA
=
~l(~,xl~)l ~ -
xEA yEB
E
~
i(uxl~l 2 =
(x,y)EAxB
i(xlu,y)t 2 = E E i(x,u*~),2 = E Ilu*yll2
(x,y)EA•
yEB xEA
yEB
The final assertion follows from Theorem 5.5.5.1 d =~ a. P r o p o s i t i o n 6.1.4.3 ( 0 ) If A is an orthonormal basis of E u Es F ) . Then the following are equivalent:
I and
a) u E s b)
E I1~11 ~
< ~.
x6_A
If these conditions are .fulfilled. then i, la
= xEA
By Corollary 6.1.2.10 and Proposition 6.1.4.2, we may assume that u is compact. Let
hEN
be a Schatten decomposition of u. Then
hEN
6.1 s
47
nEN
for every x E A (Parseval's equation), so
xEA
xEA n E N
= Z < Z i(xlxo)i" = ~ e~n nEN
xEA
nEN
(Parseval's equation), Corollary 6.1.4.4
i ( 0 ) Let u be a positive operator on E and A an
orthonormal basis of E . u is nuclear iff
xEA
and in this case ilulI1 -~-- E ( u x l x )
9
xEA
By Corollary 6.1.2.18, u E/:I(E) iff u-} E L:2(S), and in this case I1~111 - I1~ -~ I1~.
By Proposition 6.1.4.3, u89 E s
iff
y ~ I1~11 ~ < oo, xEA
and in this case
II~ll~ = ~
li~il ~
xEA
The assertion now follows from
E fl~xll ~: E(~ixl~ix> = E<~xjx> xEA
Corollary 6.1.4.5
xEA
( 0 )
Let u , v
I
xEA
be positive operators on E
such that
6. Selected Chapters of C*-Algebras
4s
If A is an orthonormal basis of E , then
xEA
xEA
(Corollary 5.3.3.7) and the assertion follows from Corollary 6.1.4.4. C o r o l l a r y 6.1.4.6 and in this case
( 0 ) Let u E s
u Es
IIU* O UII 1 =
I
iff u*ou E s
IIUII 2 .
Let A be an orthonormal basis of E . Then
xEA
xEA
xEA
and the assertion follows from Proposition 6.1.4.3 and Corollary 6.1.4.4. T h e o r e m 6.1.4.7 respectively. a)
( 0 )
Let A , B
be orthonormal bases of E and F ,
The map
xEA
is a scalar product which generates the norm o f / : 2 ( E , F ) .
b) u E / : 2 ( E , F ) , (x, y) E E x F =~
y> =
.
c) {(.Ix)yl(x, y) E A x B} is an orthonormal basis o f / : 2 ( E , F ) . d)
/f u E / : 2 ( E , F ) , then
(x,y)EAxB
where the sum is taken in /:2(E, F ) .
e) /f u, v E s
F ) , then (ulv) - (v*lu*) .
a) Now
xEA
9
xEA
6.1 s
1
1
( E Iluxll)( E xEA
49
i, l, llvi, ,
xEA
(Proposition 6.1.4.3), so the sum in a) is well-defined. The assertion now follows from
xEA
xEA
(Proposition 6.1.4.3). b) By Theorem 5.5.1.15 c), (~1(-~)y) = Z ( ~ z l ( z l x ) y ) zEA
= ~
(zl~)(~zly) =
zEA
= ~(zlz)(zl~*y)
= (xl~*y) - ( ~ x l y ) .
zEA
c) By b), given ( x l , y l ) , (x2, y2) 9 A x B, (('Ixl)YlI('Ix2}Y2)--
((X21Xl)Yl Y2)--
1
if (xl,yl) = (x2, y2)
0
if (Xl, Yl) ~ (x2, Y2)
= (x~lx~)(y~ly~)=
so {(.Ix)yi(x, y) E A x B} is an orthonormal set of /:2(E, F ) . By b), Proposition 6.1.4.3, and Proposition 6.1.4.2
(x,y)EAxS
for every u C s
(x,y)CAxB
Hence by Theorem 5.5.1.15 d ::v a, {(.[x}yi(x, y) e A x B }
is an orthonormal basis of s F). d) follows from b), c), and the Fourier expansion. e) By Theorem 5.5.1.15 a =~ c,
xEA
xEA yEB
= ~ yEB
yEB xEA
(v*yl~*y) - (~* l~*).
I
6. Selected Chapters of C*-Algebras
50
Corollary 6.1.4.8 If u, v are positive Hilbert-Schmidt operators on a Hilbert space, then (u[v) E ]R+. Let (a~)~ei be a characteristic family of eigenvalues of u. By Proposition 6.1.1.2 a =~ b, there is an orthonormal family (x,),ei in E such that
Thus
tEI
by Corollary 5.3.3.7.
I
Proposition 6.1.4.9 spaces. We set
( 0 )
k'L2(v)
Let ( S , |
>L2(#),
( T , T . , u ) b e a - f i n i t e measure
x,
>kx
for every k E L 2 (# @ u) (Proposition 3.1.6.17 a)).
a)
k Es
for every k E L2(# | u) and the map
L2(#@u)
>s
k,
>k
is an isometry.
b) /]" 9r, G are orthonormal bases of L2(u) and L2(#), respectively, and if u Es then ((uxly)y | x)(x,~)eJ=• is summable in L2(# | u) and if we put
(x,y)~y• then k
-- ~t.
6.1 LP-Spaces
51
c) I f (Xt)tEi, (Yt)tEI are orthonormal families in L~(u) and L2(#), respectively, and if f e t~2(I), then (f (t)y~ @-2~)~ci is summable in L2(# | u). If we put
(in L2(p|
k:=~f(t)y~|
then f(~)(-Ix~)y~ (in ~2(L2(u), L2(#))).
= ~ tel
a & b. (5)xeJ: is an orthonormal basis of L2(u) and (y @~)(~,y)ej:xg is an orthonormal basis of L2(# | p) (Proposition 5.5.3.8). Take k E L2(p | u). Then
(kxly) =
(k(s,t)x(t)du(t))y(s)d#(s) =
f ] k(y | x)d(# | u) = (kly | ~) for every (x, y) E 9r" x 6. We get
Z I(~xy)l ~ = ~ I(kl~| (x,y)eJ:x~ (x,y)cJ:x~
~=
Ilkll~,.|
by Parseval's equation. By Proposition 6.1.4.2 and Proposition 6.1.4.3, kEs
IIkll2= Ilkll2,,|
Now take u c ~:~(52(u),/2(p)). By Proposition 6.1.4.3 and Proposition 6.1.4.2, I(uxly)l ~ <
(z,y)~-•
~.
Hence the family ((ux]y)y | x)(x,u)~Txa is summable in L2(p | u) (Corollary 5.2.2.5). Put k :z
(x,y)e~'• By the above considerations,
(kxl~) = (kly | ~} = (~lv)
52
6. Selected Chapters of C*-Algebras
for every (x, y)
9 9v
x G, so that N k--u.
c) Put u := E
f(t)('lz~)Y~
(in s
L2(#)))
tEI
(Theorem 6.1.3.12 c)). Then (~ly~> = f(~)~
for all ~, A 9 I. By b), (f(~)y, | 5~)~e, is summable in L2(# | u) and k--u.
Remark.
The relation Ilkll < Ilkll~ = Ilkll~,.|
(Proposition 6.1.2.4, Proposition 6.1.4.9 a)) clarifies Remark a) of Proposition 3.1.6.17. Proposition 6.1.4.10
Take p 9 {0} t3 [1, oc]. For u 9 f_2(F) , define
~.L~(E,F)
>L~(E,F),
v~
>~ov
(Corollary 6.1.3.13 b)).
a) ~ 9 s b)
for every u 9 s
The map ~.(F)
~. L(L~(E, r ) ) ,
u,
>
is an injective involutive algebra homomorphism (Corollary 6.1.3.13 d), Theorem 6.1.~. 7 a) ).
c)
The representation in b) is non-degenerate.
d) If G is a closed vector subspace of E , then {v
9
s
F)[G
C
Kerv}
is a closed vector subspace of s which is invariant under {~[u 9 s In particular, if Dim E # O, then the representation in b) is not irreducible.
6.1 s
e) v 9 s f)
F) is cyclic for { ~ l u 9 s
53
iff it is injective.
has a cyclic vector iff E is separable and
{ulu 9
Hilbert dimension of E ~_ Hilbert dimension of F .
a)
follows from Corollary 6.1.3.13 b).
b) It is obvious that the map is an algebra homomorphism. Take u C/2P(F) and v, w C s F). If A is an orthonormal basis for E , then
xE A
x EA
so that ~'*
U*
whence the map is involutive. Take u C s with ~=0. Then o = ~((.Ix)y) = ~ o ((-Ix)y) = (-Ix)~y
for every (x, y) e E x F (Proposition 5.3.2.13 c)), so that ?_t - -
0 ~
whence the map is injective. c) Take (x, y) E E x F . Then ('IY}Y C s (.ly)y((-Ix)y)-
((ly)y)o
and
((.J~)y)=
(-I~)y
(Proposition 5.3.2.13 e)), so that
(.[x>y C {~vl(u, v) C s
x s
Thus
C~(E, r ) c {~1(~, v) e c,(F) • C~(E, r)} •
54
6. Selected Chapters of C*-Algebras
(Corollary 5.5.1.11) and
E2(E,F) = {'Svl(u,v ) E CP(F) •
F)}X•
(Theorem 6.1.3.12 d)). Hence the representation in b) is non-degenerate. d) is easy to see. e) If v is not injective, then, by d), it is not cyclic for {~lu E L:P(E)}. Suppose that v is injective. Let
ZOn('lXn)Yn nEN
be a Schatten decomposition of v. Since v is injective, (x,)n~N is an orthonormal basis of E . Take (x,y) E E • F and e > 0. There is an m E N such that C2
~--~ ](XnlX) 12 < 1 + Ilyll 2
nEN n>m
Define
nE1Nm
Then u E s
On
(Proposition 6.1.2.3) and o~ nElNm
nElNm
(Proposition 5.3.2.13 e)). Hence
nEN n>m
nEN n>m
nEN n>m
(Proposition 6.1.2.3). Since e is arbitrary,
' o~(.Iz~)~
=
6.1 LP-Spaces
(.Ix)y 9 {~vl~ e ~ ( F ) } •
55
.
It follows that
s
C {~vlu E s
•177
= {~vIu E s
•
(Corollary 5.5.1.11) and that
s (Theorem 6.1.3.12 d)).
f) First suppose that v is a cyclic vector for {~[u E /2P(F)}. By e), v is injective. By Corollary 5.5.2.3, the Hilbert dimension of E is less than that of F . Let
hEN
be a Schatten decomposition of v. Then (X~)~CN is an orthonormal basis of E and E is separable (Corollary 5.5.2.7 b ::~ a). Now suppose that E is separable and that the Hilbert dimension of E is less than that of F . Let (Xn)nEN be an orthonormal basis of E and (Yn)nEN be an orthonormal family in F . By Corollary 5.5.2.7 a =~ b, we may assume that N C IN. Put
nEN
Then v is injective and belongs to s
F). By e), v is cyclic for
{~1~ e ~ ( F ) } .
s
6. Selected Chapters of C*-Algebras
56
6.1.5 T h e T r a c e ( 0 )
T h e o r e m 6.1.5.1
Take u E E l ( E )
and let (xe)ee,, (Ye)ee, be families
in E such that
eel
eel
Then
xEA
for every orthonormal basis A and is denoted by t r u .
eel
of E .
This number is called the trace o f u
The operators of L I ( E )
sometimes are called trace
operators on E .
We have
xEA tEI
eEI
(Proposition 5.5.1.19), so that
xEA
tCI xEA
xEA eEI
tEI
(Theorem 5.5.1.15 a =~ c) ).
Corollary 8.1.5.2
( 0 ) C~
m
~ e ~(E, F) a ~
(~, y), (a, b)c E • F ,
t~ (~ o ((-ly)x)) = (~xl~) = tr (((.ly)x) o ~), tr (((.la)b) o ((.ly)x)) = (xia)(bly) .
Now t~ (~ o ((-ly)x)) = t~ ( ( . l y ) u x ) = ( u x l y ) ,
tr (((.iy)x) o u) -- tr ((.fu*y)x) = (xlu*y) - (uxly) ,
(Proposition 5.3.2.13 c),d), Theorem 6.1.5.1), and so tr (((.la}b) o ((.ly)x)) = ((xla)biy) = (xla)(b y) .
m
57
6.1 s
Corollary 6.1.5.3
(0)
The map LI(E)
> ]K,
u,
> tru
is a continuous involutive linear form with norm 1 (Corollary 6.1.3.1~) and t~ ~ -
II~lli
for any positive element u o f / ~ I ( E ) .
By Theorem 6.1.5.1, the map is continuous with norm 1. By Proposition 5.3.2.13 b) and Theorem 6.1.3.12 d), it is involutive. Finally, by Corollary 6.1.2.13 a :=~ b, t r u = Ilulll for every positive element u of s
I Corollary 6.1.5.4 ( 0 ) Let u be a positive operator on E and take p 9 [1, oc[. Then u 9 E,P(E) iff u p 9 L I ( E ) and in this case ml lJ
=
By Corollary 6.1.2.18, u E/:P(E) iff u p E/21(E), and in this case
By Corollary 6.1.5.3, m
I llp --
Corollary 6.1.5.5
( 0 ) If u is a self-normal operator in f~l(E), then t r u -- E O~L ~EI
for any characteristic family (c~)~ex of eigenvalues of u.
By Proposition 6.1.1.2 a =~ b, there is an orthonormal family (Xt)tE I in E such that ~.EI
Corollary 6.1.5.6
( 0 ) If u , v C f-,2(E,F) then v* o u C s
(Theorem 6.1.~. 7 a) ).
and
58
6. Selected Chapters of C*-Algebras
That v* ou E s follows from Theorem 6.1.3.1 b) and Theorem 6.1.3.21. If A is an orthonormal basis of E , then
tr(v'ou)=
~(v*uxix)
-- ~-~(uxlvx) -- (uiv )
xEA
xEA
(Theorem 6.1.4.7 a)). P r o p o s i t i o n 6.1.5.7
I ( 0 ) If u E ]~(E,F) and v E s
uov 6 s
y o u >_ 0==~ y o u 6 s
vou E s
u o v > 0 ==~ u o v E s
uov E s
v ou E s
~
then
tr (u o v ) = tr ( v o u ) .
By Corollary 6.1.3.5, there are orthonormal bases (x~)~et and (Y~)~eL of E and F , respectively, such that ((vux~lz~))~et is summable iff ((uvyxlyx))~eL is summable, and in this case
~EI
)~EL
The final implication now follows. The first two implications follow by Corollary 6.1.4.4. I Corollary 6 . 1 . 5 . 8
( 0 ) If u E f--,l(E) and v" F --~ E is an isomorphism,
then t r (v - 1 o u o v ) = t r u .
By Proposition 6.1.5.7 (and Corollary 6.1.3.13 b) ), t~ ( ~ - ~ o ~ o v) = t~ (u o ~ o ~ - 1 ) = t~ ~
Corollary 6.1.5.9
Take u,v E ReE(E)
such that one of the operators is
compact and uv E s I ( E ) .
a) vu E / : I ( E ) ,
tr (uv) - tr (vu) E JR.
b) u ~_ O, v ~ 0 =:~ tr (uv) -Ilv 89189 c) u > 0 , v > _ 0 , t r ( u v ) = 0 = ~ u v = v u = 0 .
I
~ Ilu891892
6.1 f3-Spaces
59
a) We have w
=
v * ~ * = (~v)* e L~(E)
(Theorem 6.1.3.1 b)), and tr (uv) = tr (uv)* - tr (v'u*) = tr (vu) = tr (uv)
(Corollary 6.1.5.3, Proposition 6.1.5.7), so t~ (~v) c l a . 1
.
b) By a), we may assume u to be compact. Then uv~ is compact. By 1 1 . 1 1 Corollary 5.3.3.9, v~uv~ is positive. Hence by Proposition 6.1.5.7, v~uv~ C s and 1
t~ (~v) - t~ (v 89189 - IIv89189
1
1
1
>_ Ilv-~v~ll
1
-
1
= IIv~~
II = II~v~ll ~
(Corollary 6.1.5.3, Proposition 6.1.2.4, Theorem 5.3.1.4). c) By b), 1
U~V
1
~
-
-
0,
SO
1
1
uv=u~(u~v~)v~
-0,
vu = (uv)* = O.
m
P r o p o s i t i o n 6.1.5.10 ( 0 ) Let p,q be conjugate exponents and let s be endowed with the II.llp-norm. If u ~ is a continuous linear f o r m on s
then there is a unique u C s ~'(~) - t~ (~ o v)
for every v C s
F ) . Furthermore,
Ilullq = Ilu'll.
E) such that
6. Selected Chapters of C*-Algebras
60
Given y E F , define
~-E
~,
~'
>r
(Proposition 6.1.2.3). Take y E F . uy is linear and
]~(z)i-
iu'((.lz)y)l < liu'll ii(-ix)yll, = liu'i] ] xil li~il
for every x E E (Proposition 6.1.2.3), i.e. uy is continuous and
Ilu~ii ___ Ilu'il Ilyli. By the Fr6chet-Riesz Theorem, there is a unique ~ E E such that
Then
ii~l,- il~ii < Ilu'll ilyll. Define u'F
~E,
y,
>~.
Choose y , z E F and a, fl E IF(. Then
( ~ l+~ " 7 7 " ~
= ~o~§
= ~r
= ~(~)
+ ZUz(~) -
= u'((.Iz)(~y + ~z)) = Zr
=
~ ( ~ i ~ + Z(xlz-) = ( x i ~ + Zz-)
for every x C E , so that u ( a y + flz) = a y + flz = a ~ + fl-5 = a u y + fluz .
Thus u is linear. By the above considerations, u is continuous and liu]] _< ilu'l].
We have u'((.ix)y) =
u~(x) =
(~1~ = (~y ~) = t~ (~ o ((-Ix)y))
for every (x, y) E E x F (Corollary 6.1.5.2), so that
6.1 s
61
~'(v) = t~ (~ o ~)
for every v E L s ( E , F ) (Corollary 5.5.1.11). Let (xt)tes, (Yt)<e, be orthonormal families in E and F , respectively. If we endow IK (I) with the norm induced by ~P(I), then
tel
tEI
= I E s(,>~,<<.ix:>~,)I --I~,<~ s<~)<.l,,>~)l _< tel
_<
tel
I1~'11 IIz
#.,El
s(~)<-I~>,:ll, -II~'il I Sll,,
for every f E IK (I) (Corollary 6.1.2.12). We deduce that ((uytlx~))~es E f q ( I ) ,
((~y~lx~>)~, Iq ~< 11~'11 (Proposition 1.2.2.2). Since (x~),e~, (Y~),EI are also arbitrary, it follows that
(Proposition 6.1.3.10 and Ilu[I _< Ilu'll ). Given v E / : ] ( E , F) lu'(v)l = Itr (u o v)l ~ I[u o viii ~ IlullqllVTI
(H61der's Inequality), so that
Ilu'll < Ilullq, Ilu'll = Ilullq, The uniqueness of u follows from Theorem 6.1.3.12 d). T h e o r e m 6.1.5.11
( 0 ) Take p , q , r E [1, c~] with 1 r
1 p
- <_ - + - ,
1 q
u E I2P(E,F) and v E I2q(F,G). Then v o u E s
IIv o ullr ~ Ilullpllvllq
and
(H61der's Inequality).
62
6. Selected Chapters of C*-Algebras
By Corollary 6.1.3.13 b) and Proposition 6.1.2.4, we may assume that p , q , r E [1, oo] and
1 -p
+--
1 q
1 r
--
--,
First suppose that p, q E [2, cx)[. By Corollary 6.1.3.2 and Corollary 6.1.2.18,
~, o u e L~ ( E ) ,
v o ~* e L~ ( a ) ,
IIv o v*ll~ = Ilvll'q. Let (xt)te,, (Yt)tE, be orthonormal families in E and G , respectively. Then
tel
tel
tel
(E tlu~,ll~)~( ~ iiv'~ll~):=
-<
tEI
tEI
= (s
<_
tEI
tEI
<_I1~.ooll IIv o v * I1~ = rIlull:llvllqr (Proposition 6.1.3.10). We deduce that v o u E s
G) and
llv o ~ I~ __ Ilull, I vll~. Now suppose that p < 2. Take p', r' E [1, co] with 1 1 P-+~=I
1 1 and -r + r ~
1.
Then q > 2, r < 2, and r' > 2. Choose w E I : I ( G , E ) . Since 1 -+ q
1 1 1 1 . . . . . . +-1 . . r' r p r
.
p-1 . p
1 p-1 _..p__
it follows by the above considerations that w o v E/:p-1 (F, E) and
6.1 s
I1~ o ~ 1 1 ~
63
< Ilvll~]l~ll~,
By Theorem 6.1.3.21, w o v o u 9
and
IIw o v o ~11~ _~ II~llpllw o vll~
By Proposition 6.1.5.10, v o u 9
~_ II~llpllvll~llwll~,
9
G) and
IIv o ull~ _< II~ll,llvll~.
m
Remark. By the above theorem and Proposition 6.1.2.4, /:P is a Banach category for every p 9 {0} U [1, cx~]. Hence, by Corollary 6.1.3.13 b), it is even a Banach /:-category.
C o r o l l a r y 6.1.5.12
I f p 9 [1, co[, u 9
and n 9 IN then
u ~ ~ t:~ ( E ) , and
II~nll~ __ Ilull~.
The assertion follows from Theorem 6.1.5.11 by complete induction. C o r o l l a r y 6.1.5.13
( 0 )
Take p , q , r 9 [1,cx~[ with
1 p and u 9
i
1 q
1 r
F ) . Then the following are equivalent:
a) u 9 / : r ( E , F ) . b)
There are v C/:P(E)+ and w 9
~=~o~,
Ilvl,=
such that
~
,
II~llq=
~
9 T
c)
There are v E s
and w C s
such that
r
d)
r
There are a Hilbert space G and operators v E / : P ( E , G), w E / : q ( G , F ) with ?J,~--WOV.
6. Selected Chapters of C*-Algebras
64
a ~ b. Put s := (u* o u)~. Then s is positive and, by Corollary 6.1.3.2, s
llsll~-llull~.
e s
By Corollary 6.1.2.18,
~ e t:~(E),
ll~ll~: li~lL~-~ ilu]l~
~-(~),
By Proposition 5.5.5.10, there is a t E/2(E, F) such that u=tos,
ilti]_
Define v:=s~,
w:=tos~.
Then Wo?.):U,
(Corollary 6.1.3.13 b) ). By HSlder's Inequality,
ll~Jl~< i,vl,, I~ll,_< II~fl~fluil~- iI~ll~, r
r
SO
I~wllq- Jlull~ r
a ~ c. By Corollary 6.1.3.2,
u* E s By a ==~ b, there are w E s
Ilu*ll~
= Ilull~.
and t C/:P(F, E ) , such that
r
6.1 s
65
i, ll = Ilu*ll = Ilull r
r
Put V .m t* Then U---WOV.
By C o r o l l a r y 6.1.3.2, r
v E s
Ilvllp = Ittlb =
II II u
r
b =~ d a n d c =~ d are trivial.
I
d =~ a follows f r o m T h e o r e m 6.1.5.11.
Remark.
For p -
q = 2 and
r =
1, we get c h a r a c t e r i z a t i o n s of n u c l e a r
o p e r a t o r s by m e a n s of H i l b e r t - S c h m i d t o p e r a t o r s .
Proposition
6.1.5.14
If IK = ~ ,
then u 6 s
is positive iff
re tr u > re tr (vu)
for every unitary operator v on E . If u is positive, t h e n re tr (vu) ~ Itr vu I ~ Ilvulll ~ Ilvll I1~11~ - t r ~ - re tr u for every u n i t a r y o p e r a t o r v on E
( C o r o l l a r y 6.1.5.3, C o r o l l a r y 6.1.3.13 b) ).
Now suppose t h a t re tr u k re tr (vu) for every u n i t a r y o p e r a t o r v on E . Take v C Re s
21~1 Ilvll < 1. Define
w "= i ~ v ,
a n d a C IR w i t h
66
6. Selected Chapters of C*-Algebras
U1
:=
(1 + w)(1
-
w) -1 ,
u2 := (1 + w ) - l ( 1 -
w),
u3 : - ul - 1 - 2 w . Then W* =
UlU ~ :
--W
UlU 2 --
~
U 1 --" U 2 ,
U~Ul = U2Ul = 1,
1,
i.e. ul is u n i t a r y . We have oo
u3=(I+w)
oo
oo
Ew~-I-2w=Ew~+Ew~-I-2w= n----O
n:0
-- 2 E w n
n:l
'
n--2
so t h a t oo
I1~11 _< 2 ~
n=2
Ilwll~ = 2 ~llvll~ 1 -I~1 Ilvll ~ 4~llvll~
H e n c e we see, successively, t h a t
r e t r u >_ r e t r ( u l u ) = r e t r ((1 + 2w + u3)u) =
= re tr u + 2 a re tr (ivu) + re tr ( u 3 u ) ,
2c~ re tr (ivu) + re
t r (u3?.t) ~
O,
2 a re tr (ivu) - 4c~211vl12[lu[]~ < 2c~ re tr (ivu) - Ilu311 [lull1 _<
_< 2 a re tr (ivu) - Ilu~ull~
_~ 2~
re tr (ivu) + re tr (u3u) _< 0
2 a re tr (ivu) <_ 4a211vl12]]ul[1. ( C o r o l l a r y 6.1.3.13 b)). Since c~ is a r b i t r a r y ,
6.1 s
67
re tr (ivu) = O,
im tr (vu) = re ( - i t r (vu)) = - r e tr (ivu) = 0
t~ ( ~ )
x 9 E ~
E IR,
( ~ x l ~ ) = t~ (~ o ( i - I x ) x ) ) =
= t~ ( ( ( - I x ) * ) o u) 9
(Corollary 6.1.5.2, Proposition 6.1.5.7, P r o p o s i t i o n 5.3.2.13 b ) ) . By Corollary 5.3.3.3, u is selfadjoint. By Corollary 6.1.2.13 a =~ b & d, there is an o r t h o n o r m a l family in E and a characteristic family
(O~n)nENof eigenvalues
(~)~
E e~(N),
= E
o<.lxo>xo
(Xn)nEN
of u such t h a t
nEN
c~n E ]R for every n E N . Take v E s
VXn
_{xo --X n
such t h a t
i f O~n>O if an<0
for every n E N and VX
-- X
for every x E {xnIn E N } • . T h e n v is u n i t a r y and
w = E
= E
nEN
I ol<.,xo>xo.
nEN
It follows t h a t E hEN
an -
tru =
r e t r u >_ r e t r v u = E
]anI
nEN
(Corollary 6.1.5.5). Hence an E IR+ for every n E N , and so u is positive. I
6. Selected Chapters of C*-Algebras
68
Corollary
6.1.5.15
Let IK =(~. Take u E s vu =
and v E U n E .
Then
lul
re tr(vu) >_ re tr(wu)
for every w E Un E . We have re tr(vu) >_ re tr(wu) for every w E Un E iff re tr(vu) >_ re tr(wvu) for every w E Un E . By P r o p o s i t i o n 6.12.5.14, the last relation is equivalent to
vu >_ O. If vu is positive, t h e n (~u) ~ = (w)*(~)
= u*~*~u
= ~ * ~ = I~l ~ ,
SO
vu = lul
Conversely, it follows from
vu
=
lul
t h a t vu is positive.
Corollary 6.1.5.16
I Let IK = ~ .
Take u E s
operator on E . Then re tr(vu) = Ilui]l
vu
=
lul
and let v be a unitary
6.1 f3-Spaces
69
Given any unitary operator w on E , re tr(wu) <_ Itr(wu)l <_ Ilwull 1 ~_~ Ilwll Ilutli--IlUlll
(Corollary 6.1.5.3, Corollary 6.1.3.13 b)). It follows from re tr(vu) = I171,111 that w
= lul
(Corollary 6.1.5.15). Conversely, it follows from vu = lul
that IlUlI1 --II lUl II1 -- trl?~I : tr(v?~) -~ re tr(vu)
(Corollary 6.1.3.2, Corollary 6.1.5.3).
Corollary 6.1.5.17
m
Let IK =(F, and take u, v C s
such that at least one
of the operators is nuclear. If
retr(uv) >_ re t r ( w u v ) ,
re tr(uv) >_ retr(uwv)
for every unitary operator w on E , then uv and vu are positive. uv and vu are nuclear (Corollary 6.1.3.13 b)). Given a unitary operator w
on E , re t ~ ( ~ )
> r~ t ~ ( ~ )
and re
t~(w) =
re t ~ ( ~ )
> re t ~ ( ~ )
= r~ t ~ ( ~ )
(Corollary 6.1.3.13 b), Proposition 6.1.5.7). By Proposition 6.1.5.14, uv and m
vu are positive,
P r o p o s i t i o n 6.1.5.18 for F = E and p (Corollary 6.1.5.3).
The representation defined in Proposition 6.1.~.10 b)
1 is the representation of s
associated to the trace
70
6. Selected Chapters of C*-Algebras
Take u,v E L I ( E ) . Then u, v E s orthonormal basis A of E ,
xEA
(Proposition 6.1.2.4) and for any
xEA
The assertion now follows.
I
Remark. By Proposition 6.1.4.10 f), this representation does not always admit a cyclic vector. This does not contradict Theorem 5.4.1.2 k), since f.)(E) is not quasiunital (Proposition 6.1.3.13 d)).
P r o p o s i t i o n 6.1.5.19 ( 8 ) Let n E IN and A E {]R,r and denote by A n the Hilbert right A-module e2(]Nn, A) (Example 5.6.,~.2 a)). Put n
Earr r--1
cp'An,n
>IK,
a~ ~
ifA-IK
n
2re~a,.r
iflK=IR
and A = C
r=l n
4re.art
if IK = IR and A = IH
r--1
and denote by E the Hilbert space obtained by endowing the vector space A n with the scalar product An • An
~ IK,
(~, ~ ) ~ - ~
~[r/~t],,te~n
(Proposition 5.6.2.5 a),e), Proposition 5.6.6.17 a)). For every a E An,n, tr~=~(a), where ~d'E
>E ,
~:
>a~.
In particular,
for all a, b E An,n.
Denote
B :=
{1}
if A - I K
{1,i}
i f l K = ] R and A -
{1, i , j , k }
if I K - I R and A = ] H .
6.1 f_,P-Spaces
7/
Fix r E INn and put er := [Srs]se~ E A n ,
B~ :: {~Y I Y e B } . Then <~rl~> = ~
~
a~,~r,
s:l
= a~r .
t=l
Since 0 Br is an orthonormal basis of E (Corollary 5.6.4.11 e)), r=l n
r=l
n
~76Br
r=l
n
yEB
n
--"
r-1 yEB
Y
rrY -- ~(a).
r=l yEB
The final assertion follows from Proposition 6.1.5.7.
I
72
6. Selected Chapters of C*-Algebras
6.1.6 Duals of f_.P-spaces
Throughout this subsection, p E {0} t2 [1, c~] and q is the conjugate of p. ( 0 ) Given u E f_~q(F~E), define
Definition 6.1.6.1
P
:=~ "s
--+ IK,
v,
) tr(uov)
(Theorem 6.1.3.21).
Proposition 6.1.6.2
( 0 ) Given any (a,b), (x,y)c E x F , u e LP(E, F ) , and v e f~q(F, E ) , ('ly)~(u) = ( ~ z l y ) ,
(:~((.l~)y)
= (al~)(ylb)
Now
~((lz)y)
= t~(~ o ((-Iz)y)) - ( v y l ~ ) ,
('lb)a(('lx}y) = ((('lb)a)ylx)= (ylb)(alx}
(Corollary 6.1.5.2). Corollary 6.1.6.3
I ( 0 ) Let E be a complex (real) C*-algebra and (H, (p)
the (complex) universal representation of E . Then for every x' E E' there is a u E f~l(H) such that oo
x' - ~o~. The assertion follows from Corollary 5.4.2.4 (Proposition 5.4.2.6 e)) and Proposition 6.1.6.2. I
6.1 f_.P-Spaces
C o r o l l a r y 6.1.6.4
73
If ~, ~ are filters on E and F , respectively, which con-
verge weakly to 0 and such that E # C ~, F # C ~ , then 0
<.Ix>y(x,y),~xa lim
o
in the topology of pointwise convergence.
Given u c K:(F,E) and (x,y) E E x F , 0
l(.Ix>y(u)l- l<~ylx>l < ll~y I II~II (Proposition 6.1.6.2) and the assertion follows from Theorem 5.5.5.1 a :=> c. I P r o p o s i t i o n 6.1.6.5
If u E s
then = <'1~*>.
Given v E s
F), ~(~) = t~(~ o v) = <~1~*>
(Corollary 6.1.5.6). T h e o r e m 6.1.6.6
I P
(O)
If u c L q(F,E), then ~ C E~V(E,F)' and the map P
~_~q(F, E )
> P~P(E, F ) t ,
721
>
is a norm preserving linear map. If p ~ c~ , then it is an isometry.
is linear and if v C s
F ) , then
IT(v)[- [tr(u o v)] _< ]Iu o VIII
~
Ilu[lq[IV[[p
(Corollary 6.1.5.3, Theorem 6.1.3.21). Hence ~ C s F)'. The final assertions follow from Proposition 6.1.5.10, Theorem 6.1.3.12 d), and Corollary 5.2.3.4. I
Remark.
~I(F~E)
We will often identify /C(E, F)' with
using the isometry
0
L ~(F, E )
and s
with s
> t:(E, F)'
~ ~-~
using the isometry 1
s
>E ) ( F , E ) ' ,
I ~ ( E , F ) " is thereby identified with E ( E , F )
u,
>~.
and ~:I(F,E)" with E ( E , F ) ' .
6. Selected Chapters of C*-Algebras
74
Corollary 6.1.6.7
( 0 ) EP(E,F) is reflexive for every p C]l,c~[. s and f_.(E, F) are dual spaces,
Corollary 6.1.6.8
m
( 0 ) Invoking identifications of the Remark of Theorem
6.1.6. 6, the inclusion 1C(E, F)
~ f~(E, F)
is given by the evaluation map.
Let j" K:(E, F)
>s
F)
be the inclusion map and take u E ]C(E, F). Then, given any v E El(F, E), (u, v) = ~(u) = tr(v o u) = tr(u o v) - tr((ju) o v) = ju(v) = (v, ju)
(Proposition 6.1.5.7). Hence, the inclusion is the evaluation map. Remark. Assume that E and F are infinite-dimensional. By Corollary 5.5.2.8,
]C(E, F) is not a complemented subspace of s F). By the above corollary and Corollary 1.4.2.16 c ~ a (and the Murray's Theorem) K:(E,F) is not a dual space. If fact, this was already proved in Proposition 6.1.3.23. Corollary 6.1.6.9 a)
The following are equivalent:
One of E and F is finite-dimensional.
b) K:(E, F) =/:1 (E, F) = / : ( E , F). c) One of IC(E, F), /:I(E, F), and L(E, F) is reflexive. d) 1C(E,F), L:I(E, F ) , and f_.(E, F) are reflexive. e) /C(E,F) is a dual space. a =~ b. Take u E s
F). From U - - U O l E - - 1FOU
we get that u C/C(E, F) (Proposition 3.1.1.11), so that /C(E, F) - E(E, F).
6.1 s
Take u 9
75
F) and let
nEN
be a Schatten decomposition of u (Theorem 6.1.3.1 a)). By a), N is finite, so that u 9 s F) (Corollary 6.1.2.12). Hence ;C(E, F) = s
F)
(Corollary 6.1.2.10). b =~ c. By Corollary 6.1.6.8, K:(E, F) is reflexive. c ~ d follows from Theorem 6.1.6.6 and Proposition 1.3.8.4. d =~ e is trivial. e ~ a. Let I be a predual of ;C(E,F). By Theorem 6.1.6.6, s F) is isometric to the tridual of I . Hence, by Proposition 1.3.6.19 b) (and by Murray's Theorem), ;C(E,F) is a complemented subspace of s By Corollary 5.5.2.8, one of the spaces E and F must be finite-dimensional, m Corollary 6.1.6.10 ( 0 ) If we identify s of Theorem 6.1.6.6), then the mapping s
with s
>s
u,
(Remark
>
is the evaluation map. If u e s
and v 9 s
E), then
1
cx~
<~, ~ - t~(v o ~) - t~(~, o v) - <~, v>
(Proposition 6.1.5.7).
m
Corollary 6.1.6.11 ( 0 ) The ultraweak topology (which will be introduced and studied in Subsection 6.3.1) on s F) is obtained by identification with s (Remark of Theorem 6.1.6.6). Let .~ be an ultraweak closed vector subspace of s F) and q: ~(F,E)
~ ~(F,E)/~
the quotient map. Then the map
(L'(F,E)/~ is an isometry.
'
>7,
~',
> 4x'
6. Selected Chapters o.f C*-Algebras
76
The assertion follows immediately from Corollary 1.3.5.5. 0
Proposition 6.1.6.12
Given u E s
oo
the linear f o r m s ~ and ~ are
order continuous.
Let
hEN
be a Schatten decomposition of u (Theorem 6.1.3.1 a)), jc a downward directed set in s
with infimum 0 and ~ its lower section filter. By Theorem 5.3.3.14
c), lim(vynlxn) - 0 v,~
for every n E N . An easy consequence is that oo
limE(v) = 0 . v,~ (x)
Hence ~ is order continuous. Since K:(E) is a hereditary set in s
(Corollary
0
3.1.1.13, Proposition 4.3.4.5 e)), ~ is also order continuous. C o r o l l a r y 6.1.6.13 ~(E)'=
tg(E) 'r =
If IK =(~ (IK = IR), then
{u'lu' is an order continuous linear f o r m on K~(E)}, (Re E ( E ) ' - Re K:(E) ~ =
= {u'iu' is a selfadjoint order continuous linear f o r m on K(E)}).
The inclusion ]C(E) ~ C {u'lu' is an order continuous linear form on K:(E)},
(ReK:(E) ~ C {u'lu' is a selfadjoint order continuous linear form on ~ ( E ) } ) is trivial. The inclusion {u'lu' is an order continuous linear form on K:(E)} C K:(E)'
I
6.1 s
77
{u'lu' is a selfadjoint order continuous linear form on K:(E)} c Re K:(E)')
follows from Proposition 1.7.2.4 b) and Theorem 4.2.2.9 a). Take u' E K:(E)~_. By Proposition 6.1.6.12, u' is order continuous and so it belongs to K:(E) '~ Hence K:(E)' C K:(E)"
(Re K:(E)' C ]C(E) ~)
(Corollary 4.1.2.7 d)).
m
If E is infinite-dimensional, then the supplementary hypothesis in
Remark.
the real case is necessary. Indeed, if we put 9~ "= {u e K : ( E ) I u * - - u } and consider a non-continuous linear form on ~ , then its linear extension to K:(E) restricting to 0 on Re ]C(E) is a non-continuous order continuous linear form on K:(E). P r o p o s i t i o n 6.1.6.14
a)
UV
b)
For x, y E E ,
m
Given u, v c s
the following are equivalent:
V?.t.
(~lv*y>
= (~l~*y>.
In the complex case, the above assertions are equivalent to:
c)
x C E =~ (uxlv*x) = (vx]u*x). ~
b.
<~xl~*y> =
(vuxl~> =
.
(~vxly> =
b =v a. By Proposition 6.1.6.2, 1
~
-
v~(<-Iy>x)
= (~zly>
-
(wxly>
= -
(~zl~*y)
= 0
1
for all x , y c E . Hence therefore on s
uv-
vu vanishes on s
UV
i.e.
(Corollary 5.5.1.11) and
(Theorem 6.1.3.12 d)). By Theorem 6.1.6.6, --
V?.t - -
O,
6. Selected Chapters of C*-Algebras
78
UV =
Vlt.
b ==> c is trivial. c~b.
The map
is sesquilinear. By c),
~(~, ~) = o for all x E E . By the polarisation identity, ~ vanishes identically.
m
Let ~2 be the class of Hilbert spaces and ICn the subdefined by the map
P r o p o s i t i o n 6.1.6.15
category of s
(E, F) ,
~ tg(E, F)
on $22 . If we identify tg(E,F)" with F_.(E,F) for all E , F e Y2 (Remark of Theorem 6.1.6.6), then the Arens multiplications of 1C~ coincide with the multiplication in gr~. m
6.1 f_.P-Spaces
79
6.1.7 E x t e r i o r M u l t i p l i c a t i o n a n d s
Throughout this subsection, p 6 {0} U [1, co] and q is the conjugate of p.
(0)
D e f i n i t i o n 6.1.7.1
Define
E - ~ F :r
u 6 s
E -~ F :<-->, u' e s
E)'.
p
Suppose that E--+ F , F-Y+G, H - ~ E .
We define
p
vu' . C'(a,E)
~ ~,
~ ,
~ <~ o v, r
u'w" s
> ]K,
u,
> ( w o u , u'>,
~'*. L ~ ( E , F )
> ~,
~,
,
.~ (~*, ~'>
(Corollary 6.1.3.13 b), Corollary 6.1.3.2 a :=v b)). If E - F = G - H , then the above definitions coincide with the definitions given in Proposition 2.2.7.2 and Definition 2.3.1.1. P r o p o s i t i o n 6.1.7.2
Suppose that F Y~ E . If u' C ]C(E,F) ~ , then oo
UU' -- O ~
U' V -- 0
for every u e ~ ( E , G) and v 6 K:(G, F ) , and 7rG• U ! :
U! ,
Ul7rH •
--__ U !
for all finite-dimensional vector subspaces G of E and H of F . If G ~ F
(resp. E - ~ G ) , then
<~, ~'> = <~ o
u, ~'> = 0
(<~, u'v> = (~ o ~, ~'> = o) (Proposition 3.1.1.11), so
uu' = 0
(u'v - 0).
It follows that
u' - (rG + ~c•
= rcu' + ~
u' = u'(rH + rH•
= u'~
+ u'~
u' = ~c" u', = u'~H~ .
I
6. Selected Chapters of C*-Algebras
80
Proposition 6.1.7.3
u!
( 0 ) Suppose that E - - + F . P
a) If F-~G , then E S~ G and IIvu'll <_ IIvll Ilu'll, P
b) If C ~ E , th~n a ~ F and Ilu'~ll _< Ilu'll IIvll. P
c) If F--~G-~H, then w(vu') = (w o v)u' . d) /f H-~G-Y+E, then (u'v)w = u'(w o v) . e) If F-Y+G and H--~E, then (vu')w = v(u'w).
f) 1F u !
--" U I 1 E
--- U t "
U I*
g) F~E, P
IIr
Ilu'll,
u'**- u'.
h) If F ~ G ,
then (vu')*= u'*v*.
i) /f G-Y+E, then (u'v)* = v*u'* a) Given u e s I(~, vu')l - I(~ o v, u')l _< II~ o vll,llu'll <_ II~ll,llvll Ilu'll (Corollary 6.1.3.13 b) ). b) Given u E s G), I(~, u'~>l - I(~ o u, ~'>l _< IIv o ~ll,llr
_< IIvll Ilull~llu'll
(Corollary 6.1.3.13 b)). c) ,d) ,e) ,f) ,h) ,i) are trivial. g) Given u E s I(u, ~'*>1 - I(~*, ~'>1 -< llu*l ~llu'll - II~ll~llu'll (Corollary 6.1.3.2), so FGE,
II~'*ll-< I1~'11.
P
The relation U I** - - 7./,!
is trivial. We deduce from the above inequality that il~'ll- I1~'**11 _< I1~'*11 < I1~'11, I1r
I
6.1 f~P-Spaces
C o r o l l a r y 6.1.7.4 Remark.
( 0 ) s
81
is an involutive unital f..(E)-module,
m
This assertion was already proved in Corollary 6.1.3.14.
C o r o l l a r y 6.1.7.5
( 0 ) If u' 9 s
then u*u'u 9 s
for every
e ~:(F, E). By Proposition 6.1.7.3 e),h),i), U*?2#U)
-- U*,tt#?_t.
Thus u*u'u is selfadjoint. Given v c s
<~*~, ~*~'~) = <~v*~*, u') = ( ( ~ * ) * ( ~ * ) , ~') e ~ + , and so u*u'u E s
W
C o r o l l a r y 6.1.7.6 Let F be a closed vector subspace of E , and take u' c s + such that U#7rF :
Then ~FU', 7rF•
T'F%t # ,
Es
By Corollary 6.1.7.5,
~':
~'-
~.~'~ e ~(E)~
Since ~-Flu' = (1 - WF)U' = u'(1 -- WF) = U'WF• it follows that 1rFlU' E s C o r o l l a r y 6.1.7.7 Let F be a closed vector subspace of E and take u' e T0(s such that U# TrF ~
7TF u# .
Then 7rF u# - - 0
or
7~F• ?ff - - O .
W
82
6. Selected Chapters of C*-Algebras
Using Corollary 6.1.7.6, it follows from ~t ! - - 71-F721 -~- 7 i ' F •
I
that
O <_ rFU' <_ 4 ,
O <_ rF•
<_ 4 .
By Proposition 2.3.5.4 a ::v c, there are a, 13 E IK such that 7TF?.t I =
OL~t I ,
7rF•
I --
~U I
By Proposition 6.1.7.3 c), arF~u' = r ~ ( a u ' ) = ~ F ~ r u ' =
0,
Z~Fu' = r F ( Z u ' ) = ~F~F~u' = 0,
and the assertion now follows, since a + ~ = 1.
Proposition 6.1.7.8 w E s G), then
m
( 0 ) If u e L q ( G , E ) , v
~=vou,
uw=uow,
e L(E,F),
and
(~)*=u*.
Given t e/:P(F, G), v ~ ( t ) = ~ ( t o ~) = t ~ ( u o t o ~) = t~(~ o ~ o t) = ~ z - ~ ( t )
(Proposition 6.1.5.7). Given t E s ~(t)
Finally, if t E s
F),
= ~ ( ~ o t) = t ~ ( ~ o ~ o t) = ~ " ~ ( t ) .
E ) , then
~*(t) = ~(t*) = tr(u o t*) = tr(t o u*) = tr(u* o t) = u*(t) (Corollary 6.1.5.3, Proposition 6.1.5.7).
Remark. By the above proposition, the mapping u ~-~ ~ is a functor of s modules of /:q into (s
m
6.1 EP-Spaces
Corollary 6.1.7.9
(0)
83
Themap
Lq(E) ..... >LP(E) ',
u,
P
> :u
is an involutive, norm-preserving homomorphism of involutive E(E)-modules (Corollary 6.1.7.4). It is an isometry unless p = c~.
P r o p o s i t i o n 6.1.7.10 ( 0 ) Take u E Eq(E) (IK = IR), then the following are equivalent: a)
u>0.
b)
vEE2P(E,F)~
I
(u E R e E q ( E ) ) . If IK = ~
P
~(v* ov) e lR+. P
c) ~ e ~,(E+) ~ ~(~) E ~ + . d)
(~xlx)
E
n~+ fo~ ~w~y x
E
E.
P
e) ~ is positive in the sense of Definition 2.3.~.1 (see Corollary 6.1.3.1~).
a =~ b. We have v* o v e EP(E) (Theorem 6.1.5.11), v*ov_>0,
u~1 o V* o v o u ~
1
>_ O
(Corollary 5.3.3.9), and
~(~* o ~) = t ~ ( ~ o v* o v) = t ~ ( ~
1
o v* o ~ o ~ )
e ~+
(Proposition 6.1.5.7, Corollary 6.1.5.3). 1 b ~ c. We have that v~ E s (Corollary 6.1.2.18), and so
~(v)- ~(~)
E ~+.
c :=v d. We have that (.Ix)x E f : ( E ) + , and so
(Proposition 6.1.6.2). d :=~ a follows from Corollary 5.3.3.7. b =:> e is trivial. e =:> d. We have that
(~xlx) = ~((-I~)x) = ~(((.Ix)x)* o ((-I~)x)) E ~ + (Proposition 6.1.6.2).
I
6. Selected Chapters of C*-Algebras
S4
C o r o l l a r y 6.1.7.11
( 0 ) If j is the evaluation map of L I ( E ) , then j u + = (ju) +,
ju- = (ju)-
for every u E Re/21 (E).
By Corollary 6.1.2.14, u +, u - E L:I(E) and I1~111 = Ilu+ll, + Ilu-II1.
By Proposition 6.1.7.8 and 6.1.7.10 a => e, j u E R e / : ( E ) ' and j u + , j u - E / : ( E ) ~ . Since j u -- j u + - j u - ,
llaull= llull~= llu+ll,+ llu-lll= llju+ll+ lju-ll we see that (ju) + = j u +,
(ju)- = ju-
(Theorem 4.2.8.13).
m 0
Proposition 6.1.7.12
( 0 )
The ,r
{(.Ix)~lx
E E})
is closed in the
topology of pointwise convergence.
Take 0
and let a be an ultrafilter on E , such that ~o(~) converges to some u' E K:(E)' in the topology of poinwise convergence. Since the set K:(E){~ is closed in the topology of pointwise convergence, we have u' E K:(E)~ (Corollary 5.3.3.7). By Theorem 6.1.6.6 and Proposition 6.1.7.10 e => a, there is a u E /:I(E)+ such that ~ = u'. By Corollary 6.1.2.13 a => b & c, there is an orthonormal family (Xn)neN(u) in E such that
nEN(u)
We may assume that 01 (u) =/- 0. Then
6.1 f_2-Spaces
85
0
= lim
<.lx>x(<.lxx>xl) =
lim I<xxlx>l 2
(Proposition 6.1.6.2). Take y E (x~) -L T h e n
?.ty-
~ On(U)(YlXn)X n E {Xl} -[- , nEW(u) 0
0 -- ( u Y l x l ) -- ~((.Ixl)y) -- lira ( . I x ) x ( ( . I x l ) y )
-
-lim(xixl)(yix ) x~ lim(ylx ) - 0 x,~
(Proposition 6.1.6.2). It follows t h a t
On(?.t) -- (?.tXn Xn) --"U((.IXn)Xn ) -0
-- lim ( . I x ) x ( ( . l x n ) x n ) x,~
- lim I(x x~)l 2 -- 0
x,iY
for every n E N ( u ) \ { 1 } , i.e. 0 0
U'-- U-
Proposition
6.1.7.13
0 1 ( U ) ( - ~ ~ X ~- ( . I ~ I ( ? s 1 8 9
I
( 0 )
0
~) ~o(~(E))- {<.lx>xlx E E, Ilxll- 1}. b)
To(K:(E)) is closed in T ( ~ ( E ) ) . a) Given x E E , define 0
:= (.l~)x. Take x' E ~-o()U(E)). By T h e o r e m 6.1.6.6, there is an u E L I ( E ) , such t h a t o x' -- ~. By Proposition 6.1.7.10 e ==~ a, u E s so by Corollary 6.1.2.13, there is an o r t h o n o r m a l family ( X n ) n c y
in E and an f E g l ( N ) + such t h a t
6. Selected Chapters of C*-Algebras
86
n E N ==~ f(n) ~ O,
= 3-: hEN
Take n E N . Then
f (n)xn < x' , so t h a t
x'= ~ e {<.I~>~ Ix e E, llxll= 1 } . (Proposition 2.3.5.4 a =:> b). Take x E E with Ilxll = 1. By Proposition 6.1.2.3 and Proposition 5.5.5.7
d), (.]x)x E LI(E)+,
ll<.Ix>~ll = i.
Hence 5 E r ( E ) . Let x' E /C(E)~ with x' _ 5 . By Theorem 6.1.6.6, there is 0
an u E s
such that x' = ~. By Proposition 6.1.7.10 e =v a u E s
so by Corollary 6.1.2.13, there is an orthonormal family (Xn)neg in E as well as an f E e 1(N)+ such t h a t
f ( n ) ~ O for n E N ,
u:E:(,,)<.l,o>xo. hEN
By Corollary 5.3.3.7, given y E {x} • ,
E
f(n)l(Ylxn)12= (uyly) <- ((('Ix)x)ylY)= O,
nEN
so that
{xnln E N} C {x} •177= {c~xlc~ E IK}. Hence Card N _< 1, and therefore there is an c~ E IR+ such that u = ~(.Ix>x.
6.1 s
87
We get 0
x' = ~ = aS. By Proposition 2.3.5.4 b ::v a, 5 C T0(E). b) follows from a) and Proposition 6.1.7.12. Corollary 6.1.7.14
ll
( 0 ) Assume E # {0}. We identify s
involutive s
with an via the norm-preserving homomorphism
of s
of involutive s (3O
~'(E)
>~(E)',
~,
>~
(Corollary 6.1.7. 9).
a)
The map
~(E)
>~I(E)',
~,
1
>~ :
is an isometry.
b) /:(E) is a W*-algebra with s c) Every element of s
as predual.
has a left and a right carrier and a polar repre-
sentation.
d)
Take u C /:I(E). The left and the right carrier and the polar representation of u as element of s
coincide with the corresponding notions
for u as an element of the predual of s (Corollary ~.~.2.17, Theorem ~.~.3.5). In particular, the moduli of u in f~l(E) and s coincide.
a) follows from Corollary 6.1.7.9. b) follows from a). c) By b) and Theorem 4.4.1.8 j), every element of /:(E) has a left and right carrier. By b) and Theorem 4.4.3.1, every element of /:(E) has a polar representation. d) follows from b) and the definitions, m T h e o r e m 6.1.7.15
( 0 ) Let u be a self-normal operator on E, j ' a ( u ) - - +
IK the inclusion map, and B the Banach algebra of bounded Borel functions on o(u)
.
6. Selected Chapters of C*-Algebras
88
a)
There is a unique involutive unital algebra homomorphism cp : 13 -~ f~(E) such that ~j -- u and
nEIN
nEIN
for every decreasing sequence (fn)nEiN in B+.
b) ~f - f(u) for every f E C(a(u)). Given f E B, define f(u) := ~ f . c) If f E B, then a(f(u))
C f(o(u)),
IIf(u)ll < I l f l l ~ ,
{u} ~ c { f ( u ) } c.
d) If f E B and if g is a bounded Borel function on f ( a ( x ) ) , then g(f(u)) = g o f ( u ) . e)
For every ~ > O, there is a finite family (a,),e, in a(u) as well as a family (p~)~e1 of orthogonal projections in E such that
f) /f IK = r
(IK = JR), then the vector subspace of s generated by the orthogonal projections is dense in s (in Re L:(E)).
g) Assume that IK = IR
(IK =q~ ) and let jr be the hereditary closed (and involutive) subalgebra of f~(E) generated by u. If f E B satisfies OEa(u):::=V lim f ( a ) = 0 ou--+O
then f (u) E Y .
h) Let IK = IR (IK = ~ ) . Let jr be a a-faithful (involutive) closed subalgebra of E,(E) containing u, ~ a a-faithful 5nvolutive) closed unital subalgebra of s containing u, and put
/30:={fEBIOEa(u)~
lim f ( a ) = 0 } .
a---~0
~a(u) Then
~(Bo) c .r,
~(B) c G.
6.1 s
89
i) If F:=Keru,
G:=Imu,
then ~(u) - ~:o~ ( ~ ) ,
~
j)
~(u) - ~(~)\:o~ (u).
~
If f E 13 and a C IK, then 7FKerf(u )
-
-
e~_~u) (U), f (0)
Ker (al - u) C Ker ( f ( a ) l - f ( u ) ) . In particular as(u ) is the set of atoms of u.
a) to h). By Corollary 6.1.7.14 b), /2(E) is a W*-algebra, and the assertions follow from Theorem 4.4.1.8 c),f),g), Corollary 4.3.2.5, and Proposition 4.3.4.11. i) The assertion follows from Corollary 4.3.3.14 and Propositions 5.3.4.1 and 5.3.3.13. j) By i) and d), 7FZerf(u)
--
e{o}(f(u)) -- (e{o} o f ) ( u ) -- e),(o ) (u).
Hence
(U) ~
--- e{c~}
7FKer(al--u)
e-: f (f(o~))
(?%) ---- 7 r K e r ( f ( a ) l _ f ( u ) ) ,
Ker (al - u) C Ker ( f ( a ) l - f ( u ) ) (Corollary 5.3.3.8.)
m /
C o r o l l a r y 6.1.7.16
~
0
\
) Let u be a self-normal operator on E . Then the
following are equivalent:
a) u is not compact. b)
There is an f E C(a(u)) such that
0 ~ Supp f and f ( u ) ~ 12f(E).
6. Selected Chapters of C*-Algebras
90
There is a Borel set A of or(u) such that O ~ A and
c)
a =~ b follows from Corollary 5.5.6.6 b =~ a. b :=~ c. f is a linear combination of a finite family (fL)~eI in C(cr(u))+ such t h a t
L _~ Ifl for each ~ C I .
By b), there is a c E I
such that
L ( u ) ~ E l ( E ) . We may
assume that fL _< 1. Then there is a Borel set A of a(u) such that 0 ~ A and f, < eA(~) By Theorem 6.1.7.15 a),
o < f~(~) < ~(~)(~), and by Corollary 5.5.6.7, eA(U)(u) ~ E I ( E ) . c =~ a. Assume that u is compact. Then a ( u ) \ { 0 } is discrete (Theorem 3.1.5.1 c)) and so eA(u) is continuous. This and c) contradicts Corollary 5.5.6.6 a ::~ b.
1
Corollary 6.1.7.17
( 0 )
The following are equivalent for u E s
a)
u is compact.
b)
Every closed vector subspace of I m u is finite-dimensional.
F) "
a ~ b follows from Proposition 3.1.1.16. b =:v a. Assume t h a t u is not compact. Case 1
E-
F and u positive
By Corollary 6.1.7.16 a ~ 0~A
c, there is a Borel set A of a(u) such that
and
p - ~)(~) r ~:(E). Define
a :-dA(O) > O.
G "- I m p .
6.1 f~P-Spaces
91
p is an orthogonal projection in E and pu = up,
a2p <_ u 2
(Theorem 6.1.7.15 a)). Thus, given x E G,
Iluxll~ = = > ~2(pxlx > -
o?(xlx>
-
~=llxll ~
(Corollary 5.3.3.7), so that ___ ~11~11.
II~zi
By Proposition 1.2.1.18 a),c), the map G---+G,
x~
> ux
is injective and u(G) is closed. In particular, u(G) is a closed infinitedimensional vector subspace of Im u and this contradicts b). Case 2
E, F , and u are arbitrary
By Proposition 5.3.2.2 d :=v a, u o u* is not compact. By Case 1 (and Corollary 5.3.3.9), there is an infinite-dimensional closed vector subspace of Im (u o u*). Since Im (u o u*) C Im u, this contradicts b).
Corollary6.1.7.18
l
( 0 )
The following are equivalent for every u C
Sn~(E) " a) b)
u is atomic. There is an orthonormal basis A of E
as well as an f C g~
such
that u x - f (x)x for every x E A . In particular, every compact self-normal operator is atomic and every atomic operator is determined by its eignevalues and the corresponding eigenspaces.
92
6. Selected Chapters of C*-Algebras
a :=~ b. For a E ap(u), let A~ be an orthonormal base of E~ : - Ker ( a l - u ) and 7r~ - - ~rEo. By Corollary 5.3.4.5, A:=
U A~ aeap(u)
is an orthonormal set of E . By Theorem 6.1.7.15 j),
for every a E ap(u). Since u is atomic, 1E is the supremum of (Tra)~e~p(u). Take x E A z . Then ~'~x = 0 for every a E ap(u) (Corollary 5.5.1.16), so that
for every B E ~f(ap(u)) (Corollary 4.2.7.5). Thus,
where B ranges over gtj.(ap(u)) (Theorem 5.3.3.14 b =~ c ). Hence, A is an orthonormal basis of E . The existence of f
with the required property is
obvious. b =~ a. We have -1
Ker ( a l - u) = f (a) •177=: E~ for every a E IK. By Theorem 6.1.7.15 j), =
for every a E IK. Since A is an orthonormal basis of E ,
: V o~EIK
V o~EIK
Hence u is atomic. The final assertion follows from Theorem 5.5.6.1 a =~ d and Proposition 5.5.5.7 a).
I
6.1 s
93
C o r o l l a r y 6.1.7.19 Let u be a self-normal operator on E and f a bounded Borel function on or(u). -1
a)
If f (O) is not dense in a ( u ) , then f (u) 7~ 0 .
b)
If c~ E IK and if for every c > 0 the interior (with respect to a(u))
of
-1
f ( U ~ ( a ) ) is nonempty, then (~ e a ( f ( u ) ) . c)
If c~ e a(u) and if f is continuous at c~, then f(c~) E a ( f ( u ) ) .
a) Assume that f(~) = 0
Then Ifi2(u) = f * ( u ) f ( u ) = o, so we may assume that f ( a ( u ) ) C [0, 1].
Define
~ S 1 if f(a)-r
/ j:c~(u)
>IK,
0 c~,
if
f(ct)--O. >c~,
h:=j-jg.
( f l ) n ~ is an increasing sequence of bounded Borel functions on a(u) with g as supremum. Since f ~ (u) -- ( f ( u ) ) 1 -- 0
for every n E IN, g(~) = 0
6. Selected Chapters of C*-Algebras
94
-1
o(u) c h(o(~))c ~(~)n f (0)~ o(~) (Theorem 6.1.7.15 a),c)), which is a contradiction. b) Assume t h a t c~ ~ a ( f ( u ) ) . Then there is an ~ > 0 with U~(o~) M a ( f ( u ) ) = O . Let g be a continuous function on IF( equal to 1 on U ~ ( a ) and equal to 0 on I K \ U ~ ( f ( a ) ) .
Then g = 0 on a ( f ( u ) ) , s o
that
g o f(u) = g(f(u)) = 0 --1
(Theorem 6.1.7.15 d)). But g o f = 1 on f ( U ~ ( a ) ) . Thus, by a),
g o f(u) # o. Hence, oL E a ( f ( u ) ) . c) follows from b).
Corollary 6.1.7.20
Let u be a self-normal operator on E ,
a E a ( u ) , and
e > O. Then there is an x E E with
Define f " a(u)
> ]K,
/~,
)
S 1 if
Ia
L0
la-/3l _> ~,
j'a(u)-----+IK,
if
a:
-/31 <
>a.
By Corollary 6.1.7.19 a), f(u)#O. Since f ( u ) is an orthogonal projection in E , there is an x E E with
f(u)x=~,
i1~11= 1.
We have
I(J- af)fl
E
< -~
on a ( u ) , so that c
If(u -
ozf(u))f(u)[l "-
[I((J -
af)f)(u)][ -<2 -
(Theorem 6.1.7.15 a)) and c
llux- o~xl] -- l](u- o~f(u))f(u)xll ~
~]lxll < ~llxll .
I
6.1 s
C o r o l l a r y 6.1.7.21
95
I f u is a s e l f - n o r m a l operator on E , then
~(~)
c
((~xlx)l~ E E , I1~11 = 1),
I1~11 = sup I ( ~ l x ) l. xEE#
Take a E a(u) and c > 0. By Corollary 6.1.7.20, there is an x E E with
I1~-~11
< ~,
I1~11 = 1
We get that I(~1~)
- ~1 = I ( ~
- ~xl~)l
< ~.
Since c and a are arbitrary,
~(~) c { ( ~ l x ) l ~ E E , Ilxll = 1}. The second relation follows from the first one. Proposition 6.1.7.22
I
Let u be a self-normal operator or, the Hilbert space
E . I f 0 is a non-isolated point of or(u) then:
a)
There is a sequence (Pn)nc~ in {u} c n P r / 2 ( E ) \ { 0 }
such that
PmPn = 0 for all distinct m, n E IN and u-- EPnU. nE IN
b)
There is an orthonormal sequence (Xn)nEIN in E such that
lim uxn
= O.
n--~ oo
a) Put
~0 := Ilull. Since 0 is a non-isolated point of or(u), there is a decreasing sequence (rn)nEIN of real numbers with infimum 0 such t h a t for every n E IN,
c~(u) n An =/0,
6. Selected Chapters of C*-Algebras
96
where A~ "= {c~ 9 IKIr~ <
I~1 ~ m - l } .
Given n 9 IN, put
By Theorem 6.1.7.15 a),c), (pn),e~ is a sequence in {u} c M Pr s
such that
PmPn = 0
for all distinct m, n 9 IN. By Theorem 6.1.7.15 c),
II~- ~-~p~ull < rn k=l
for every n 9 IN, so that ~pnU.
U--
nE IN
By Corollary 6.1.7.19 a), pn Ts
for every n C IN. b) For n e IN, let xn 9 I m p n , Iixnll = 1. Then (xn)ne~ is an orthonormal sequence in E with
lux~ll = lup~x~ I< Ilupall for every n E IN. Thus lim uxn = O.
I
n---+ (x)
Theorem
6.1.7.23
(H. Weyl) If u is a self-normal operator on an infinite-
dimensional Hilbert space E then for every c~ C IK the following are equivalent:
~) ~ e ~(~) b)
There is an orthonormal sequence (xn)ncr~ in E such that
lim
n---+ o o
(uxn -
o l x n ) = O.
6.1 s
97
We may assume t h a t c~ = O. a ~ b. First suppose that 0 is an isolated point of a ( u ) . Then eo(u) is an orthogonal projection and u - e o ( u ) is invertible. Since 0 E ere(u), it follows that eo(u) is not compact, i.e. Ime0(u) is infinite-dimensional. Since ueo(u) = o,
the existence of (xn)~c~r is obvious. If 0 is not an isolated point of a ( u ) , then b) follows from Proposition 6.1.7.22 b). b => a. By Theorem 3.1.3.10, u is not Fredholm, so t h a t by Proposition 3.1.3.25 a), 0 E oe(u).
I
If u is a self-normal operator on an infinite-dimensional Hilbert space, then every nonisolated point of a(u) belongs to ae(u). Corollary 6.1.7.24
Let c~ be a nonisolated point of or(u). By Proposition 6.1.7.22 b), there is an orthonormal sequence (x~)ne~ in E such that lim(uxn
-
OLXn)
--- O .
By Theorem 6.1.7.23 b => a, a c a e ( u ) . Proposition
6.1.7.25
Let A be a W*-algebra, u and v the canonical pro-
jections A • s
>A ,
A • L(H)
>s
respectively, and B a unital W*-subalgebra of A • s
(Proposition ~.~.~.21
a), Corollary 6.1.7.1g b)) such that v(B) = Z.(H). Then either {0} • 1 6 3
C B
or uiB is injective. By Proposition 4.4.4.21 b), {0} • f_.(H) is a W*-subalgebra of A • s Hence B A ({0} • f_.(H)) is a W * - s u b a l g e b r a of A • s
and therefore a
W*-subalgebra of B (Corollary 4.4.4.10). Hence B 1"3({0} x ~:(H)) is a closed ideal of B B . By Corollary 4.4.2.16, there is a p C Pr B such that B n ({0} • L ( H ) ) = p B = B p
and the map
(1-p)B(1-p)
>A ,
x,
>ux
6. Selected Chapters of C*-Algebras
98
is injective. P u t q:--up,
r "= v p .
Then for every x E B ,
(qua, r ~ ) = ((~,p)(~,~), (vp)(v~)) = (u(p~), v(pz)) =
= p x E p B = B M ({0} x s
so t h a t qux=O,
(O, r v x ) E B .
Assume t h a t r r {0, 1}. Then there are r E Ker r , 77 E I m r , with I1~11= 1, lit/l] = 1. Take x E B with
We have
(Proposition 5.3.2.13 c)). By the above ,
(0, ('1~)~)e B. Since
(.1~)~--((.1~)~)((.1~)~)--((-i~)~)*((.1~)~) (Proposition 5.3.2.13 b),e)) we have
(0, <1r162 = (o, (.l~>v)*(o, <-Ir
e B.
From
<-I~>~:
(1 - r ) o
<-I~>~o
(1 - r )
(Proposition 5.3.2.13 c),d)) we deduce t h a t (0, (-]~}~) = (1 - p)(0, (-I~/~)(1 - p) E (1 - p)B(1 - p). Since the map
6.1 f_2-Spaces
(1-p)B(1-p)
>A ,
y,
99
> uy
is injective, we obtain the contradiction
(.10~ = o. Hence r C {0, 1}. If r = 0, then p = 0 and u i B is injective. Assume t h a t r = 1 and take x E f_.(H). There is a y E B with x = v y . By the above (0, x) = (0, rvy) C B .
Hence {0} x s C o r o l l a r y 6.1.7.26
C B.
I
Let (H~)~ei be a finite family of Hilbert spaces and A
a unital W*-subalgebra of H L(H~) such that for every ~ E I and every x c tEI
E.(H~) , there is a y C A with y~ = x . Then there is a J c I such that the map
A
~ HE'(H~)'
x,
~ (x~)~cj
LEJ
is an isomorphism of W*-algebras.
We prove the assertion by induction on Card I . If L:(H~) x
H
{0}CA
for every L C I , then
A = H L(H~) tEI
Assume that there is a ~ E I such t h a t
C(H~) •
1-[ {0} r A. XEI\{/,}
By Proposition 6.1.7.25, the m a p ~.
A
~ H
C(H~),
x~
(~)~i\f~
)tEI\Ir}
is injective, u is obviously a unital W * - h o m o m o r p h i s m , whence u(A) is a unital W*-subalgebra of
H
s
(Corollary 4.4.4.8 b)). By the hypothesis
of induction, there is a J C I\{~} such t h a t the m a p
6. Selected Chapters of C*-Algebras
100
u(A)
;I-Ic(H~),
;(~)~
x,
)~EJ
is an isomporphism of W*-algebras. It follows that the map A
>1-Is
x,
> (xx)xEg
)~EJ
is an isomorphism of W*-algebras. P r o p o s i t i o n 6.1.7.27
I
( 0 ) For every ~ E/:Z:(E)(K:(E)) (Example 5.6.1.5,
Definition 5.6.1.7}, there is a u E s
such that
qPV - - U V
for every v E K.(E).
Let ~ be an ultrafilter on K:(E) which is an approximate unit of K:(E). By Theorem 6.1.6.6, Z:(E) may be identified with the dual of Z:I(E), so that by Alaoglu-Bourbaki Theorem (and Proposition 5.6.1.8 b)), qo(~:) converges to some u in ~.(E)c,(E). Take v E/C(E) and w E/::I(E). By Proposition 6.1.7.3 a), Corollary 6.1.7.14 a), and Proposition 5.6.1.8 d), (uv, w) = (u, vw) = lim(qDt, vw) = t,ld
-- l~,~l(~at)v , w) = lim/~a(tv / w / - /~av, w) t,~
'
so that uv = ~ v .
I
T h e o r e m 6.1.7.28 ( 0 ) For n C IN, let E ( E ) n denote the Hilbert f~(E)module e2(INn, K:(E)) defined in Example 5.6.4.2 c). Take m , n C IN and for a C f-.(E)m,~, put -d" ~ ( E ) n
, ~ ( E ) m,
~ ~ ~. ( ~ a i j ~ j l \ j=l
9 ] ~'iEINm
Then for all a E s ~d e ~.C(E)(1C(E)n, IC(E)m),
~d* = a*,
n sup Ila~yll~
iEINm jEINn
sup jEINn
).
Ila.,jll ~ I1~11~ ~ Ila.,yll~ ~ j=l
"=
i=1
Ila~jll2
1
6.1 Z.p -Spaces
101
and the map
~(E)m,~
," ~ ( . ) ( r ( E ) ~, r(E)m),
~,
>
is an isometry of f_,(E)-modules (Proposition 5.6.1.8 g)). If m = n , then this map is an isomorphism of C*-algebras (Theorem 5.6.1.11 a), Theorem 5.6.6.1
f)). By T h e o r e m 5.6.6.1 a),b),c) (and P r o p o s i t i o n 5 . 2 . 3 . 2 2 ) i t is sufficient to prove t h a t the m a p is surjective for m - n - 1 and this was done in P r o p o s i t i o n 6.1.7.27.
m
6. Selected Chapters of C*-Algebras
102
6.1.8 The Canonical Projection of the Tridual of K: Definition 6.1.8.1 tridual of E ( E , F )
( 0 ) Let PER denote the canonical projection of the (Proposition 1.3.6.19 b)) and put PE := PEE. u !
Proposition 6.1.8.2
( 0 ) Take E - + F . oo
a) PGE(VU') = VpFEU' whenever F-Y+G. b) pfe(U'V) = (PFEU')V whenever G-Y+E.
c) p~(~'*)= (p~u')* Let iEF and jEF be the evaluations of IC(E,F) and 1C(E,F)' ( - s respectively. Then iEF is the inclusion map 1C(E, F)
>s
F) (= IC(E, F)")
(Corollary 6.1.6.8) and .!
PEF
=
jEF
0 ~EF
"
a) Given u 6/C(G, E), (u, i~s(VU')) = (iCEU, VU') = ((iGEU) o v, u') = 9!
!
= < i ~ . ( u o v). u'> = <~ o ~ , , ~
> = .
SO that ZOE" (VU') = Vi'FEU'
By Proposition 6.1.7.8 and Corollary 6.1.6.10, PGE(VU') = jGE (i'GE (VU')) = jGE(Vi'FEU' ) = V3FEZFEU" "
' = VpFEU' 9
b) Take u 6/C(F, G). Then (u, i~Fc(U'V)) = (iFVU, U'V) = (V o iFVU, U') = 9!
= < i ~ ( v o ~), ~'> = <~ o ~ , , ~ u
!
> = <~, (i'~.~')~>,
6.1 s
103
SO that ZFG" (U'V) = (i~FEU')v"
By Proposition 6.1.7.8 and Corollary 6.1.6.10, p~(u'v)-
3 ~ ~'
9
(~'v) = 3~((~ '
.!
!
= (3~~)v
' )~) =
= (p~.~')v.
c) Given w E/C(E, F), W
,~(~'*)> "!
~.
9!
u'*> ~ < i , ~ ( ~ , ) , ~,>
!
- (w*, ZFEU > = <W, (i'FEU')*>, so that 9I
I*
[
By Proposition 6.1.7.8 and Corollary 6.1.6.10, = 3" ~ . ~" ( u
PEF(U'*)
,
,
~3FE?~FEU
Corollary 6.1.8.3
( 0 )
'*
) = j~((,'~')
,,, )
(p --
*
)
-
-
')* FE u
PE is an involutive homomorphism of s
modules (Corollary 6.1.7.~).
Theorem 6 . 1 . 8 . 4
m
m
( 0 ) Invoking the identifications of the remark of Theo-
rem 6.1.6.6:
a) PEF(s
= s
b) Ker PEF -- ]C(E, F) ~ c) s
= s
e )~(e,F) ~ .
d) IIr = IIPEF~'II + IIr
PEFU'II, fO~ ~w~y ~' e ~ ( E , F)'.
e) (/:(E, F)/IC(E, F))' and s
F)'/E,I(F, E) are canonically isomorphic.
6. Selected Chapters of C*-Algebras
104
a),b),c), and e) follow from Proposition 1.3.6.19. d) Define u "= PEFU' E / : I ( F , E ) ,
v' "= u ' -
PEFU' E ~ ( E , F ) 0 .
Let
nEN
be a Schatten decomposition of u. Take r > 0. There are m E N , t Es
F ) , and w E s
F) such that c hEN n>m
5, c
Iltll_
~(t) >llull,
Ilwll ~ 1,
v'(w) > IIv'l
3' c
3
Let p be the orthogonal projection of E onto the (finite-dimensional) vector subspace of E generated by
{z~ln e N , n < m} U t*(F), q the orthogonal projection of F onto the (finite-dimensional) vector subspace of F generated by
{ynI n E N , n <_m} U t(E), and put
~:=(1F--q) owo(1s--p). Then
n E N, n < m ~ so that
Yxn = ~*yn = 0,
6.1 s
105
I~(~)1- 1 ~ o~(.ly/~xn(~) - [~o.(~x.lu.)
nON n>m _
6.1.6.2). F r o m
(Proposition
w--~--w--(1F--q) owo(1E--p)-=wop+qow-qowopC
E.f(E,F),
it follows t h a t
v'(~,)- v ' ( ~ ) - v ' ( ~ - ~) = o. Thus
~'(~) - v'(w) > IIv'll
3
We deduce that
l~'(t + ~ ) 1 __~
I~(t) + ~ ( ~ ) + v'(~)l _~ ~(t) + v'(~) - I ~ ( ~ ) 1
U
1 --
C
g
g
~ +t~'tl
3
3 = I1~ 11 + I~'11- C.
___
Now Kert-
( I m t*) -L
( P r o p o s i t i o n 5 . 3 . 2 . 4 , so t h a t for a n y x C E ,
II(t + ~ ) x -
-
[(q o t o p)x +
~ -[tx
+
~zll ~ ;
(1F -- q) o w o (1E --
p)~ll ~ -
l(q o t o p ) x 2 + II(1F - q) o w o (1E - p ) x l ~ <_
_~ IIp~ I~ + II(1E - p)~ I= - I~ll = (Pythagoras' Theorem). We deduce successively that
It+~ll_
I1~'11 ~ I~'(t + ~)1 ___I~111 § I1~'11- ~, 114 _~ I1~111+ Iv'l -IIp~411 + I1~'- p~'ll I~11- I p ~ ' l l + I ~ ' -
p~'ll
9
_~ I~'11, II
106
6. Selected Chapters of C*-Algebras
C o r o l l a r y 6.1.8.5 a)
Let IK = r
(IK = I R ) . A n e l e m e n t u' 9 s
(u' 9 ReL(E)') is order
continuous iff u I = pE ul .
b)
If IK-r L(E) = =
then {u' e L(E)'Iu'
= pEU'} =
- {u'lu' is an order continuous linear f o r m on s
a) First suppose that UI =
pEU
I .
By Theorem 6.1.8.4 a), u' E s Hence by Proposition 6.1.6.12, u' is order continuous. Now suppose that u' is order continuous. Put V ~ :=
UI
_
pE ul
"
By the above, v' is order continuous and by Theorem 6.1.8.4 b), v' c ]~(E) ~ . Moreover, v' E Re L(E') if IK = ]R (Corollary 6.1.8.3). Take u E s By Corollary 5.3.3.17, there is an upward directed set ~" of K:(E)+ with supremum u. Then v' (u) = lim v' (v) = 0 where ~ denotes the upper section filter of ~ . By Theorem 4.2.2.9 a) (and Theorem 5.3.1.13), v' vanishes on R e s Hence v ' = 0 and u I = pE ul .
b) Let u' be an order continuous linear form on s 1.7.2.4 b) (and Theorem 4.2.2.9 a)), u' is continuous. By a),
Now take u' E s
Ut =
pEU
UI =
pE ul
! .
with .
By Proposition
6.1 s
107
By Theorem 6.1.8.4 a), this is equivalent to u' E / : I ( E ) . Since every element of s
is a linear combination of positive elements of ~ I ( E ) , it follows from
a) that u' E s
~.
The inclusion
L(E) ~ c {~'lu' is an order continuous linear form on L:(E)} is trivial.
II
C o r o l l a r y 6.1.8.6
( 0 ) If ~' e R~ f_.(E)', th~n p,~u '+ = (p,~u') +
~'+ - pE,~ '+ = ( , ~ ' - p~,~')+,
p,~u'- -(p,~u'),~'- - p,~,~'- = ( , ~ ' - p ~ , u ' ) - .
First suppose that u' is positive. Then
I1~'11- (1E, u')
=
IIp~'l
I1~' - pEU'II - I1~'1
§
(Corollary 4.1.2.7 c), Theorem 6.1.8.4 d)), so IIp~'ll -
I1~'- p~'llSince pEU' and u ' -
<1~, ~'- p ~ ' > .
pEU' are selfadjoint (Corollary 6.1.8.3), it follows from
Corollary 4.1.2.7 c), that pEU' and u ' - p E u~ are positive. Now suppose that u' is selfadjoint. By Theorem 6.1.8.4 d), I1~'11- IIp~'l
+ I1~'- p~'ll,
I1~'+11 = IIp~ '+11 § II~'+ - PE~ '+11,
I1~'-II = IIp~'-II
+ I1~'- - p~'-II
9
Thus
IIP~'II + I1~'-- PE~'II- I1~'11- I1~'+11+ I1~'-II= IIp~'+ll
+ IIp~'-II
+ I1~ '+ -
pEU'+II +
I1r
--
pEU'-II
6. Selected Chapters of C*-Algebras
108
(Theorem 4.2.8.13). Since
IIPEU'II <--IIPEU'+II + IIPEu'-II, Ilu'-
PE411 ~
II 4 + --
pEU'+II +
114- -
pEU'-II,
it follows that IIPE411 =
114-
PE411 =
IIPEu'+II +
IIPE~'-II,
II 4 + - PE4+II + 114- -
pEu'-II.
By the first part of the proof p~u '+ = ( p ~ 4 ) + , ~'+ - p~u '+ = ( u ' - p~u') + ,
C o r o l l a r y 6.1.8.7 that u' < v'. Then
( 0 )
p~u'- = ( p ~ , , ' ) - , r
- p~u'- = ( u ' - p ~ u ' ) - ,
L~t ~',~' b~ po~iti.~ linear fo~m~ on L ( E ) ~ h
u'e LI(E),
whenever
u' e_ IC(E) ~
whenever v' E IC(E) ~
v'E
LI(E),
and
Since v ~ - u ~ is positive, we have, by Corollary 6.1.8.6, U ' - - pEU' ~__ 0,
p ~ , , ' - u' + ( v ' - p J )
pE u' ~_
0,
- ( v ' - ~') - p ~ ( ~ ' - ,,') > o,
pE v ' -
pE u' -- p E ( V -- U') ~> O.
By Theorem 6.1.8.4 a),b), it follows from
~' e ~ ( E ) that pE vl -- V l ,
m
6.1 s
109
and hence that u'=
s
pEU' 9
Moreover, if v' 9 K:(E) ~ , then p E r t ---
O,
so that fiE ut -- 0 ,
and hence ~' e ~:(E) ~ U !
C o r o l l a r y 6.1.8.8
( 0 ) if E - , r , v w'=
p~(v~')
m
~ ~C(F, a), a~d ~ 9 IC(a, E), th~ 9 f~(e, a),
~'w = p~,~(~'~) e ~ . ' ( a , F ) .
By Theorem 6.1.8.4 b), u' - PFEU' C
Ker PFE = ~ ( F , E ) ~
If ~ e ~ ( a , E), then ~ o v e ~(F, e ) , s o that 0 = (~ o v, ~' - p ~ ' )
= (~ o v, u') - (~ o ~, p w ~ ' ) = (~, v~') - (~, v p ~ ' ) ,
(~, v~') = (~, v p , ~ u ' ) = (~, p ~ ( v ~ ' ) )
(Proposition 6.1.8.2 a)). Since u is arbitrary, ~'=
p~(~')
e L~(E, a )
(Theorem 6.1.8.4 a)). It follows that (~'~)* = ~* u'*
= p~(w* ~'*)
(Proposition 6.1.7.3 i)), ~'w - p v ~ ( u ' w ) e L ~( a , e )
(Proposition 6.1.8.2 c), Proposition 6.1.7.3 h), Theorem 6.1.8.4 a)).
m
6. Selected Chapters of C*-Algebras
110
Let ~ and ~ be the set of finite-dimensional vector subspaces of E and F, respectively, ordered by inclusion, and let r and ~ be the upper section filters of ~ and ~, respectively. Then, for each u' E s F)',
Proposition
6.1.8.9
PEF u
,
---
limTrau
!
=
GO
Ut
--
PEF ut
~
lim
7rGUtTrH ,
( G , H ) , O x.r
lim 7fez u t - lim UtTrH• G,O
lim
UtTrH - -
H,.~
--
U,~
lim
7rG• U ' 7 r H •
(G,H),Ox~
in the topology of pointwise convergence. Let
on (.ly,,)x, nEN
be a Schatten decomposition of PEFU' (Theorem 6.1.8.4 a)). Take v E s
F)
and ~ > 0. Choose m E N with
211vll~-~s
<
~.
nEN n ~ rrL
Let Go be the vector subspace of E generated by {xn]n E N N INto}. Given any G E if, with G D Go,
I(v, PEV~') -- (~, ~G~')I = I(v, PV.V~') -- (V~G, ~')1 --
o~ ( v x . l y . )
-
nEN
~ o. (wc~n y.) hEN
InEN
hEN
~n>m
n>m
nEN
n>m
]hEN In>m
nEN n>rn
(Proposition 6.1.6.2), so that !
PEF u
!
--
lim
G,O
7rGu
6.1 EP-Spaces
111
in the topology of pointwise convergence. We deduce that PFE(U'*) - lim 7rH(U'*) H,~
and so
PEFU'= (PFE(U'*))* - (lim~rH(U'*))* -lim(1rH(U'*))* = lim U'TrH H,.~
H,~
H,~
is the topology of pointwise convergence (Proposition 6.1.8.2 c), Proposition 6.1.7.3 h)). Take v E s
and c > 0. Since PEFPEF ut -- PEF2t t
there is a (Go, Ho) E ~ x ~ such that
r I < ~ , I
(v, 7rcp~Fu'
E
I(~, ~'=H
- pEFu')
I
PEFU')I < -4
-
-~
<
for all (G,H) C ~ x ~ with
GocG,
HoCH.
Take such a (G, H ) . Then
I(v, =cu'=~ - PEFu')I _<
I(~ , = G u
7rH
- 7rcPEFu')l + I(v, 7rOPEFU'-- PEF~')I ~_ C
C
C
C
<- I(~, ~'=H -- PEFU')I + -~ < ~ + -4 -- -2 < C
I(v, ~-~1,,' - (~' - PEF~'))I -- I(v, p~FU' -- = J )
C
l < ~ < ~, C
I
- (~'-
p~u')>l-
- 7OH) -- u' + 2 p E F U ' - - PEFU'>I <
112
6. Selected Chapters of C*-Algebras
_< I
- = ~ ' > 1 + Il + I<~. ~-~.'~-. C
~
PEFU')I <
E
<~+~+ff=~ Hence PEFU'
Itt -- P E F u t --"
=
lim~e•
lim
( G,H),@ x.r
r v u 71"H ,
- lim U'rH• --
G,O
H,~
lim
(G,H),O x.~
rc•177
in the topology of pointwise convergence, T h e o r e m 6.1.8.10 given x E E , define
m
( 0 ) Let j denote the evaluation map of Z,t(E) and
: - j(<.lx>x).
a) To(f-..(E)) C f.,l(e) (_JK~(E)~ . b) To(E,(e))CI f_..l(e)= {xlx G E ,
llxll =
i}.
c) T(E(E)) is the closed convex hull of {E]x e E , the topology of pointwise convergence.
I1~11=
1} with respect to
d) If we endow 7-(E(E))NI:I(E) with the topology induced by T(E(E)), then To(f~(e)) N/~l(e) is a closed set of 7-(/:(e)) A l : ' ( e ) . The topologies induced on ~-0(L:(e)) N s E(E) coincide and the map {x 9
Elllxll
-
i}
> To(L(E))
by T(s
A L'(E) ,
E ' ( E ) , and
x ~
"~
is a homeomorphism.
a) Take u' E To(E(E)). By Corollary 6.1.S.6, pEn' and u ' - p E U ' positive. Hence pEU' = (1,pEU')U',
are
u ' - pEU' = (1, u ' - pEU')U'
(Proposition 2.3.5.4 a =:v b, Corollary 2.3.4.7, Theorem 5.3.1.13). By Theorem 6.1.8.4 a),c), either pEn' or u ' - - p E ul vanishes. Hence ~' e C ' ( E ) u lC(E) ~
6.1 ~_,P-Spaces
113
(Theorem 6.1.8.4 a),b)). b) Take x E E with IIxll - 1. By Proposition 6.1.7.10 a ::~ e (and Corollary 5.3.3.7), 5 is a positive linear form on s Since
II~ll
II<-Ix>~lll
=
= 1
(Theorem 6.1.6.6, Proposition 6.1.2.3), e ~(~(E))
Take u', v' E "r(s
n L:(E).
and a, fl E ]0, 1[ with c~+/~=l,
By Corollary 6.1.8.7, u', v' E
au'+/~v'=5.
Z: i (E),
so that
~'I~(E), v ' l t : ( E ) e ~(~:(E)). By Proposition 6.1.7.13 a),
~'lr(E) - ~'lr(E) : ~ l r ( E ) . Since K:(E)is dense in s163
(Corollary 1.3.6.5),
and ~ E T0(E(E)). Now suppose u E T0(/:(E)) N / : I ( E ) . By Proposition 6.1.7.10 e ==~ a, u E s and so by Corollary 6.1.2.13, there are an orthonormal family
(Xn)nEN in E and an f E gi(N)+, such that u = ~-~. S(n)(.Ix,,)x,, . nEN
Choose n E N with f(n) --fiO. Then
f (n)~,~ < u, SO
f (n)'xn -- f (n)u (Proposition 2.3.5.4 a ==> b), and ,'tt---Xn
"
6. Selected Chapters of C*-Algebras
114
c) We have
I1~*~11 = Ilull ~ = sup I1~11~ = sup ( ~ x l = x ) = xEE
xEE
Ilzll= 1
Ilzll = 1
sup ( u * u x l x ) = xEE
Ilxll= 1
= sup (u'u,5) xEE
Ilzll-1 for every u E s
(Proposition 6.1.6.2). By Proposition 2.3.5.13, T(s
is
the closed convex hull of {Six E E , Ilxl] = 1} with respect to the topology of pointwise convergence. d) Let ~ be a filter on T(~.,(E)) C1s
converging to a u' E T(s
M
/:1 (E) such that
{~1~ e E , Ilxll = 1} E ~ . converges to u' in the topologly of pointwise convergence on ]C(E). By Proposition 6.1.7.12, there is an x E E such that
u'ltg(E) = <.lx>~. Since u' E s
it follows U~ _
~"
(Theorem 6.1.8.4 a),b)). Since u' E T(E.(E)), we deduce that
Ilxll~ = < 1 , ~ > -
= 1.
The assertion now follows from b). e) Let us denote the mentioned topologies by ~1, ~72, and ~73. The inclusions ~1
are obvious. Take x, y E E with
C
'~"2,
'~3
C
I1~11- Ilyll-
~'2
1. Then
I1~- ~11---II<.l~>x- <-Iy>yll~ II<-Ix>~- <.lx>yll~ + II<.lx>y- <-Iy)yll~ =
= II<-Ix>(~ - y)ll~ + II<-Ix- y)yll~ ~ 21Ix- yll
6.1 s
115
(Proposition 6.1.2.3). Hence %.2 C %'3. Let x 9 E with Ilxll = 1 a n d let ~ be ~ filter on {y c EIIlYll = 1} such that lim ~ Y,~
in %.1 Since (.Ix)x 9 s
it follows
1 - 5((.Ix)x) - lim~((.lx)x ) - lim I(xly>l~ Y,~
(Proposition 6.1.6.2). Take y e E with ]IY] - 1. Since I l y - (ylx)~ll ~ + I(xly)l ~ -
1,
it follows lim Ily - (ylx>xll = o . y,~ Since I I x - yll ~ I1 - lllxll + I I ( y l x > x -
yll
we deduce
lim IIx- Yll- O. y,~ Hence %.3 C %.1-
9
Corollary 6.1.8.11 no~h~
d~r
T(s
~ is a convex set of s
K ( E ) ~ is a vector subspace of s
convergence. Hence 7 ( s set of 7 ( s 7(s T(s
and a closed
~ t of ~ ( ~ ( E ) )
closed in the topology of pointwise
1C(E) ~ is a convex set of / : ( E ) ' and a closed
By Theorem 6.1.8.10 b),c), T ( s Since it is contained in ~-(s
N K ( E ) ~ is a nowhere dense set of T ( s
N / : I ( E ) is a dense set of ~ (Theorem 6.1.8.4 c)), the set .
m
6. Selected Chapters o.f C*-Algebras
116
6.1.9 I n t e g r a l O p e r a t o r s on Hilbert Spaces P r o p o s i t i o n 6.1.9.1 ( 0 ) Let T be a a-compact (i.e. the union of a sequence of compact sets) Hausdorff space, # a positive Radon measure on T , and A an orthonormal basis of L2(#). Take f 9 e2(A), and put
k := ~ f(~)~ | ~ (i~ L~(, o ~)). xEA
~) IIk(t,.)ll~,,, = E If(~)~(t)l ~ fo~ ~-~tmo~t ~ y
t e T.
x6A
b) If T is compact, Supp # = T , and k is continuous, then
Ilk(t,-)11~,, = ~
II(x)~(t)l ~
x6A
for every t 9 T .
c) If A C C,(T), Suppp = T , and there is a #-null set S such that
sup Ilk(t, )11~,~ < o~,
t6T\S
then for every y 9 L2(#), (f(x)(ylx>x)xeA is summable in C(T).
a) (resp. b)). By Proposition 3.1.6.16 (by Corollary 3.1.6.4) ,-almost everwhere (kz = f (x)x)
kx = f (x)x
for every x 9 A. Since {z 9 A I f ( x ) # 0} is countable, we deduce
E
)~(~)x(t)) ~ = ~
xEA
Ikx(t) ~ = ~
x6A
= ~
x6A
i f k(t, ~)~(~)d#(~)l ~ = J
I1~ = Ilk(t, )11~,~
x6A
for #-almost every t 9 T (for every t 9 T), by Parseval's Equality. c) Take c > 0. Put c~ := sup IIk(t,')ll~,.. t6T\S There is a B 9
~3I(A) such
that 62
Z x6A\B
ll ~ _< 1 + ~ 2 "
6.1 s
117
We deduce, by a), that for every C c ~3I(A\B), ](ylx}12)} ( E
If(x)@ x)x(t)l <- ( E xCC
xEC
If(x)x(t)[2) 89<
xCC
--
cc~ V/r~ -~ Oz2 ~ C
for p-almost every t C T. The above inequality holds for every t E T since the functions in C are continuous and since Suppp = T. It follows that the family (f(x)(ylx}X)xeA is summable in C(T) (Proposition 1.1.6.6). I T h e o r e m 6.1.9.2 ( 0 ) Let (T, ~, p) be a ~-finite measure space and take k E f_2(p | p). Assume that the operator k" L2(p)
>L2(p),
x,
>kx
is self-normal (Proposition 3.1.6.17 a)). Then k is a mlbert-Schmidt operator, and there is an orthonormal basis A of L2(p) and an f E IK A such that kx - f (x)x for every x C A . If A and f have the above properties, then:
a)
f C e~(A) and
- ~ f(x)(.Ix)x
(in
L2(L2(p))).
x ~2A
b)
k ~ iZp(L2(p)),
If p E [1, co[ U{0} and f E g ' ( A ) , thr -- E
f(x)('lx)x
(in s
xcA
and
Ik l p - l f l p . c)
k -
E f(x)x |
(in L2(p | p)).
xCA
d)
c~p(k)- f ( A ) .
e) ky-- E /(z)(YI~)~
for every y C L2(p).
xEA
f) If ~ e H<\{0}, t h ~ -1
Im (c~l - k) - f (c~)• ,
6. Selected Chapters of C*-Algebras
118
and -1
1
1
f(x)(ylx) ~ x E a l
F.
a-- f(x)
--I
- k ( y )
yeA\ f (a)
for every y E Im (al - k).
g)
if T is a a-compact Hausdorff space, # is a Radon measure on T with
Supp # = T, A C C ( T ) , and there is a #-null set S such that sup tET\S
Ilk(t, ")112,. <
~,
then the families in e) and f ) are summable in C ( T ) .
By Proposition 6.1.4.9 a), k is a Hilbert-Schmidt operator. By Corollary 6.1.3.18, there is an orthonormal basis A of L2(#) and an f E IK such that kx - f (x)x
for every x E A. a) ,b) ,d) ,e) , and f) follow from Corollary 6.1.3.18 a) ,b) ,c) ,d) . c) follows from a) and Proposition 6.1.4.9 a),c). g) By Proposition 6.1.9.1 c), (f(x)(ylx)x)xeA is summable in C(T). Take a E ]K\{0}. Put -1
B.=A\y(a).
Then inf l a - f(x)l > 0
xEB
f (x) ce. -
) ~ . e e2(B) .
f(x)
') By the above considerations, the family (l(x)/vlx) Ik o~--f(x) X l x
P r o p o s i t i o n 6.1.9.3
( 0 )
is summable in C(T) I
Let T be a Hausdorff space and # a positve
Radon measure on T such that S u p p # in s
EB
T . Let k be a continuous function
| #). Then the following are equivalent:
6.1 s
a)
119
The operator L~(,)
~ L~(,),
~,
~ k~
(Proposition 3.1.6.17 a)) is selfadjoint (positive). b)
For every finite family (t~)~eI in T , the sesquilinear form
t,AEI
is Hermitian (positive). Case 1 Selfadjointness By Proposition 5.3.2.1, a) is equivalent to
s, t 9 T ==v k(s, t) = k(t, s) , and by Example 2.3.3.6 this is equivalent to b). Case 2 Positivity a =~ b. By Corollary 5.3.3.7,
E
/ k(s,t)x~(t)x~(s)d(#|
#)(s,t)
t,AGI
for every family (x~)~e~ in Le(#). It follows (by approximation) that
~,AGI
for every (a~)~ei 9 IKI , i.e. the sesquilinear form of b) is positive. b ==)>a. Take x 9 L2(#). Let K be a compact set of T such that the restrictions of x and kx to K are continuous. By b),
(egkX]eKX) 9 IR+ (using approximation). By the Vitali-Lusin Theorem (of integration theory), we see further that
(kzlx) e Ia+. By Corollary 5.3.3.7, the operator of a) is positive,
m
6. Selected Chapters of C*-Algebras
120
Theorem 6.1.9.4
( 0 ) Let T be a compact space and # a positive Radon measure on T with Supplt = T . Choose k E C(T • T) such that the operator k" L2(#)
~ L2(#),
x,
~, kx
(Proposition 3.1.6.17 a)) is positive. Let A be an orthonormal basis of L2(it), and take f E IK A such that x E A ==r kx = f ( x ) x (Theorem 6.1.9.2). -1
a) A\f(O) C C(T),
ap(k)= f(A).
b) f E t~'(A). c) k -
E f(x)x|
(in C ( T |
xEA
d) k is nuclear, = E
f(x)('ix)x
(in s
xEA
and
IikiI1-
/
k(t,t)d#(t) = Iifi]l (J-Mercer, 1909).
a) By Corollary 3.1.6.4, f (x)x = kx E C(T)
for every x E A, so that -1
A \ f (0) r C(T).
By Proposition 5.5.5.7 c), crp(k) - f (A) .
b)
Step 1
t E T =~ E f(x)ix(t)l 2 <- k(t, t). xEA
Take B E ~f(A). Define h "= E xEB
f ( x ) x | 5,
6.1 s
121
:= E:(,)<.ix>x. xEB
Since k is positive, f is positive, and so k c)). It follows that
h is positive (Theorem 6.1.9.2
0 <_ k(t, t) - h(t, t) (Proposition 6.1.9.3 a =~ b) and
f(x)lx(t)l 2 = h(t, t) <_ k(t, t). xEB
Since B is arbitrary, it follows that
~-~f(x)lx(t)l 2 <_ k(t, t). xEA
Step 2
f E g'(A), E f ( x ) =
f(E
xEA
f(x)lx(t)12)d#(t)
xEA
By Step 1,
f(x) = ~ f(x) /Ix(t)12dp(t) xEA
xEA
--S (~ f(x)"(t)')d#(')~-I k(',')d#(')9 Step 1
s E T =~ (f(x)x(s)~)xr
is summable in C(T) .
Choose c > 0. By Step 1 in b), there is a B E ~ J f ( A \ B )
E
f(x)lx(s)[2 <
9~A\B
such that
s 1 + Ilkl
Let C E ~ s ( A \ B ) and t E T. Then, by Step 1 in b),
Ef(x)x(s)x(t)
f(x)lx(t)l 2
xEC
c v/k(t, t ) -
V/1 + IikiI~
<_
122
6. Selected Chapters of C*-Algebras
Now the assertion follows from Proposition 1.1.6.6. k0" T • T -+ IK,
(s,t) ~-~ E f(x)x(s)x(t)
Step 2
xEA
is continuous in every variable The assertion follows from Step 1. Step 3
k = k0
By Proposition 6.1.4.9 c),
k=Ef(x)x@-~
(in L 2 ( # |
xEA
Hence k=k0
#|
By continuity, at every s E T , the assertions
k(~, .)
-
k0(~, .),
k(s,.) = ko(s,.),
#-a.e.
are equivalent. By Fubini's Theorem, we deduce that #({s e Tlk(s, .) ~ k0(s,-)}) : O. Hence k mk 0 9
Step4
k= E f(x)x|
(in C ( T x T ) ) .
xEA
By Step 3,
k(t, t) "- ko(t, t) - E f (x)lx(t)I2 xEA
for every t E T. By Dini's Theorem, this sum converges uniformly. Take s > 0. There is a B E ~/(A) such that f (x)ixl 2 <
xEA\B
--c.
6.1 s -Spaces
123
We get that
[f(x)x(s)x(t)l <
f(x)lx(s)l 2
xEA\B
x
f(x)lx(t)] 2 \xE
\
for all s,t E T . Hence the family ( f ( x ) x | 5)xeA is summable in C(T • T ) , and therefore k = E
f(x)x | 5
(in C(T x T)) .
xEA
d) By Step 2 in b) and Corollary 6.1.2.13 d), k is nuclear and
By Step 3 in c), Iiklll - / k(t, t)dp(t). By Theorem 6.1.3.12 c), - E xEA
f(x)('lx>z
(in /21(L2(#))).
124
6.2
6. Selected Chapters of C*-Algebras
Selfadjoint
Linear
Differential
Equations
This section contains examples of selfadjoint ordinary linear differential equations as well as examples of selfadjoint linear partial differential equations of second order on compact Riemannian manifolds.
6.2 Selfadjoint Linear Differential Equations
125
6.2.1 Selfadjoint Boundary Value Problems We use the notation from Subsection 3.2.1 throughout this subsection and assume that u is injective (except in Proposition 6.2.1.1). We also use the following additional notation: a)
A is Lebesgue measure on [a,b].
b)
r is a strictly positive continuous function on [a, b] (we shall work in the Hilbert space L2(r-A) and lr u will be our basic differential equation).
c) g is the Green function of the given boundary value problem (Theorem 3.2.1.3). g is assumed to be selfadjoint, so that u is selfadjoint (Corollary 3.2.2.6 d =v c). d)
v'L2(r.)~)
e)
Given 7, 5 ~ IK, define
>L2(r.~),
u~:D
x,
>
)C,
g(.,t)x(t)r(t)dt.
x~
)ux+hrx,
A~:={Te~l~'~,~#{0}},
A:=A0.
If u5 is injective, then g~ denotes the Green function associated to u5 (and to the given boundary value problem) and we put ~a b
v~: L2(r-A)
x,
g~(.,t)x(t)r(t)dt.
[
( 5 ) injective. If % 5 c IK, then
Proposition 6.2.1.1
> L2(r.A),
In this proposition we do not assume u to be
jqZ'~,9, z .~'~ - 5
126
6. Selected Chapters of C*-Algebras
and A~=A+5.
Take x E 7:). Then (x E .~,~) ~
(u6x -- 9/rx) ~
(ux + 6rx = 7rx)
, = . (u~ = ( 7 - 6 ) ~ ) ~
(z e 7~_~).
Hence 9~ ' 6 , 7
---- - ~ ' 7 - 6
9
It follows that (~/E A6) ~
(~6,~ :/= {0}) ~
($'~_6 # {0}) ~
A~=A+6.
('~- 6 E A), I
Remark. In fact this proposition is contained in Proposition 3.2.2.1 a),b) applied for !u instead of u. r
Proposition 6.2.1.2 ( 5 ) a) v is injective and selfadjoint. b)
/ xEV~v(~ux)=-x
(
x E C =~ vx E T), l u v x = - - x .
c) ~ E ]K\{0} ::~ ~ - {x E n 2 ( r . A ) l v x - - ~ x } . 1 E A} . d) a , ( v ) = {-~17 e)
AcIR.
b) By Corollary 3.2.1.5,
7"
for every x E 7). By Proposition 3.2.1.4, vx E 7:) and
6.2 Selfadjoint Linear Differential Equations
1
rUVX -- rU(
127
g(., t)x(t)r(t)dt) = - x
for every x E C. a) By Proposition 5.3.2.1, v is selfadjoint. By b), 7) C Im v, so that Imv is dense in s By Proposition 5.3.2.4, Ker v = (Im v) • = {0}. Hence v is injective. c) Take x C L2(r.A) such that VX
1 ---- - - - - X
.
By Corollary 3.1.6.4, x E C, so by b), x c 7:). It follows by b), that {x e L 2 ( r . , ~ ) i v x - - l - x } = {x e ~ l v x -
= {x e 9 l ~ = ~ }
-lx}
=
e s~.
d) follows from c). e) follows from d) and a). f) By a) and b),
!
1
-~x + ~ly)
= -(r-UX v ( u y ) )
!
= ( ~ x y) + ~(xly) =
+ 5(xly ) - - ( v (
uX)lrluy"~ + 5(xly}
- (xl!~y)+ (xi~y)- (xl~u~)
-
"
T h e o r e m 6.2.1.3 ( 5 ) v is an injective selfadjoint Hilbert-Schmidt operator and there is an orthonormal basis B of L2(r.)~) and an f E IK B such that vx - f (x)x for every x C B . If B and f have the above properties, then: a)
f C g2(B) and v = ~f(x)(.Ix)x xEB
(in s
128
b)
6. Selected Chapters of C*-Algebras
g -
(in L2((r.,~) @ (r.A))).
E f(x)x | xEB
c) a p ( v ) = f ( B ) . d)
f is real and vanishes nowhere.
e)
x E B => x E T), u x - - --f-~)rx.
f)
IIvll-
1
Ilflt~
- sup ~ ,
Ilvll~ - Ilfl2.
TEA
g) y E L 2 ( r - A ) = > v y -
E f(x)(y x)x
(in C).
xEB
h)
If a E IK\{0}, then -1
Im (al - v) = f (a) jand -1
1 1 - Y + -a o~
E
f(x)(ylx) ~ x a-f(x)
E
~~-v(y)
yE B\ -f'( o )
for every y E Im (c~l - v) , where the above series converges in C.
i)
The following are equivalent for every y E L2(r.)~) 9
il)
yET).
i~) (<~-~
E ~(B)
\ f(z) ]xEB
(ylz> z xEB
If these conditions are fulfilled then: i3)
v
i4) y -
7-~
-
y
E (YIx) x (in C ). xEB
i5) uy = - r j)
~
(/n L2(r-A)).
xEB f----~X
If y E v(D) then r(ylx>~,~ f--~-~]xE~ is a summable family in C.
6.2 Selfadjoint Linear Differential Equations
129
v is injective and selfadjoint (Proposition 6.2.1.2 a)). By Theorem 6.1.9.2, v is a Hilbert-Schmidt operator and there are an orthonormal basis B of L2(r.,~) and an f C t~2(B) such that
vx = f (x)x for every x C B. a) follows from Theorem 6.1.9.2 a). b) follows from Theorem 6.1.9.2 c). c) follows from Theorem 6.1.9.2 d). d) follows from c) and Proposition 6.2.1.2 a). e) By d), Corollary 3.1.6.4, and Proposition 6.2.1.2 b), x c 79 and 1
1
ux-~-~uvx-
f(x)
~
/
'
X
.
f) By Theorem 6.1.9.2 b), I1~11 = Ilfll~,
I1~11~ = II/11:.
By c) and Proposition 6.2.1.2 d),
1
IIf~ll - sup TEA"~ g) and h) follow from e) and Theorem 6.1.9.2 e),f),g). il =~ i2 &; i3. By Proposition 6.2.1.2 b), y E Im v. By Theorem 5.5.5.4 a =a i,
Since v is injective and v(C) = 79 (Proposition 6.2.1.2 a),b)), it follows x~B f----~x E C. i2 =~ il & ia & i4. By g),
f-~
) = ~ ](~) f ( ~ ) ~ - ~Y~(YI~)~ (in C).
i3) and i4) follow from the Fourier expansion of y. By i3) and Proposition 6.2.1.2
b), y ~ . i2 &: ia ::a i5. By Proposition 6.2.1.2 b),
130
6. Selected Chapters of C*-Algebras
uy = uv
(~e~B ( y [ x ) )
(Y]X) x (in L2
j) There is a z E T) with vz = y . By d) and Proposition 6.2.1.2 a), f (x) =
f (x) =
f (x------~ = (zlx)
for every x E B Hence by i) /{N~x~ 9
' \ f(x) ]xEB
is a summable family in C.
I
Proposition 6.2.1.4 ( 5 ) Take 5 E IR. Let B be an orthonormal basis of L2(r.A) and f E ] K B such that vx = f (x)x for every x E B
(Theorem 6.2.1.3) and put h'= 5f -1 f
(Theorem 6.2.1.3 d)).
a) x E ~'6,h(x) whenever x E B . b) h ( B ) = Aa = A + 5. c) I f ua is injective, then ga and v~ are selfadjoint, 1 - 5 f
vanishes nowhere,
f(x) v~x = 1 - 5 f (x) x f o r every x E B , and
1
ap(v~) = { - ~ 3' E A + 5}. a) By Proposition 6.2.1.2 b),
1 ttX
--
--
-
-
r
x
.
f(x)
It follows that ?.t6X
=
1 ---rx f(x)
5f(x)- 1 + 5rx = - - ~ x f(x) x E ~,h(x).
= h(z)x,
131
6.2 Selfadjoint Linear Differential Equations
b) By a) and Proposition 6.2.1.1, h ( B ) C Aa - A + 5.
Take 7 c A + 5. Then - y - 5 E A. By Proposition 6.2.1.2 d) and Theorem 6.2.1.3 c), there is an x C B such that 1
Thus h(x) -
5 f (x) -
I =5-
f(x)
- -
= 5-
(5-~)-
~.
f(x)
Hence
Zeh(B) and A + 5 C h(B).
c) By Proposition 3.2.2.2 (applied to the differential operator lr u ) g5 is selfadjoint and so v5 is also selfadjoint (Proposition 5.3.2.1). Since u5 is injective, 0 ~ As. By b), h vanishes nowhere. Take x c B. By a), x C $'5,h(x)Thus 1 v~ = -h(x---~x -
f(x) 1 - 5f ( x ~ x
(Proposition 6.2.1.2 c)) and ap(v~)- {-11- ), e d + 5} 7 by b) and Proposition 6.2.1.2 d). P r o p o s i t i o n 6.2.1.5
( 5 )
orthonormal basis of L2(r.A)
Take 5 C ]K and y C L2(r-A). Let B and f C ]K B such that x E 7) and usx - f ( x ) r x
for every x C B .
a) For every x C B , - f(~) e ~\{0}
and 1 VX--
X.
5 - f (x)
1 be an
132
6. Selected Chapters of C*-Algebras
-1
b)
f (C) is finite whenever C is a bounded set of ]K.
c)
The following are equivalent:
c~) y e / ) . c2)
(f(x)(Ylx))xcB 9
E
g2(B),
f ( x ) ( y l x ) x 9 C (where the sum is considered in L2(r./~)).
xEB
If these conditions are fulfilled then:
%,
v
C4)
y =
c5)
u~y - r E f ( x ) ( y l x > z
E (YIx) x xCB
~n
C).
(in L2(r.)~)) .
xcB
d) If y 9 v(T)) then ( f ( x ) ( y l x ) x ) ~ c , is summable in C. e)
If ( f ( x ) ( y l x ) b e B c g2(B) and if the coefficients of u are C~-functions, then
xff B
in the sense of L2-distributions.
a) We have ux - ( f (x) - 6)rx
so that f ( x ) - 6 9 A\{0} c IR\{0} (Proposition 6.2.1.2 d),e)). By Proposition 6.2.1.2 b), v
so that VX -- - - X .
6 - f (x)
b) Put
6.2 Selfadjoint Linear Differential Equations
133
o~ "-- sup I/~1 < oo. fiEc
Then - f ( ~ ) l _< 15 + ~ ,
1 Ia -
1
>
Ial +
f(x) l
-1
for every x E f (C). By a) and Theorem 6.2.1.3 a), 1
1
-1
(15l + c~)~Card f (C) _< x--"2..., ]5 - f(x)l 2 < o o . --1 xE f (C)
-1
Hence f (C) is finite. c) First assume that i4 & i5,
holds. By a) and Theorem 6.2.1.3 il => i2 & ia &
el)
( ( 5 - .f(x))(ylx))xeB E g2(B), ~-~(5-
f ( x ) ) ( y x ) x e C,
xEB
y= E(ylx)x
(in C),
xEB
usy - uy + 5ry - - r ~-~(5 - f ( x ) ) ( y i x ) x xEB
= rEf(x)(y
x)x
( in L2(r.A)).
xEB
From this it follows easily (f(~)(v
~))x~. E e:(B),
+ 5ry -
6. Selected Chapters of C*-Algebras
~3~
f(x)
Now assume c2) holds. By Theorem 6.2.1.3 g),
9eB 5 - f ( x ) x -- v
Z
f(x)(ylx)x
9 C,
5 - .,,-,~,~ = vy ~ c .
xEB
Hence
~--~(ylx)x= 5~_, t5- f(x) x xEB
~- f(x) x 9 C,
xEB
~-~(5- f(x))(ylx)x 9 C. xEB
Since
( ( 5 - f(x))(ylx))~eB e E2(B) it follows y E 79 by Theorem 6.2.1.3. i2 ::> il. d) By a) and Theorem 6.2.1.3 j), ( ( 5 - f ( x ) ) ( y l x ) x ) ~ s is summable in C. By Cl ::> c4, (.f (x) (y]x)x)xeB is summable in C. e) Since u is selfadjoint, ua is also selfadjoint. Let ~o be a C~ on ]a, b[ with compact support. For every x C B ,
lu~. )
x~-~dt =
(u6x)~dt =
b
fa
: f(x)
x~rdt-
f(x)(x]qo).
Hence (Theorem 5.5.1.5 a =~ c)
(ylx)(xl u6~)=
y ~ - ~ d t = (Yl u ~ ) = xEB
6.2 Selfadjoint Linear Differential Equations
135
- ~ ( y x)f(x)(xl~o)- (~-~(ylx)f(x)xl~)xcB
xEB
=
r
(y x ) f ( x ) x
~dt.
Since 9~ is arbitrary, ~-"~l
l
\t'l
\
xcB
in the sense of L2-distributions. T h e o r e m 6.2.1.6
( ~ ) Assume that A is bounded below. Let B be an orthonormal basis of L2(r-A) and f E IK B such that vx - f (x)x
for every x E B .
a) f e g~(B). b) v is nuclear,
Ilvll~- Iflll, and (in /21(L2 (r-/~))).
v-
xEB
c) If y C D , then
v/If(~)' d) g - E f (x)x |
e el(B). x~.
(in C([a, b] x [a, b])).
xCB -1
e)
f (IR+) is finite.
a & b & d & e. By hypothesis, there is a 5 E IR such that A + 6 C]0, oc[. Put
h._ 5f-1 f
136
6. Selected Chapters of C*-Algebras
(Theorem 6.2.1.3 d)). Then, by Proposition 6.2.1.1, Aa = A + S c ]O, c~[.
Thus ua is injective. By Proposition 6.2.1.4 c), va is selfadjoint, 1 - 5 f vanishes nowhere, f(x) vex -- 1 - ~ f ( x ~ x
for each x E B, and 1 crp(-va) = -crp(va)= {~17 E A + 6} C 10, c~[.
Hence -v6 is positive. Define
h:
=
1 -Sf"
By the above considerations, v ~ = f~(~)~
for each x E B. By Theorem 6.1.9.4 a),b),c), fa E el(B),
Beg.,
(in C([a, b] x [a, b])), xEB
sup Ilfa(x)x o ~11~ < ~ . xEB
It follows from l+hfa=l+
5f 1 -hf
=
1 1 -hf
that 1 + 5fa vanishes nowhere. Define e "= infl 1 + afal > O.
Then it follows from s f
1 +Sfa'
6.2 Selfadjoint Linear Differential Equations
that
Ifl ~ ~lf~, C f 9 gl(B). Hence the series
~-~ f (x) f ~(x)x @ -~ xCB
is absolutely convergent. From f = f~(1 - 5f) - f~ - 5ffa it follows that the series
~-~ f (x)x | -~ xCB
converges in C([a, b] x [a, b]). Thus, by 6.2.1.3 b),
f(x)x | -~ (in C([a, b] x [a, b])).
g= Z xCB
By Theorem 6.1.9.2 b), v is nuclear. (in s (L2(r.k))) xcB
and
IVll = II/11. Since -va is positive, fa is strictly negative. From
f-
fa l+Sfa
it follows f(x) 9 IR+ ==> 1 + 5fa(x) < 0. -1
Since fa 9 el(B), f (IR+)is finite. c) By Theorem 6.2.1.3 il => i2, f(*)
~.
c e~(B).
By a), f c g~(B) and so x/qfl C g2(B). Hence (/Y'x/) el . v/If(z)[ x~B e (B)
137
138
6. Selected Chapters of C*-Algebras
Corollary 6.2.1.7
( ~ ) Take 5 E IR and y E 7). Let B be an orthonormal
basis of L2(r.A) and f E IK B with B C T) and u:
f (x)rx
=
f o r every x E B . I f A is bounded below then:
a)
E
xEB
~<~-
S(z)#o b)
(~f(x)l(ylx))~eB
e s
By Proposition 6.2.1.5 a), 1
VX =
- f(x)
X
for every x E B. a) By Theorem 6.2.1.6 a), 1 xEB
I<~- :(x)l
Hence 1
E If(x)l xEB
S(x)#0
b) By Theorem 6.2.1.6 c),
(v/I:(~) - <~l@lx>):~ e :(B). Hence by a),
(v"iS(:~)lmlx)):~. e e'(B).
I
6.2 Sel/adjoint Linear Differential Equations
6.2.2 T h e R e g u l a r
Sturm-Liouville
139
Theory
In this subsection, we set ]K = ]R and use the following notation: a)
a, b, a,/3 are real numbers with a < b.
b)
)~ is Lebesgue measure on [a, b].
c) p is a strictly positive continuously differentiable function on [a, b]. d)
q is a continuous function on [a, b].
e)
r is a strictly positive continuous function on [a, b].
f)
7:) is the set of twice continuously differentiable functions x on [a, b] such t h a t x(a) sin a - x'(a) cos a = 0
x(b) sin/3 + x'(b) cos/3 = O. ~ -(px')'+qx.
u : 7:)
h)
9r~ := {x 9 T)lux = ~rx} for every ~ / 9 IR.
i)
A := { ~ / 9 ]Rigr~ =/= {0}}.
Proposition
)C([a,b]),
x,
g)
6.2.2.1 ( 5 )
For any ~/ C IR, ~
is a one-dimensional vector
subspace of 7). The Wronskian d e t e r m i n a n t of any two functions of 7:) vanishes at a and b. Hence, given any x, y E S"v , the Wronskian d e t e r m i n a n t of x and y vanishes identically, and x and y are linearly dependent.
6.2.2.2 ( 5 )
I
Let y , z be twice continuously differentiable functions on [a, b], which are not identically O, such that Proposition
-(py')' + q y = O ,
y(a)sina-y'(a)cosc~=O,
-(pz')' + qz = O,
z(b) sin /3 + z'(b) cos/3 = 0.
If u is injective (equivalently, if 0 r A ) then:
6. Selected Chapters of C*-Algebras
140
a)
p ( y z ' - zy') is a constant 5, and 5 r O.
b)
The function
g" [a,b] • [a,b] -----+lR,
f (
(s,t),
<_ t
if
8
if
~ >__ t
~{
~y(t)z(~)
is the Green function of the differential operator u (for the given boundary value problem) (Theorem 3.2.1.3). c)
If we define v.L2(r.)~)
~L2(r.)~),
x,
~
g(.,t)x(t)r(t)dt,
then v is an injective selfadjoint IIilbert-Schmidt operator and 9 e ~ ~ ~(~)
el)
/
=
-~
x E C([a,b]) =~ vx e V , ~uvx = - x .
c~) 7 9 ~ \ { o } ~ 7~ = {~ 9 L~(~-~)l~ = - ~ - z }
c~) o~(v)= {-~1~ 9 A} c~) I I ~ l l = s u p ~ , ~EA
Ilvll,.-
@
9
a) We have (p(yz' - zy') )' = (pz')' y + pz' y' - ( A ) ' z
- py' z' =
= q z y - qyz = 0 i.e. p ( y z ' - zy') is a constant 5. y z ' -
zy' is the Wronskian determinant of
y and z. 5 = 0 implies that y and z are linearly dependent, i.e., y , z c :D, which contradicts the injectivity of u. b) Take t C]a,b[. g ( - , t ) i s continuous on [a,b] and twice continuously differentiable on [a, b]\{t}. Moreover, 1 limg'(s, t) - limg'(s, t) = ~y'(t)z(t) - -~y(t)z'(t) = s~t
s~t
-(pg'(., t))' + qg(., t) = 0 on [a, b]\{t},
1
6.2 Selfadjoint Linear Differential Equations
141
and g(a, t) sin a - g'(a, t) cos a = 0
g(b, t) sin ~ - g'(b, t) cos/~ - 0. Hence g is the Green function of u for the given boundary value problem (Definition 3.2.1.2). c) By b) and Proposition 6.2.1.2 a), v is injective and selfadjoint, and by Theorem 6.1.9.2, it is a Hilbert-Schmidt operator.
Cl)
By Corollary 3.2.1.5,
~ab v( rl u x )
g(.,t)ux(t)dt
x
for every x C 7:). By Proposition 3.2.1.4, vx C 1:) and
rUVX = -Ur
g(., t ) x ( t ) r ( t ) d t
= -x
for every x E C([a, b]). c2) Choose x C L2(r.&) such that 1 VX
.~
----X
.
7 By Corollary 3.1.6.4, x E C([a,b]), and so, by cl), x E T). It follows by El) , that 1
1
{x ~ L2(r.A)lvx = ----x} = {x ~ ~lvx - -~x}
7
--
7
c3) follows from c2).
C4)
Since v is selfadjoint, we get
Ilvll = s u p
1
")'EA~ '
by c3) and Theorem 6.1.9.2 d).
(r 1)~ m
6. Selected Chapters of C*-Algebras
1r
L e m m a 6.2.2.3 ( 5 ) function on [a, b] , then Ilxll~ < ~
Take 5 > O. If x is a continuously differentiable
/ i ~' ( 1 x"dt + a
1 ) /ib + (2 + -~)~ ~ x2dt .
We may assume that
f b x 2dt
1
1
9
"r - I l x l l ~
1)2 5 "
+(2+
> b-a
Define
B :- {t~ [a,b]ll~(t)l > ~ } , ::
Io, l
Then
7A(B) <_ s
<_ 1,
so that A(B) _< -1< b - a . 7 Hence, there are tl, t2 C [a, b] with Ix(t~)l = w,
Ix(t2)l = w89
It~ - t21 _< ,,/ 1 m
.
Then
f l t2 x'dt
7 ~'+ 7
-
~f
Jc
x '2dt
and therefore z'2dt >_ ~
z'2dt
>_&~ (~ 1
_ 2)=.y2
2~ 89 - ~"~(~89 - 2)
>__ m
6.2 Selfadjoint Linear Differential Equations
Lemma
( 5 )
6.2.2.4
Take xCTP\{O}
and 7 E I R
such that
~tX = ")'rx. Define
5 := inf p(t),
(~' := inf r(t)
tc[a,b]
tc[a,b]
&-
a
i f cos a # 0
0
if c o s a - 0 ,
r
'
/ /3 i f cos/3#0 / 0
if c o s / 3 - 0 ,
and assume that q is positive.
a)
3' +
/a
rx2dt - P(a)x2(a)tg& + p(b)z2(b)tg~+ (px '2 + qx2)dt.
b) c~, /3 C [0, 2] + 7 r ~ = - 7 > c)
O.
1
7 >_ --5;(P(a)ltg&[ + p(b)ltg31)x
•
1
(2
b---~ +
p(a) tg&[+p(b)[tg~[)2)2 +
6
a) We have
~x~dt -
x(~x)dt -
x ( - ( p x ' ) ' + qx)dt -
-- - fabz(px')'dt + fabqz~dt -
= x(a)p(a)x'(a) - x(b)p(b)x'(b) +
= p(a)x2(a)tg& + p(b)z2(b)tg~ +
z'(pz')dt +
Z
(px '2 +
b) follows from a). c) By b), we may assume that [sinal+lsin/3 I r
[cosal+[cos/3 I r
qz2dt= qz2)dt.
1r
6. Selected Chapters of C*-Algebras
~44
Put p(a)ltg~l + p(b)ltg~l "
By Lemma 6.2.2.3, x'2dt + b - a + (2 + _)2
Ilxll~ -< r
x2dt <
c
~b x'2dt + ~i1( b -1 a + (2 + __)2 1 )2fb rx2dt ' e
and so, by a), 7
• (~
~b
jfab
rx2dt >_ -(p(a)ltg& I + p(b) tg~l) x
11
~,~d~+ ~ ( ~
1 )2~b
+ (~ + ;)~
)
rx~d~ + ~
~b
x,~d~
1 1 ) 2 ~ brx2dt, 5'(P(a)ltg&l + P(b)ltg~l) ( ~ -1 a + (2 + _)2e 1 (p(a)ltg&l + p(b)ltg~l ) • "Y>--5;
b_--Z-a+
Theorem 6.2.2.5
+
p(a) tg& I + p(b)ltg~l) 2) 2 5
m
( ~ ) (Sturm, 1836; Liouville, 1837)
a) A is infinite and bounded below, E
1
-~ < c~ ,
3~eA\{O} and there is an orthonormal basis (X~)~A of L2(r.A) such that x~ e jz for every 7 C A.
b) If x C D, then (7@ x~))~cA C g2(A),
( X / ~ ( x x~))~A C gl(A),
6.2 Selfadjoint Linear Differential Equations
(where the sum is considered in
145
L2(r-A)),
,),cA
x - E(xIx~)x~
(in C([a,b])),)
TEA
ux = r E
T(xlx~)x~
(in
L2(r.A)).
TEA
c) If u is injective (equivalently, if 0 ~ A ), g is the Green function of u (for the given boundary value problem), and v "L2(r.A) ~
L2(r. A),
x. ~ ~
g(.,t)x(t)r(t)dt,
then: C1)
V
is nuclear and
IlVlll = ~ I~l'. "yEA 1
c2) v -- - E ~(.]x~)x~
(in
Z:I(L2(r-A))).
~EA
c3) g = - E -1 | X.y (in C([a, b] • [a, b])) -ycA ,X.y C4)
X E
L2(r-A) =v v x - - E
(xlx~)x~
( in C([a,b])).
d) If q is positive and if a,/3 e [0, ~] + 7r~, then A C ]0, co[.
e) If a,/3E 2 then TEA\{0}
E g~(A\{0}),
and
(~T~-/<x Ix.~)IIxIIoo) "yEA c g2(A),
((xlx~)llx~ll~)~A e for every x C 7).
t~(A)
6. Selected Chapters of C*-Algebras
146
a) Given "),,5 E IR, define u~'T)
~C([a,b]),
x,
>ux+Srx,
~-~,~ := {z e Z)lu~x = ~ r x } , A~ := {'1' E IRI.T',~,~,:/: {0}}. If x E 7:), then
and so 9~'5,~/ ---- ~'-),-5
9
If ~/E fit, then (3, E A~) r
(gv~,~ =/= {0}) ~
(~'~-6 ~: {0}) r
('1'- 5 E A) r
('t' E A + 5),
so that A~=A+5.
Take 5 = IR such that q + S r >_ O.
By Lemma 6.2.2.4 c), A~ is bounded below. Hence A is bounded below, and there is an rl E IR, such that A, C ]0, co[. Let g~ be the Green function of u~ and consider v, " L2(r.)~) ---, L2(r 9 ~),
x,
~
g,7(',t)x(t)r(t)dt.
By Proposition 6.2.2.2 c), v,1 is selfadjoint and 1
Hence -v~ is positive. By Theorem 6.1.9.4 (and Theorem 6.1.9.2), v~ is nuclear and there is an orthonormal basis B of L2(r.,k) and an f E gl(B) such that
6.2 Selfadjoint Linear Differential Equations
147
(in /:l(L2(r./~)),
--v, -- E f(Y)('IY)Y yEB
f (B) = a p ( - v , ) C ]0, co[,
B C C([a, b]), -g, - ~
(in C([a, b] x [a, b])).
f (Y)Y | y
yEB
By Proposition 6.2.2.2 c3), 1
and by Proposition 6.2.2.1, the mapping B
> A,,
y,
>
1 f(y)
is bijective. Since An = A + r / , the mapping h "B
>A ,
y,
1
> 7;---c-r~ SLY)
is bijective. Since L2(r.A) is infinite-dimensional, B, and therefore A, is infinite. Given 7 E A, define x~ := h -~(~). (X~).yeA is an orthonormal basis of L2(r-,~), and
% - - - E TTEA +r/
1
(.[x,)xv
(in ~l(L2(r..~))).
If 7 E A , then V rl X ,), .~. _
and so, by Proposition 6.2.2.2 c2),
1 X,), 7q-r/
6. Selected Chapters of C*-Algebras
148
We have that
Ev+rl
1
=
~EA
~ f(y)< oc, BEB
so that 1
")'EA\{0}
b) By a) and Proposition 6.2.1.5 cl =~ c2 & C4 ~; C5,
(~(~lx,))~A e e~(A),
3,EA
(where the sum is considered in L2(r.)~)),
x=E(xlxT)x.
Y (in C),
"yEA
ux - r E
3`(YlX'y)x~
(in L2(r.A)) .
?EA
By Proposition 6.2.1.7 b), (X/~(YIx-y})-ycA E ~' (A). c) By a) and Proposition 6.2.1.5 a), ?J X,,/ - -
1
-3` x 7
for every 3' E A. By Theorem 6.2.1.6 a),b),d), v is nuclear, 1
jvjl, = "yEA
V ---
-
(in s
-
9A))),
"yEA 3`
g --
_
E _1 TEA "7x7
|
x7
(in C([a, b] x [a, b])).
6.2 Selfadjoint Linear Differential Equations
149
By Theorem 6.2.1.3 g), (in C)
1 (xlx,y)x.r "yEA
for every x E L2(,~). d) followsfrom Lemma 6.2.2.4 b). e) Take 5 > 0 such that q + 5r is strictly positive. Put 51 : - - tE[a,b] inf p(t)
52 "= tc[a,b] inf (q(t) + 5r(t))
>0
.
Let 7 E A. By Lemma 6.2.2.3, IIx~ll2 _< 5 ~ x~,2 d t +
5
<- 5--[fa bpx;2dt +
(5 <-
( 1
~ +
b -1a
( 1b - a +
2+
x~dt <
( 2 + ~1)2)2~1 j~ab(q + Sr)x~dt
_<
~_)2)a1 ) ~ "b
(
b- a +
+
2+
-~2
(pz'.f dt + (q + 5r)x~)dt.
We have -(px;)' + (q + ~ ) ~ - (~ + 5)~x~,
and so, by Lemma 6.2.2.4 a), + ~ = (~ + ~)
~x~dt =
t , ~ at + (q +
Thus IIx~ll~_
(5
~+
(1
~+
(1)2)21) 2+~
~
(~,+~).
It follows that
(llx~l~)
e e~(A\{O)). 3'EA\{0}
By b), "yEA
(<~lx~>llx~lloo)~A C ~I(A).
m
6. Selected Chapters of C*-Algebras
150
6.2.3 S e l f a d j o i n t L i n e a r D i f f e r e n t i a l E q u a t i o n s on I'
In this subsection we use the following notational conventions. a)
Given j E IN U {0}, write C (j) for the set of j - t i m e s continuously differentiable functions on I ' .
b)
n is an even natural number.
c)
(qj)je{o}uIN~_l is a family in C (~ such that qj E C (j) for every
j E IN~_I. d)
u" C (n) ~
C(~ ,
x,
~ ( - 1 ) } x (n) +
qjx(J) ,3, j=O
e) A is the Haar measure on ]F with A(]F) = 1. f)
Given ~/E IK,
g) A := {~/E IKIgV~ r {0}}. By Theorem 3.2.2.15 b =~ c, u is selfadjoint. If u is injective, then we write g for the Green function associated to u (Theorem 3.2.1.3, Proposition 3.2.2.10 g)) and define v'L2(A)
P r o p o s i t i o n 6.2.3.1
(
rL2(A),
5
x,
~fg(., J
t)x(t)dA(t) .
) A is a subset of ]R which is bounded below.
We may assume that IK - ~ .
By Propositions 3.2.2.10 f) and 6.2.1.1, we
may assume u to be injective. By Proposition 6.2.1.2 e), A c IR and by Example 3.2.2.12, A is bounded below. P r o p o s i t i o n 6.2.3.2 a)
v and g are selfadjoint and v is injective. v(~x)
b)
( 5 ) I f u is injective, then:
/
-
- ~ yo~ ~ w ~ y x E C (~)
vx E C (n) and u v x =
-x
for every x E C (~
I
6.2 Selfadjoint Linear Differential Equations
c) Fw : {x ~ L2(A)lvx - - 8 8
151
for every ~ ~ ]K\{0).
1 e A} . d) a p ( v ) = {-~17
By Theorem 3.2.2.15 b =~ d, g is selfadjoint. The assertions now follow from Proposition 6.2.1.2 and Proposition 3.2.2.10. I T h e o r e m 6.2.3.3
( 5 ) If u is injective, then v is an injective, selfadjoint, nuclear operator and there is an orthonormal basis B of L2()~) and an f E IN B such that vx - f (x)x
for every x C B . If B and f have the above properties, then:
a) f e el(B) and v = E
f(x)('ix)x
( in s
xcB
b)
Ilvll~ = tlfll~-
c) o-p(v)-
f(B).
d) f is real and vanishes nowhere. e)
If x c B , then x C C (n) and u x -
f) g = E f ( x ) x |
i-~) x .
(in C(V •
xEB
g) If y e L2()~), then vy = E f ( x ) ( y i x ) x
(in C).
xEB
h) If y c C (n), then
xeB E - ~
'
v/if(x) i ~eS
C (~ (where the sum is considered in L2(a))
xEB
y = Z(yl~)x
(i~ c),
xEB
(ylx)~ xEB
(in L2(r.A)).
6. Selected Chapters of C*-Algebras
152
i)
If ~ 9 IK\{0}, then -1
Im (c~1 - v) = f (a) • and -1
--1
yeA\ I (a) for every y E I m ( a l - v) , where the above series converges in C.
By Proposition 6.2.3.1, A is lower bounded and by Proposition 6.2.3.2 a), v is injective and selfadjoint. The assertions follow from the Theorems 6.2.1.3 and 6.2.1.6 and from Proposition 3.2.2.10 e). m
6.2 Selfadjoint Linear Differential Equations
153
6.2.4 Associated Parabolic and Hyperbolic Evolution Equations We continue with the notational conventions from Subsection 6.2.1 in this subsection, replacing the assumption "u is injective" with the assumption "A is a lower bounded subset of IR". By Theorem 6.2.2.5 a) and Proposition 6.2.3.1, A is bounded below as a subset of IR whenever the hypotheses of Section 6.2.2 or of Section 6.2.3 are fulfilled. Let B denote an orthonormal basis of L2(r.•)
and f a function B - + IK for which x E :D and
ux = f (x)rx for every x C B (by Theorem 6.2.1.3 and Propositions 6.2.1.1 and 6.2.1.4 a), there always are such B and f ).
Proposition 6.2.4.1
( 5 ) Let S , T
be topological spaces, (x,),c, a sum-
mable family in C(S) and (y,),~I a bounded family in C(T) . Then (x~ | Y,),cI is surnrnable in C(S • T ) . Take c > 0. Put II
II
5 : = ~up lly, ll. There is a J C ~S(I) such that c
IIEx~II < 4(1+6) ~EK
for every K E ~3f(I\J) (Proposition 1.1.6.6). By Proposition 1.1.1.5,
x~(~)y,(t)
<
~EK
for all K E ~f(g\J) and (s, t) C S x T. Hence
~cK
for each K e ~ i ( J \ J ) . C(S x T).
Corollary 6.2.4.2
By Proposition 1.1.6.6, (x~ | YL)~EI is summable in m
Let T be a topological space, (x~)~I a summable family in C(T), and (y~)~eI a bounded family in C(T). Then (x~y~)~zi is a summable family in C(T).
6. Selected Chapters of C*-Algebras
154
By Proposition 6.2.4.1, the family (x, | y,),eI is summable in C(T • T). Hence the family of restrictions of the functions to the diagonal of T x T is summable. Thus (x~y~)~, is summable in C(T). I T h e o r e m 6.2.4.3 ( 5 ) vanishes nowhere. Define
Take c E]0, co] and p,q E C([0, c D such that p
p . [O,c[
~IZ
s,
~r/~ da
'
Q'[O,c[
~IK
'
Jo p ( o ) '
s,
q(a)da
) f/s Jo
p(o)
and assume that
sup r e P ( s ) < 0 se[,o,c[ for every So E ]0, c[ and that
inf reQ(s) > - c o . se[o,c[ Take h E 79. For each x E B , define
h.: : [a, b] • [O, c[
)IK,
(t,s),
~ (hix)x(t)eS(x)P(s)-Q(s)
Put
Bo := {x e Blf(x) e IR+}. a)
{x E B I f ( x ) < a} is finite for every c~ E IR. In particular, B \ B o is finite.
b)
(hx)~eBo is a summable family in C([a, b] • [0, c D . We set (by a)) y'[a,b]•
) IK ,
(t, s) ,
)E
h~(t, s) .
xEB
c)
Given so E ]0, c[, the family (f(x)hxl([a, b] • [So, CD)xe, o is summable in C([a, b] • [so, c D . Define
z "[a,b]• ]O,c[
~ IK,
(t, s),
~ Ef(x)hx(t' xEB
d) y and z are continuous.
s).
6.2 Selfadjoint Linear Differential Equations
155
e) If A c IR+, then y in bounded and zi([a , b] • [so, c]) is bounded for every so ~ ]o, c[.
f) y(., o) = h. g) y(t, .) is differentiable on ]0, c[ for every t 9 [a, b] and
Oy P-~s + qy = z o~
h)
[a, b] • ]0, 4-
y(-,s) 9 7) for all s 9 [0, c[ and
on
[a, b] • ]0, 4.
a) By hypothesis, there is a 5 9 IR such that A + 5 C ]0, cx~[. By Proposition 6.2.1.1, A~ = A + 5 c]0, oo[. Hence u~ is injective. Take x 9 B with f(x) < a. Then
f(z)+5 < a+5, -1
x e f + 5(]0, ~ + 5[).
Hence -1
{x 9 B I f (x ) < a} c f + 5(]0, a + 5D, so by Proposition 6.2.1.5 b), {x 9 B[f(x) < a} is finite. b & c. By Proposition 6.2.1.5 Cl =~ c4, ((hlx}x)xes
C([a,b]). Take so E]O,c[. Since (e I`x,P-#) \
/ xEBo
is summable in
(resp. (f(x)eS(x)P-Q)
) xEBo
is a bounded family in C([0, c D (resp. a bounded family in (:([so, cD), it follows from Proposition 6.2.4.1, that (h~)xeBo is summable in C([a, b] x [0, c D (resp. (f(x)hx)~CSo is summable in C([a, b] x [So, cxD[). d & e follows from a),b), and c). f) follows from Proposition 6.2.1.5 c: =vc4.
6. Selected Chapters of C*-Algebras
156
g) Given x E B, Ohx (f(x) p~ + qhx = ph, P
q) P + qh, = f (x)h, .
By b) and c), it follows that y(t, .) is differentiable on ]0, c[ for every t e [a, b] and that Oy P--~s + qy = z
on [a, b] • ]0, 4 . h) Let 5 be a real number such that u6 is injective. Then, given x C B , ~ ( ( f ( ~ ) + 5)z) = ~ ( l;( u x + S r x ) ) = v ~ ( ! u ~ x ) = - x
(Proposition 6.2.1.2 b)). Using a),b), and c), we deduce that
~ ( z ( , ~) + ~ ( , ~)) = v ~ ( Z ( f ( ~ )
+ ~)h~(, ~)) =
xEB
xEB
xEB
xEB
By d) and Proposition 6.2.1.2 b), y(., s) E :D and
~y(., ~) = ~z(, ~) + ~y(., ~ ) Hence by g),
on [a,b]• ] 0 , ~ [
m
T h e o r e m 6.2.4.4
( 5 ) Tak~ c ~ ]0, ~ [ , p, q e C([O,c]), f~, f~ e D, ~ d ~, ~/ E IK 4 . Let C denote the set of those ~ C lit for which there are uniquely twice differentiable functions y~ and z~ on [0, c] such that y" + A
+ (q + ~)yo - o
Z~y.(o) + Z~y'(o) + Z~yo(c) + Z~y'(~) = 1
71y~(o) + 7~y'(o) + 7 ~ ( ~ ) + ~y'(c) = o, z~ + pz~ + (q • ~)z~ -- 0 fllZa(O) + ]~2z~(O) + ]~3Za(C) + ~4Z~(C) -- 0 ~'lZa(O) -~- ')'2z~(O) -~- ')'3Z~(C) + (~4Z~(C) ---~ 1.
6.2 Selfadjoint Linear Differential Equations
157
We assume that A c C , that the coefficients of u are C ~ - f u n c t i o n s , and that there are
so, 5 C IR+ with
Ily~ll _< 5,
IIzoll _< 5,
Ily'll __ 5v/-d,
I z ' l l _ 5v/-d
whenever ~ c C and ~ >_ so.
a)
There is a unique y E
C([a, b] x [0, c]) such that
z'[O,c]~L2(r.A),
s,
>y(-,s)
is twice differentiable, ( f ( x ) ( z ( s ) l x ) ) ~ e B e e~(B) fo~ every s 9 [O,c],
{ ~lZ(O) -J- ~2z'(O) -~- ~3Z(C) -Jr-~4Zt(C) --- fl ')'lZ(O) --{-'~2z'(O) -~ '~3Z(C) + ')'4Z'(C) -- f2, and r(z" + pz' + qz) + uz - 0 on
]0, c[, where uz is understood in the sense of L2-distributions. Given x E B , define
h~ "[a,b] x
[O,c]
> ]Z,
(t, 8),
> ((f~lx)y:(~)(8)+ (f2l~)z:(~)(s))x(t).
Then (hx)xcB is summable in C([a,
b] x [0, c]) and
y = ~-~ hx. xEB
b) If p and q are C~ - functions and IK = IR, then r \-~s2 + p-~s + qy on
)
=l=uy = O
]a, b[ x ]0, c[ in the sense of distributions.
c) I f the families (Y~)~ec,~>~o, (z'~)~ec,~>~o are bounded in C([O,c]), then is summable in C([a, b] x [0, c]) , y(t, .) is differentiable for every t C [a, b] and Oy ~ as =
Ohx ~ 9 C([a,
b] x [o, c]).
6. Selected Chapters of C*-Algebras
158
d) /f ufl, u f 2 9 7), then the families ( ~ )
xEB
, (~)
xEB
are summable
in C([a,b] x [0, c]), y(-, s) 9 7) for every s 9 [0, c], y(t, .) is twice differentiable for every t 9 [a, b] and Oy ~ Oss =
Oh~ ~ 9 C([a, b] x [0, c]),
02y - ~ Os2
02 h~ ~ 9 C([a, b] x [0, c]).
(Y' ) f(x)
a) Stepl
The families (Yf(~))xcs, (Zf(~))~eS, v/If(x)] xeS,f(~)r ( , ) Zf(x) a r e bounded in C([0, c])
V/If(x)[ ~.,f(x)~o -1
By Proposition 6.2.1.5 b), f (]-c~, a0]) is finite and so the assertion follows directly from the hypotheses. Step 2
(hx)xes is a summable family in C([a, b] x [0, c])
By Proposition 6.2.1.5 cl :=re4, the families ((fl[x)x),es, ((f2[x)x)xeB are summable in C([a, b]). By Step 1 and Proposition 6.2.4.1, (h,),eB is summable in C([a, b] • [0, c]). We put y :----E h x , xEB
z'[0, c]
Step 3
>L2(r.A),
s,
>y(.,s).
If S 9 [0, C], then (v/[f(x)[h~x(.,s))xes and (f(x)hx(., s))xem are summable in L2(r-A)
By Proposition 6.2.1.5 Cl
::~
C2,
(f(x)(fl[X))~B,
(f(x)(A[X))~m e E2(B)
and the assertion follows from Step 1 and Corollary 5.2.2.5. Define
6.2 Selfadjoint Linear Differential Equations
~'" [0, c]----+ L2(r-A),
s,
> Ef(x)hx("s)" xEB
Step 4
{ (f(x) II(zlx)II)x~Be g~(B), z i s
twice
differentiable,
and z" + p z ' + q z - :t=~
By Proposition 6.2.1.5 cl => c2, ( f ( x ) ( f l X))xCB,
(f(x)(f2lx))xcB e e~(B).
Since I(z(s)[x)l <_ I(fl[x)[(~ q-I(f2tx)]6 for every s C [0, c], and for every x C B with f ( z ) _> a0,
(f(x)ll(z[x)ll)~B c e2(B). Take x E B . Then d
i
1
ds(Zlx) - (fl x)yf(x) + (f21x)zf(x), d2 so that if f ( x ) >_ C~o,
II~d (zlx)l _< 5v/If(x) (l(fl]x)] + [(f2lx)[), d2 lids2 (z[x)] I < 25( (fl[x)[ q-I(f2lx)[)([ p[Iv/if(x)] + [[q[] + [f(z)[). It follows that
(.) xEB
xEB
By Proposition 5.5.1.20, z is twice differentiable and z" + pz' + qz =
--
fllx) xEB
("Yf(x) + PYI(z) ' + qYf(x) ) x +
159
160
6. Selected Chapters of C*-Algebras
, + qzs(x )) x = zs(~) + pzs(~) xEB
= T ~/(~)y:<.>z
T y}~/(x)zs<.lx
xEB
=
x6B
= T ~/(x)h~
= T~.
x6B
r(z" + p z ' + qz) + uz = 0, with uz
Step 5
in the sense of L2-distributions By Proposition 6.2.1.5 e), for every s E [0, c] ~z(,) =
r
Zf(x):~ = r ~ :(x)h,(., ~) = r~(,) xEB
x6B
in the sense of L 2-distributions. By Step 4,
~z(~) = Tr(z" + pz' + qz).
Step 6
The Boundary Condition
We have
Az(O) + Z~z'(O) + Z~z(~) + Z~z'(c) =
xEB
x6B
x6B
- E <:, ),> (., ,,<,> + . ) < ~ >
s6B
+
:,,~<.,
xEB
xEB
- Z<s~ix>x
= s~,
xEB
wz(O) + o,~z'(O)+ ~,~z(~)+ ~,~z'(~)- S~.
161
6.2 Selfadjoint Linear Differential Equations
Step 7
Uniqueness
Assume that r/ has the described properties. Define C'[O,c]
>L2(r.)~),
s,
>~?(-,s).
Take s E [0, c]. Then
xEB
By Proposition 6.2.1.5 e),
xGB
in the sense of distributions. Hence E(~(s)tx}f(x)x
- l u g ( s ) -- T ( ~ " ( s ) + p ( s ) ~ ' ( s ) + q(s)~(s)) .
xE B
It follows that for every x E B ,
(C"(~) + p(~)C'(~) + q(~)C(~)lx) - ~=f(~)(C(~) x). Thus d2 d ds 2 (CIx) + p--~s (~[x ) + (q + f ( x ) ) ( C [ x ) - O,
(C x) - (~ ~). Hence ~ - z
and ~/= y.
b) Let w and w* be the differential operators
02r 02~ ~ > Os 2
0r
0 Os (pC) + qr
on ]0, c[, respectively. Let 9~ be a Ca-function on ]a, b[ x ]0, c[ with compact carrier and take x c B . Then
fe b ~ ( t ) ~ ( t ,
~)dt = (xl 7lug(',
S)) --
(luxI~(', S))
-
-
6. Selected Chapters of C*-Algebras
162
j~ab = f(x)(xl~(.,
(z(t,
s)) - f ( x )
s)x(t)r(t)dt
for every s E ]0, c[ (Proposition 6.2.1.2 f)) and
/o ~w*~(t, s)yl(x)(s)ds = /o ~:(t, s)wy:(x)(s)ds
- :Ff(x)
/ocw*:(t, s)z/(z)(s)ds - /oc:(t, s)wz/(~)(s)ds -
/o ~:(t, s)yf(x)(s)ds,
Tf(x)
/oc:(t, s)z:(,)(s)ds
for every t C ]a, b[. It follows that ]"
h~(rw*
+ u(p)dsdt =
]a,b[x]o,~[
=
+
+
/~
(fllx)x(t)r(t)
b(f2]x)x(t)r(t)
(/c
w*~(t, s)yl(~)(s)ds
w *r
)
s)zl(~)(s)ds
dt +
dt 4-
/o~((fllx)y/(~)(s) + (f21x).-l(~)(s)) (/~ x(t)u:(t,s)dt ) ds
=
= :~f(x)fab(fllx)x(t)r(t)(fo~P(t,s)yf(x)(s)ds)dtT
:Ff(x)
if(x)
/~ (f2]x)x(t)r(t) (/o ~:(t,
((fllx}yf(x)(s) + (f21x)zf(x)(s))
)
8)Zf(x)(S)ds dt i
(z(t, s)x(t)r(t)dt
Hence /
y(rw *~ + u~)dsdt ]~,b[x]o,~[
= ~//l~,bt• xcB
hx(rw*~ 4- u(p)dsdt - 0
d s - O.
163
6.2 Selfadjoint Linear Differential Equations and
) -t-uy=O
r \~s2+P-~s+qy
on ]a, b[ x ]0, c[ in the sense of distributions. -1 c) By Proposition 6.2.1.5 b), f ( ] - oo, C~o]) is finite and the families ((flix)x)~eB, ((f2 X)X)~eB are summable in C([a,b]). Hence the families '
are bounded. By Proposition 6.2.4.1, the family (~-~)
C([a,b] x
is summable in xEB
[0, c]). By Proposition 1.1.6.26, y(t,-) is differentiable for every t E [ a , b ] and
Oy ~ Ohx O---s= ~ e C([a, b] x
[0, c]).
d) Take 5 C IR such that ue is injective and take k E {1, 2}. Then 1 v~x = - 5
+
f(z)z
for every x C B (Proposition 6.2.1.5 a)) and
fk =--Ve(!Uefk) C v~(Z)) (Proposition 6.2.1.2 b)). By Theorem 6.2.1.3 j), ((5 + f(x))(Alx)x)~e, is summable in C. Hence (f(x)(fk X)x)zeB is summable in C and therefore (f(x)h~(., s))xeB is summable in (J for every s 9 [0, c]. By Proposition 6.2.1.5 C2 ~ Cl, Y(', 8) e ~) for every s E [0, c]. By Step 1 of a), the families
f(x)
v/If(x)l
v/iX(x) I -~s,:(~)r
y"S(x)~\
,
- ~ , ] x~B,S(x)@o
xEB,f(x)#O
( zs(x) " ~
f (x),] xEB,f(x)r
are bounded in C([0, cl). Hence by Proposition 6.2.4.1, the families (O_~)
(~ -
are summable in C([a,b] x [0, c]). It follows that xEB
differentiable for every t C [a, b] and
y(t .)
xEB
,
is twice
6. Selected Chapters of C*-Algebras
~64
Oy = E Os
Oh~
~
E C([a, b] • [0, c]),
xEB
02y Os 2 = E
02hx
E C([a,b] x [0, c]).
~
m
xEB
L e m m a 6.2.4.5
( 5 )
Take n E IN and c E]0, cr
Let (aij)i,jer%
be a
family in C([0, c]) and (xi)ielN= a family of differentiable functions on [0, c] such that 7'1,
'
X i --
E
aijxj
j-1
for every i E INn. Then I1~,11oo < ~ ',~='
I~(0)1 j=l
for every i E INn.
Take ]Kn with the usual scalar product and define x [ o , c]
a ' [ 0 , c]
~ IK n,
t,
~(z,(t)),~o,
" IKn,n,
t,
~ [aij(t)]~,jeINn.
Then d
d
-dtllx(t)[I 2 = - ~ ( x ( t ) l x ( t ) ) = (x'(t)lx(t)) + (x(t)lx'(t)) =
= 2re (x'(t)lx(t)) - 2re (a(t)x(t)lx(t)) <
<_ 211a(t)x(t)ll IIx(t) I _< 211a(t)ll IIx(t)I ~ 9 It follows that IIx(t)ll ~ < dZ'l~(s)'ld~llx(O)ll~
and therefore
2c ~ Ila~klloo
_< e ,,~=,
\ oo
c E II~klloo~ IIx~lloo < ~ ~,~='
Ixj(0)l j=l
for every i E lNn.
m
6.2 Selfadjoint Linear Differential Equations
Lemma6.2.4.6
( 5 )
Take c 9
165
cxz[ and p,q 9 C([O,c]). For a 9 IR+,
let x~ be the twice differentiable function on [0, c] such that x~I I + px~I + (q + a2)x~ = 0 and
xo(0)
=
1,
x'(0)
=0
(as(O) = O,
x~(O) = 1).
Then there is a ~ E IR+ such that
IIxoll < z,
IIx'l] _< ~z,
Iix"]] -< ( a2 + Iipila + Iiq ])/~ for every a > 1.
Let a 9 [1, c~[ be fixed. Let y, z be the solution on [0, c] of the system of differential equations { y'=(-psin2at+~sinatc~176
(,)
z' - (psin at c o s a t - q cos 2 a t ) y - (pcos 2 at + q sin a t c o s a t ) z satisfying the initial conditions y(0)-l,
z(0)=0
(y(0)--0,
z(0)--l).
(This system was obtained by the m e t h o d known as "variation of constants".) We put x - y cos a t + z sin a t . Then y' cos a t + z' sin a t -- O, so that x ~ = -c~y sin at + a z cos a t . Furthermore, - y ' sin at + z' cos at - - p ( y sin a t -
z cos at) - q (y cos a t + z sin a t ) = a
166
6. Selected Chapters o/C*-Algebras
1 --(px' a
+ qx) ,
so that x" = a ( - y ' sin a t + z' cos at) - a 2 (y cos a t + z sin at) = = - p x ! - qx - a2x,
x" + px' + (q + a2)x = 0
and x(O) = 1,
x'(O) = 0
(x(O)=O,
x'(O)=l).
Hence xo = x and therefore x~ = y cos a t + z sin a t , !
x~ = - a y sin a t + a z cos a t .
By Lemma 6.2.4.5, there is an upper bound for of a > 1. The desired inequalities follow,
llyll
and
llzll
independent m
T h e o r e m 6.2.4.7 ( 5 ) Take c 9 ]0, c~[. Let p, q be continuous functions on [0, c[ and take fl, f2 9 T). For a 9 JR, let ya and z~ be the twice differentiable functions on [0, c[ satisfying y~+py~+(q+a)y~=O,
II
I
z~+pz~+(q+a)z~=O,
y~,(O)=l,
y~(O)=O,
z~(O)=O,
z~(O)=l.
I f the coefficients of u are C a - f u n c t i o n s , then:
a)
There is a unique continuous function y on [a, b] x [0, c[ such that
z ' [ 0 , c[ is twice differentiable,
>L2(r.A),
(f(x)(z(s)[x))xeB z(O) = f l ,
and
s,
>y(.,s)
e g2(B) for every s 9 [O,c[,
z ' ( O ) = f2
6.2 Selfadjoint Linear Differential Equations
167
r(z" + pz ! + qz) + uz = 0 on [0, c[, where uz is understood in the sense of L2-distributions. Given x 9 B , define
Then (hz)xes is summable in C([a, b] • [0, d]) for every d 9 ]0, c[ and y(t, ~) - Z
h~(t,
~)
xEB
for all (t, s ) 9 [a, b] • [0, c[.
b)
{o2h~)x~B
If u f l , uf2 9 T), then for each d 9 ]0, c[, the familes (-O~-~)~eB , \ ~
are summable in C([a, b] • [0, d]), y(-, s) 9 7) for every s 9 [0, c[, y(t, .) is twice differentiable for every t 9 [a, b] and Oy Os = ~
xEB
Oh~ Os '
02y 02h~ Os ~ = ~ zeB Os 2 "
By Lemma 6.2.4.6, if d 9 ]0, c[, then there is a 5 9 IR+ such that
llyoll < 5,
ilz~ii < 5,
Ily'li ___5 v ~ ,
II~'ll < 5v/-~
for every a > 1, where the norms are cconsidered on [0, d]. The assertion now follows from Theorem 6.2.4.4 a),d), m Lemma6.2.4.8
( 5 ) Take cC]O,c~[ and ~,'7 C [0,~]. Let q be a conti-
nuous, positive real function on [0, c] and x a twice differentiable real function on [0, c] such that x" = qx,
x(0) sin fl - x'(0) cos fl = 0, x(c) sin '7 + x'(c) cos'7 = 1.
a) x and x' are increasing and positive. b)
I f / ~ r ~, then x(O) > O.
c) If ~ = ~, then x'(O) > O.
168
6. Selected Chapters of C*-Algebras
Case 1
~-~
x(0) < 0 implies that x'(0)_~ 0, so that x and x' are decreasing. Then
9 (~) _< x(O) < o,
~'(c) _< ~'(o) _< o,
so that 1 = x(c) sin')' + x'(c) cos~ ~ O, which is a contradiction. x(0) = 0 implies that x'(0) = 0, and so x vanishes identically and we again obtain a contradiction. Hence x(0) > 0. It follows that x'(0) > 0 and that x and x' are increasing. Case 2
/3 =
71"
We have x ( 0 ) = 0. x ' ( 0 ) < 0 implies that x and x' are decreasing, so that
9 (~) ___ ~(o) = o,
~'(c) <_ ~'(o) < o
and we get the contradiction 1 = x(c) sin 7 + x'(c) cos 7 _< 0.
If x'(O) - 0 then x vanishes identically, which also leads to a contradiction. Hence x ' ( 0 ) > 0 and x and x' are increasing. Lemma6.2.4.9
( 5 )
Take c c ] O , c x ~ [ , c > O , / 3 E
I [0,~], and 7 C [0,~[.
Then there is a 5 > 0 such that for each continuous real function q on [0, c] and for each twice differentiable real function x on [0, c] such that q > c,
x"=qx,
x(O) sin ~ - x'(O) cos ~ - 0
x(c) sin'), + x'(c) cos ~ / - 1, we have
IIxlloo < ~,
IIx'll~ < ~
6.2 Selfadjoint Linear Differential Equations
By Lemma 6.2.4.8 a), x and x' are increasing and positive, so that
x(c),
I1~11~ -
Step 1
II~'li~ = x'(~).
1 -- ~ = 0 =~ Ilxlloo <_ c + -[, IIx'll~o = 1
We have x'(O) = O,
x'(c) = 1,
so that IIx'll~ = 1
and 1 - x'(c) - x'(O) --
/o c x " ( t ) d t
=
fo c q ( t ) x ( t ) d t
>_ ~x(O),
1
9 (o) < _ - , c
]]xi]oo - - x(c)
=
x(0)+
~0cx ' ( t ) d t
< -1 +
c .
C
Step 2
~ E ]0, 2[, ~/-- 0 => ]]xl]oo _< c + ctg/~, IIx'll~ - 1
We have (Lemma 6.2.4.8 a),b)) that
0 < ~(0) < x(~),
0 <_ x'(O) <_ x'(c) = 1.
Hence
]lx'll~-
1,
x(0) - x'(0)ctg fl < ctg ft. It follows that Ilxl]~o = x(c) = x(O) +
~0c x ' ( t ) d t
<_ ctg fl + c.
169
6. Selected Chapters of C*-Algebras
170
Step 3
7r
~ r ~
~ r 0 ~
I1~11~ <- - ~
1
1
~
I ~'11~ <- - COS '
We have ( L e m m a 6.2.4.8 a),b))
~'(c) > ~'(o) __ o,
0 < x(0) < x(c) _
~'(c) <
sin ?
cos 3'
Hence 1 Ilxll~ <
Step 4
71"
/3 = ~ ~
1
sin 3'
,
IIx'll~ <
cos 3'
Ilxll~ <- r = ' Ix'll~ <- co~
We have
IIx' I~ - x'(c)
I xll~ =
Proposition
6.2.4.10
x(c)( 5
)
fO C
x'(t)dt
cos 3, <_ cx'(c)<_
cos .7
II
Take /3,7 C [0,~-]2 . Let c C]O, oc[ and q a
continuous real function on [0, c]. Given a > ~/~[q , let x~ be the twice differentiable real function on [0, c] such that x" + (q - a2)x~ = 0
I
!
x~(O) sin fl - x~(O)cos 3 = 0 x~(c) s i n 7 + x~(c) cos7 ----- 1
(Theorem 6.2.2.5 d), Corollary 3.2.1.7 a ==v c). Then there is a 5 C IR+ such that
Ix~ll ~ ~,
fo~ ~ y
~ > v/I qll + 1.
and IIx'~lI ~ ~
6.2 Selfadjoint Linear Differential Equations
17/
By L e m m a 6.2.4.9, we m a y assume t h a t 3' = ~- By L e m m a 6.2.4.8 a), x~ and x~I are increasing and positive. It follows t h a t !
[Ix~ll = x~(c) = 1,
Ilx~ll = x,~(c).
Put ~0 : = v / l l q l l ,
~'[0,
c]
>IR,
t,
>sin/3sh(a-a0)t+(a-a0)cos/3ch(a-a0)t,
and X~
y~ "=
qo~
for every a > O, where !
x~(0) = limy~(t) y~(O) = a - ao t-~0 ' t>0
if /3 -- ~. T h e n
y~ is strictly positive ( L e m m a 6.2.4.8), continuous, twice
differentiable on [0, c] (on ]0, c] i f / 3 = ~ ), and yo(~) =
~o(~) "
T h e n on [O,c] (or on ]O,c] if f l = 2 ) !
y(~ ---
9 "s~ - x~"
Hence
y'~(O) = x'~(O)(a - ao) cos/3 - x~(O)(a - ao) sin
=0
( ~ - ~0 )~ cos~
if 3 - r 7" If /3 = 7~ , then !!
9 ~(o) = xo(O) = o
and limy'~(t) = lim x~(t)sh(a t-+O t-~o t>O
t>O
-
ao)t
-
(a
-
ao)x~(t)ch(a
s h 2 ( a - ao)t
-
ao)t
6. Selected Chapters of C*-Algebras
172
= lim
x" ( t ) s h ( a - ao)t - (a - a o ) 2 x ~ ( t ) s h ( a - ao)t 2 ( a - a o ) s h ( a - ao)t c h ( a - ao)t
t---+O t>O
9 " (t~ - ( ~ - ~ o ) ~ o ( t ) = lim ~ ' ' t~o 2 ( a - a o ) c h ( a - ao)t
_ o
t>O
by de l ' H 6 p i t a l ' s r u l e . Now x~ = 99~y~,
"
+ 2 ~ yI ~ I + ( a
~y~
a0)2~y~
so t h a t 0 = xoI I + (q - a 2 ) x ~ = ~ y "
+ 2 ~ aI y ~I + (q - 2 a 0 a + a2)~o~y~,
!
y" + 2 ~-~y~ + (q - 2aoa + a~)y,~ = O. Since q - 2.o.
+ .]
_< .~o~ - Z , ~ o . + .~] = 2 . o ( . o
- -)
< o.
it follows t h a t y"(O) = ( 2 a o a - q(O) - ao2)Ya(O) > O, if 13 ~ ~ ( L e m m a 6.2.4 . 8 b)). A s s u m e t h a t /3 ~9~
t
I
(x~sh(a = ( a - a o ) c h ( a - ao)t
i
X~ (p~ ~Pa -- x ~
-~- Y o ~ ----
'~ T h e n 3" t 2
99a
~3
- ao)t - (a - a o ) x ~ c h ( a - ao)t ) s h 3 ( a - ao)t '
so t h a t lt-~oqo~(t) i m ~ ( t ) y~(t) = (a - a o ) l i m x " ( t ) s h ( a - ao)t - (a - ao)2x~(t)sh(a - ao)t t--~o 3 ( a - a o ) s h 2 ( a - ao)t c h ( a - ao)t t>O
t>O
= l l i m ( a 2 - q(t) - (a - a o ) 2 ) z ~ ( t ) 3t~o sh(a-ao)t t>O
6.2 Selfadjoint Linear Differential Equations
1 3(a
OLO)
173
( 2 a o a - q(O) - ao2)X'~(O),
by de l'H6pital's rule. Hence !
!
limy~ (t) = _ 2x~ (0) ( 2 a o a - q(0) - a~) - (q(0) - 2 a o a + a~) t-+o ~ 3(a , t>O
= ( 2 ~ o ~ - q(O) - ~o~)
xo (0) Ot - -
OL o
~'(o) > o 3 ( a - ao)
by L e m m a 6.2.4.8 c). We want to show t h a t y'~ is strictly positive on ]0, c[. A s s u m e the contrary. By the above, limy~(t)t__~o= 0 ,
li
t>O
t>O
1l tlmoY~(t) > 0
Hence there is a t e ] 0 , c[ such t h a t y ' ( t ) = 0 and y~ is strictly positive on ]0, t[. T h e n y'~(t) = (2C~oC~- q - c~)y~(t) > 0,
so t h a t y" is strictly increasing on a n e i g h b o u r h o o d of t, which is a contradiction. T h u s y~ is strictly positive on ]0, c[ and y~ is increasing. We have 7~2
~r
I )I
.
r 99a
SO t h a t
~ ( c ) y " (c) -
(~(t)y'(t))'
dt -
(2c~oc~
-
q(t)
-
c ~ ) ~ ( t ) y ~ ( t ) d t <_
_< ( 2 ~ o ~ + ~ ] - ~ ) ~ ( ~ ) y ~ ( ~ ) ~ ,
y'(~) < 2aoaC ~(~)
,
t yt 7/ IIx'll = x~(c) = ~ ( c ) ~(c) + o(c)y~(c) < _
sin fl c h ( a - ao)C + (a - ao) cos fl s h ( a - ao)C < 2aoaC + (a - ao) sin fl s h ( a - ao)C + (a - ao) cos fl s h ( a - ~oojC"
6. Selected Chapters of C*-Algebras
Since lim ~oo
sin/3 ch (oL - ao)C + ( a - ao) cos/3 sh (o~ - ao)C sin ~ sh(c~ - ao)C + (a - c~0) cos ~ ch(c~ - ao)C
=1,
we m a y take 5 := 2aoC+ Lemma6.2.4.11
sin fl c h ( a - ao)C + (a - ao) cos fl s h ( a - ao)C sup ,~>~o+1 sin ~ s h ( a - ao)C + (a - ao) cos/3 c h ( a - ao)C" ( 5 )
Take c E]O, oo[ and p,q E C([O,c]). Let ~ be a
continuously differentiable function on [0, c]. Take ~, 7, ~', 7' E lR such that tg ~' = tg ~ + ~ ( o ) ,
tg.r' = t g . r - ~(~) ,
where we assign the value +co to the tangent at the poles. Let further x be a twice differentiable function on [0, c] such that x" + px' + qx = O, x(O) sin ~ - x'(O) cos ~ = 0
x(c) sin 3' + x' (c) cos 3' = 1, and define tr
y "= xe$o Then y is twice differentiable and
v" + (p - 2 , ) v ' + (q + ~: - ~ ' - p v ) v = o,
y(O) sin ~' - y'(O) cos 3' = 0
y(c) sin 7' + y'(c) cos 7' = c~ cos where cos 7' :=1 cos 7
if tg T = oo . We have successively
x = ye- 1o v(~)d~
7
~,(~)d~
6.2 Selfadjoint Linear Differential Equations
175
~, = (y,_ ~y)~-So ~(~)~, x" = (y" -- 2~y' + ~2y _ p, y ) e - f~ V(~)d~ Thus 0 = (x" + px' + qx)efo v(s)ds =
= y"-
2~y' + ~2y _ p,y + p(y, _ ~y) + qY _
= y" + (p - 2~)y' + (q + 9~2 - 9~'- P~)Y. If tg ~ / ~ cxD, then tg 0'1 ~ cx~ and it follows t h a t y(c) sin ~/' + y'(c) cos 3" =
= (x(c)(sin 7' + ~(c) cos 7') + x'(c) cos 7')e f~ ~(~)d~ =
= (x(~)tg~' + ~(c)) + x'(~)) co~.~'~z ~(~)~ =
= (~(c)tg~ + x'(~))cos~'~z
~(~)~ =
= cos ~Y'efo ~(~)d~ cos "), If tg ~ / = co, t h e n tg 0" = co, so t h a t y(c) sin 7' + Y'(C) cos 7' - y(c) = x(c)ef~ ~(s)d~ = efo ~(~)d~ =
= cos ~/'efo ~(s)d~. cos 3' A similar calculation can be carried out for /~. Proposition
6.2.4.12
(
5
)
Take c E]0, cx~[. Let p
m (resp.
q ) be a
continuously differentiable (resp. a continuous) real function on [0, c]. Take ~, 7 E IR, such that
tg Z +
p(~ 2)
, tg ~ -
p(~)
- - T e [0, ~ ] ,
where the tangent takes the value +c~ at the poles.
176
6. Selected Chapters of C*-Algebras
For each a > IIq- ~__ 4 v'9 II there is a unique twice differentiable real-valued function xa on [O,c] such that
a)
x~I I + px~I + (q - a)x~ = O, !
x,(O) sin 13 - x~(O) cos B = 0 !
x~(c) s i n 7 + x~(c) c o s 7 = 1
( {
x,~(0) sin lff - x,~ (0) cos t3 -- 1
rasp.
!
x.(c) sin 3' + x.(c) cos 3' = 0
)
There is a 5 E IR+ such that
b)
IIx~ll <__~,
IIx~ll _< ~v/~
for every c~ > IIq- ~4 - ~11 + 1" 2 c)
tg ~ + P---~), tg-y- ;(c) --y- e ~ + , then ther.x is a 5 C IR+ such that
for every ~ > IIq- ~4 - P'II + 1" 2 a & b. Let /3',7' E [0, ~] such t h a t
p(c) '
2
By T h e o r e m 6.2.2.5 d) and Corollary 3.2.1.7 a ~
c, there is a unique twice
differentiable function y~ on [0, c] such t h a t
,, Y~+
( q
p2
p,
4
2
) a
y,~=0,
y~ (0) sin 13' - Y" (0) cos B' = O, i y~(c)sinT' + y~(c) c o s ."vI
=
COS 7 ~e~foP(S)ds 1 COS 7
6.2 Selfadjoint Linear Differential Equations
177
where cos "),~ "-cos "),
1
if t g 7 = c~. P u t 1
x~ "= y a e - ~ fo p(s)ds
By L e m m a 6.2.4.11, x~ is twice differentiable and XI!,~ + px~I + (q - a ) x o
= O,
!
x~ (0) sin/3 - x~ (0) cos 3 = 0 !
x~ (c) sin 7 + x~ (c) cos 7 -- 1. The uniqueness of x~ follows from the uniqueness of y~, using L e m m a 6.2.4.11 again. By Proposition 6.2.4.10, there is a 5' such t h a t IlY~II-~ 5' and [[Y'I[-< 5'v/-~
for every ~ > IIq
p2
T - ~11 + 1. Put
Then
e
_<
The conclusion in parentheses can be derived from the other one by defining ~ ' [ 0 , c]
~ IR,
~-[0,~] - - ~ ~ ,
~o-[0,~]--~,
t,
,~ - p ( c -
t),
t,
~ q ( c - t),
t,
~xo(c-t),
178
6. Selected Chapters of C*-Algebras
3:="/,
and 5 " = ~ .
Then tg~+~=tgT-~E[O,
co]
E[0, co],
tg~-~--tg~+~
{ 5~(0)sin ~ - -' x~(0) cos~ = x~(c) sin7 + x~(c) cos7 = 1 ! Y~(c) sin~ + Y'~(c)cos ~ = x~(0) sin/? - x~(0) cos 7 = 0
and for any t E [0, c], Y~(t) + ~(t)Y'(t) + (~(t) - a ) 5 ~ ( t ) =
= ~ " ( c - t) + ; ( ( c
- t)~'(c
- t) + ( q ( ~ - t) - . ) ~ o ( c
- t) = o .
c) follows from the above proof and Lemma 6.2.4.9. T h e o r e m 6.2.4.13
( 5 ) Take c E ]0, co[. Let p (resp. q ) be a continuo~
differentiable (resp. a continuous) real function on [0, c]. Take fl, f2 E 7) , t3, ~/ E ]R such that tg~ +
L~ , t g T - p(c) ~ E [0,~] ,
p2 where the tangent takes the value +co at the poles, and given a > [Iq--u
t
,
let y~ and z~ denote the twice differentiable real functions on [0, c] such t ~"o + py" + (q - ~ ) y o = 0
z~I! + pzo! + (q - a)z~ = 0
y . ( 0 ) sin 3 - y~(0) cos/~ = 1
z~(O) sin ~ - z'ot (0) cos ~ = 0
y.(c) sin7 + y'(c) cos7 = 0,
z~(c) sin 7 + z~(c) cos 7 = 1
!
(Proposition 6.2.4.12 a)) 9 Assume that A C ] [ [ q - ~4 - ~lI ' co[ and that coefficients of u are C~-functions.
a)
There is a unique y E C([a, b] x [0, c]) such that
z ' [ 0 , c]
~ L2(r-A),
s,
~ y(.,s)
is twice differentiable, (f(x)(z(s)Ix))xeB E t2(B) for every s E [O,c],
179
6.2 Selfadjoint Linear Differential Equations
z(O) sin fl - z'(O) cos/3 = fl z(c) sin'7 + z'(c) cos-7 - f2, and r(z" + pz' + qz) - uz = 0 on ]0, c[, where uz is understood in the sense of L2-distributions. Given x 9 B , define
hx: [a, b] • [0, c]
> IK,
(t, s) ,
; ((fllx)ys(~)(s) + (f21x)zs(~)(s))x(t) .
y=Eh~. xEB
b) I f p and q are C ~176functions and 1K = ]R, then r \ ~
+ p-~s + qy
- uy = O
on ]a, b[ • ]0, c[ in the sense of distributions.
c) If
p(o)
p(~)
is summable in C([a, b] • [0, c])
then the family (ah~)
y(t, .) is dif-
xEB
ferentiable for every t 9 [a, b] and Oy _ x--" Oh• Os - ~ -~s 9 C([a, b] • [0, c]).
,
,
, k~
ble in C([a, b] • [O,c]),y(-,s)
xCB
differentiable for every t 9 [a, b] and Oy _ ~
02y _ ~ Os 2 --
are s u m m a xEB
e 7) for every s 9 [O,c],y(t,-)
Ohx
C C([a, b] • [0, c]),
02hx ~ e C([a,b] • [O,c]).
is twice
180
6. Selected Chapters of C*-Algebras
By Proposition 6.2.4.12 b), there is a 5 c IR+ such that
Ily~ll < 5,
Ilzol] < 5,
for every c~ > ] l q - p2 u
P']] + 1 If 2 tg~ +
I1r
_< 5 v ~ ,
p(c)
, tg7-
~
liz'il _< 5 v ~
6 IR+,
then, by Proposition 6.2.4.12 c), there is a 5 C IR+ such that
I1~'11_ 5,
IIz'il <
for every c~ > ilq- a~ _ p'2 II + 1 " The assertions now follow from Theorem 6 2.4.4. 4 i
L e m m a 6.2.4.14
( 5 )
Take c 6]0, c~[. Let q be a strictly positive con-
tinuous real f u n c t i o n on [0, c] and y, z twice differentiable real functions on
[0, c] such that y" = qy
z" = qz
y(c) - y(O) = O,
z(c) - z(O) = 1,
y'(c) - y'(O) = 1,
z'(c) - z'(O) = O.
Then
l i y i l -< c +
1
c inf q(t) .
iiy'l <. 1 .
.
l i z. i l < l .
1
i l z ' i i < - +c i l q l l c .
tE[O,c]
Step 1
y vanishes nowhere.
Take to 6 [0, c] such that y(to) = 0. Then y'(to) ~ O, since otherwise y would vanish identically. If y'(to) > 0 (y'(to) < 0), then y is strictly increasing (strictly decreasing) on [0, c], which contradicts the fact that y ( 0 ) = y(c). Step 2
y is strictly positive
Assume the contrary. By Step 1, y is strictly negative. Hence y' is decreasing and we obtain the contradiction
1 = ~'(c) - y'(0) ___ 0. Step 3
]]Yl]-< c + c
1
inf q(t)' t6 [o,c]
IlY']I~ 1
6.2 Selfadjoint Linear Differential Equations
181
By Step 2, y is strictly positive and therefore y~ is strictly increasing. Since y(O) = y ( c ) , it follows that y has a minimum on [0, c[ at a unique point to
and u'(0) < o < u'(c).
Since y'(c) - y'(O) = 1,
we see that Ily'll _~ 1 and that
/o
1 -- y'(c) - y'(O) =
y"(t)dt -
/o
q ( t ) y ( t ) d t > cy(to) inf q ( t ) . te[o,c]
Hence
Ilyll =
y(~)
-
y(to) +
y'(t)dt < c inf q(t) + c . te[o,c]
Step 4
3to e ]0, c[, z ( t o ) = O, z'(to) > 0
If z does not vanish on ]0, c[, then z t is strictly monotonic, which contradicts the fact that z'(0) = z ' ( c ) . Hence there is a to E ]0, c[, such that z(to) = O. Then zt(to) 7~ O, since otherwise z would vanish identically, z'(to) < 0 implies z is strictly decreasing, which contradicts the fact that z(~) - z(O) = 1.
Hence z'(to) > O. Step 5
II~ll-< 1
By Step 4, z is strictly increasing and vanishes at a single point of ]0, c[. Hence
z(O) < o < z(~). It follows from z(c) - z(O) = 1
that Iizi] _< 1. Step 6
[[z'l[ __ ;1 + [IqIlc
6. Selected Chapters of C*-Algebras
182
By Step 4, z' is increasing on [to, c] and decreasing on [0, to]. By Step 5, 1 >__z ( c ) -
z'(t)dt >_ z'(to)c,
and IIz'll = z'(c) = z'(to) +
q(t)z(t)dt < -1 + llol,_c
I
c
Theorem
6.2.4.15
( ~ )
Let q be a continuous real function on T and
take fl, f2 E I ) . Given a > sup q(t), let y~ and z~ be twice differentiable real tET
functions on [0, 1] such that
y~ + (q - ~)yo = o
z" + (q - . ) z .
= o
y ~ ( 1 ) - y~,(0) = 1
z~(1) - z~(0) = 0
y'(1) - y~(0) = 0,
z'(1)-
z'(0) = 1
(Proposition 3.2.2.11 and Corollary 3.2.1.7 a :=~ c) and assume that the coeffiand A C ]l]q]l, c~[.
cients of u are C~
a)
There is a unique y E C([a,b] x T) such that
z'T
>L2(r.A),
s,
>y(-,s)
is twice differentiable, ( f ( x ) ( z ( s ) l x ) ) x e B E e2(B) for every s E [0, c],
z ( O ) - f~,
z'tO)= s
and r(z" + qz) - uz - 0 on T , where uz is understood in the sense of L2-clistributions. We identify each s E [0, 1] with e 2~is E T and define h~ " [a,b] x T
~ IR,
(t,s),
> ((fllx)y:(~)(s) + (f21x)z:(~)(s))x(t)
for every x E B . Then the families (hx)xEB, (oh_b_~) Os ~CB a r e s u m m a b l e i n
C([a, b] • T), y(t, .) is differentiable for every t E [a, b] and y=Ehx, xEB
Oy
Ohx
o~ = Z
~
xEB
e C([a, b] • V ) .
6.2 Selfadjoint Linear Differential Equations
b)
183
If q is a C ~-function and ]K = IR, then r
~+qy
-uy-O
on ]a, b[xT in the sense of distributions. C([a,b]
N
xEB
r ) , y(., s) C D for every s C r ,
y(t, .) is twice differentiable for every
t c [a, b] and 02y _ ~
02hx
o~
~
c c([~, b] • r ) .
xEB
Since A is discrete (Theorem 6.2.1.3 c),f)), there is an r > 0 such that A c ][[q[[ + 5, c~[. By Lemma 6.2.4.14, there is a 5 c IR+ such that
I]wll _< ~,
IIv'll _< 6,
II~'~ll <_ 6
for every a E [[[q[[ + 5, ~ [ . The assertion now follows by Theorem 6.2.4.4.
9
6. Selected Chapters of C*-Algebras
IS4
6.2.5 Selfadjoint Linear Partial Differential Equations Throughout this subsection, we abide by the notation and hypotheses of Subsection 3.2.3. In addition, we assume T to be compact, u selfadjoint, r
:-- --Ue T
strictly positive, and
/ rdA = 1. (J.-M. Bony, Principe du maximum, indgalitd de Harnack et unicitd du probl~me de Cauchy pour les opdrateurs elliptiques ddgdndrds, Ann. Inst. Fourier 19.1 (1969) 277-30~, Thdor~me 6.1), eT is a potential on T (Lemma 3.2.3.2), and It follows that the Green function g of u is selfadjoint
eT = / g(', t)r(t)dA(t) .
L e m m a 6.2.5.1
( 5 )
Every x E Ll(r.)~)+ is eT-integrable and
First suppose that x is bounded. By Tonelli's Theorem
/(x.eT)rd)~ = / ( / g(s,t)x(t)r(t)d)~(t)) r(s)d)~(s) =
=/(/g(t,s)x(t)r(t)dA(t)) r(s)d,~(s) =
-- f ( / g(t, s)r(s)d)~(s)) x(t)r(t)dA(t) = = /eT(t)x(t)r(t)d)~(t)- /xrd)~. Given arbitrary x, we deduce that
/ xrd)~ - n-~lim/ ( x A neT)rd)~ --
6.2 Selfadjoint Linear Differential Equations
185
-li+moo/((xAner)'er)rdA=/(x.er)rdA. Hence x.eT is r.k-integrable and therefore finite on a dense set. This implies that X ' e T is superharmonic, i.e. x is eT-integrable. Lemma6.2.5.2
( 5 )
Let U be an open set o f T ,
I K
a compact set of U
and take g > O. There is a 5 > 0 such that for every x E Ll(r.A)+ vanishing on U , we have ([,emma 6.2.5.1)
/
xrdA < 5 ==> s u p ( x . e T ) ( t ) < g. tcK
Assume the contrary. Then there is a sequence (xn)nc~ in Ll(r.A)+ such that the following holds for every n E IN"
Xn vanishes on U, 1 xnrd)~ < 2n,
sup(xn.eT)(t) _> c. tGK
Put X "~ EXn. nE IN
Then x is r-)~-integrable and vanishes on U. By L e m m a 6.2.5.1, x is eT-integrable and harmonic on U. Since X'e T ~
~
X n'C T
nE IN
it follows by Dini's Theorem t h a t (Xn'er)ncIN converges uniformly to 0 on K , which is a contradiction.
I
/
L e m m a 6.2.5.3
~ 5 )
Take t E T and c > O. There is an open neighbour-
hood U of t and a a > 0 such that for every Borel set A of T
s u p { ( ~ . ~ ) ( ~ ) I ~ c ~} < whenever fn r dA < 5.
6. Selected Chapters of C*-Algebras
186
Given n E IN, put Un := U ! ( t ) and xn := ev~. n
Then (xn'eT)neIN is a decreasing sequence of continuous real functions on T converging pointwise to 0. By Dini's Theorem, it converges uniformly to 0. Hence there is an n E IN such that g
sup{(x~.eT)(S) l S e T} < -~. We set g
"~ g2n.
By L e m m a 6.2.5.2, there is a 5 > 0, such t h a t for every Borel subset A of
T\U~ ,
sup{(eA.eT)(S)[S E U} <
E
whenever fA rdA < 5. Let A be a Borel set of T with
Ard)~ < 5. Then
eA'eT ----eAnu,'eT + eA\U,'eT <_ xn'eT + eA\U~'eT. Thus
sup{eA.eT)(S) I s E U} < <_ sup{(xn'eT)(S)[S E U} + sup{(eA\U'eT)(S) I s E U} < e.
I
[
Lemma 6.2.5.4
[ 5 ) For every 6 > O, there is a 5 > 0 such that for every
Borel set A of T , sup{(eA.eT)(t) l t E T} < 6, whenever fA rd,k < 5. By L e m m a 6.2.5.3, for every t E T , there is an open neighbourhood Ut of t and a 5t > 0 such that for every Borel set A of T ,
sup{(eA.eT)(S) I S E Ut} < 6,
6.2 Selfadjoint Linear Differential Equations
187
whenever fA rdA < St. Since T is compact, there is a finite subset B of T such t h a t
T=Uut. tEB
Put 5 := inf St. tEB
Let A be a Borel set of T with
ArdA < ~. Take s 9 T . There is a t 9 B such t h a t s 9 Ut. Then
(~-~)(~)
< s u p { ( ~ ~)(~') I ~' e u~} < ~.
Hence
m
sup{(eA.eT)(S) i S 9 T } < c. T h e o r e m 6.2.5.5 a)
( 5 )
Take ol C ] l , c ~ ] .
If x e L~(r.)~) then x . e T e La(r.)~) and
IIx-~lio _< lixll~. b)
The operator v : L~(r.A)
> L~(r . A),
x,
> x'eT
is compact. a) follows from L e m m a 6.2.5.1 for c~ ~ c~. For c~ - c~, the assertion is trivial. b) Let B be the Banach space of bounded Borel functions on T (with the supremum norm). By Proposition 3.2.3.1, {x.eT I x E B #} is a precompact set of C(T) and therefore a precompact set of L ~ ( r . , ~ ) . This proves the assertion for
o/ ---- (:x:).
Assume now t h a t a ~ c~. Take c > 0. By L e m m a 6.2.5.4, there is a 5 > 0 such that for every Borel set A of T ,
ii~lil < 5 ~
ii~~ll~
<
(~)
6. Selected Chapters of C*-Algebras
188
Let x be a Borel function of L~(r.)~)#+. Put A := {Ixl > 5-~ }. Then
_>
/
J(AI ~ l ~ d ~ _ > ~I / A
I~l~
rd~,
IleAII,= ~ rd)~ < 5, I1<~'~11oo< (2)'~' By H61der's Inequality,
v(~)
1
~--1
_< (~o.~1-(~. ~ ) - -
< (~.~1"~,
~ ( ~ ) ~ < (~) (~~ Hence by Lemma 6.2.5.1,
IIv(x<,,)ll,,= I J" v(xeAl~rd'~l • < - -2( Since v(5 - •
(x~.eT)rd)~)• = ~ ( ~ d ~ )
<--
= ~11~11~ --- ~.
#) is a precompact set of L~(r.,~), there is a finite subset B
of L~(r-,~) such that
v(,~-~B#) c U u~(y). yEB
Since x -
xeA E 5 - •
# , there is a y E B with
v(x- ~)
e v~(y).
Thus C
6
I ~ - yll~ < I1~(~)11o + IIv(~ - ~*A) -- YlI~ < ~ + ~ -- ~, w 9 v~(y) and hence
~(L~
#)
c U
u~(y).
yEB
Thus v(L~(r.A) #) is a precompact set of LO(r.A) and so v is compact.
6.2 Selfadjoint Linear Differential Equations
T h e o r e m 6.2.5.6 a)
189
(s)
The map
v: Ll(r.)0
~ Ll(r./~),
x,
> X'eT
is an operator with ]lv]] _< 1.
b)
For every x 9 Ll(r.A), ~(vx)
=
- ~
and
(xEImv) ~ c)
(ux 9 Ll(r.)~)) ~
(v(lux)) = -x)
.
Given ~ 9 IR, put
9~'a :-- {x 9 Ll(r.A) l u x - - -c~rx}. Then 1 . ~ = {x 9 nl(r.A) l vx = -~x} for every a 9 IR\{0}, and
{~ e ~IY=o # {0}} c ]0, ~[, ~(~) c ]o, oo[. a) follows from Lemma 6.2.5.1. b) By a) and Lemma 3.2.3.4 a), ~(vx)
=
-~x
.
Assume that ux C Ll(r.)~). Then ! u x is eT-integrable (Lemma 6.2.5.1) By Lemma 3.2.3.4 b) (and Lemma 3.2.3.2), r
x = -v(!ux)
c Imv.
Assume that x C Im v. Then there is a y C LI(r-A) with x:vy.
6. Selected Chapters of C*-Algebras
190
Then, by the above, ux = uvy = -ry
C Ll(r.)~).
c) By b), if a e IR\{0}, then
(~ e ~ ) ~
(~z = - ~ )
1
r
(vz = ~ x ) .
Since 7-/(T) = {0} (Lemma 3.2.3.2), ~'o = {0}. Take a > 0 and x e jr
.
Then o~vx + x -- 0,
so that = -,~
= -~(,~vo~
+ ~)
= -~v~(,~w
+ x) = o .
Hence
{~ 9 n~ I J=~ # {0}} c 10, o~[. It now follows that
o,(v) c ]0, o~[. T h e o r e m 6.2.5.7
(5)
m
The m a p
v : L2(r.A)
~L2(r.~),
x,
> x.eT
( T h e o r e m 6.2.5.5 a)) is an injective, positive, compact operator.
Step 1
v is injective
This follows from Theorem 6.2.5.6 c). Step 2
v is selfadjoint
Take x, y C L2(r-A). Then, by Theorem 6.2.5.6 b),
<wi l iv ,
-/Ivxl rx(vy)d)~-- @Ivy).
6.2 Selfadjoint Linear Differential Equations
Step 3
191
v is compact
This was proved in Theorem 6.2.5.5 b). Step 4
v is positive
This follows from Theorem 6.2.5.6 c).
Corollary 6.2.5.8 a)
I
( 5 )
There is an orthonormal basis B of L2(r./~) and an f E IR B such that ux = - f ( x ) r x for every x E B .
b)
If B and f are as above, then f is strictly positive and 7 c co (B).
c)
Given ~ c IR, define
~-~ := {x e L 2 ( r . A ) ] u x = - ~ r x } . Then ~-'a is finite-dimensional and
{~ e n~ l.r,~# {0}} c ]0, ~ [ . By Theorem 6.2.5.7, v : L2(r.)~)
>
L2(r.A),
x,
>
x'eT
is an injective, positive, compact operator. By Theorem 6.2.5.6 c), {x e L2(r.)~) l ux = - a r x } = {x E L2(r.)Q l v x -
1
-~x}
for every a E JR\{0}. a) and b) follow from Corollary 6.1.3.18. c) By Theorem 6.2.5.6 c),
{~ e n~ {.to _/={0}} c ]0, ~ [ . Since v is compact, each 9v~ is finite-dimensional.
I
192
6. Selected Chapters of C*-Algebras
6.2.6 Associated Parabolic and Hyperbolic Evolution Equations
This subsection retains the notation and all the hypotheses from the preceding subsection except that the hypothesis "r is strictly positive" is replaced by the weaker one "r is strictly positive, strictly negative, or it vanishes identically". We put r if r is strictly positive p
,__
-r
if r is strictly negative
any strictly positive C~
on T if r vanishes identically.
Given c~ E IR, put J:~ : - {x e L2(A) l u x -
-~px}.
Finally define A := {a e IR ] ~'~ =/= {0}}.
Proposition 6.2.6.1
( 5 ) If (Xn)nC~ is a sequence in L2(p.A) such that 1 - - U X n --__ X n + 1
P for every n E IN, then each xn (n E IN) is a C~
on T . In particular,
any x C L2(p.)~) is a C ~ - f u n c t i o n iff ux is a C~-function.
The assertion follows from Theorem 18 in L.P. Rothschild and E.M. Stein, Hypoelliptic Differential Operators and Nilpotent Groups, Acta Math. 137, 1-2 (1976) 2~7-320.
Proposition 6.2.6.2 a)
I
(5)
There are an orthonormal basis B of L2(p .•)
~x = - f ( z ) p ~ for every x C B .
and an f C ]Rs such that
6.2 Selfadjoint Linear Differential Equations
b)
193
If B and f are as above, then f (B) is bounded below and has no point --1
of accumulation, each f (a) is finite (a E IR) and every element of B is a Ca-function.
c) A is bounded below and each ~ d)
is finite-dimensional (a E A ) .
If B and f are as in a) and if h E I R B satisfies h f E g2(B), then
E h(x)x E C(T), xEB
(h(x)ux)xEB is summable in L2(p.A) and
xEB
xEB
Given a,/3 E IR, define u n "- u - / 3 p ,
J~z,~ - {x c L2(p.A) l u z x -
-c~px},
AZ "- {a E IR I 9rZ,~ --/={0}}. Take a, fl E IR. Then for each x E L2(p 9 ,k),
(~ 9 m,,.) r
(~-
-~p~) ~
(~-
- ( ~ - Z)p~) r
(~ e m . _ , ) .
It follows that ~z,. _ )c_z,
(a E An) e====v(gr~,, r {0}) ~
(gc,_Z r {0}) r
A-An-/3.
We have
?_t2e T - -
--r-
2p-
-3p
if
r>O
-p
if
r
-2p
if
r-O.
( a - t3 E A),
6. Selected Chapters of C*-Algebras
194
a) By Corollary 6.2.5.8 a), there is an orthonormal basis B of L2(p.A) and a k C IP~B s u c h t h a t
u2x = - k ( x ) p x ,
~x = (2-
k(z))px
for every x E B . b) Given x E B ,
u2x = - ( f (x) + 2)px. By Corollary 6.2.5.8 b) , f + 2 is strictlv~ positive and y-j5 1 E co(B). It follows -1
that f ( B )
is bounded below, has no point of accumulation, and f (a) is finite
for each a C JR. By Proposition 6.2.6.1, each element of B is a C~-function. c) follows from the above considerations and Corollary 6.2.5.8 c). d) We write T for the harmonic space constructed on T with respect to u2. Since U2eT is strictly negative, if x C L2(p.A), then we may write X'eT for multiplication with respect to T . By Theorem 6.2.5.5 b), the map v "L2(p.)~)
> L2(p.A),
x,
~ x'eT
is an operator and by Theorem 6.2.5.6 b), ~(~)
= (~)~
for every x r L2(p.A). Take x C B . Then U2X
= -(f(x)+
2)px = ( f ( x ) +
2)u-~U2(VX )
and so - ( f (x) + 2)
p vx = x ?.t2eT
(Theorem 6.2.5.6 c)). Then h(~)x
xCB
-
P ~t2eT
~h(x)(f(x)
+ 2)vx =
xCB
__v(hlxlIixl+2x)u2e ciTi
6.2 Selfadjoint Linear Differential Equations
195
so that
--
--
-
-
~
2
V
~2eT
h ( x ) ( f ( x ) + 2)x + 2p E
= -P E xEB
h(x)x =
xEB
- -p ~
h(x)f(x)x - ~
x6B
h(x)ux.
I
xEB
P r o p o s i t i o n 6.2.6.3 ( 5 ) Let B be an orthonormal basis of L2(p.)~) and take f c ]R B such that ux -- - f (x)px for every x c B . Take n C IN and let y be a 2n-times continuously differentiable function on T . Then (f~(x)(Ylx))~cB e g2(B).
We prove the assertion by induction on n. Assume that the assertion holds for n - 1. Then fn(x)(yix)-
fn-~(x)(ylf(x)x ) ---f"-l(x)(yl~ux)-
1
= -f~-~(z)(~ylx) for every x E B. Since l u y is 2 ( n - 1)- times continuously differentiable
by the inductive hypothesis. Hence (fn(x)(Ylx))x~B e e2(B).
I
196
6. Selected Chapters of C*-Algebras
P r o p o s i t i o n 6.2.6.4
( 5 ) Let B be an orthonormal basis of L2(p-A) and
take f E IR B such that ux - - f ( x ) p x for every x C B . Take h C g ~ ( B )
and let y be a C ~176
on T . Then
~ h(~)(ylx)x xEB
is a C ~ - f u n c t i o n on T .
By Proposition 6.2.6.3, if n C IN, then ( f " ( x ) h ( x ) ( y l x ) ) ~ B e g2(B) ,
and we define y.
= (-1)~-~ ~-~ f"-'(x)h(x)(ylx)x. xEB
Then by Proposition 6.2.6.2 d),
1 1 = -~y. = (- 1).-1 ~ fn- '(~)h(x)(~ ~)-~x P
xeB
= (-1)n E
P
f ' ~ ( x ) h ( x ) ( y l x ) x = y,+l
xEB
for every n C IN and the assertion now follows from Proposition 6.2.6.1. T h e o r e m 6.2.6.5
( ~ )
I
Take c C]O, co] and let p,q be continuous real
functions on [0, c[ with p strictly positive. Then for each Coo-function h on T , there is a uniquely continuously differentiable map
y: [0, c[ ---+ L2(p.A) such that y(O) = h, and y(s) is a C~176
on T and
whenever s E [0, c[. If we define
z'T•
[0, c[
> IR,
(t,s),
then uz = -P(P~ss + qz) in the sense of distributions.
>(y(s))(t),
6.2 Selfadjoint Linear Differential Equations
197
By Proposition 6.2.6.2 a),b), there is an orthonormal basis B of L2(p.)~) such t h a t every element of B is a C a - f u n c t i o n on T and an f C IR B such -1
that
f (] - c~, 0]) is finite and u x -- - f ( x ) p x
for every x C B . Define P'[0, c[~IR,
~ ~o s p do (a) '
s,
Q-[0, c[-----+ IR ' s,
) fo x q(a)da p(~)
and for each x C B ,
a~ "[0, c[
) L2(p.A),
s,
> (hlz)e-S(~)P(~)-Q(s)z.
By Proposition 6.2.6.3, ( f n ( x ) ( h l x ) ) x e u e g2(B) for every n C IN. For each n E IN, define
yn" [0, c[
~ L2(p.)~),
s,
) ( - 1 ) n-1 E f n - l ( x ) a x . xEB
Then Yl(0) -- h and by Corollary 5.2.2.6, each Yn is continuous. Take s E [0, c[. Since
1 -uax(s) - -f(x)ax P we see that
1 P for every n c IN. By Proposition 6.2.6.1, yl(s) is a C a - f u n c t i o n on T . Given any x c B , d f ( x ) + q(s) ds (ax(s)[x) = p(s) (ax(s)ix) " By Proposition 5.5.1.20, yl is differentiable and dyl ds
1 E(f(x) P xEB
+ q)ax
1 q = -Y2 -- - Y l 9 p p
198
6. Selected Chapters of C*-Algebras
Hence P(P~s + qyl ) = uyx .
Now we prove the uniqueness. Assume that 77 has the properties described. Given x E B , define
~-[o, c[--~ ~ ,
~, > (~(~)x)
Take s E [0, c[. Then ~(~)-
E<~(~)lx>x- ~ ~(~)x, xEB
xEB
u~(s) = - p ~ ,7~(s)f (x)x xEB
(Proposition 6.2.6.2 d)). Take x E B . Then f (x)rl~ (s) - - ( ~ ur/(s)Ix> dr/ = -(p(s)--~s(S ) + q(s)rl(s) x) - -p(s)rl.(s) - q(s)rlx(s), p(s)rf~(s) + (q(s) + f(x))rlz(s) -- O.
Since ~ ( 0 ) - (hl~),
r/x is uniquely determined. It follows that r], too, is uniquely determined. I
T h e o r e m 6.2.6.6 ( 5 ) Take c E]0, c~] and let p,q be continuous real functions on [0, c[. Then, given any C~-functions hi,h2 on T , there is a unique twice continuously differentiable map y" [0, c[
> L2(p.A)
such that y(O)=hl, and for each s E [0, c[, y(s) is a C~~
dy ~(O)=h2 on T and
6.2 Selfadjoint Linear Differential Equations
d2y uy(s) - p \ ~ ( s )
199
dy ) + p(s)-~s (s ) + q(s)y(s) .
Define z : T • [o, c[
) ~,
Then
uz = p
(t, ~),
) (y(s))(t).
( 2z z ) + P-~s + qz
in the sense of distributions. By Proposition 6.2.6.2 a),b), there is an orthonormal basis B of L2(p-A) -1
consisting of C~-functions on T and an f e ]R B such that
f ( ] - c~, 1]) is
finite and ~x = -f(x)px
for every x E B . By Proposition 6.2.6.3,
(f"(x)
(f"(x)
for every n C IN. Given ,:~, let y~ and zo denote the twice differentiable real functions on [0, c[ which satisfy
y~ + py~ + (q + c~)y~ -- O, z~I I + pz~l + (q + ~)z~ = O, y~(O) = 1,
y'~(O) = O,
z,~(O)- O,
!
z~(O) = 1.
Given x E B , define ax: [O,c[
~ L2(p.A),
s,
> ((h~lx>yi(~)(s) +
Take c' E ]0, c[ and define II
II
ll~J,(~,) := sup i~(s) l s~[0,c']
for every continuous real function ~ on [0, c[. By Lemma 6.2.4.5, there is a E lR+ such that
6. Selected Chapters of C*-Algebras
200
II(q~lx)llr
~ :~(l(glx>l + I(hlx)l),
d
ll~ll(~,)~/~(ll+ ll)v/f(x) d2
ll~s2(a~Ix>l(=')~/~(ll)(f(x)+ llPll(=,)v/f(x)+ llqll(=')) for every x E B with f ( x ) > 1. Hence ( f n ( x ) a , ( s ) ) , e B
is summable whenever
s E [0, c[ and n E IN U {0}. For each n E 1N define Yn "[O,c[
) L2(p.)~),
s,
~ (-1)"-' E
fn-l(x)ax(s)"
xEB
By Corollary 5.2.2.6, yn is continuous. Take s E [0, c[. Since 1
-Uax(S) -- -f(x)ax P for every x E B , it follows that 1 P
for every n E IN (Proposition 6.2.6.2 d)). By Proposition 6.2.6.1,
yl(8)
is a
C ~ - f u n c t i o n on T . If x E B , then d
,
d---~(a~(s)lx) = (glx)YS(~)(s)+
,
(hlx)z:(~)(s),
d2
ds 2 = yy(~)(s) + z~(~)(s) and so
d2
d
ds 2 (a~(s)lx > + p
(a~(s)lx) + q(ax(s)lx > -
= -f(x)((glx)Y/(~)(s ) + (hix)zs(~)(s)) = -f(x)(a~(s)Ix ) . By Proposition 5.5.1.20, Yl is twice continuously differentiable, yl(0) - h i , dAa (0) -- h2 and ds
d2yl ~8 d82 + p + qyl - y2. Hence
6.2 Selfadjoint Linear Differential Equations
P ~+P
+qYl
201
=uyl.
We now prove the uniqueness. Assume that ~7 has the properties described. Given x E B , define
~-[o,~[
>~,
~,
><~(~)lx>.
Take s E [0, c[. Then
xEB
xEB
~(~) - _ p ~ , ~ ( ~ ) f ( ~ ) ~ xEB
(Proposition 6.2.6.2 d)). Take x E B . Then
f (x)q~(s) -- -<~url(s)lx )
d2r/ dq = -<~-fi~ (~) + P ( ~ ) ~ ( ~ )
-
-
+ q(~)~(~)l~> -
= - v ~ ( ~ ) - p ( ~ ) ~ : ( , ) - q(~)v~(~),
rl"x + prl" + (q + f(x))rlx = O. Since ~(0)
= ,
,~'(o)= ,
fix is uniquely determined. It follows that ~, too, is uniquely determined.
I
6. Selected Chapters of C*-Algebras
202
6.3
Von
Neumann
Algebras
This section is devoted to the study of von Neumann algebras. A v o n Neumann algebra on the Hilbert space H is a W*-subalgebra of / : ( H ) . We first introduce a number of topologies on s
and establish several properties of these
topologies, such as Kaplanski's Theorem. With the universal representation, this theorem shows that the bidual of a C*-algebra has a natural W*-algebra structure. This enables us to once again extend the functional calculus, this time to W*-algebras. This extension is used to decompose self-normal operators with respect to bands of measures, in analogy with the manner employed with measures. The von Neumann algebras can be characterised by means of the abovementioned topologies, or by the order relation, or by commutants (von Neumann's Theorem). In the commutative case the last characterization leads a very elegant statement. The results allow us to characterize the irreducible representations (Segal's Theorem), to list all finite-dimensional C*-algebras, and to prove that every reflexive C*-algebra must be finite dimensional. Finally, representations of W*-algebras are examined. We prove that every W*-algebra admits a faithful representation whose image is a von Neumann algebra.
In this section E is always a C*-algebra and H is always a Hilbert space. We identify s
with s
by means of the involutive isometry 1
s
>s I(H)',
u,
>
(Corollary 6.1.7.9). According to the notation introduced in Definition 1.2.6.1, /:(H)~,(H) and s
) denote the set /:(H) endowed with
the topology of pointwise convergence in s
and s
respectively.
These topologies are usually called the ultraweak operator topology and the weak operator topology of / : ( H ) , respectively. This terminology is rather unfortunate, since the weak operator topology is weaker than the ultraweak operator topology. The topology of ~.(H)H is called the strong topology of s topologies of s
and s
The
are finer than the topology of f_.(H)Ls(H)
and
H)V.!(H)
(U) '
6.3 Von Neumann Algebras
since s
is dense in s
203
(Theorem 6.1.3.12 d)). By the Alaoglu-
Bourbaki Theorem, this space is compact. The usual norm topology of s is finer than the topologies of s and s 6.3.1 T h e S t r o n g T o p o l o g y
If H is infinite-dimensional, then the maps
P r o p o s i t i o n 6.3.1.1
f-.(H)#1(H) s
>s
>~(H)s
s
>s
u)
>u,
U)
) U,
u,
> u*,
are not continuous. Let (xn)~e~ be an orthonormal sequence in H . For p E IN, put
up'H
>H,
x,
>E{xlxn)x,+p. nE IN
Then (Up)pE~ converges to 0 in s s . Hence, the map
~(H)~I(H )
but does not converge to 0 in
>s
u,
>u
is not continuous. Put ?.t "-- E ~ ('[xn)xn E nE IN
(H)
and for every A E ~ f ( H ) , ~ ( A ) "= {v E s and let ~ denote the filter on s
]x E A ~
vx = 0}
generated by the filter base
{$-(A) [ A E q3f(H)}. Then ~ converges to 0 in s Take A E q3f(H). There is a p E IN, such that xp q~ A •177. Given c~ E IK, define
6. Selected Chapters of C*-Algebras
2o4
~o := ~ < . 1 ( - ~ ) ~ > ~ . Then v~ 9 ~'A and 1
v~H)(u) = o~ ~
1
-~(v~x.lx.} =
n6 IN 1
O~
nEIN
(Proposition 6.1.6.2, Theorem 5.5.1.15 c)). It follows that ~ does not converge to 0 in E.(H)s
Hence the map
s
~ s163
,
u,
~u
is not continuous. Let u~ and ur be the left and the right shift on t~2 , respectively. Then u; - ur (Example 5.3.1.6), (u~)ne~ converges to 0 in s does not converge in s
and (u~)n~rq
Hence the map s
--+ s
u ~-+ u*
is not cor.tinuous.
!
The traces on s with respect to the topologies s s
of the O-neighbourhood filters ) , and s coincide.
P r o p o s i t i o n 6.3.1.2
Take x 9
and r
Put I II~xll < ~}.
/4 "= {u 9 s ]; := {u e s
< c2}.
If u 9 l;, then 0 = < ~ 1 ~ > ___ <~xl~> = <.l~>z(~) < ~
(Corollary 5.3.3.7, Proposition 6.1.6.2). Hence, u 6 / 4 and ]2 C / 4 . Since x and c are arbitrary, it follows that the traces on s filter with respect to the topologies s sertion now follows from
s
and s
- s
# of the 0-neighbourhood coincide. The as-
I
6.3 Von Neumann Algebras
Proposition a)
s
6.3.1.3
• s
d)
)s
• s
u, --+ s
s
s
The maps
(u,v) ~
~ f-.(H)u,
b) Nof_.(H)H c) s
(0)
uv,
) u*, ),
(U~ V) ~-+ V*~t
u ~ u*u.
are continuous.
a) Take (u0, v0) e s
# x / : ( H ) . Then
I I ( u v - uovo)xll ~ I l u ( v - vo)xll + I1(~- uo)voxll II(v - vo)~ll § I1(~ - ~o)voxll for every (u, v) e / : ( H ) # • s and x e H . The assertion now follows. b) Take u, v C No f_.(H) and x E H . Then I ( u - v)xll = = < ( u - v)x I ( ~ -
v)x> - <(~* - v * ) ( ~ - v)~ Ix> =
= <(u*u + v * v ) x l x > - - ,
so that I1(~* - v*)xll = - I 1 ( ~ -
v)xll ~ =
= (vxl~x> + < ~ x l v x > - < x l v u * x > - .
Since ~ i m ( ( v x l u x ) + (uxlvx) - (xlvu*x) - (vu*xlx)) =
= (~xlu~) + (~xl~x) - ( x l ~ * x )
- (~*~lx)
we get that lim II(u* - v*)xll : o. v-.--+u
= 0
205
206
6. Selected Chapters of C*-Algebras
c) Take Uo, V0 E E ( H ) , x,y E H , and ~ > 0. Put b := inf
1, 3(lluoxll + IlvoYll + 1)
U : - {u e L ( H ) I II(u- uo)xll < 6}, V : - {v e Z:(H)I II(v- vo)xll < ~},
Then U and V are neigbourhoods of uo and vo in /~(H)H, respectively. For (u, v) E U • V ,
(~*u - ~;uo, ( l y ) z )
= I<(v*u- v~uo)xly>l = I ( u x l v y > - l I<(u -
uo)x I voy>l
+
I
- vo)y>l + I((u -
uo)x I(v
- vo)y>l
II(u - uo)xll Ilvoyll + Iluoxll I I ( v - vo)yll + II(u - uo)xll II(v - vo)yll
~llvoyll + Iluoxll~ + ~ < ~,
d) follows from c), since the diagonal map s
) C(H)H • f~(H)H,
x,
) (x,x)
is continuous. P r o p o s i t i o n 6.3.1.4
II ( 0 ) (Vigier, 1946) If 3v is a nonempty downwards
directed set of L(H)+ and ~ is its lower section filter, then the following are equivalent:
a) 0 is the infimum of ~ . b)
Given x E H , hm(~.l.) = in~(~*l*> = o. u,~
c) ~ converges to 0 in ~.(H)H.
6.3 Von N e u m a n n Algebras
207
We have O4) {.]x)x E s
(Proposition 6.1.7.10 a =, e, Corollary 5.3.3.7) and oo (~zlx)-
for all x 9 g
(-Ix)x(~)
and u 9 Z:(H) (Proposition 6.1.6.2).
a ~ b follows from Theorem 4.4.1.8 b) (and Corollary 6.1.7.14 b)). b ~ c. Take x 9 H . Then lim { . ] x ) z ( u ) -- l i m { u x ] x } = O. u,;~
u,;~
By Proposition 4.2.4.12, 0-
lim (-I~~x(u 2) - - l i m ( u 2 x l x }
--
= l i m ( ~ x l ~ z ) - lim I I ~ l : u,~ u,~
Since x is arbitrary, ~ converges to 0 in s c => a. Let v be the infimum of 9c ( T h e o r e m 4.4.1.8 c), Corollary 6.1.7.14 b)). Take x 9 H . By a =, c, limll(u - v)xll = 0. u,N We deduce t h a t l i m ( ( u - v ) x l u x I - l i m ( ( u - v ) x l v x ) - O. u,;~
u,;~
Since
I1~11 ~ = (vxl~x) = ((~ - ~)~l~x) + ( ~ x l ( ~ - ~ ) ~ / + ( ~ l ~ x ) for every v C 9c , it follows t h a t v x - O.
Since x is arbitrary, v = 0. Proposition
6.3.1.5
( 0 )
I Take u C s
/7 -- I m u .
and put
6. Selected Chapters of C*-Algebras
208
a) 0 < a < / 3 ~ u ~ < _ u
~
V u~-limu~-rE
b)
a>O
in L ( H ) H .
a--~O a>O
a) follows from Corollary 4.2.7.2 (and Corollary 4.2.1.17 a =v b). b) By a) and T h e o r e m 4.4.1.8 b) (and Corollary 6.1.7.14 b)), the s u p r e m u m of {u~la > 0} in L:(H) exists and is equal to limu ~ c~--+0 c~>0
in ~(H)s
) . Put V
9= \u ] u ~ - l i m u
c~--~0 ct>O
a>O
~.
By Proposition 6.3.1.4 a =v c, =
v
limu ~ cx---+0 o~>0
in
s
By Proposition 6.3.1.3 a), v 2 = limu 2~ = v a---~0
in L ( H ) H . In particular, v is an orthogonal projection. We have
vx = limu~x = 0 ~--~0
for every x E Ker u and 0 = Ilvxll2 = (vxlx) = lim(u~xlx) >_ ( u x l x ) = Ilu89 2 c~--r 0
a>0
for every x C K e r r
( a ) and Corollary 5.3.3.7). Hence Ker u = Ker v
(Proposition 5.3.3.10
a))
and
Im v = (Ker v) • = (Ker u) • = Im u = F (Proposition 5.3.2.4). It follows t h a t V
---
TrF
.
m
6.3 Von Neumann Algebras
P r o p o s i t i o n 6.3.1.6
( 0 ) Let A be a closed set of IR. Define 9r ' -
{u E R e g ( H ) ] a ( u )
c A},
and let f 9 A -+ ]R be a continuous function such that
lim sup[ ~f ( a ) I < cc f_s
r oo aEA\{0}
whenever A is unbounded. Then the map
9v.
>C(H).,
u,
> f(u)
is continuous.
Let G be the set of continuous functions f 9 A --+ ]R for which the map 9t-H
>L:(H)H,
u,
>f(u)
is continuous. Then G contains the maps
Step 1
A
>IR,
oL ~--+ 1,
A
>IR,
a~
>a,
f , g E G, g bounded ==> f g C G
Follows from Proposition 6.3.1.3 a). Step2
UriC(A) is a closed set of C (A) .
Take f E ~ M C(A) and c > 0. There is a g C G MC(A) such that IIf-gll
II(f (u) II(f(~)
:
I((f
- g(u))~ll
-
+ ll(g(~)
g)(u))xll +
II(g(u)
- f (v))xll _~
- g(v))~ll
-
+ II(g(v)
g(v))xll +
II(f -
-
f(v))xll-
g)(v))xll ~_
209
210
6. Selected Chapters of C*-Algebras
_ 2~llxll
+
II(g(~) - g(v))xll.
It follows that f E g; M C(A). Step 3
f " A --+ JR,
a~_+ 1__4_ ~a belongs to G
Take u, v E ~-. Then
f(u) - f(v) = (1
+ u2)-lu-
v(1 + v2) -1 --
= (1 + u2)-1(u(1 + v 2) -- (1 + u2)v)(1
+ v 2 ) - 1 "-
= (1 + u 2 ) - l ( u - v)(1 + v2) -: + (1 + u 2 ) - l u ( v - u)v(1 + v2) -1 . Since
I1(1 + u2)-~ll ~ 1,
I1(1 + u2)-'ull <_
1
for every u E ~ , we get that 1
ll(f(u)- f(v))xll~ ll(u- v)xll+ ~ll(u- v)~ll. Hence, lim f (u) - f (v) , u-.'-~ v
and f E 6 . 0 E A and A unbounded Step 4
f E C(A) &
::~ f(O)=
aim f ( a )
~
1
I :=>f E ~.
A3c~oo
We set A := A if A is bounded, and denote by A the Alexandroff compactification of A whenever A is unbounded. Define
G - {g E C(A) I glA c G}. By Step 1 and 2, G is a closed subalgebra of the real algebra C(A). By Step 3 and Theorem 1.3.5.14, f E G. Step 5 The assertion Let f be the function in the proposition. We may assume that f(0) = 0.
6.3 Von Neumann Algebras
211
Given n E IN, define
gn:A
>IR,
f (Ol)O~n-1 1 +a 2
olt
By Step 4, gl E G. By Step 1, we get successively that g2 E G and g3 E G. Hence 1
f =g~+gzE6. P r o p o s i t i o n 6.3.1.7 ( 0 ) following are equivalent: a)
u' E s
Let u' be a linear form on s
(where we invoke the identifications Cs
s b)
Cs
u' is continuous on s
a ==~ b
is trivial.
b =v a. There is a finite family (Xt)tE I in H such that
I~'(~)1 _< 1 for every u E s
with supilux~l] < 1. ~EI
It follows that !
I~'(u)l _<
II~x~ll ~
for every u E s Given u E s
put
~EI
and let F denote the closure of {(ux~)~e, I u E s in C)H. Take u, v E / : ( H ) such that
Then the
6. Selected Chapters o] C*-Algebras
212
Then
~'(u)
= u'(~).
Hence, there is a linear form x' on F such t h a t x'((ux~)~e,) = u'(u) for every u E L:(H). Since x' is continuous, there is, by the Fr~chet-Riesz Theorem, a family (Y~)teI in H , such t h a t
x'((~)~,)
= ~(u~ly~/ tEI
for every u E s
It follows that
tel
tGI
for every u E / : ( H ) . Hence
tEI
Coronary 6.3.~.S s and s Let ~
( 0 ) I] a: ~ ~ ~ o ~ coincide.
~ t oS L(H), t h ~ ~t~ d o ~
be the closure of ~" in f-.(g)g and take u E s
~
There is a
finite subset A of H such that u + A
C/:(H)\~',
where
A := {v E s
lx E A =:=>Ilvxll < 1}.
By Proposition 1.3.1.6, there is a linear form u' on / : ( H ) , which is bounded on .4, such that c~ := sup re u'(v) < re u'(u). vE2 c
u' is continuous in the topology of f-.(H)H. By Proposition 6.3.1.7, u' is continuous in the topology of f_.(H)v.I(H). Hence
{v E s
l reu'(v ) <_ o~}
is a closed set of /:(H)t:I(H ) . Thus u does not belong to the closure of 9v in /:(H)s
and this closure is contained in .T'. The reverse inclusion is trivial. m
6.3 Von Neumann Algebras
( 0 )
P r o p o s i t i o n 6.3.1.9
s
213
and f~(H)+ are closed sets of
Res
. Given x, y E H , define
fx,y
"s
>~ ,
gx" s
~,
> < ~ l u > - <~l~u>,
~, ]K,
u,
> .
These functions are continuous. The assertion now follows from -1
Re s
= n
fx,y(O)
x,yEH
and =
N-1 xEH
(Corollary 5.3.3.7).
I ( 0 )
T h e o r e m 6.3.1.10
s
(Kaplanski, 1951) IS E is a C*-subalgebra of
then #,
ReE#=ENRes
Ee+ - -E N s
,
E # = E N f_.(H) #
where closures are taken in /2(H)H. The inclusions from left to right are trivial, since the sets s and s
are closed in s
# , Re s
#,
(Proposition 6.3.1.9).
Take v E E n Re s # (v E -E n s There is a filter ~ on s with E E ~ converging to v in s Since the map
s
>s
,
u,
>reu
is continuous, r e ~ converges to v in f_.(H)cs(H). Hence v belongs to the closure of Re E in s of Re E in s
By Corollary 6.3.1.8, v belongs to the closure Hence, there is a filter ~5 on s
with Re E C
converging to v in s Define f'IR
>IK, a ,
, s u p { - 1 , inf{a, 1}}
(resp. sup{0, inf{a, 1}}).
By Proposition 6.3.1.6, f(~5) converges to f(v) in s
Since
6. Selected Chapters of C*-Algebras
214
f (v) = v, and since R e E # (resp. E+# ) belongs to f(~b), it follows that
veReE#
(veE+*).
Take u C E f l f_.(H)# and define H (2) :=
(I) H . iEIN2
Let E2,e be the C*-subalgebra of s (2)) , comprising the 2 x 2-matrices with entries in E (Theorem 5.6.6.1 f)). Thus
[0.]
E E2a A Re s
u* 0 By the above result, there is a filter ~ on E ( H (2)) containing ReE~2 and converging to
u* 0 in /:(H(2))H(2). Define
~a" C(H (2))
s
v ~, '~ v12
(Theorem 5.6.6.1 f)). Then ~(~) is a filter on /2(H) containing E # (Theorem 5.6.6.1 a)) and converging to u in s
Hence
ucE#.
m
p
C o r o l l a r y 6.3.1.11 Let E be the closure in f~(H)H of the C*-subalgebra E of s Take u C C~(H). Then 0(3
OO
__
I~IEII : II~IEII.
Since L ( H ) #CI(H) - ~( H)c.~(H) # co
(~
the restriction of ~ to ~(H)#s s
is continuous. Hence, the restriction of ~ to
is continuous. By Kaplanski's Theorem, (X)
(~
I[~lEI -I[~IEII.
m
6.3 Von Neumann Algebras
215
Corollary 6.3.1.12 ( 0 ) The closures of a C*-subalgebra E of s in s s s ) coincide and are the W*-subalgebra of L(H) generated by E (Corollary 6.1. 7.1~ b)). Let F (resp. G ) b e the closure of E in s Corollary 6.3.1.8, G is the closure of E in s
(in f-.(H)LI(H)). By Take
u 9 G n I~(H) #. By Theorem 6.3.1.10 and Corollary 6.3.1.8, u belongs to the closure of E # in s163 Since s
# )= s Ls(H
ucF. By Corollary 4.4.2.9, F is a C*-subalgebra of I:(H) and by Corollary 4.4.4.12 a), it is the W*-subalgebra of s generated by E . m
Corollary 6.3.1.13
Take A 9 {C, IH}. Let H be a Hilbert right A-module and H the real Hilbert space obtained by endowing the underlying real vector space of H with the scalar product H•
-~]R, (x,y)~-+re(xly) N
(Proposition 5.6.2.5 a),e)). Then s
is a unital W*-subalgebra of s
By Proposition 5.6.2.5 b),c),e), s is a strongly closed unital C*subalgebra of s By Corollary 6.3.1.12, s is a unital W*- subalgebra of s m
Corollary 6.3.1.14
Let IK = ~ . If E is a unital C*-subalgebra of s
then {u 9 E lu is unitary } C {u 9 E lu is unitary}, where the closures are taken in f~(H)H. By Corollary 6.3.1.12, E is a W*-subalgebra of s Let u be a unitary element of E . By Corollary 4.3.2.7 (and Theorem 4.4.1.8 c)), there ia a v C Re E such that U
--"
e iv
.
6. Selected Chapters of C*-Algebras
216
By Kaplanski's Theorem, there is a filter ~ on s
containing Re E and
converging to v in ff-,(H)H. By Proposition 6.3.1.6, u = e iv = lime iw C {w C E ]w is unitary} w,~
(Corollary 4.1.3.10).
I
P r o p o s i t i o n 6.3.1.15
Let u be a self-normal operator on H , if3 the (r-
algebra of Borel sets of or(u), B the Banach space of bounded Borel functions on or(u), and # the spectral measure of u (Corollary 6.1.7.1~ b), Corollary ~.~.1.11 a)). Given ~,~7 E H , define
P~,n: ~3
) IK,
a)
#~,~ is a measure for all ~,71 C H .
b)
If f c B, then
A,
>(lz(A)51O).
(f(u)SI77) - f f dlze,o '
IIf(u)~l[ 2 = /Ifl2dp~,~ for all ~, ~7 C H .
c)
If (f~)nelN is a bounded sequence in B converging pointwise to an f c 13, then (f~(u))n~Ii converges strongly to f ( u ) .
d)
A closed vector subspace K of H reduces u iff 7rK C #(~3) c . In particular, Imp(A) reduces u for every A E ~ .
a) #r
is obviously additive. Let ( A n ) n ~ be a decreasing sequence in ~3
with empty intersection. Then (p(A~))~e~ is a drecreasing sequence in s with infimum 0. By Theorem 5.3.3.14 b :=v c, lim #~,, (An) - 0 .
n--~ (X)
Hence #~,, is a measure. b) The first equality follows from Theorem 4.4.1.8 f) (and Corollary 6.1.7.14 b)). Then IIf(u)~ll2 = ( f ( u ) g l f ( u ) g ) = ( f ( u ) * f ( u ) g l g ) =
6.3 Von Neumann Algebras
=
(Ifl2(u)~l~)
f
= j
217
Ifl2@r162 .
c) follows from the second equality of b). d) K reduces u iff 7rg C {u} c (Corollary 5.2.3.12 b)). The assertion now follows from ~(~)
c
~(~)~ = {u} ~
(Corollary 4.4.1.11 d)).
m
P r o p o s i t i o n 6.3.1.16 If E is a C*-subalgebra of E(H) acting non-degenerately on H , then any approximate unit of E converges strongly to 1H. The assertion follows immediately from Proposition 5.3.2.21.
m
6. Selected Chapters of C*-Algebras
218
6.3.2 Bidual of a C*-algebra T h e o r e m 6.3.2.1 ( 0 ) (Takeda, 1954) Let E be a (unital) C*-algebra, (H, r a faithful representation of E such that every continuous linear form on ~(E) is continuous on ~(E)s (e.g. the complex universal representation of E ), and ~v(E) the strong closure of ~ ( E ) . a)
The map E
~(E),
x,
~x
is an isometry of C*-algebras.
b)
The Arens multiplications of E" (Definition 2.2.7.13) coincide, E" endowed with this multiplication is a W*-algebra. jE is an involutive (unital) algebra homomorphism, ~(E) is the W*-subalgebra of s generated by ~ ( E ) , and there is a W*-isometry u : E" ---+eft(E) such that UjEX -- 99X for every x E E .
c) If E is Gelfand, then so is E" and a(E") is a compact hyperstonian space. d) E ''# , Re E ''# , and E~ # are the closures in E~, of j E ( E # ) , j E ( R e E #) and jE(E#+), respectively. Moreover jE,(E+) C
I~, III
...,+.
If (H, ~) is the complex universal representation of E then by Proposition 5.4.2.6 b), ~ is faithful, and by Corollary 6.1.6.3, every continuous linear form on ~v(E)is continuous on ~(E)LI(H). a) follows from Theorem 4.2.6.6. # b) By Kaplanski's Theorem, ~(E) # is dense in p ( E ) g , and by Corollary 6.3.1.12, ~(E) is the closure of p(E) in ff-,(H)s and the W*-subalgebra of s generated by ~(E). By Theorem 4.4.4.18 h), the Arens multiplications on ~(E)" coincide, ~(E)" endowed with this multiplication is a W*-algebra, and there is a W*-isometry v : ~v(E)" --+ ~(E) such that v o j~(E) is the inclusion map ~(E) -+ ~(E). Define r
~(E),
x,
>~x,
6.3 Von Neumann Algebras
219
?.t : ~ V O r
By a), the Arens multiplications of E" coincide, E" endowed with this multiplication is a W*-algebra, jE is an involutive (unital) algebra homomorphism, and u" E" -+ ~ ( E ) is a W*-isometry such that u ojE = V o r
o j E = V ojv(E) o r
(Proposition 1.3.6.16). Hence UjEX -- r
for every x C E . c) follows from Theorem 4.4.4.18 a) and Corollary 4.4.1.10. d) By a) and b), we may indentify E and E" with ~(E) and ~ ( E ) . The first assertion follows from Kaplanski's theorem (Theorem 6.3.1.10), since on bounded sets of s the strong topology is finer than the topology of f~(H)s which induces on E the topology of EE,. Take x' E E~. Then (jEX, jE, X') -- (jEX, X'} = (X, X'} C IR+
for every x E E+. By the above, (x", jE, X') C 1R+
for every x" e E~, i.e. 2E, ' e~+~'" and jE,(E~+) C ~+ .
I
Take (x', x") C E' • E " . The multiplication x'x" defined in Chapter 4 using the above b) coincides with the multiplication defined in Chapter 2. Indeed, let us denote by x'-l-x" and x'_l_x" these two multiplications, respectively. Then for every y" C E " , Remark.
(y", x'-l-x"} - (y", (jE, x')x") -- (x" y", jE, X'} =
= (x"y",x')- (x"~ y " , z ' ) - (y",x'• so that x ' T x " - x'_Lx". T h e o r e m 6.3.2.2
( 0
)
Assume
E
is a W*-algebra and let F
be its
predual. Denote by P2 the set of downward directed sets of Pr E with infimum 0 and put (Theorem 6.3.2.1 b) and Theorem
4.4.1.8 i))
6. Selected Chapters of C*-Algebras
220
:= {A E 9 1 1 - A is well-ordered}, R := { A JEql A E Ql} C P r E " , qEA
S := { A JEql A E ~ } C P r E " . qEA
a) J'F " E" --~ E is a unital W*-homomorphism (Theorem 6.3.2.1 b)), jR is its pretranspose, and the canonical projection of the tridual o ] F (= E") (Proposition 1.3.6.19 b)) is an involutive unital algebra homomorphism.
b) (ImjR)~
Kerj~ is a closed ideal of E'~,.
c) There is a p E E ''~ M PrE" such that (Im jR) ~ = pE"p . d)
a" E F" =~ pa" = a" p .
e)
The map F"
>F",
a" ~ >a" - pa"
is a projection of F" onto Im jR. E " ( 1 - p) is a closed ideal of E~,, with predual (1 - p)F" = Im jR and
F"
>ImjR,
a" ,
> a" ( 1 - p)
the pretranspose of the inclusion map E"(1 - p ) --+ E " . The map 9 E"(1 - p)
>
E,
xH '
>
9I
3F x
II
is an isomorphism of W*-algebras with pretranspose
F
>ImjR,
a,
>jFa.
f) p is the supremum of R and of S in PrE" and in E~ # {
g) ImjR --
themaps E--+E',x~--~xx'(resp. x'x) } x' E E' are order continuous.
i.e. ImjF =
E',
where E' is considered an E-module (Definition
~.~.1. ~). h)
The following are equivalent:
6.3 Von Neumann Algebras
hi)
221
E is reflezive.
h2) p = 0 . h3) jE is a W*-homomorphism. h4)
The canonical projection of the tridual of F is a W * - h o m o m o r p h i s m .
i) (Takesaki, 1959) The following are equivalent for every x' E E'+" il)
x'p-
x'.
i2) For every q C P r E \ { 0 } , there is an r E P r E with 0
x'(r)=0.
(Takesaki, 1959) Assume IK =(~ (IK = IR). The following are equivalent for every x' C E'
jl)
(x' E Re E') :
x'p = O .
j2) x ' E I m j F . j3) x' is order continuous. j4) x'iG is order continuous for every maximal Gelfand C*-subalgebra GofE.
jb) For every A C f8, lim (q, x') -- 0 q,~
where ~ denotes the lower section filter of A .
k) /f x' e Re E' is order constinuous then x '+, x ' - , and Ix'] are also order continuous.
a) By Corollary 4.4.2.10, F is an involutive E-submodule of E'. By Theorem 2.2.7.15 f4) (and Theorem 4.4.1.8 h)), j~ and the canonical projection of the tridual of F are algebra homomorphisms. By Proposition 2.3.2.22 d),e), jF and j~ are involutive, so that the canonical projection of the tridual of F is also involutive. It is easy to see that j~ is unital. Hence, by Theorem 6.3.2.1 b), the canonicaml projection of the tridual of F is also unital. Since the map j'F " E'~,,
~ EF
is continuous, it follows that j~ is a W*- homomorphism (Proposition 4.4.4.6). It is obvious that jF is the pretranspose of j~ (Corollary 4.4.4.8 c)).
6. Selected Chapters of C * - A l g e b r a s
222
b) follows from a) since (Im j R ) ~ = Ker j~.. C) follows from b) and Corollary 4.4.4.12 d). d) follows from c). e) By c), for a E F , p ( j F a ) = O,
SO that jFa = jFa-
p(jFa) .
Let a" E F" such that p a " = O.
By c), for every x" E (Im j R ) ~ ,
(~", x") = (a", x"p) = (pa", x") = o. Hence a" E ~
jF) ~
= Im j F
(Proposition 1.3.5.7). It follows that the map F"
> F" ,
a" ,
> a" - p a "
is a projection of F" onto I m j F . The second assertion follows from c) and Corollary 4.4.4.12 c),d). By a), is a W*-homomorphism with the given pretranspose and by b) and c), qo is injective. From 9!
9
x -- 3F3EX
for every x C E (Proposition 1.3.6.19 a)) it follows that qD is surjective. f)
Stepl
rcR=:c,r
By c), we have to show r E (ImjF) ~ . Take a E F . We must prove (r, j F a ) = O .
In the real case we have
6.3 Von N e u m a n n Algebras
223
(r, j F a -- j F a *) -- (r, jFa* -- j F a )
and therefore (r, jFa -- jFa*) -- 0,
so t h a t we may assume a C Re F if IK - IR. Let A E 91 with
r - A jEq qEA
and let ~ be the lower section filter of A. By Corollary 4.4.4.3 a r
b,
(r, j r a ) -- lim(jEq, jFa) -- lim(q, j F a I = q,~
q,~d
= lim(q, a) = 0. q,~
Step 2
P-
V s, where the supremum is taken in Pr E" sES
By Theorem 4.4.1.8 i), ~,! s is well-defined. P u t sES
Po'-P-
Vs sES
and assume p0 ~ 0. By Step 1, P0 is positive. By Corollary 4.4.1.5, there is an a" C F~ with (P0, a"/ > 0. By Corollary 4.3.1.5 d ::v a, poa"po > O.
Take A c ~ and put s "- A jEq. qEA
Then spo -- 0
(Corollary 4.2.7.6 a :=> c), so that
224
6. Selected Chapters of C*-Algebras
0 - (s, poa"po) = lim(jEq, poa"po) q,~
where ~ denote the lower section filter of A. By Theorem 4.4.4.2 c ~
a,
poa"po E I m j F . Hence 0 = (p, poa"po) = (PoPPo, a") = (Po, a"),
which is a contradiction. It follows
po=O,
p=
V s 9 sES
Step 3
p is the supremum of S in P " #
Take So E ~ s ( S ) . By Proposition 4.2.7.21 c), the supremum of So in Pr E" is also the supremum of So in E~_# . By Theorem 4.4.1.8 i), p is the supremum of S in ~-,+ ~" # Step 4
~--,+ p is the supremum of R in ~ "#
The assertion follows from Step 1 and Step 3. g) Take x' E E' such that the maps E
~E',
x,
(resp. x ' x )
>xx'
are order continuous. Let A E 92 and put r "-- A jEq E Pr E " . qEA
By Corollary 4.4.3.10, rx' - l i m ( j E q ) x ' -- lim qx' -- 0 q,~
q,~
x ' r = l i m x ' ( j E q ) = limx'q = 0 q,~d
q,~
: '
where ~ denotes the lower section filter of A. Put C := {z" E
E " l x " z ' = z"*z'= x'x"= x'z"* =
0}.
By Proposition 4.3.4.14, G is a hereditary C*-subalgebra of E " , which is obviously closed in E~,. By the above, R C G . By f) and Proposition 4.4o4oll c), p E G. By c),
6.3 Von Neumann Algebras
(Im jR) ~ -- p E p
C
225
G
SO that x' e ~
c ~
~ = ImjF
(Proposition 1.3.5.7). The reverse inclusion follows from Corollary 4.4.3.10. hi =ah2. By Proposition 1.3.8.4, F is reflexive. Hence (ImjF) ~ = E '~ = {0}. Byc), p=0. h2 =~h3. By c), (Imjg) ~ = {0} so that Im jR = ~
jF) ~ = F"
(Proposition 1.3.5.7). Take a" E F ' . By the above, there is an a c F with jFa = a". Then = <jFa, jEX> = (jFa, X> =
for every x C E and the map jE " EF
>
E~,
is continuous, i.e. jE is a W*-homomorphism (Proposition 4.4.4.6). ha ==> h4 follows from a). h4 =v hi. By Corollary 4.4.4.8 b) and Proposition 1.3.6.19 b), I m j ~ is a W*-subalgebra of E ' , i.e. it is closed in E~,. But I m j E is dense in E~, (Corollary 1.3.6.5), so that jE is surjective. Hence E is reflexive. il =~ i2. We may assume x'(q) > 0. There is a y' C I m j F with
y'(q) > z'(q) (Corollary 4.4.1.5). Put P := {r e P r E I r
< q, y'(r) < x'(r)}.
Take a totally ordered subset Q of P and define
6. Selected Chapters of C*-Algebras
226
ro "-- V r 9 rEQ
Then
y'(ro) = supy'(r) _< supx'(r) _< x'(ro) rEQ
rEQ
(Theorem 4.4.1.8 c),i)), so that r0 E P . Hence P is inductive. By Zorn's Lemma, there is a maximal element r of P . Since q ~ P , it follows q - r
> 0.
By the maximality of r ,
y'(s) > x'(~) for every s E P r E , 0
< s < q-r.
Since ( q - r ) E ( q - r )
is aW*-algebra
(Proposition 4.4.2.2), it follows
0 <_ (q - r ) x ' ( q - r) <_ ( q - r ) y ' ( q - r) (Theorem 4.4.1.8 f)). By Corollary 4.4.2.13,
( q - r ) x ' ( q - r) E I m jF. By e),
(q - r)x'(q - r) = (1 - p)(q - r)x'(q - r) = O, so that
x ' ( q - r) = (1, ( q - r ) x ' ( q - r) > = O. i2 =:V il. Bye), x ' ( 1 - p ) E I m j g . Let q be the carrierof x ' ( 1 - p ) (Corollary 4.4.2.17). Then
(~, ~'(1-p))=o for every r E P r E , r _< q. By i2), x'(1 - p) = 0 so that x' = x'p. jl ==vj2 follows from e). j2 =v j3 follows from Theorem 4.4.1.8 c). j3 =v j4 follows from Corollary 4.4.2.7. j4 =~ js. Since A is commutative (Corollary 4.2.7.6 a =v b), there is a maximal Gelfand C*-subalgebra G of E containing A (Corollary 4.1.2.3). Hence lim (q, x') = O. q,~
6.3 Von Neumann Algebras
j5 =~ j l . We may assume x' E R e E . By e), x'(1 - p )
227
E I m j F , so that, by
j2 ~ js, lim(q,x'(1-p)) qE;~
--0
for every A E ~3, where ~ denotes the lower section filter of A. The same assertion follows therefore for x'p. Let q E Pr E and let q3 be the set of well-ordered subsets P of Pr E such that (r, lx'ip ) = 0
r < q,
for every r E P . We order q3 by P<_Q
iff P C Q and (r,s) E P x Q , s <_ r = = , s E P .
By Zorn's Lemma, gl has a maximal element P . By il =~ i2,
V r=q.
rEP
For every r E P ,
o <_ (~,x'+p)<_ (~, I~' ; ) - o ,
0 <__(~, x'-;)<_ (~, Ix'Iv)- 0, so that (r, x'p) = O .
Let ~ be the upper section filter of P . Then {q--rr
E P } E fB
and so 0-
lim ( q -
r x'p) - (q, x'p).
r ,qd
Since q is arbitrary, it follows x ' p - 0 k) follows from ja Remark.
::V j2
(Theorem 4.4.1.8 f)).
and Theorem 4.4.3.9.
E is reflexive iff it is finite-dimensional (Theorem 6.3.6.15).
I
6. Selected Chapters of C*-Algebras
228
\
I
C o r o l l a r y 6.3.2.3 ( 0 ) If E , F are C*-algebras and u " E --+ F is an involutive algebra homomorphism, then u" is a W*-homomorphism and Imu' is closed. If F has a predual G , then j ~ o u " is the unique W*-homomorphism E" --+ F such that j'aou" ojE--U.
By Theorem 6.3.2.1 b), E" and F" are W*-algebras and by Proposition 4.4.4.6, u" is a W*-homomorphism. By Theorem 4.2.6.6, Imu" is closed, so that by Proposition 1.4.2.11, Imu' is also closed. By Theorem 6.3.2.2 a), j~ is a W*-homomorphism, so that j~ o u" is also a W*-homomorphism. By Proposition 1.3.6.16 and 1.3.6.19 a), 9!
2G o
?./,11
-I
o
jE = ]G
o
jc,'
o
u = u.
The uniqueness follows from the fact that ImjE is dense in E~, (Corollary 1.3.6.5). W C o r o l l a r y 6.3.2.4 a) If F is a C*-subalgebra of the C*-algebra E and i : F -+ E is the inclusion map, then F ~176is the W*-subalgebra of E" generated by j E ( F ) , Im i " = F ~176 , and the map F" ---+ F ~176 y" ~ >i"y" is an isometry of W*-algebras. In particular, if F is also Gelfand, then F ~176 is also Gelfand. -1
b)
If G is a W*-subalgebra of E " , then ~176= jR(G) is a C*-subalgebra of E and (~176176176 is the W*-subalgebra of E" generated by G N j E ( E ) .
a) By Corollary 6.3.2.3, i" is a W*-homomorphism. By Proposition 1.3.6.14, F ~176 is the closure of jE(F) in E~, and by Proposition 1.3.6.17, Imi" = F ~176 and the map F"
~ F ~176 y" D ~ i"y"
is an isometry of W*-algebra. By Corollary 4.4.4.12 a), F ~176is the W*subalgebra of E" generated by j E ( F ) . If F is Gelfand, then by Theorem 2.2.7.15 g), F" is Gelfand. Thus, F ~176 is Gelfand. b) Since (~176 = G
6.3 Von Neumann Algebras
229
(Proposition 1.3.5.4), we have -1
~176= jE(G). -1
By Theorem 6.3.2.1 b), jE(G) is a C*-subalgebra of E . It follows that --1
G N jE(E) = jE(jE(G)) = jE(OOG).
and so, by a), (~176176176 is the W*-subalgebra of E" generated gy G N j E ( E ) . I
C o r o l l a r y 6.3.2.5
If E" is simple (Theorem 6.3.2.1 b)), then E is simple.
Let F be a closed ideal of E and q " E -+ E l F the quotient map. Then q" is a W*-homomorphism (Corollary 6.3.2.3, Theorem 4.2.6.5) and F ~176 = (Ker q)OO _ Ker q" (Theorem 4.2.6.6). Hence F ~176 is a closed ideal of E" and therefore F ~176
or E " ,
F = {0} or E . Thus E is simple.
I
E x a m p l e 6.3.2.6
( 0 ) Let T be a locally compact space. Put E := Co(T),
and let B be the C*-algebra of bounded Borel functions on T . We identify E' canonically with the involutive Banach space of bounded Radon measures on T and for each x E B, we define ~" E'---+ IK,
x'~
>/ xdx'
a) "~ E E " , for every x E B. b) ~ x ' = x.x', for every x E B and x' E E ' . c)
The map B
> Ell :
X,
)X
N
is an injective involutive algebra homomorphism (Theorem 6.3.2.1 b)) such that
A ~n = 0 for any decreasing sequence ( X n ) n ~ in B whose n E IN
infimum is O.
6. Selected Chapters of C*-Algebras
230
If (x~)~e, is an upward directed family of bounded lower semicontinuous real functions on T whose supremum x is bounded, then
d)
5= Vh~. tEI
e)
If A is a closed set of T and G "= {x E E IxeA = 0}, then
{x" ~ E" I x"~A = 0} is the order faithful vector subspace of E" generated by jE(G). f)
E" is the order faithful vector subspace of E" generated by j E ( E ) .
a) is obvious. b) Given y E E ,
(y, yx') = (~, y~') = (~, y.x') =
=f xa(y.x')=f
f
yd(x.x') = (y,x.x') .
Hence 5X
t --- X.X
I "
c) The map is obviously injective and linear. Take x C B. Then
x-,(x')=f x*dx'=f xdx'*=~(x'*)=~*(x') for every x' C E' and thus X* ---- X*.
Hence the map is involutive. Take x, y E B. Then by b) (and Theorem 6.3.2.1 b),c)), for every x' E E ' ,
= (~', ~)=
(x', ~ .
6.3 Von Neumann Algebras
231
Hence xy = xy
and therefore the map is an algebra homomorphism. The final claim follows from integration theory. d) follows from integration theory. e) Let Go be an order faithful vector subspace of E" containing j E ( G ) . By c) and d), Go contains {~I x E 13, xeA = 0}. Let if3 be the set of Borel sets of T which are disjoint from A. Let U be a clopen set of cr(E") (Theorem 6.3.2.1 b),c)) such that eue-A -- 0
and take x', y' C E+' with Suppx'cU,
UMSuppy'-0,
where we have identified E' with (E") ~ (Corollary 4.4.4.3 a r
b). Then
x' A y ' = O . Thus there is a Be E ~ such that x'(T\Br
= y'(By,)=0
(Hahn's Theorem). By c), e--Be C Pr E", and so {~'B~, # 0} is a clopen set of cr(E"). Hence
Suppx'C{FBy, 7~0},
{e-~r
Let V~, denote the clopen set of a(E") such that
y~
where y' runs through the set of elements of E~_ with U M Supp y' = 0. Then ev~, C G , Supp x' C Vx,, and by Example 1.7.2.14 d),
6. Selected Chapters of C*-Algebras
232
Vx, C U .
Again, by Example 1.7.2.14 d), eu
-- V
ev~, ,
xt
where x' runs through the set of elements of E~_ with Supp x' c U. Hence eu C G . Since U is arbitrary, it follows that {x" ~ E " l x " ~ n
= O} c a .
Thus {x" 9 E" I x"~A --O}
is the order faithful vector subspace of E" generated by j E ( G ) . f) follows from e) by putting A := O. E x a m p l e 6.3.2.7
( 0 ) Let T be a compact space, and put E "- C ( T ) .
We canonically identify T and a ( E ) 7) " a ( E " ) --+ a ( E ) = T ,
(Example 2.~.3.1) and define x'",
9t
~ 2E x
m
--
Xttl
o jE
(Theorem 6.3.2.1 a), Proposition 2.~.1.17 a)),
s : : {x"' e o(E") I {z'"} i~ op~}. For t c T we write t ~ for the Dirac measure on T at t and define
t":E'
>IK,
x',
)x'({t}),
t'" := jE' t~ 9 a) t " x ' = t"(x')t' for every (t,x') C T x E ' . b) t'x" = x"(t')t' and t"x" = x"(t')t" for every (t,x") e T x E " . c) t'" E S and ~(t'") = t, for every t E T . d)
-1
~(t)\{t"'} is meager for every t C T .
6.3 Von Neumann Algebras
e)
233
The map S
>T,
s,
>~(s)
is bijective. f)
Given x' c E ' , let x~ denote its atomic part (as Radon measure). Take f c g ~ ( T ) , and define x" " E '
> IK,
x' ,
Then x" 9 E " , I]x"ll = Ilfl]oo,x" _
_
a(E")\S,
-
>/ f o~
f dz'a. on S ,
and x" = 0 on
A
if f is continuous then x " = e ~ ( f o ~ ) .
a) Take x E E . Then (x, t"x'> - (xx', t"> = ( x x ' ) ( { t } ) = x ( t ) x ' ( { t } ) = t"(x')(x, t'> , so that t"x'-t"(x')t'. b) Take x C E . Then (x,t'x"> = ( x t ' , x " > -
(x(t)t',x"> -
= x(t) - x"(t')<x, t'>, so that t'x" = x"(t')t'. By a), (x',t"x"> = (x't",x"> = t"(x')(t',x"> = x"(t')(x',t"> for every x' C E ' . Hence t"x" = x"(t')t". c) By b),
<x"y", t'"> - <x"y", t'> - = x"(t') = = <x", t"'>
6. Selected Chapters of C*-Algebras
2~
for all x", y" E E " . Thus t'" E a ( E " ) .
Then t ' ( ( t } ) = 1.
(t",t'") = ( t ' " , t ' ) =
Take s E a ( E " ) , s =/=t'". There is an x " E E " such that
t'"(z")=l,
~(~")=o
(Theorem 2.4.1.13). By b), 0 = s(t")s(x") = s(t"x")=
x"(t')s(t") - s(t").
Hence
{t"'} = {t"# 0}, t'" E S .
Take x E E . Then (x, 5v(t,,,)) = ( x , j ' ~ t ' " ) ( j E x ,
t'") = (jEx, t ' ) =
(x,t').
Hence ~(t") = t.
d) Assume that ~ ( t ) \ { t ' " }
is not meager. Since a ( E " )
is hyperstonian
(Theorem 6.3.2.1 c)), there is an x'" E ~ ( E " ) ~ such that Suppx'" c ~ ( t ) \ { t ' " } ,
and IJx"'Jl = 1.
By Corollary 4.4.4.3 b ~ a, there is an x' E E ' such that x,~ _ jE, X ~ .
Then I"
"' x '"'2 = (jEX, X'"} = / ( z o~)d~" = (~,~') = (x, 3~jE, ' 9 z') = (z, 3E J
= x(t) = (~, t') for every x E E (Proposition 1.3.6.19 a)), so that X ! --- t !
6.3 Von Neumann Algebras
235
x"' : jE't' : t'", which is a contradiction. e) By c), the map is surjective, and by d), it is injective. f) It is easy to see that x" 9 E " , and Iix"lI = IiftI~. By c) A
x"(t'")
= (~", t'") = (~", t') = y(t) = f o ~ ( t " )
for every t 9 T. Thus, by e), x" = f o ~ on S. Assume that x" does not vanish identically on a(E")\-S. Since a(E") is hyperstonian (Theorem 6.3.2.1 c)), there is an x'" 9 (E") ~ such that x"dx'" ~- O,
and S M Supp x'" = 0.
f^
-
By Corollary 4.4.4.3 b ==~ a, there is an x' 9 E' such that xlI!
=
jE'X I
Since S n Supp jE, X' = ~, x r is atomless, and we get the contradiction P
0 = (x", x'> = (x", jE, X'> = <x",X'"> = ] x3'dx ''' # O. The last assertion follows by continuity. E x a m p l e 6.3.2.8
I
( 0 ) Let T be a locally compact space, d " - C 0 ( T ) ,
12
the (order complete) lattice of clopen sets of a(C") (Theorem 6.3.2.1 b),c)), and fit the (order complete) lattice of bands of C'. We identify Imjc, with (d") ~ (Corollary ~.~.~.3 a r
b) and for each U 9 12 we define U'-
{# 9 C ' l S u p p # ' c U}.
Then U c fit for every U E ~l and the map N
12
>fit,
U ~ >U
is an isomorphism of lattices. If we identify C(a(C")) with C" via the Gelfand transform, then for every U E 12 and every # C d' the measure eu# is the projection of # on the band U.
236
6. Selected Chapters of C*-Algebras
By Theorem 6.3.2.1 b),c), C' is a Gelfand W*-algebra and a(C') is a hyperstonian compact space. The first assertion follows from Example 1.7.2.16, while the second one follows from the relation jc,(eu#) = eu(jc,#)
(Corollary 4.4.2.10).
I
P r o p o s i t i o n 6.3.2.9 Let T be a locally compact space. Let ffJt be the set of all sets .M of positive bounded Radon measures on T such that ]]pi] = 1 for every # E .M and p A u = O for all distinct # , u E . M . Let further J~4 be a maximal element of ff)~, where ff)~ is ordered by inclusion. Put
L :- {x E H L'(U) I ~ IIx.II < ~} ~E.M
~E.M
and denote by M the C*-direct product of the family of C*-algebras ( L ~ ( # ) ) u e ~ .
a)
L is an involutive vector subspace of
H Lx(#) and L is an involutive trEAd
Banach space with respect to the norm
#E.A4
b)
If we identify Co(T)' in the usual way with the involutive Banach space of bounded Radon measures on T , then the map L
)Co(T)',
x,
~ Exu'# I.tE Ad
is an isometry of involutive Banach spaces. We shall identify L and Co(T)' using this map.
c)
Given y E M , define
tt E.M
Then ~ E Co(T)" .for every y E M and the map M
>Co(T)",
y,
>Y
is an isometry of C*-algebras (Theorem 6.3.2.1 b)).
6.3 Von N e u m a n n Algebras
237
a) is easy to verify. b) The map is obviously linear and involutive. Take x E L and c > 0. There is a finite subset A/" of A/I, such t h a t c
Let B be the Banach space of bounded Borel functions on T . Given # E A f , there is a y, E B # such that
IIx.II = /
y,d(x,.~).
By Hahn's Theorem, there is a y E B # such that Y = Yt,
#-a.e.
for every # E A/'. Since Co(T) is dense in Co(T)~o(T), , there is a z E C(T) such that <
(y. - z ) d ( . . . . )
c 3(1 + Card Af)
for every p EAf. We may assume t h a t z E C(T) # , replacing z by T -----+ IK ,
t ~
>
z(t) sup{lz(t) ], 1}"
Then
>_ ~ ( y . , x . . ~ )
- ~
t~N
_>,~~ IIx.II-
I(z- y.,x..~)l -
t~EX
~
I(z,*.-p)l _>
tL~\N
E Card.Af
3(1 + CardN)
-
-
~
.e~\x
I1~,11> .~e ~ IIx, ll-~.
Hence
il
x,.,lt _> Z iix, tl. #EA/I
#EA/I
The converse inequality being trivial, the m a p in b) preserves norms.
6. Selected Chapters of C*-Algebras
238
We now show that the map is surjective. Take v 9 Co(T)r+. By the R a d o n Nikodym Theorem, for every # 9 j~/[, there is an x t, 9 LI(#)+ such that u A # = xu. # . Then
I1~ ^,11 = I1~.II for every # 9 .M. Since .M is maximal,
~ E 2r
I~E M
hence the map of b) is surjective. c) It is easy to see that ~ 9 Co(T)" for every y 9 M and t h a t the map M
> C0(T)",
y,
>
is linear, involutive, and injective. We show that this map is surjective. Take
x" 9 Co(T)". For every # 9 Ad, the map LI(#)
>IK,
x,
is linear and continuous with norm at most
> <x.p,x")
I1~"11. Hence
there is a Yt, 9 L ~ ( # )
such that
(x.#, x") = / y~,xd# for every x 9 L I ( # ) . We have IlY,II-< IIx"ll for every # 9 A d , s o Y := (Yt,)ue~ 9 M . For every x E L,
(~, ~") = ~ (~.~, ~") = ~ f y~d(~.~) and therefore ~ -
x " . This proves the surjectivity of the map. By Theorem
4.2.6.6, the map is an isometry of C*-algebras.
I
Assume ]K = IR. Let T be a locally compact space and jz an order faithful set of Co(T)" containing Co(T) 5dentified with a subset of Co(T)" ). Then .T = Co(T)". C o r o l l a r y 6.3.2.10
6.3 Von Neumann Algebras
239
If we identify the C*-algebra B of real bounded Borel functions on T with a C*-subalgebra of Co(T)" in the usual way, then B c 9v . Let 92 be the set of all sets A// of positive, bounded Radon measures on T with I1,11 = 1,
for every # 6 .A4 and #Ap=O for distinct #, v c A J . If we order 92 by the inclusion relation, then by Zorn's Lemma, 93~ has a maximal element .h//. Take x" 6 Co(T)~. By Proposition 6.3.2.9 (using the notation of this proposition), there is an y C M such t h a t y.~
X t'
.
For every finite subset Af of jk//, the set {z e S [ # e.N" ~
z e y,}
is downward directed. Its infimum z x in Co(T)" belongs to $-. We have (zx)t, -- Y. for every # 6 A/" and
(zz), = 0 for every # E A4\Af. The set { z x [Af c ~f(.A//)} is upward directed and its supremum in Co(T)" is x". Hence x" C .~ and :
to(T)".
-,
Proposition 6.3.2.11 ( 0 ) L~t ~ b~ a C*-~lg~b~a an~ tak~ x' C ~' a ~ y' 6 E'+. Then the following are equivalent: a)
y' = ]x'[, where x' is considered as an element of the predual of the
W*-algebra E" (Theorem 6.3.2.1 b)). b)
[Ix'I[ = [[y'[] and
I~'(~)l: < IIz'lly'(~*) for every x 6 E .
6. Selected Chapters of C*-Algebras
240
a ~ b. Let (x", y') be the polar representation of x' (Theorem 4.4.3.5 a)). Then ] l x ' l l - IlY'll and
I.'(x)l~ = I(x.z"y')l'
= I ( x x " . y ' ) ] ~ < ~'(zz*)y'((~")*x")
<__ !*'ll~'(x**)
for every x C E (Proposition 2.3.4.6 c)). b ==~ a. Let (H, ~) be the complex universal representation of E . By Proposition 5.4.2.6 d), there is a ~ e H , such that ]0~]] = ]]y']]89 and
v'(z) = ((~*)~1~) for every x E E . We identify E with ~ ( E ) (Proposition 5.4.2.6 b)) and E" with the closure of E in s
(Theorem 6.3.2.1 a)), which is the same
as its strong closure (Corollary 6.3.1.12). By b),
Ix'(x*)] = ~ IIx'lly'(x*~) = II~']l{~*x~l~> - ]]~' I ]1~11 = for every x C E . Take x" C (E") # . By Kaplanski's Theorem , there is a filter on s
converging strongly to x" such that E # E ~. Hence ~ converges
to x in ~(H)s
) and we get I*'((*")*)1 ~ < I1-'11 I1-"~11 ~ 9
Let (x", I~'l) be the polar representation of x". Then I(*, Ix'l)l ~ = I(**, I~'1) ~ = I(x*, ( * " ) * z " l x ' l )
~ =
= I(x*(*")*. x') ~ < I]*'11 IIx"x~ll ~ <__ IIx'll I1<11 ~ for every x E E . Define
u'E Let q
E ~
E/Keru
~H,
x,
>x~.
be the quotient map and v 9 E / K e r u
~ I m u the
associated algebraic isomorphism of u. By the above inequality K e r u C Ker x']. Let w be the factorization of x' I through E / K e r u . Take r/E I m u . There is an x C E with ~7 = u x .
6.3 Von Neumann
Algebras
We get r] = v q x ,
q x = v-1~7 ,
(x, Ix'l) = w q x = w v - l r l ,
Hence, w o v -1 is continuous with
il~ ov-~i -~ ~'lJ89 We extend w o v -1 to a linear form z' on H with the same norm. By the Fr~chet-Riesz T h e o r e m , there is a ( E H , such that
I1(11-
Ilz'll and
z'(~) = (~IC) for every w C H . We get
(X, ]Xtl} :
wy-lux
--- Z ' ( U X ) ~-
(X(](}
for every x C E . If ~3 is an approximate unit of E , then, by Proposition 5.3.2.21 (and Proposition 2.3.4.9),
(Proposition 2.3.4.10 a)). By Proposition 5.1.1.11 b ==> c,
Thus
~l (~, I~'1)-~
i~J, (~l~)= iI-~y (~)
for every x E E , and hence
I;ll, J~'i-- ii-~y 9 It follows
I~ff Ily' I- I1~' I- II Ix'l II- ii-~lly'll, 11~11_ 1, I1~11 Ix'l- v'.
i
6. Selected Chapters of C*-Algebras
242
Proposition
6.3.2.12
( 0 ) Let E be a C*-algebra and take n E IN. Given
r~(p) , define p c iN and a E --n,n "5" E n(p-i) ,n
> IK ~
b,
>~
(bij, aji)
i,j--1
~(P) with the norm endow ~-Jn,n n(p)
>IR+
a,
> Ilall
defined inductively for all p E IN, and identify ~n~CP),nwith (En,n) (") (Proposition 2.3.6. 7 and Theorem 5.6. 6.1 a)). We have for every p E IN : a)
The above identification map E n,n (2p) -+ (En,n) (2p) is an isomorphism of C*-algebras (Theorems 5.6.6.1 f ) and 6.3.2.1 b)).
b)
(2p-1) and x E E n,n (2p) , then If a E E n,n (ax)ij=~aikxkj,
(xa)ij=~xikakj
k=l
k=l
for all i, j E INn. We pove the assertions by induction on p. a) The only thing to be proved is that the products in ~,nP(2P) and (E,~,~)(2p) are identified too (Proposition 2.3.6.7 b)). The assertion is obvious for the E n,~ (2p-2) . This set being dense in ~(E(2P))E(2.~-I) (Corollary elements of jS(2~-2) n,n , 1.3.6.5) the assertion holds for any elements of E n~n (2p) by Theorem 2.2.7.15 d) ~(p-2) 9 Then, by Proposition 5.6.6.14, b) Take y E ~n,~ n
(y, ax) = (ya, x) = E
((ya)jk, Xkj) =
j,k=l
:
Yjiaik, Xkj j,k=l
=
--
= E ( Yji, aikXkj) = i,j,k--1 i,j--1
(Yjiaik, Xkj) = i,j,k--1
Yji,
aikXkj k--1
=
=
6.3 Von Neumann Algebras
243
so that n
(ax)ij = E
aikxkj.
k=l
The relation
(xa)ij -- ~
xikakj
k--1
is proved similarly.
I
Corollary 6.3.2.13
( 0 ) Let H be a Hilbert space and n E IN. We set 0
a ~(H)n,n ~
IK,
b~
(bij, ~dji> i,j--1
for every a E s
and ~d" El(H)n,~ ~
]K,
b:
>
(bij, ~dji> i,j=l
for every a E s (Definition 6.1.6.1). Then (with the identification of Proposition 6. 3. 2.12) the maps Ll(H)n,n
> (K:(H)n,n)'
s
~
(s
a,
>a
a,
>"d
are isometries of ~(H)-modules. Moreover, the W*-algebras s (K:(H),~,n)" (Theorem 5.6.6.5 c) and 6.3.2.1 b)) may be identified.
and I
E x a m p l e 6.3.2.14 Let E be the C*-direet sum of the family (Et)tE T Of C*algebras and G the C*-direct product of the family (E~')t~T (Example ~.1.1.6, Theorem 6.3.2.1 b)). Define
F
-
{.'
c I I E ; I ~ II*',ll < oo} tET
tET
and endow F with the norm F
> ]R+,
x'~ ~ ~ 11411. tET
Given x' E F and y E G, define
244
6. Selected Chapters of C*-Algebras
x"
E
> IK,
x,
xt, xt)
~ tET
tET
T h e n 7' C E ' f o r every x' E F , ~ E F' f o r every y C G , the m a p s u "F
v
9
) E' ,
G
~ F'
x' ~ ) x' ,
y'
,
,
are i s o m e t r i e s of involutive B a n a c h spaces, predual, and v -1 o u' is a W * - i s o m e t r y
~ y-
G
is a W * - a l g e b r a with F
as
(i.e. G is the bidual of E ).
The first assertions are easy to see (Proposition 1.2.2.13, Proposition 2.3.2.22 b)). It follows that v -1 o u ~ is an isometry of involutive Banach spaces (Proposition 2.3.2.22 d)). Now we prove that v -1 o u' is an algebra homomorphism. Given t E T , let Pt and qt denote the natural projections E - + G -~ E~', respectively, and it the natural injectien E~ -+ F . Step 1
t c T ~ u o it-
P't
i Take x It C E t. Given x E E ,
(~, ui,x',> -
= (~, p',~',>,
so t h a t 9
,
!
I
uztx t = PtXt.
Step 2
t c T =:v qt o v -1 o u' = p'[ !
!
Take x" C E " . By Step 1, if x t c E t , then (xtt, q t v - l u ' x '') - ( i t x l t , ulx '') - ( u i t x ~ t , x '') -
-
I I xll (p,z,,
I __
(z,,I p tIIx
so that qtv-1
IXll __ Ptx l l I! .
II\
),
Et and
6.3 Von Neumann Algebras
Step 3
V -1
o
245
u' is an algebra homomorphism
Take x", y" E E" and t E T . Pt and qt are involutive algebra homomorphisms. By Corollary 6.a.2.a, p',' is an involutive algebra homomorphism too. By Step 2, qt((v-lutxt/)
(v-lutYt'))
---
--,,,,(qtv-lu'x"~(qtv-lu'y''~ = " ~Pt,,x ,,,, ,, ,,,) = Pt,,,(x ,, y ,,,,) )[PtY -
-
qtv - l ? f f {x,ty,~' ~,
,
Since t is arbitrary, (V-I"u, t Ttt)(V-I?./,ty It) = v--l?ff(xttyt').
Hence V - 1 0 "/_t' is an algebra homomorphism and therefore an isomorphism of C*-algebras. Since E" is a W*-algebra (Theorem 6.3.2.1 b)), G is also a W*-algebra. By the above and Corollary 4.4.4.4, F is the predual of G. Using again the above result and Proposition 4.4.4.6, we deduce that v -1 on' is a W*-isometry. I
E x a m p l e 6.3.2.15
Let ( H t ) t E T
be a family of Hilbert spaces, E
the C * -
direct sum of the family (K(Ht))tET, and G the C*-direct product of the family (s
(Example ~.1.1.6). Define
F "- {u E H / 2 1 ( H t ) I ~-~-l[utl[1 < c~}, tET
tET
and endow F with the norm
F
> 1R+, tET
For each v E F and w E G , define o tET 1 tET
(Definition 6.1.6.1). Then "~ E E' for every v E F , ~ E F' for every w E G , the maps
6. Selected Chapters of C*-Algebras
~6
~'F
~ E'
r
>
~V~ ""
Vl
F'
Wt
)W
are isometries of involutive Banach spaces, G is a W*-algebra with predual F , and r o r is a W*-isometry (i.e. G is the bidual of E ).
The assertion follows immediately from Example 6.3.2.14 and Theorem 6.1.6.6. 1 P r o p o s i t i o n 6.3.2.16
( 0 ) Let E be a real C*-algebra. For (x',y') 9
E'
o
and (x", y") 9
(~,, r (~,,, r
E"
put
~
~r
(~)'
~r
(x, y) ,
(~', y')
,
~ ~'(~) - ~'(y) + i(~'(y) + y ' ( ~ ) ) ,
~ ~"(~') - y"(y') + i ( ~ " ( r
+ y"(~')),
o
where we identified (E)' with
E'
E'
using the map
~ (E)',
(x',y'),
~ (x',y') o
o
(Corollary ~.3.6.2 b)). Then (x", y") 9 (E)" for every (x", y") 9 E"
and the
map ,A..,
o
(z" y")
E"
~ (E)",
,
~ (~",r
,
is a W*-isometry (Theorem 6.3.2.1 b), Theorem 4.4.2.21). o
By Corollary 4.3.6.2 gl),g2), (x",y") 9 (E)" for every (x",y") 9 E" and the map ~'
E"
~(~)",
(~",y,,) , ~(~",r
is an isomorphism of involutive complex vector spaces. Take o
II
Eli
6.3 Von N e u m a n n Algebras
2~7
o
By Corollary 4.3.6.2 g3), for every (x', y') E E' ,
I
=
ll~!
tt
x'x~-yyl,xyl
-- (- x ' x ' ~
Y '" Y l , x 2")
-
+i( (~'~'1'__ I X I, --
+i((~',
-
II~'
" ~x ' " Yl +
(Xlt
_ t/~
t 2,Y2)
=
Y ' X l", Y 2") +
y ' y l" , y~) -3- ~x " ' yl" -3t-y ' Xl, " x~}) ---
II
II
II
II\
Xl x2 - Yl Y2 ~ -
"
I
+YXl),
"
"
""
T2
--
(Y', Yl" x2" -+- Xl" Y22 "" -4-
x l Y 2 + YlX2~ +
,,,,^,,,,
-
--
,
~Xl
Yl Y2, x l y2 + yl x2)
=
II
Hence ( ~ , y~'),~x~, ,, y ~,,, _ (~', y~')(~', y~';
and the map o
E"
~ (E)",
(~",y"),
~ (~", y")
is an isomorphism of C*-algebras. By Proposition 4.4.4.6, this map is even a W*-isometry. m P r o p o s i t i o n 6.3.2.17
Put
A "- {x c E+ I Ilxll < 1}. a) A is u p w a r d d i r e c t e d a n d its u p p e r s e c t i o n f i l t e r ~ is a n a p p r o x i m a t e u n i t orE.
6. Selected Chapters of C*-Algebras
248
For every x" e E~, supremum.
b)
( x"~1(jEx)x ''~1)
is upward directed and x" is its xEA
C) For every x" C E",
x" = lim(jEX)X" = lim X" (jEX) x,i~
x,i~
in E~,. a) was proved in Proposition 4.2.8.1 and Theorem 4.2.8.2. b) By a ) a n d Corollary 4.2.2.3, (x"89(jEx)x ''89 is upward directed. Let xEA
y" be its supremum in E" (Theorem 4.4.1.8 c), Theorem 6.3.2.1 b)). We want to prove x" = y". Step 1
x' E E~ =~ (x", x ' ) = (y", x')
By Proposition 2.3.6.4, 1
i
x ''89 ''~ C E+ so that by a), Proposition 2.3.4.10 a), and Theorem 4.4.1.8 c) (and Theorem 6.3.2.1 b)), (x" x'> = (XE,,' x"89
"89 =
IIx"89189
= lim(x,x" 89 x~
= lim(jEX, X"89 ''89 --Iim(x" 89 z,~d z,td Step 2
''89 =
''89' X') = (y",X')
x' e Re E' :=v (x", x ' ) = (y", x)
The assertion follows from Step 1 and Corollary 4.1.2.7 d). Step3 x"-y" In the complex case the assertion follows immediately from Step 2. So assume ]K = IR and let x' C E'. Put y,.
1
= ~ ( ~ ' + ~"1,
1
z' := ~(~'- ~'*).
By Step 2,
(~", y'> = (y", r Since
<x", z') = o =
6.3 Von Neumann Algebras
249
it follows
(x", ~') = (y", x'), x" - y". c)
Case 1
x " E E ~ , x" invertible
Take x' E E ' . By b), (x",x') -- (x"2, x'x ''-1) = l i m ( x " ( j E x ) x " x'x ''-1) - - l i m ( ( j E x ) x " x') z,i~
'
~X",X) -- ~X"2, X " - l x t) - - l i m ( x " ( j F x ) x " , x " - l x x,i~
Case2
x,i~
'
'
') = lim(x"(jFx) x'). x,~ '
x"EReE"
The assertion follows from Case 1 since x" is the difference of two invertible positive elements of E " . Case 3
IK - ~
Follows immediately from Case 2. Case4 IK=IR Follows from Case 3 and Proposition 6.3.2.16. P r o p o s i t i o n 6.3.2.18 E,i.e.
I
Denote by p the canonical projection of the tridual of
.!
fl "-- jE, OJE.
a)
I m p is an involutive E"-submodule of E ' " , Ker p volutive E-submodule of E ' , E-modules.
b)
(x'",y"') E E"' x I m p , 0 < x " < y"' :=~ x " E I m p .
c)
(x'", y'") E E"' x Ker p, 0 < x'" < y"' ==> x'" E Ker p.
d)
(Im jE) ~ is an in-
and p is a homomorphism of involutive
If Im jE is an ideal of E" then Ker p is an involutive E"-submodule of E "~ and p is a homomorphism of involutive E"-modules.
e) jE,(E~+) - ~'" N I m p
6. Selected Chapters of C*-Algebras
250
f) jE,(T(E)) = ~-(E") fq I m p .
The map
g)
9I
lit
is bijective and continuous with respect to the topologies of 7-. jE,(T(E)) is a dense set of T(E").
h)
i) jE,(To(E)) = To(E") M I m p . a) By Corollary 4.4.2.10, I m p is an involutive E"-submodule of E"'. By Proposition 2.3.2.22 b),d),e), jE,jE,, and j~ are involutive. Hence p is involutive. By Proposition 2.2.7.5, jE and jE' are homomorphisms of E modules so that by Proposition 2.2.7.6, j~ is also a homomorphism of E modules. Hence p is a homomorphism of E-modules. By Proposition 1.3.6.19 c), Ker p = (Im jE) ~ . b) follows from Corollary 4.4.2.13. c) By a), for every x E E + ,
0 <_ (jEX, X'") < (jEX, y " ' ) = 0 SO that
{jEX, X'") = O . It follows x"' E (ImjE) ~ = Ker p. d) Take (x", x"') E E" x Ker p. By a), for every x E E ,
{jEX, X" X'") = ( (jEX)X", X"') = 0, {jEX, X'"X") = (X"(jEX), X"') = O, so that
x" x'", x"' x" E (ImjE) ~ = Ker p. Hence (by a)), K e r p is an involutive E t! -submodule of E'". Take (x", x'") C E" • E'". By a) and the above,
x"(px'"), (px'")x" e Im p, Since
x"(x'" - px'"), (x'" - px'")x" e Ker p.
6.3 Von Neumann Algebras
p(x"x"'), p(x"'x") 9 Im p,
x"x'" - p(x"x'"), x'"x" - p(x'"x") 9 Ker p,
it follows p(x'"x") 9 I m p n Ker p
x"(px'") - p ( x " x ' " ) , ( p x ' " ) x " i.e.
x" (px'") - p(x"x'")
= o
(px'")z"
= o,
- p(~'"z")
p(z"~") - ~"(p~"'),
p(x"'z") = (px'")x".
Hence (by a)), p is a homomorphism of involutive E"-modules. e) Take x'" 9 ~_.+ P"~ A I m p Then
(X~JE x "I
III\
)
__
xlII 1
(jEX,
9 ]l:[+
for every x 9 E+ (Theorem 6.3.2.1 b), Corollary 4.2.2.11), so that 9I
2E x
III
x'"= px'"-jE,
I
9 E+ ,
J'E x ' ' 9 j E , ( E + ) .
Hence EIII + n Im ~ c
j=,(E~_).
The reverse inclusion follows from Theorem 6.3.2.1 d). f) follows from e). g) follows from f). h) follows from Corollary 4.4.1.9. i) Take x' C To(E). By f), jE, X' C T(E u) n I m p .
Let y"', z'" c T ( E " ) and o~,/~ e ]0, 1[ such that c~ + ~ = 1,
ay"' Jr- ~z'" -- jE, X' .
By b), y'", z "~ C I m p . Put .I
Y' :-- 3EY
Ill
,
ZI
.I
:= 3E z
III
251
6. Selected Chapters of C*-Algebras
252
By g), y',z' 9 T ( E ) . From X t
=
-t
9
]EJE'X
I
-~- o l Y ! +
~ Z!
(Proposition 1.3.6.19 a)) it follows y' = z', Y'" = PY"'
=
J E ' ] E"
Y"' = jE' Y'
=
3E" Zl _. ]E z ' J"E .t
Ill _ . p z ! ! l
__ Z111 .
Hence jE, X ' 6 To(E") and jE,(7-o(E)) C 7o(E") n Imp.
Take x'" C T o ( E " ) N Imp. Put Xt
.t m "= ] E x 9
By g), x' 9 ~-(E). Let y',z' 9 T(E) and a, fl 9 ]0, 1[ such that a + /3 = l ,
ay' + /3z' = x'.
Then xlll
--
pxlll
9
-- ]E'JE
"I
x
lit
-- jE'
X I
9
-- a]E'y
I
+ /3jE'Z'.
By f), jE, y',jE, z' 9 7 ( E " ) . It follows jE'Y' = jE' Z' , yl
=
ZI "
Hence x' e T0(E), x'" 6 jE,(To(E)),
T0(E") AIm p C jE,(To(E)) . Take E : = c . Then E ' ~ g l • E"~t~ ~• Let ~, ~ be two distinct free ultrafilters on IN, A 6 ~ \ ~ , Remark.
W
'•
z" := (eA, O) c E", and x'"" Z"
> IK,
(y",a),
>limy"-li~ny".
Then x'" C Ker p and x"x'" ~ Ker p. Hence in this case d) does not hold.
6.3 Von Neumann Algebras
P r o p o s i t i o n 6.3.2.19
253
Denote by p the cononical projection of the tridual of
E,i.e. .!
p:=jE'O3E
and assume that for every x" E E~ there is an upward directed family (x,)~e , in E+ such that
x,,= tEI
(by Proposition 6.3.2.17 b), this condition is fulfilled whenever I m j E is an ideal orE"). b-: m :=> 0 < _ ~+
px
m
a)
x" C
b)
If x'" C Re E"' then
<
X m
.
IIx"'ll-
Ilpx'"ll + I1~'" - px'"lt,
(px'") + - pz '''+ ,
( x ' " - px'") + = x '''+ - px '''+ ,
c)
For every x"' c Re E'" ,
d)
~-0(E") C I m p U Ker p.
(px"')- - p x ' " - ,
(z"'- px'")- = z"- - pz'"-
x'" is order continuous iff x'" c Im p.
a) Take x " e E~ and let (x~)~ei be an upward directed family in E+ such that X
"
~
Vu
EX~
9
LEI
By Corollary 4.4.4.3 a =~ b, (X", px'") -- (X", jE'j'EX'") = (X", j'EX") =
-- sup(jEx~, j'EX"') = sup(x~, j'E x'"\/ -- sup(jEX~, X"> .
Since
254
6. S e l e c t e d
Chapters
o] C*-Algebras
for every ~ E I , it follows
o <_ (x", p~'")___ (:~", :~'"), i.e. 0 _< px'" _ x'". ~'" then by a) and Corollary 2.3.4.14, b) If x'" E --+,
I1~'"11 = IIp~'"ll § I1~'"- p~'"ll 9 Now assume x "~ E Re E " . By Proposition 6.3.2.18 a), p is involutive so that p x "~ E
Re
E'".
By a),
px m+
,
px "l- , x "l+
-
px m+
,
x m-
-
px m-
b-? m E ,_,+ .
Since X m -- X"l+
_
X m-
,
p x "1 - - p x " l +
pX m-
_
9 ' " - p ~ ' " = (x ''+ - p.'"+) - ( ~ ' " -
,
- pz'"-),
it follows from the above,
I1~'"11 ~ IIf'll
+ I1~'" - p~'"ll
_~ Ilpx"'+ll + IIp~'"-II + IIz '''+ - p~'"+ll + I1~'"- - p~'"-II-
= I1~'"§
+ I1~'"-II = I1~'"11
Hence
I1~'"11 = IIp~'"ll + I1~'" - p~'"ll,
Ilpx"'ll = Ilpx"'+ll + Ilpx'"-II,
I1~'" - p~'"llso that
II ~'''+ - p~'"+ll + IIx'"- - p~"'-II,
6.3 Von N e u m a n n Algebras
(px'") +
(x'"-
p x '''+ ,
=
px'") + = x '''+ - p x '''+ ,
255
(px"')- = px'"-,
(x'"-
px'")- = z"'- - px'"-.
c) If x ' " E Im p then by Theorem 4.4.1.8 c), x"' is order continuous. Assume now that x'" is order continuous. By the above, p x ' " is order continuous, so that x " ' - p x ' " is order continuous. Since p is a projection, x" - px'" C
Ker p = (Im j E ) ~ .
Let x" C E~ and let (x~)~ci be an upward directed family in E+ such that x"--
V(jEX~).
Then ( x " , x "l -- p x "l) -- 0
and therefore x "l -- p x "l = 0 ,
x "t = p x "~ C
Imp.
d) Take x" e % ( e " ) . By a) and b), ~III , p x III , X I l l - p x Ill C ~_~+
IIp~'"ll + I1~'" - p ~ ' " l l -
I1~'"11- 1.
By Proposition 2.3.5.4 a =~ b, px'"--IIp~'"llx'",
If p x ' " r 0 and x " ' 1
IIJ'll
px'" ~ 0
px'"
~"'then 1
'
p~"'--IIx'"- px'"ll~"'.
IIx'"- p~'"ll
(z'"-
pz'")
We get z"' = px'" + ( z " ' -
px'") =
e ~-(E").
6. Selected Chapters of C*-Algebras
256
1
=
px'" + I1~'"- p~'"ll
IIp~'"ll Ilpx'"ll
1 (x'"- px 'r') I1~"' - px'"ll
'
which contradicts the hypothesis x'" E To(E"). Hence
px'" = O or
x"' - px'" = O
and
x'" E I m p U Ker p.
I
Remark. In the above proposition we cannot simply drop the initial hypothesis as the example E " - c shows. Indeed, in this case E r ~ gl
x
IK, E" ~ goo
x
IK, E ''r ~ (goo)r x IK.
Let ~ be a free ultrafilter on IN and define x III 9 E l i ----+ IK,
(x",ol),
> limx"(n). n,~
Then x " r E - +~~II! ,
(x, j'EX'") = (jEX, X'") = lim x(n) n ---+o o
for every x E E , pX m
9 .t ttt -- 3E,3E x --
5,
where
5-E"
>IZ,
(z",a),
>o~.
Hence
px III ~ x III
Theorem
6 . 3 . 2 . 2 0 ( 8 ) L e t E be a C*-algebra and S , T sets. We put H0"-
(D E - g 2 ( T , E ) ,
K0:=
tET
(D E - g 2 ( S , E ) , sES
W
H:=
(DE", tET
W
K'-
(DE", s6S
6.3 Von Neumann Algebras
s := s
lc - - IC.(Ho, Ko),
257
K)
and fro "--(~ttol)teT 6 H ,
g~o "-- (Sssol)~cs e K
for every to E T and So 6 S . Moreover, for every x E E (s• (E') (Txs) define
> ( ~ x~t~tI
>Ko, ~ '
~" Ho
\t6T
~'H
X'' {xix -
e
>K,
and x' 6
,
/ s6S
~,
E (S•
> IK,
x -
<x,,, z',~>
E (s,t),eSxT
(Proposition 5. 6.1.8 h), Proposition 5. 6.~. 16 b)), and
x~,=
~
ix'~,s~,g~.
(t,s)6TxS
a)
The map
{xix E E (sxT)}
>
f_,,
x,
>
(Proposition 5.6.1.8 h), Proposition 5.6.~.16 b)) is linear and preserves the norms.
b)
The map
x ' e (E') (~•
>/],
z',
>x'
is well-defined, linear, and preserves the norms; we denote by ...
~" IC'
>H
its continuous extension (Proposition 5.6.5.19).
c) Let y' 6 E ' , (s,t) C S x T , x",y" 6 E " , and x' := (Sss, Stt, x"y'Y"*)(t,,s,)eT•
C (E') (T•
Then 9~
= "(y',f,x", gsy"~
.
6. Selected Chapters of C*-Algebras
258
d) ~ is an isomorphism of Banach spaces. e)
~' 9 tO' ~
(~')* - ~'*
f) If we identify )C' and [t by 99, then s is identified (by ~ ' ) to the bidual of ]C (Theorem 5.6.3.5 a)), jlcx = for every x E E (S•
(Propositions 5.6.5.6 d) and 5.6.5.9 b)), and
A:(~('IS>) -(jE~Ts)seS(l(jE~t)teT) for every (~c,7/) E Ho x Ko.
g) Assume T is a one-point set and put ~" E
> Ko,
x,
/C, r/,
>
> ~Tx
for every ~7 C Ko. Then
Ko
>
K
>s
rl<.{1),
>~"<'ll>,
are isometries of Banach spaces, so that (by f )) K may be identified with the bidual of Ko, in which case JKo r] -- ( j E r l ) s e T
for every ~7 C Ko.
h) Put 9 2 - - {A • B I A e ~ : ( S ) ,
Be~:(T)},
order 9)l by inclusion, and denote by ~ its upper section filter. Then for every u' C H there is a unique x' C (E') T• such that .
.
.
u'
lim A•
i
TxS
X eBx A
is the topology of pointwise convergence on s
6. 3 Von N e u m a n n Algebras
259
a) Take x C E (SxT) . There is a suitable n c N such t h a t x may be considered an element of En,n. Then II jE,~,,~X E ~/ E n,nj~" = E n,n (Proposition 6.3.2.12 a)). Our notation identifies ~ and ~ with the corresponding operators associated to the matrices x and JEn,~x, respectively. The assertion follows from the fact that jE~,~ is linear and preserves the norms. b) The map is well-defined since the map x' ~-+ x' is linear and injective. Take x' c ( E ' ) (Txs) . There is a u E / 2 # such that (u. x') = IIx'll (Corollary 1.3.3.8 a), Theorem 5.6.3.5 a)). P u t
9 ':, - ( ~ f ,
lg,) c E"
for all (s,t) E S x T . Then
(u,~') -
Xt
~
u, ( - , s , , g , )
-
(s,t)ESxT
(s,t)CSxT
(s,t)CSxT
Let c > 0. By Theorem 6.3.2.1 d) and Proposition 6.3.2.12 a), there is an x C E (sxT) such that II~ll < 1 and
E
(~':~.
~',) - ~ <
(s,t)CSxT
E
( x ~ . x'.) -
(s,t)ESxT
It follows A
~
II~'ll- ~
N
<__ (~, ~') _< I~'!
and so A
IIx'll _< Ix'll. Take y C E (SxT) For every (s, t) C S x T , (Yftlgs) = E ( j E Y s t ' ) h t t ' ft, = jEYst ff ET
and so (by the above),
(~, ~,). -
6. Selected Chapters of C*-Algebras
260
<~, x") =
E
E
(s,t)6SxT
(jsY~t, x't~) =
(s,t)6SxT
= <~, ~,>.
By a), ~
A
A
I<~,~'>1 = I<~',~'>1 _< I1~11 I1~'11 = IlYl I1~'11 , A
IIx'll < IIx'll, .-...
.~.
II~'ll = II~'ll.
c) By Proposition 5.6.3.1 d),
"(y', Ax", g~y") = (z" y' y"*, It, gs; J%
--
E
-
.,~
(5581c11111" ott, x y y , ft', gs') -- x' .
(t',s')6 T x S
d) Take u 6 • with u = 0 on I m p . W e deduce successively for x' 6 E', s 6 S, and t 6 T ,
<,~'> = = 0,
<~f, lgo> = o , uft = 0 ,
u=O (Propositions 5.6.4.6 c) and 5.6.3.4 c)). By b), ~ is an isomorphism of Banach spaces. e) Take x' 6 (E') (T• . By Proposition 5.6.3.1 b) and Proposition 5.6.5.18 b), x, =
((~,~), g~, I, = ~'*, (t,s)6TxS
"
6.3 Von N e u m a n n Algebras
Xl
--
Xl*
261
.
It follows ~(~*) = ~(~)
= x,m = x3* - ( ~ ) . .
By continuity,
~z'* = (~oz')* for every x' E )E' (Proposition 5.6.5.19). f) Let y' C E', (s,t) C S x T, x",y" E E", and x' :=
(5.,st~,x"y'y"*)(~,,~,)~• e (E') (~•
By c),
(~, ~') = {~, ~(y', f~x", e~y"Y) = ((2f~x"lg~y"), y') = = (y"*(~.f~lg~)x",y')-
((~f~lg~), x"y'y"*)-
= (jEX~t, x"y'y"*} -- (Xst, x"y'y"*} -- (~, ~'} =
= (5, x') = (j~:~, x'}. By Proposition 5.6.5.19, ~= j~. Put 9)t"- {A • B IA e ~!(S), B e ~s(T)},
order 9)l by inclusion, and denote by ~ the upper section filter of 9//. By the above, Proposition 5.6.5.9 a), and Proposition 5.6.5.2 m), j~:(n(.15>)=
lim
A x B fld
j~(ve~(.l@T>)-
= lim ((jEr]s)eS(s))ses(.I ( 0 ~"4 ~ ) ~ .T ( t ) ) ~ > AxB,~ = (jEris)seS('l(jE~t)teT).
-
6. Selected Chapters of C*-Algebras
262
g) By Proposition 5.6.5.6 d),e), the maps -+
>K:, r/,
qO'Ko
r
>rl,
>Z:, ~", >r/'(.ll),
are isometries of Banach spaces with
>K,
~b- 1 " / :
>u"l.
u",
By f),
JKorl = r
7 -- ~)-ljtc- ~ = ~2-1( (jE~% )seS(.Jl ) ) -= (jErls)seS,
for every r/ E E (s) , since in this case rl - ~. The general assertion follows by continuity (Proposition 5.6.4.1 e)). h) The uniqueness of x' follows from
X t's " E
~ ]K ,
x ,
~
ss' ~tt ' x
(s',t')ESxT,
?2t
for every (t, s) E T x S. Put "ITA " K
~
rrB'H
K,
rl ,
> ~esA ,
~H,
~,
>{e r
for every A C S and B C T . By Proposition 5.6.4.20 i), 7rA E Pr s
7rB E Pr s
for every A c S and B C T and u' =
lim
A x B,~d
7rBU'7( A
--
lim X'CBxATXS
A x B ,~
in the topology of pointwise convergence on /2, with the above defined x' E (E,) r• II
6.3 Von Neumann Algebras
263
6.3.3 Extension of the Functional Calculus Throughout this subsection, E will denote a W*-algebra. Theorem 6.3.3.1
( 0 ) Denote by F the predual of E and take x C SnE.
Let j " a ( x ) --+ IK be the inclusion map. Put C "= C(a(x)) and consider ~C----~E,
f,
>f(x).
a) There is a unique unital W * - h o m o m o r p h i s m u 9 C" --+ E (Theorem 6.3.2.1 b)) such that u j c j = x . We define f ( x ) := u f for f c C".
b) U = fFOCP"
uojc=~
,
9
c) If f is a bounded Borel function on a(x) and f " C_,'
-
~ IK,
#,
)
1
f d#,
then u(f) = f(x), where f ( x )
was defined in Corollary 4.3.2.5 b).
d) Im u is the unital W*-subalgebra 9f E generated by x . e)
a ( f ( x ) ) C f ( a ( C " ) ) for every f c C".
f) g o f e C(a(C")) = C" and g ( f ( x ) ) = (g o f ) ( x ) , whenever f e C" and g e c(f(~(c"))).
g) Take f e C", 7:):= C ( f ( a ( C " ) ) ) , and r
>C" = C(a(C")) ,
h,
>h o f .
If g c T:)", then 3c, "~ YJ - " g E C" and g(f(~))
-,
= 0e,r
,,
g)(z) ;
for every ~ e f ( a ( C " ) ) we have that 9, _,, (~) 3c, V2 g = g -1
on the interior of f ( a ) , where 5~ denotes the Dirac measure on f ( a ( C " ) ) at a .
6. Selected Chapters of C*-Algebras
264
h)
If the predual of E is separable, then for every f E C" there is a bounded Borel function g on a(x) such that f(~)
= g(x);
in particular {h(x) l h bounded Borel function on a(x)} is the unital W*-subalgebra of E generated by x .
i)
If p denotes the spectral measure of x (~.~.1.11 a)), then x' o p is a measure for every x t E E ~ and
(f(~), ~') = (f, ~' o .) for every f E C". Moreover, if 91 (resp. ~3 ) denotes the a-algebra of Borel sets of a(C") (resp. a ( f ( x ) ) ) A'C'
~ IK,
and if we set A,
> (jc,)~)(A)
for every A E 9.1, then -1 A
>P:E,
B,
~ f (B) (x)
is the spectral measure of f ( x ) .
a & b. First we prove the uniqueness, u o jc is an involutive algebra homomorphism (Theorem 6.3.2.2 a)) such that u ojc(j) - x.
Hence uojc - ~.
Since the map U " C~l,
> EF
is continuous and since Imjc is dense in C~', (Corollary 1.3.6.5), the map u is uniquely determined. Now we prove the existence. Put
6.3 Von Neumann Algebras
U
3F
265
.
o
By Corollary 6.3.2.3 and Theorem 6.3.2.2 a), u is a unital W*-homomorphism. By Proposition 1.3.6.16 and Proposition 1.3.6.19 a), uojc--fFo~"
ojc=JtFOjEo~=~.
In particular, ujcj = cpj = x .
c) Let B be the set of bounded Borel functions on a(x) and put 13o "= { f 9 B iu(f) = f ( x ) } .
By b) and Corollary 4.3.2.5 b)), C C/3o. Let ( f ~ ) n ~ be an increasing sequence in Bo with the supremum f in B. Since
/-V/o nEIN
we get
= V u(L)= V :o(-)= :(x) nC1N
nc1N
by Proposition 4.4.4.8 d) and Corollary 4.3.2.5 a). Hence, f E B0. It follows that B0 - B. d) By a) and Corollary 4.4.4.8 b), I m u is a unital W*-subalgebra of E containing x. Let G be a unital W*-subalgebra of E containing x. By Corollary 4.4.4.9 a :=> b, G is a W*-algebra. By the uniqueness of u, we get that I m u c G. Hence I m u is the unital W*-subalgebra of E generated by x. e) By a) and Proposition 2.4.1.17, there is an injective continuous map ~o: a(Im u) --+ a(C") such that A
A
f O~o - f ( x ) .
By Corollary 2.4.1.7 a), o ( f ( x ) ) = f(x)(cr(Im u)) = f o ~0(cr(Im u)) C f ( o ( C " ) ) .
f) By Example 2.4.3.1, we may canonically identify C(o(C")) with C". First suppose that there is a P E IK[s, t] such that
6. Selected Chapters of C*-Algebras
266
A
for every a E f ( a ( C " ) ) . Corollary
By b), Theorem 6.3.2.1 b), and Corollary 6.3.2.3 (and
4.1.3.8),
g(f(x)) =
P(f(x), f(x)*) = P(j'FqO" f ,JFqO" " "J) = A
A
= j'FqO"P('f, f ) = j'FqO"(g o f ) = g o f ( x ) .
By the Weierstrass-Stone Theorem, the above equality holds for an arbitrary g. g) Define w'7:)
>E,
h,
>h(f(x)).
By b) and f),
~=J'Fo~" o r so that
~ " = j~' o ~"" or Take (a, h) E F x C". Then
(h~ qO
III
= (r
"11
9
~
I
"1
I1~
3F3Fa~, = ~ 3 F ~ I~, j F a )
jFa)=
I
"1
lit
= ~ 3 F ~ 12, a ) =
(h, c p ' j F a ) = (h, jc, qo'jFa).
We deduce successively that ~'" o j ~ o jF = jC, O ~O' o j f ,
J'F o j'~' o qO"" = 3 F' O ( P " 03C,, '
j~.ow"
= 3"' F O 3""' F O ~ "" ~ - " :
3"' F 0
g ( f (x)) = J'F o w"(g) = J'F o ~O"(j~,r
~o" 0 3C' "
0
r
,
-- (j'c,r g ) ( x ) . -1
Now we prove the final assertion of g). Let U be the interior of f (c~), and take x ' E C with Suppjc, z' C U .
6.3 Von Neumann Algebras
267
Then
(y, r
x') - (r
jc, x') = (y o f', jc, x') -
= (y(a)e~(c,,),jc, x ' ) = (y, 5~)x'(cr(x)) for every y E 79, so that
r
x' = x'(a(x))5~.
It follows that
(x',j'c,r
= (r
x',g) = x'(a(x))g(5~).
Since a(C") is a hyperstonian space (Theorem 6.3.2.1 b),c)), -, r 3c,
= g(~o)
on U. h) Let (an)ne~ be a dense sequence in the predual of E . Then r for every n C IN. Define
# :- ~
nEIN
1
2n(ll~,jFan)l + 1) I~"jFanle
E C'
CI"
The map
LI(#)
~ IK,
h,
~ (h.p, f )
is continuous and so there is a bounded Borel function g on a(x) such that (h-p, f) = f hgdp for every h E LI(p). Take n C IN. By the Radon-Nikodym Theorem, there is a gn E L~~ that
qJjFan -- gn'#. Then, by a),b), and c),
(an, f ( x ) ) = (an, u f ) = (an,j~Fcp" f} -- (qg'jFan, f) =
such
6. Selected Chapters of C*-Algebras
268
f
f
<.o.., :> = ],oge,
= ],e(:'j,ao)=
(:'j,ao.
=
= (an,j~Fcp"g'g") = (an, ug~ =(an, g(x)). Since n is arbitrary and (an)ner~ is dense, it follows that
f(x) -- g(x). The final assertion follows by d). i) x' o # is obviously a measure. Put ~" := {: E C " l ( f ( x ),x') = ( f , x ' o # ) }. By a), Theorem 4.4.1.8 c), and Corollary 4.4.4.8 d), :" is an order faithful vector subspace of C". Since it contains C (Corollary 4.4.1.11 e)), ~" = C" follows from Example 6.3.2.6 f). Take v E C"'. Then, using the notation from g), for every h E 79, h) = (u, Ch) = (v, h o f) = (f(v), h>
(r
so that v:', = f(,.,).
Take B E ~ and define B'Z)'
> ]K,
p,
> p(B).
By the above considerations, given )~ E C',
(j~,r
= (r
jc, A ) = (B,r
A)=
= (B, f'(jc, A)) = (f"(jc, A))(B) = -1 A
-1 A
= (jc, A ) ( f ( B ) ) =
f (B)(A).
Hence -1
j'c, g2"B = f (B). By g), -1
if3 .. > P r E , is the spectral measure of f ( x ) .
B,
~ I(B)(z)
I
6.3 Von Neumann Algebras
269
Let F be the predual of E . Take a c F (resp. z E E ) and x, y C Sn E , such that
C o r o l l a r y 6.3.3.2
xa-
ay
(resp. xz = z y ) .
Put c :=
c(~(x) u o(y))
Then f(x)a = af(y)
(resp. f ( x ) z -- z f ( y ) )
for every f C C". Define ~" := {f e C" l f ( x ) a = a f ( y )
(resp. f ( x ) z = z f ( y ) ) } .
Let G be a nonempty upward (downward) directed subset of 9v with supremum (infimum) f in C" and let ,~ be its upper (lower) section filter. By Theorem 6.3.3.1 a) and Corollary 4.4.4.8 d), f ( x ) is the supremum (infimum) of (g(x))ge6 in E and f ( y ) i s
the supremum (infimum)of (g(Y))yca in E . By Corollary
4.1.3.10,
f(x)a=af(y) (by Theorem 4.4.1.8 c),
(b, f ( x ) z ) - (zb, f ( x ) ) = lim(zb, g(x)) gcqd = lim(b, g(x)z) - lim(b, zg(y)) = lim(bz, g(y)) = g,iY
g,~
g,~
= (bz, f ( y ) ) = (b, z f ( y ) ) for every b C F , so that
f(x)z-
zf(y)).
Hence f e 3c. Since C ( a ( x ) U a(y)) c ~" (Theorem 4.1.4.1), we get 9v = C". m
6. Selected Chapters of C*-Algebras
270
E x a m p l e 6.3.3.3 Let K be a closed vector subspace of H such that neither K nor K • is separabel. Then for any self-normal compact operator x on H 7rK r { f ( x ) l f 9 C(a(X))" }.
By Theorem 5.5.6.1 a :=> e, there is a separable closed vector subspace L of H such that X =
7TLXTr L .
Since 7rLf-.(H)TrL is a W*-subalgebra of Z:(H) (Corollary 4.4.4.12 c) and Corollary 6.1.7.14 b)), it follows that {TrLyzrL + a l l y
e L ( H ) , a e IK}
is a unital W*-subalgebra of E(H). By Theorem 6.3.3.1 d), {y(x) l y e c(o(~))"} c { - ~ u ~ + . 1 l u e s
e ~}.
Assume that
~
e {Y(~) I Y e c(o(z))"}.
Then there are y E f..(H) and a E ]K such that 7r K
---- 7 r L y T r L
+
al
.
Since L is separable and K is not separable, a 7rK• -- 1 -
7r K
=
1. Hence
7rLyTr L ,
which is a contradiction, since K l is not separable.
I
T h e o r e m 6.3.3.4 ( 0 ) Take x E S n E , C ' = C ( a ( x ) ) , and let G be the unital W*-subalgebra of E generated by x . Identify C" with C(a(C")) using the Gelfand transform (Theorem 6.3.2.1 c)). a)
Given f, g e C" with {g :/: 0} C {f # 0}, g(x) = 0 whenever f ( x ) = O.
b)
There is a greatest clopen set U of a(C") such that ~ ( ~ ) = o. Its complement in a(C") is denoted by S(x) and is called the support of x .
6.3 Von Neumann Algebras
c) S(x) is a compact hyperstonian space and C(S(x)) is a W*-algebra. d) If f 9 C(S(x)) and f l , f 2 9 C" such that
flls(x)- f, lS(~)- f then fl (X)
--
f2(x) 9
Define
f(x) e)
:= f l ( x ) .
The map C(S(x))
>G,
f,
> f(x)
is an isometry of W*-algebras.
f)
es(x)(x) is the carrier of x in G .
g) a ( f ( x ) ) = f ( S ( x ) ) , whenever f 9 C(S(x)). h) g(f(x)) = g o f ( x ) for all f e C(S(x)), g 9 C ( f ( S ( x ) ) ) . a) We may assume that Ifl __ 1. Given c~ > 0, Ifl~(x) =
If(x) ~ -- 0
(Theorem 6.3.3.1 f), Proposition 4.2.5.4). Define U'- {fr U is a clopen set (Theorem 6.3.2.1 c)). Then
oL>O
and so, by Theorem 6.3.3.1 a) and Corollary 4.4.4.8 d),
eu(x)
-
V
I f l ~ ( x ) - 0.
o~>o
Thus g(~) = (g~.)(~)
= g(~)~.(~)
= 0.
271
272
6. Selected Chapters of C*-Algebras
b) Put t/"-- { V clopen set of a(g")ley(x ) -- 0},
u.=Uv. VEIl
Take V, W 9 il. By a),
=o,
ev\w(x)
so that
evuw(x) = ev\w(x) + ew(x) = O,
VuW 9 By Theorem 6.3.3.1 a) and Corollary 4.4.4.8 d),
~(~)
:
V
~"(~) : o.
V EgI
Hence U 9 H. U is obviously the greatest element of H. c) C(S(x)) may be identified with es(x)C", which, by Corollary 4.4.4.12 c), is a W*-subalgebra of C". By Corollary 4.4.1.10, S(x) is a compact hyperstonian space. d) We have
{f~ :/: f2} C o'(C")\S(x) and so, by a), fl(X)
-- f 2 ( x )
-- (fl
-- f 2 ) ( X )
= O.
e) By Theorem 6.3.3.1. a), it suffices to prove that the map is injective. Take
f 9 C(S(x)) with f(x) --0. Define
I f(x") g " a(C")
> IK,
x" ,
if x" E S(x)
)
0
if x" 9 a(C")\S(x),
273
6.3 Von Neumann Algebras
v.-
{9#0}.
Then g(~) = 0
and by a), ~.(~) = 0
Hence (by b)) v n S ( x ) - ~,
V=0,
f=0. f) follows immediately from e). g) By e), a(f(x)) = a(f) = f(S(x)) .
h) First suppose that there is a P 9 IK[s, t] with g(a) = P(a, ~) for every a 9 f ( S ( x ) ) . Then g(f(x)) = P(f(x), f(x)*) - g o f(x).
By the Weierstrass-Stone Theorem, the above equality holds for an arbitrary g. I P r o p o s i t i o n 6.3.3.5
( 0 )
Take x E SnE
and put C "= C ( a ( x ) ) . Let
U be a clopen set of a(C"). We identify C" with C(a(C'))
.via the Gelfand
transform. Define j " a(x)
> IK,
a,
>a ,
y "= ( ( j c j ) e v ) ( X ) , and let A and # denote the spectral measures of x and y , respectively. Let fB be the a-algebra of Borel sets of a(x) and for p C A/Ib(a(y)), define ~fi" f8
) IK,
A,
~ p ( A M (a(y)\{0})).
6. Selected Chapters of C*-Algebras
274
~) o(y) c o(z) u {o}. If O ~ a ( x ) , then for every x' E E ~ and f " E C",
b)
(f", x ' o #) = ( f " e u , x' o ,~).
If 0 E a(x) and if 50 denotes the Dirac measure on a(x) at O, then for every x' E E ~ and f" E C" with (f", 5o) = O,
c)
(f" x' o It) -- (f"eu, x'o ~) Supp jc,(x' o #) C U for every x' E E ~ .
d)
e) If x is selfadjoint and f : a(x)
~ ]K,
a,
} sup{a,O},
g : a(x)
} IK,
a,
} sup{-a,O},
h : o(x)
>~,
c,,
r lc, I
then y is selfadjoint and y+ = ((jcf)ee)(X), lyl =
y- = ((jcg)eu)(X)
((jch)eu)(x).
a) By Theorem 6.3.3.4 g),
a(x) = (jcj)(S(x)), a(y) = ((jcj)ev)(S(x)). Hence
o(y) c o(~)u {o}. b & c. Define Co := {f E C IO E a(x) ~
f(O) = O}
and for f E Co, define
i f a E a(x)
N
f " o'(y)
>IK,
Oil
~ { f (a)O i f a ~ a(x)
6.3 Von Neumann Algebras
275
N
Step 1
f 9 Co, p e M b ( a ( y ) ) =:> f e C(a(y)) , ( f , p) -- (f, p~
The assertion follows from a). Step 2
f 9 Co ==>f o ( ( j c j ) e v ) = ( j c f ) e v
The maps C (:
;(:", )C",
f' f,
>jcf, >f o ( j c j )
are involutive algebra homomorphisms (6.3.2.1 b)). Since they take the same value at j 9 12, they coincide. The assertion follows. Step 3
f 9 Co =~ ( f , x ' o #) = ( ( j c f ) e v , x' o A)
By the first two steps and Theorem 6.3.3.1 f),i),
(f, x' o"~) = (f, z' o p) = (f-(y), x') = (f(((jcj)ev)(x)),x')
= (f o ((jcj)ev)(z),x')
=
= (((jcf)eu)(X),X') -- ( ( j c f ) e v , x' o A).
Step 4
b &c
Let ~" be the set of f" C C" such that ( f " , x ' o #} = ( f " e ~ , z ' o A).
Then 9v is an order faithful vector subspace of C". By Step 1, it contains jc(Co). If 0 ~ a(z) then C = C0, jv contains jc(C), and so ~ - = C". Assume
that 0 C a(z). By Example 6.3.2.6 e), { f " C C " [ ( f " , 50)--O} C 9~.
d) follows from b) and c). e) By Theorem 6.3.2.1 b), (jcj)eu E Re(:", ( j c f ) e u , (jcg)eu 9 C~_, j c f - jcg = j c ( f - g) = j c j , ( j c f ) ( j c g ) = j c ( f g) = O.
276
6. Selected Chapters of C*-Algebras
Hence y = ((jcj)ev)(X) E Re E ,
( ( j c f ) e v ) ( X ) , ((jcg)ev)(X) E E+,
( ( j c f ) e v ) ( X ) - ((jcg)ev)(X) = ((jcj)ev)(X) - y ,
(((jcf)ev)(X))(((jcg)ev)(X)) = 0
(Theorem 6.3.3.1 e)). Thus ( ( j c f ) e v ) ( X ) = y+,
((jcg)ev)(X) = y - ,
((jch)ev)(x) = ( ( j c f ) e v + (jcg)(ev))(x) = y+ + y - - l Y l . Remark.
m
If x is not selfadjoint then the relation
lyl
=
((jch)ev)(x)
in e) may fail (see Remark of Corollary 6.3.3.7). T h e o r e m 6.3.3.6
( 0 ) Take x E Sn E . Let G be the unital W*-subalgebra
of E generated by x . Put C := C(a(x)) and let (Mt)tE~ be a finite family of pairwise orthogonal bands of C' (identified with A/Ib(a(x))) such that C' is the vector subspace of C ~ generated by U .A/tt. Let ~
tel
denote the a-algebra of
Borel sets of a(x) and for every y E G, let py be its spectral measure. Define ~ty " ~
a)
) Pr E,
A,
) #y(A (-l (a(y)\{O})) .
There is a unique family (xt)t~, in G such that
X---EXt~ tel t,A E I , t # A ~
t E I , x' E E € ~
xtx~ = O,
x~o ttx, E M t .
xt is called the c o m p o n e n t o.f x on .hdt for every t E I .
6.3 Von Neumann Algebras
277
b) ~(x) c U~(x~) c ~(~) u {0}, ~EI
Card I r 1 ~
U a(x~) =
a(x) U {0}.
tEI
c) For every ~ E I , let p~ be the projection of C' onto All, and define s
C'
; IK,
j " a(x)
f' , > IK,
> {e,,(z),p~f'} , oz.
>o~.
Then f[' E C", f['(x) E Pr E , and x~ = ((jcj)f[~)(x) .for every ~ E I .
d)
We identify C" with C(a(C")) using the Gelfand transform. Take ~ E I and let U~ be the clopen set of a(C") such that
Ad, = {f' E C' I Supp jc, f ' C U,} (Example 6.3.2.8). Then
f: 1 __
eu~ .
d) follows from Example 6.3.2.8. Given ~ E I , put x~ := ( ( j c j ) f [ ' ) ( x ) . Step 1
(~ E I =~ f['(x) E Pr E ) and E f['(x) - 1 ~EI
The assertion follows from d) and Theorem 6.3.3.1 a). { ~,,~EI,~r
x~x~--O
Step 2 E X LEI
t ~X
By d) and Theorem 6.3.3.1 a),d), x~ - ((jcj)eu~)(x) E G ,
6. Selected Chapters of C*-Algebras
278
x,x)~ - -
0,
and
X'--~Xe. tel Step 3 b) By Theorem 6.3.3.4 g),
a(x) = (j c j )(S(x )), a(x,) = ((jcj)ev,)(S(x)). It follows that
o(~) c Uo(~,) c o(~)u (o} eEI
and
U o(~)= o(x) u {o}, whenever Card I r 1. Step4
~ E I , x~ E E " : = v x ' o p x ~ E M ~
By d) and Proposition 6.3.3.5 d),
Suppjc,(x'o Ft.,) C U~, so that
Step 5
Uniqueness
Let (Y,)~eI be a family in G satisfying the properties described in a) for (x,),el. By Theorem 6.3.3.1 d), for every L E I there is a g~' E C" with y~ = g : ' ( ~ ) .
We have
6.3 Von Neumann Algebras
279
and II
II \
g~ g~)(x) -- y~y~ -- 0 for all distinct ~, A E I . Hence II
fl
ff
g~ g~ = 0
g~ = j c j , tEI
on S ( x ) for all dinstict c,A E I (Theorem 6.3.3.4 e)). Therefore there is a partition (V~)~E, of S ( x ) into clopen sets such t h a t g:'= (jcj)ey~ on S ( x ) for every c E I . It follows t h a t
~(~) c U~(x~) c ~(x)u {0} tEI
by Proposition 6.3.3.4 g). If 0 E or(x), then write 50 for the Dirac measure on or(x) at 0. Take x' E E ~ . By Proposition 6.3.3.5 b),c),
(h",x'o ~ )
~I h " ~
(h", x' o ~ , )
-
l~h l l e
~ XI
~
~ Xt
o ,~), o
~)
for every ~ E I and h" E C" satisfying the condition
(h", ~0) = 0 if 0 E a ( x ) . It follows t h a t
(h,,,
o
~EI
tEI
for every h" E C" satisfying the condition (h", ~o) = 0 whenever 0 E a ( x ) . Since fi~ ( { o } ) = ~ ( ( o } )
= o,
6. Selected Chapters of C*-Algebras
280
for every ~ E I whenever 0 E a ( x ) , it follows t h a t
Ex'o , eel
eel
From z' o ~ . , , z' o ~y, e M , for every ~ E I , we see t h a t
~' o~, =~' o~, for every ~ E I . Since x r is arbitrary, we deduce t h a t #x, = #u, for every ~ E I . Take ~ E I . If A is a Borel set of a ( x e ) \ { 0 } (of a(ye)\{0} ), then
#~, (A) = ~ , (A) = ~u, (A) , (#u,(A) = ~y,(A) = ~ x , ( A ) ) . It follows t h a t
o(~)\{0} = o(y~)\(0} and #x,
-- #y,.
Hence xe = Y e . Corollary a)
6.3.3.7
( 0 )
1
Take x E S n E .
There are uniquely determined y, z E S n E
such that y is atomic, z has
at most one atom at O, x=y+z, and
~o(~) =~o(y), for every a E IK\{O}. y (resp. z ) is called the atomic (resp. atomless) part o f x .
6.3 Von Neumann Algebras
b)
281
y and z belong to the unital W*-subalgebra of E generated by x .
c) yz = zy = O. d)
a(x) C a(y) U a(z) = a(x) U {0}. Given p 9 A/lb(cr(x)), we write Pa and Pb for the atomic and atomless part of p, respectively. Define: C := C ( a ( x ) ) ,
f " : C' ---+ I[4,
P'
> Pa(O(X)),
g": c'
p.
> p~(~(.)).
> IK.
j : o(x)
> IK,
~,
>~.
Then f " , g" C C" , f " ( x ) , g"(x) 9 P r E , f " ( x ) -+- g"(x) = 1,
f)
y -- ( ( j c j ) f " ) ( x ) ,
z -- ( ( j c j ) g " ) ( x ) .
If x is selfadjoint then [Yl (resp. y+, resp. y - ) is the atomic part of Ixl (resp. x +, resp. x - ) a n d Izl (~e~p. z+, reap. z - ) i~ the atomless part of Ixl (resp. x + resp. x - ) .
Let A/J1, J~42 be the bands of C of atomic and atomless measures on a ( x ) , respectively, and let y and z be the components of x on the bands A/t1 and M2, respectively. By Theorem 6.3.3.6 (and Corollary 4.4.4.3 c r d), y and z have the properties described in a) ,b) ,c) ,d) , and e). Uniqueness follows from the fact that y is completely determined by the properties in a). f) follows from a),e), and Proposition 6.3.3.5 e). I Remark.
Take E := C(T)", j:T
>~,
as
>r
x := jc(r)j. Then I x ] - 1, so that the atomic part of Ix I is 1, while the atomic part of x is 0. C o r o l l a r y 6.3.3.8
( 0 ) Assume that H ~ {0} and let u be a self-normal
operator on H . Then there are uniquely determined self-normal operators v, w on H such that:
282
a)
6. Selected Chapters of C*-Algebras
u--v+w.
b) ~ . ( ~ ) c o . ( ~ ) c ~ . ( ~ ) u {0},
o.(~)
c
{o}.
c) If a e ap(u)\{0}, then the eigenspaces of u and v corresponding to the eigenvalue a coincide.
d)
There is an orthonormal basis A of H and an f E g ~ ( A ) such that
v~ = f(~)~ for every ~ E A . In addition: e)
v is atomic.
f) v and w belong to the W*-subalgebra of E ( H ) generated by u (Corollary 6.1.7.1.4 b)).
g) vw = wv - O. h) a(u)
C
a(v)
U
a(w)
=
a(u) U {0}.
i) v (rasp. w ) is the atomic (rasp. atomless) part of u . By Corollary 6.3.3.7, there exist uniquely v, w C L:(H) such that e ~<\{0} ~
~o(~) - ~o(v),
~ e ~<\{0} ~
~o(~) = o
and a),e),f),g),h), and i) are fulfilled, b) and c) now follow from Theorem 6.1.7.15 j). d) follows from Corollary 6.1.7.18. I Corollary 6.3.3.9
( 0 ) Let x E ReE and G the unital W*-subalgebra
of E generated by x . Let ~
be the a-algebra of Borel sets of a ( x ) , A the
restriction of the Lebesgue measure to ~ ,
and for every y C G , let #y be its
spectral measure and define
fly'~ a)
>PrE,
A,
> #~(A N (a(x)\{O})) .
There are uniquely determined x l , x 2 , x 3 C RaG such that X =
X 1 Jr- X 2 -~ X 3 ,
i , j C IN3, i # j ~
xixj = O,
and for every x t E E ~ , x' 0 x l is atomic, x' o x2 is absolutely continuous with respect to A, and x' o x3 is atomless and singular with respect to A. 3
b) a(x) C U a(xi) = a(x)U {0}. i--1
I
6.3 Von Neumann Algebras
283
6.3.4 Von Neumann-Algebras D e f i n i t i o n 6.3.4.1 ( 0 ) The (unital) W*-subalgebras of s (unital) yon N e u m a n n algebras on H (Corollary 6.1.7.14 b)).
are called
Every von Neumann algebra on H has a unit, which is an orthogonal projection in H (Corollary 4.4.4.9). Proposition
6.3.4.2
( 0 ) Let E be a C*-subalgebra of F_,(H). Then the
following are equivalent:
a)
E is a yon N e u m a n n algebra on H .
b)
E is closed in /:(H)t:I(H).
c)
E is closed i n ~ . ( H ) s
d)
E is closed in L ( H ) H .
) .
If these conditions are fulfilled then: If p c P r E ,
e)
~(H),,
f)
then p C P r E ,
where E
denotes the closure of E
in
.
A linear form u' on E belongs to the predual of E (Corollary 4.4.4.9 a b), iff there is a u C s
g)
such that u ' = T I E .
If 1H, V C E , K "- I m v , L ' -
(Kerv) • then 7rK and 7~L belong to E
and are the left and the right carrier of v in E , respectively. In the complex case e ~
a r
b r
cr
d.
d follows from Corollary 6.3.1.12.
d :=re is trivial. e :=v d. Take u c R e E and c > 0. By Theorem 4.4.1.8 f) (and Corollary 6.3.1.12), there is a finite family ((c~,p~))~c, in e(u) x P r E such that
u - 2_., a~p~ LCI
By e),
E a,~p~ C E . t~CI
6. Selected Chapters of C*-Algebras
284
Since c is arbitrary, u E E . Hence E is closed in E ( H ) H . c :=> f follows from Corollary 6.1.6.11. a =r g. By Proposition 5.3.2.4, 7rk and of v in s
71"n
are the left and the right carrier
By Proposition 4.4.4.11 a), 7rg and
71"L
belong to E and are
the left and the right carrier of v in E . Remark.
If IK -
I
IR and H is infinite-dimensional then the implication e
=> d does hot hold. Let us consider an orthonormal basis of H consisting of two families (e,),~1 and (f,),ei and denote by /g,V, and W the set of u,v,w E s
repsectively, for which there is an a E g o , a b E Co, and a
c C go such that ue~ = a~e~ ,
ve~ = b~f~ ,
u fi = a,f~ ,
weL = c~ft
vie, = - b , e , ,
w f~ = - c , f ~
for every c E I . P u t
E : = {u+v I(u,~) e U x V}. Then E is a C*-subalgebra of s
and
E = {~ + w I(u, w) e u x w } # E , so t h a t d) is not fulfilled. On the other side PrE-
{u E b / [ u 2 = u} C E ,
i.e. e) is satisfied. Proposition
6.3.4.3
(
0
)
(G.K. P e d e r s e n ) L e t
E
be a unital C * -
~ubalge~a of C(H) ~uch that th~ ~ p ~ m ~ m in C(H)#+ oS a~y ~omm~tative increasing sequence in E#+ belongs to E
(Corollary 6.1.7.1~, b), Theorem
~.~.1.8 b)) and let E be the closure of E in f-.(H)H. Put Po :-- 1, u0 :-- 0. Then, for every p E P r E
(Corollary 6.3.1.12), any x c H , and any E > O,
there is a sequence (Un)nON in E#+ and a sequence (P,~),~ON in Pr E such that the following hold for every n E IN "
1) I ( u n - p)p.pxl 2 < ~ - ,
1
1
Ilunz- Pzll < ~, I l u , p z - pxll < -~.
2) pn(Un - un-1)2Pn <_ 2.1-~-~1.
6.3 Von N e u m a n n Algebras
285
3) p~ ~ p~_~, II(p~-x- p~)pxll ~ < ~s Let F be a maximal Gelfand C*-subalgebra of E , sequence in F+# , and v its supremum in s
(vn)ne~ an increasing
By hypothesis, v 9 E . By
Corollary 4.2.2.20 a), v 9 F . In particular, F is order o.-complete. By Corollary 4.3.2.5, any u 9 Sn E admits a functional calculus with Borel functions. We construct the sequences (un)~e~ and (p~)~e~ inductively. P u t Pl "- 1. There is a Ul 9 E ~ , such that 1) is fulfilled (Theorem 6.3.1.10). 2) is then automactically fulfilled. Take n 9 IN, and assume that the sequences have been constructed up to n. We have 0 <_ p . ( u n - p)2pn <_ 1
(Theorem 4.2.2.1 b :=> a, Corollary 4.2.2.3, Corollary 4.2.1.16) and the map C(o-(pn(un - p)2p~))
) IK,
f,
> (f(p.(u.
- p)2p.)pxlpx )
is a positive linear form (Corollary 4.2.1.12, Corollary 5.3.3.7). Hence, there is an f 9 C([0, 1])+ and an open nonempty interval I C [2-~r+~,2Jzl, such that f-
1 on I and ( f ( p n ( U n -- p)2pn)pX p x ) <
C
4n+2 "
Take c~0 E I , put g := e]_~,~o[ , and let h be a real function on IR, such t h a t S u p p h C I , 0 < h < 1, and k " - g + h is continuous. Define Vu :-- Pn (U -- U n ) 2 p n ,
Wu :-- Pn (u -- p)2pn ,
for every u E E+# . Then 0
0<w~
(Theorem 4.2.2.1 b ==> a, Corollary 4.2.2.3, Corollary 4.2.1.16). Hence (using the notation of Corollary 4.3.2.5 b)), g(vu)wug(v~) - (k(v~) - h ( v ~ ) ) w ~ ( k ( v ~ ) - h(vu)) -
- k(v~)w~k(vu)+
h(v~)wuh(v~) - k(v~)w~h(v~)-
h(v~)w~k(v~) <
<_ 2 k ( v u ) w u k ( v u ) + 2 h ( v u ) w u h ( v u ) < 2 k ( v ~ ) w ~ ( v ~ ) + 2h(v~) e <
6. Selected Chapters of C*-Algebras
286
< 2k(vu)w~k(vu)+ 2f(Vu) for every u C E+# (Corollary 4.2.2.4, Corollary 4.2.2.3, Corollary 4.3.2.5 a)). Let ~: be the trace on E+# of the n e i g h b o u r h o o d filter of p in ~(H)H (Theorem 6.3.1.10). We have lim k(v,,)w,,k(vu) = 0
lim w~ = 0 u,~
'
u,i~
lim f(vu) = f(Pn(P - Ztn)2Pn) u, qd
in s
(Proposition 6.3.1.3 a), Proposition 6.3.1.6). By 1), lim(pr,(Un - u)2pnpxlpx) = lim((un - u)p.pxl(Un -- u)pnpx) = u,~ u,~
=
lim u,~
II(un
-
lim
U)PnpXll 2 = II(un - p ) p , ~ p x l l 2 <
.4 n
sup(g(Vu)Wug(Vu)pxlpx ) <_ u,~
< 2 lim sup(k(vu)w~,k(vu)px]px) + 2 lim sup(f(vu)pxlpx ) = u,i~
u,i~
= 2 ( f ( p n ( p - Un)2pn)px[px) < 2 9 4 n+l (Corollary 5.3.3.7). Hence there is a
l Un+lX - pxll <
1 n+l'
Un+ 1 E
E#+ such t h a t
IlUn+lpx - Pxll <
(pn(Un - Un+I)2Pnpx[px) <
1 n+l'
.4 n ,
2 . 4 n+l " Since
cr(g(v,n+l) ) C g(a(vun+l)) C {0, 1} (Corollary 4.3.2.5 c)), g(vun+l) e Pr E (Proposition 4.1.2.21 c =~ a). Put
Pn+l :--g(vun+x)Pn"
6.3 V o n N e u m a n n
Algebras
287
Since Pn E {Vu~+l}c C { g ( l ) u ~ + l ) } c
( C o r o l l a r y 4.3.2.5 c)), we have 2
Pn+l -- Pn+l ,
i.e., P n + l C P r E . Since Pn+lPn = P n + l ,
we deduce t h a t P~+I __~ Pn (Corollary 4.2.7.6 c ==~ a ) . We have II(Un+~ -- p ) p n + l p X l 2 -- ((Un+~ -- p ) p n + l p X l ( U n + l -- P ) P n + l P X ) --
= (Pn+l (Un+l -- p ) 2 p n + l p x l p x )
-
z ( g ( V u n + l ) p n ( I t n + 1 -- p)2png(Vu,~+l)PX p x ) --
--(g(Vun+~)Wun+lg(Vu,~+l)PXlpX) <
C 2 9 4 n+l "
Since 1 g(~)c~ < ~T,
1 -- g(o/) ~ 2n-FIoL
for every a C [0, 1], we deduce successively t h a t g(Vttn+l)Vttn+l
<
1
-~1,
1 g(vu,~+~)~ -
-
2 n-F1 VUn+I
(Corollary 4.3.2.5 a)),
Iip~+i(uo§ uo)2pn.ll -IIg(vUn+l)PO(~o+l-u~)2pog(vu~+i)lJ1
(Corollary 4.2.1.16 b =~ a ) ,
288
6. Selected Chapters of C*-Algebras
1
Pn+l(Un+l - Un)2pn+l < -~1 (Corollary 4.2.1.16 a ==> b),
Pn
II(pn
-
Pn+l
--
Pn( 1
__
ff(Vun+l))~__ 2n+l
V ttn_t.
1
p n + l ) p x l l 2 - ( ( p . - p.+l)px (Pr, -- p.+I)px) =
= ( ( P n - P.+I)pxlpx) < 2"+t(vu.+~PxlPx)_.._ 2 n + l
C (pn(U n -- u.+l)2pnpxlpx) < 2---~
(Corollary 5.3.3.7).
I
P r o p o s i t i o n 6.3.4.4 ( 0 ) Let E be a unital C*-subalgebra of ~,(H) such that the supremum in E,(H)#+ of any commutative well-ordered set of E#+ belongs to E (Corollary 6.1.7.1~ b), Theorem ~.~.1.8 b)). Further take x c H , and put A := {p C P r E i p x = 0}. Then A possesses a greatest element.
Take p,q C A and let K be the closed vector subspace of H generated by I m p U I m q . Then 7r~r is the supremum of {p,q} in PrZ:(H) (Corollary 5.3.3.8). Since xEK
• ,
7rKX - 0 .
By Corollary 4.2.7.15 (and Proposition 4.2.7.14), 7ru C E , so that 7rK C A, and A is upward directed. By Proposition 4.2.7.17, the supremum p of A in Z:(H) belongs to E . Since 7r{x}• is an upper bound of A in s successively that
we have
p _< 7r'{~}_t_, px=0, pCA,
p is the greatest element of A.
I
6.3 Von Neumann Algebras
289
( 0 ) Let E be a unital C*-subalgebra of r.(H) such
P r o p o s i t i o n 6.3.4.5
that the supremum in s
of any commutative well-ordered set of E#+ be-
longs to E (Corollary 6.1.7.14 b), Theorem 4.4.1.8 b)) and let E be the closure of E in s Then, for every p C P r E (Corollary 6.3.1.12) and every x E H , there is a q 6 P r E such that px-
qx-
qpx.
Take c > 0. Put P0 := 1, u0 "= 0. By Proposition 6.3.4.3, there is a sequence (U,~)nC~ in E+# and a sequence ( P n ) n ~ following hold for every n 6 IN" 1)
I1("ttn -- P ) P n P x 12 <
2-~4n ,
I1~
- p~ I < 1n
~
in P r E ,
such that the
I I ~ p ~ - pxl[ < & n "
2) ; . ( ~ . - ~._1)~;. _ 2%_11. 3) Pn <_ Pn-1, II(pn-1- pn)pxl] 2
<
2n-E-1 9
By 3) and Corollary 4.2.7.6 a =~ b, ( P n ) n ~ is a commutative decreasing sequence in E + . Put
P~'-
A Pn 6 Pr E n6IN
(Corollary 4.2.7.10). (Pn),.~ converges to p ~ in s
(Proposition 6.3.1.4
a => c), and so by 3), 0(3
(p~px px) - l i m (pnpxlpx) - ~ ( ( p n -
p~-l)pxlpxl + (pxlpx) -
n-I oo
= I p~l = - ~-~<(p~_~ - pn)pXl(p~-x
- pn)pX}
n=l
o()
IIp~ ~ - Y2~
C 2n-1
Ilpxll ~ - 2c,
=
n=l
We have (Un+ 1 --
'l.tn)Pcx:~('l.tn+ 1 -- Un) ~ ( U n + l -
Un)Pn+l('l.tn+ 1 -- Un)
(Corollary 4.2.2.3), so by 2) and Corollary 4.2.1.18, I(Un+l-
'Ltn)poo[ 2 - - I I ( ? . t n + l -
L/,n)poo(Un+
1 - - ?_tn)Jl
290
6. Selected Chapters of C*-Algebras
II(u~+~ - ~n)PnWl(Un+l
--
II(u.§
Un)ll--"
- un)p~+ll] 2 = 1
= I l P . + l ( u ~ + l - u.)2pn+lll < 2~ for every n E IN. Hence, (unP~)nEIN converges, and so (unp~u,~)nE~ also converges. Put u := lim u n p ~ u n E E # + . n----~(:x:)
By 1), 1
Ilu.px - px [ < -, n
2 I l u ~ ( x - px)ll ~ I I ~ -
pxll + I l p x -
u.pxll < n
for every n E IN. Thus lim Un ( x
- px)
-
0,
n---+ (X)
lim UnpX
-
px ,
n---+ (:x:)
u ( z - p z ) - 0,
(upxlpx) = lim ( u n p ~ u n p x l p x ) -- lim ( p ~ u n p x l u n p x ) -n ---.~~
n ---+o o
= ( p ~ p x l p x ) > Ilpxll 2 - 2c.
By Proposition 5.3.3.10 a), uo(z - px) - o,
and by Proposition 6.3.1.5 a), Corollary 5.3.3.7, u ~ >_ u ,
(u~px]px) >_ (upx px) >_ Ilpx ]2 _ 2e
for every a E ]0, 1[. Put v'-
Vu~EPrE a>0
(Proposition 6.3.1.5 b)). By Proposition 6.3.1.5 b). v ( x - px) = lim u ~ ( x - px) - O. (~ - + 0
6.3 Von Neumann Algebras
291
< v p x l p x ) - lim > IIp~ll = - 2~. o~---+0
Define A "- {r 9 P r E I r ( x -
px)-
0}.
By Proposition 6.3.4.4, A has a greatest element q. We have qx-
qpx.
Since v c A, q~_v,
(qpxlpx) >_ (vpx px) >_ Ilpxll 2 - 2c
(Corollary 5.3.3.7). Since c is arbitrary, it follows that ]lpx[[ 2 <_ (qpx[px) < Ipx]l 2,
qpx -- p x
(Proposition 5.2.3.1). T h e o r e m 6.3.4.6
m ( 0 ) Let IK = ~
and let E be a C*-subalgebra of s
such that the supremum in I:(H)#+ of any commutative well-ordered set of E#+ belongs to E . Then E is a yon N e u m a n n algebra on H .
Let E be the closure of E in L ( H ) H
and take p C P r E
(Corollary
6.3.1.12). For x C H , take Px E Pr E such that px = p~x -- p~px
(Proposition 6.3.4.5), and let qx be the infimum in Pr s {
y c
of
}
By Proposition 4.2.7.17 (and Proposition 4.2.7.14), qx E Pr E . Take x C I m p . Then pyX -- pypx ~ py -- x -1
-1
for every y C P ( x ) . Take y , z C P ( x ) and let pyApz be the infimum of py and Pz in Pr E . By Corollary 4.2.7.13,
292
6. S e l e c t e d C h a p t e r s o f C * - A l g e b r a s
(p~ A Pz)X = 0. By Proposition 6.3.1.4, qxx :
x.
Take z E Ker p and put y:=x+z. -1
Then y E P (x) and pyz - p~y - p~x - py - x -
O.
We deduce that
IIq~zll <~ IIp~zll = 0 (Proposition 5.2.3.13 a =~ d). Since z is arbitrary, it follows t h a t Ker p C Ker q , , and so Imq~ = (Kerq~) • C (Kerp) • = I m p ,
qx<_p
(Corollary 5.3.3.8). By the above considerations, p is the supremum of
{qxix
E H} in Pr s
By Proposition 4.2.7.17, p E E . By Corollary 6.3.4.2 e =~ a, E is a W * subalgebra of s Remark.
1
The example from the remark of Proposition 6.3.4.2 shows that the
above theorem does not hold for IK - JR.
6.3 Von Neumann Algebras
293
6.3.5 T h e C o m m u t a n t s
Proposition 6.3.5.1 that
( 0 ) Let F be an involutive subalgebra of s
such
for every x E H . Then F is dense in F~~. Take u c F cc . Step 1
For every x C H ,
there is a sequence lira u~x = u x .
(un)~e~ in F with
Define K := {wlv 9 F } K is a closed vector subspace of H containing x. K is invariant for every v C F . Hence, by Proposition 5.3.2.9,
7r g
~K u =
C
F c . Thus
~K-
We get that ~t X =
?s 7r K X =
7r K ?s X E t ( .
Hence, there is a sequence (un)ne~ in F such t h a t lim
?s x
--- ~ t X .
n---+oG
cc
Step 2
v E F iEINk
for every k C IN
i
Take
vEF}
Identifying s
H) with the involutive unital algebra of k • k-matrices over iEINk
s
c
(Theorem 5.6.6.1 f)), we obtain WijV
--- V W i j
6. Selected Chapters of C*-Algebras
294
for all i , j E ]Nk and v E F . Hence, wij E F r and WijU
--- ~ W i j
for all i , j E INk. Thus
w u):
u)
Since w is arbitrary, it follows that co
vEF iEINk
Step 3
.
i
u belongs to the closure of F in F~c
Take x l , x 2 , . . . , x k
E H . Define x := (xi)ie~k E
(I) H . iEINk
By Step 2,
(DuE iEIN k
{2~
v
vEF
i
/~
By Step 1, there is a sequence (un)ne~ in F such that
n-~c~
i
i
Thus lim
U n X i --~ U X i
n--~ oo
for every i E INk. Since x l , x 2 , . . . ,xk are arbitrary, u belongs to the closure of F in F~ ~ .
I
Corollary 6.3.5.2 ( 0 ) Ls(H)
~ ~-~
We have 1
]lxll2 (.Ix)x E s
in CIH)..
6.3 Von Neumann Algebras
295
=
for every x C H \ { 0 } . By Corollary 5.5.1.11 and Proposition 5.3.2.13 b),e), s is an involutive subalgebra of /:(H) and by Corollary 5.3.2.14, s
= s is dense in s
By Proposition 6.3.5.1, s
m
Corollary 6.3.5.3 ( 0 ) If H is separable then s f_,(H)#H is metrizable.
is separable and
By Proposition 5.5.2.12 c), K:(H) is separable. Hence IC(H)H is separable. By Corollary 6.3.5.2, /C(H) is dense in s ,so that s is separable. Let A be a countable dense set of H . Since s # is equicontinuous, s Hence, s
= s
#.
is metrizable.
P r o p o s i t i o n 6.3.5.4
m
( 0 ) If .T is a subset of s
such that
9v C ~'* tO Sn s
then yc is a unital von Neumann algebra on H . The assertion follows immediately from Corollary 4.4.4.12 e).
m
Corollary 6.3.5.5 ( 0 ) (von Neumann, 1929)A unital C*-subalgebra E of s is a yon Neumann algebra on H iff E
=
E CC "
E
:
E CC '
then, by Proposition 6.3.5.4, E is a von Neumann algebra on H . Now suppose that E is avon Neumann algebra on H . By Corollary 6.3.4.2 a =~ d, E is closed in ~.(H)H. By Proposition 6.3.5.1, E = E~ .
1
296
6. Selected Chapters of C*-Algebras
C o r o l l a r y 6.3.5.6
( 0 ) Let jc be a subset of s
such that
9~ C f * U Sn s Then ~
is the unital von N e u m a n n algebra on H generated by ~ .
By Proposition 6.3.5.4, ~ c
is a unital von Neumann algebra on H . Let G
be a unital von Neumann algebra on H containing .T'. Then
by Corollary 6.3.5.5. Hence ~c~ is the unital von Neumann algebra on H generated by ~'. C o r o l l a r y 6.3.5.7 and x E s
I I f ]K = (~, E
is a unital von N e u m a n n algebra on H ,
then the following are equivalent:
a) x c E . b)
y*xy = x f o r every unitary element y of E ~.
c) px = xp f o r every p E P r E a=vb&c.
c.
Since x E E c~, xy = yx,
px = xp,
y*xy - y*yx - x .
b ~ a. By Corollary 4.1.3.7 (and Proposition 6.3.5.4), every element of E c is a linear combination of four unitary elements of E c . Hence x E E ~ follows from b). By Corollary 6.3.5.5, x E E . c =~ a. By Proposition 6.3.5.4, E ~ is a unital von Neumann algebra on H . Hence, by Theorem 4.4.1.8 g), the vector subspace of E ~ generated by Pr E c is dense in E ~ . Thus x E E ~ follows from c), so that x E E . (Corollary 6.3.5.5). I P r o p o s i t i o n 6.3.5.8
( 0 ) Let E be a C'-subalgebra of s
the following assertions:
a)
E acts irreducibly on H .
b)
Re (E c) : IR1.
and consider
6.3 Von Neumann Algebras
c)
297
E is dense in s
Then c ~ I f IK = C ,
a r
a c==> b.
then all these assertions are equivalent.
b. An orthogonal projection p in H belongs to E c iff I m p is E -
invariant (Corollary 5.3.2.9). If a) holds, then Pr (E ~) C {0, 1}. Since E c is a yon N e u m a n n algebra on H (Proposition 6.3.5.4), we get Re (E ~) = IR1 by Theorem 4.4.1.8 g). Now assume that b) holds and let K be a closed E - i n v a r i a n t vector subspace of H . Then ~'K C Re (E ~) so that K C {{0}, H } . c ::v b. By Corollary 5.3.2.14, 1R1 c Re (E c) = R e s
~ C Re {<.Iz>zlz C H} ~ : ]R1,
so that
ae (E ~) = m . a & b =a c. Take z C H \ { 0 } . By a) and Proposition 5.3.2.20 b), z is cyclic for F so t h a t x E { u x l u E F } • 1 7 7 {uzlu C F }
(Corollary 5.2.3.9). By b), E c =el so that
Ec~: L(H). By Proposition 6.3.5.1, E is dense in s Remark.
The implication b ::v c does not hold for IK = IR. Take H :=IR 2 ,
I
298
6. Selected Chapters of C*-Algebras
a,Z c ~}. Then Ec=E, but E is not dense in s
Re(E c ) = R e E = ] R 1 ,
6.3 Von Neumann Algebras
299
6.3.6 I r r e d u c i b l e R e p r e s e n t a t i o n s T h e o r e m 6.3.6.1
( 0 )
(Kadison, 1 9 5 7 ) A s s u m e IK = (~. Let E be a be a linearly independent finite family in H .
C*-subalgebra of f~(H) acting irreducibly on H and let ( ~ ) ~ i
a)
For every ~TE H I there is an x E E with (x~)~,
b)
= ~.
{0} and H are the only E-invariant vector subspaces of H .
a)
Step 1
3y E s
( y ~ ) ~ , = r/.
Let K be the (finite-dimensional) vector subspace of H generated by (~)~EI-There is a z E f__,(K,H) with (z~)~e1 = ~ .
If we set y: H
then y E s
>H ,
~,
> ZTTK~
and (Y~L)~EI = T].
Step 2
3 x E E , (x~)~el = r ]
Put I4~ := H I
and define u :s
>K ,
Y'
~ (Y~)~eI.
By Step 1, u is surjective. By the Open Mapping Principle, the null point of K is an interior point of u(E,(H)#). By Proposition 6.3.5.8 a :=> c, E is dense in •(H)H so that by Kaplanski's Density Theorem (Theorem 6.3.1.10), E # is dense in /:(H)#H . Hence the null point of K is an interior point of u ( E # ) . By Proposition 1.4.2.2, the null point of K is an interior point of u ( E # ) . Hence ~(E) = K
and there is an x E E with
b) follows from a).
I
6. Selected Chapters of C*-Algebras
300
Theorem 6.3.6.2
( 0 ) (Segal, 1947) The following are equivalent for
x' ~ ~ ( E ) ~)
x ' e ~0(E).
b)
The representation of E associated to x' is irreducible.
If IK = C , the above conditions are fulfilled, and (H, 99) denotes the representation of E associated to x', then E/Nx, = H .
We use the notation from Theorem 5.4.1.2. a ~ b. Take v ~ ~(E) ~n L(E/F) # +.
By Proposition 5.4.1.6 a) (and Theorem 4.2.8.2), the map !
is a positive linear form less than x'. By Proposition 2.3.5.4 a =~ b,
y ' = Ily'llx'. Thus, given x, y C E ,
= @(y*~)~( I ~> - y'(y*x) - Ily'l x'(y*~) = = (Lly'll~iw)
(Theorem 5.4.1.2 k)), so that
v-[
y'lll.
Hence 99(E)C A / : ( E ~ ) + ~
C
IK1.
By Corollary 4.1.4.2 b), Re (99(E) c) c IK1. By Proposition 6.3.5.8 b =~ a, 99(E) acts irreducibly on E / F , i.e. the representation of E associated to x' is irreducible.
6.3 Von Neumann Algebras
301
b =v a. Take y' c E~_ with y' < x ' . By Proposition 5.4.1.6 b), there is a v Cs
A ~ ( E ) c , such t h a t y'(x) = ( ( ~ x ) v ~ i ~ )
for every x E E . By b) and Proposition 6.3.5.8 a =v b, Re ~ ( E ) ~ = 1R1. Hence there is an a C IR+ such that v-
al.
We get y'(x)
--
~((~x)~i~)
-- ~ ( ~
i~) = .~'(x)
for every x C E (Theorem 5.4.1.2 k)), so t h a t !
y = ax'. By Proposition 2.3.5.4 b =v a, x' E 70(E). Assume now that the above conditions are fulfilled and IK = ([;. Since E/Nx, is a ~ ( E ) - i n v a r i a n t vector subspace of H (Theorem 5.4.1.2), we get E / N x , = H by Theorem 6.3.6.1 b).
Corollary 6.3.6.3
I
For every x C E , there is an irreducible representation
(H, ~) such that li~ll
= ii~li.
We may assume that x 7(= 0. By Corollary 4.2.8.5 c), there is an x' C To(E) such that 9 '(x*~) = ii~*zii.
Let (H, p, ~) be the G N S - t r i p l e associated to x'. By Theorem 6.3.6.2 a :=~ b, (H, ~) is irreducible. We have lizil ~ = li~*xil =
x'(x*x) -
(~(z*x)~
i ~) = ( ( ~ ) ~
I (~)~)
-
= il(~x)~ii ~ < f i ~ l i ~ < llxll ~
(Theorem 5.4.1.2 k)), so t h a t Ilxil = ii~xii,
m
6. Selected Chapters of C*-Algebras
302
Corollary
6.3.6.4 ( 0 ) There is a faithful representation (H, ~) of E which is the Hilbert sum of irreducible representations. If E is separable, then
we may take H separable. It is sufficient to take a dense set A of To(E) and to apply Corollary 5.4.2.5 and Theorem 6.3.6.2 a ~ b.
Corollary 6.3.6.5
I
( 0 ) (J.M Wedderburn, 1908)Assume
E is finite-
dimensional. If IK =q~ then there is a finite family (nL)Lel in IN such that E is isomorphic to 11~n~,,~ and LEI
Dim E ~ = Dim Re E ~ = Card I .
If IK = IR then there are finite families (n~)~en, (Pu)ueM, (q~)~eN in IN such that E is isomorphic to
and Dim E c = Card L + 2Card M + Card N ,
Dim Re E c = Card L + Card M + Card N ,
These families are unique up to permutations. In particular, if E is simple and IK = C (IK = IR), then E is isomorphic to Cn,n (to IRn,,~, ~,~,n, or ]Hn, n ) for some n E IN ; moreover, in the complex case, E is the complexification of a real C*-algebra and there are at most two non-isomorphic real C*-algebras, the complexification of which are isomorphic to E . First suppose that IK : C. By Corollary 6.3.6.4, there is a faithful representation (H, p) of E which is the Hilbert sum of a family ((H~,~p~))~E, of irreducible representations. We may assume that for distinct L, ~ E I , the representation (HL, p~) and (H~,p~)
are not equivalent. Since E is finite-
dimensional, it follows that I is finite and H~ is finite-dimensional for every L E I . By Proposition 5.5.6.9,
{~(E) = ~(H~) for every c E I . By Corollary 5.5.6.13,
6.3 Von Neumann Algebras
303
~(E) : 1-I ~(g~). LEI
Hence, putting n~ "= dim H~ for ~ E I , E is isomorphic to l l ~ , n ~
9 Let us identify E and 11 r
this isomorphism and for every ~ E I let 1~ be the unit of q ~ , ~
using
and ~
the
canonical inclusion Cn~,n~ -+ E . Then (p~l~)~ez is a linearly independent family in E ~, so that Dim E c ~> Card I . Take x C Re E ~ . Then for every ~ C I ,
so that x~ C (FI~ (Corollary 5.6.6.9). It follows Dim E r < Card I , Dim Re E r = Dim E ~ - Card I . For IK = JR, the assertion follows from the above result and Corollary 5.6.6.10. The equalities Dim E c = Card L + 2Card M + Card N , Dim Re E c = Card L + Card M + Card N , follow as in the complex case using Corollary 5.6.6.9. By Corollary 5.6.6.9, IR,~,n,r
and IHn,n are simple for every n C IN.
The uniqueness of the families follows from Proposition 4.3.5.5. We prove now the final assertion. By the above results, if E is simple then it is isomorphic to Cn,n (to I ~ , ~ , ~ , n ,
or ]Hn,~ ) for some n C IN. The sup-
plementary assertion in the complex case follows from Corollary 5.5.7.13 a),b).
I
Remark. If E is finite-dimensional then by the above corollary and by Example 4.3.3.23, Un E is the set of extreme points of E # .
6. Selected Chapters of C*-Algebras
3O4
C o r o l l a r y 6.3.6.6 Let E be a finite-dimensional C*-algebra, A' the set of non-degenerate algebra homomorphisms of E into ]K, and 79 := {p 9 E c ;3 Pr EIEp a)
is one-dimensional}.
For every p 9 79 , there is a unique xp'
9 A'
such that
for every x 9 E .
b)
The map 79
~AI,
p~
~ xpI
is bijective.
c)
Every x' 9 A' is positive.
By Corollary 6.3.6.5 (and Corollary 5.6.6.9), E is isomorphic to a C*-direct product of a family of unital simple C*-algebras, so that the assertions follow from Proposition 4.2.8.29. I C o r o l l a r y 6.3.6.7 If E is a Gelfand C*-algebra and (H, ~) is an irreducible representation of E , then H is one-dimensional and
(~x)~ = ((v~)~l~)~ for every x 9 E and ~ 9 H with
Take ~ 9 H with b) ). Define
I1~11:
I1~11=
1.
1. Then ~ is cyclic for (H, ~a) (Proposition 5.3.2.20
x':E
~,
~,
><(:x)~l~).
By Proposition 5.4.1.3 a), x' C T(E) and (H, ~, ~) is equivalent to the G N S triple of E associated to x'. By Theorem 6.3.6.2 b =v a, x' C To(E) and by Corollary 4.2.8.11, x' c a ( E ) . By Theorem 5.4.1.2 j),k), it follows that H is one-dimensional and that (~)~
for every x C E .
= ~'(z)~ = ( ( ~ ) ~ l ~ ) ~
I
6.3 Von Neumann Algebras
C o r o l l a r y 6.3.6.8
305
If F is an involutive commutative complex Banach algebra
and (H, qD) is an irreducible representation of F , then
Dim H < 1. Let E be the C*-hull of F and r the factorization of ~ through E (Proposition 4.1.1.22 f)). Then ( H , r is a representation of E . Since E is commutative, the assertion follows from Corollary 6.3.6.7. II Proposition 6.3.6.9
A s s u m e IK - (~. Given x', y' c ~-o(E), the following
are equivalent:
a)
The representations of E associated to x' and y' are equivalent.
b)
There is an x C E such that x' = x y ' x * .
Let (H, qo, ~) and (K, r r/) be the G N S - t r i p l e s of E associated to x' and y', respectively. a =~ b. Let u" H --+ K be an isometry such that
for every y C E . By Theorem 6.3.6.2 a :=> b and Theorem 6.3.6.1 a), there is an x c E such that (r
- u~.
By Corollary 5.4.1.4 a), X I -- x y l x *
b ==~ a. By Theorem 5.4.1.2 k), if y E E , then ((~y)~ I 0 - (y, x') - (y, xy'~*) = (~*yx, y') = = (r
I ~) - ( ( r 1 6 2 1 6 2
- ((r162
I (r
Hence (r
#
o.
Since (K, r is irreducible (Theorem 6.3.6.2 a ==v b), (r is cyclic for (K, r (Proposition 5.3.2.20 b)). By Corollary 5.4.1.4 b2 :=v bl, (H, ~) and (K, r are equivalent. II
6. Selected Chapters of C*-Algebras
306
P r o p o s i t i o n 6.3.6.10 Let F be a C*-subalgebra of E , (H,p) an (irreducible) representation of F , and ~ a cyclic vector for (H, ~). Then there is an
(irreducible) representation ( K , ~ ) of E , such that: 1) H is a subspace of K . 2) H is r
3) ~ is the compression of e l F to H . 4) ~ is a cyclic vector for ( K , r
If F is hereditary and E unital, then (K, r
with the above properties is
unique. We may assume that I1r = 1.
We define v':F
~,
x,
~ ((~z)~l~).
By Proposition 5.4.1.3 a) (and Theorem 6.3.6.2 b ~ a), y' 9 7 ( F ) , (y' 9 T0(F)) and (H, ~,~) is the GNS-triple of F associated to y'. Bv Corollary 4.2.8.9 (by Corollary 4.2.8.10), there is an x' 9 7(E) (x' 9 T0(E)) such that x'lF = y'. Let (K, r r/) be the GNS-triple of E associated to x'. By Theorem 6.3.6.2 a =, b, (K, r
is irreducible if x' 9 T0(E). Put L := {(~x)~ix 9 F } .
L is r
and so L • is also r
(Proposition 5.3.2.9).
We have 71"LT] C
L,
r/-
TL?] 9
L• ,
and so, given x C F ,
(r (r
7rLr/) = ( r
C L, (r
e L n L• ,
6.3 Von Neumann Algebras
(r
307
= (r
Hence 7rLT/ is a cyclic vector for the representation (L, e l F ) . Moreover,
((r
I ~,)
= ((r
I ,) = x'(=) = y'(x) - ((~x)r I ~)
for every x C F (Theorem 5.4.1.2 k)). By Proposition 5.4.1.3 b), we may assume that L = H,
71Hr] = ~ ,
and 99 is the compression of e l F to H . From II~H'll
= I1r
= 1
=
I1,11
we get that ,=~H~=~. Now suppose that F is hereditary and E unital. Define z':E
>Z,
x,
> <(r
Then z' C T(E) (Proposition 5.4.1.3 a)) and z ' ( x ) = ((r
= y'(~)
for every x C F . By Proposition 4.3.4.12, Z! ~
and this proves the uniqueness of (K, r
X ! ,
(Corollary 5.4.1.3).
m
P r o p o s i t i o n 6.3.6.11 Let F be an involutive Banach algebra and (H, 99) a non-degenerate representation of F such that 99(F) C 1C(H). Then (H, 99) is the Hilbert sum of irreducible representations.
99(F) is a C*-subalgebra of K:(H). Let (H0, r be an irreducible representation of 99(F) (Theorem 6.3.6.2 a ~ b). By Proposition 6.3.6.10, there is an irreducible representation of K:(H) extending (H0, r By Theorem 5.5.1.24, this representation is the identical representation. Hence, there is a
6. Selected Chapters of C*-Algebras
308
~a(F)-invariant closed vector subspace K of H such that the compression of to K is irreducible. Let ~ be a maximal set of ~a(F)-invariant closed vector subspaces K of H such that the compression of ~a to K is irreducible. Assume that
Then, by the above considerations (and Proposition 5.3.2.9), there is a cp(F)invariant closed vector subspace L of
K
such that the compression of
~a to L is irreducible. This contradicts the maximality of ~ . Hence K
-H,
and (H, ~) is the Hilbert sum of irreducible representations. T h e o r e m 6.3.6.12
I
(A. Rosenberg, 1953)
a)
If all irreducible representations of E are faithful (e.g. they are equivalent), then E is simple.
b)
If IK = (~, all irreducible representations of E are equivalent, and E is separable, then there is a separable Hilbcrt space H such that E is isometric to ~ ( H ) .
a) If all irreducible representations are equivalent, then by Corollary 6.3.6.4, they are faithful. Let F be a closed proper ideal of E and let q : E ~ E l F be the quotient map. Then E l F
is a C*-algebra (Theorem 4.2.6.5). Let (H,~)
be an irreducible representation of E l F (Corollary 6.3.6.3). Then (H, ~aoq) is an irreducible representation of E . Since it is faithful, we see that ]]xll = ][~aqxil = 0 for every x E F . Hence F = {0} and E is simple. b) Let (H, ~a) be an irreducible representation of E . Let F be a maximal Gelfand C*-subalgebra of E . Take x ' E a(F) (= 7o(F) Corollary 4.2.8.11) and let (/tx,,~ax,) be the representation of F associated to x'. By Theorem 6.3.6.2 a =~ b, (/tx,,~a~,) is irreducible. Take ~x, E H~, with [l~x'I[ = 1. By Corollary 6.3.6.7, /Ix, is one-dimensional and
6.3 Von Neumann Algebras
for every x E F
309
(Theorem 5.4.1.2 k)). By Proposition 6.3.6.10, there is an
irreducible representation ( K , r
of E such that Hx, is a r
subspace of K and ~x, is the compression of e l F to H~,. By hypothesis, we may assume that (K, r
(H, ~). Thus
(~x)~, = z ' ( x ) ~ , for every x E F . Take x', y' C a ( F ) , x' ~ y'. Then there is an x C F such that 9 '(x) = 0 # u'(z)
(Theorem 2.4.1.13). We get 0-
(x'(~)~, I~,) = ((~x)~,l~,)
= (r162
= (~, I (~x*)~,) =
= y'(x)(r162
G' • Hence
(~z')x'ea(F) is an orthonormal family in H . Since E is separable, H is
also separable. Thus a(F) is countable. Hence there is an x' E a(F) such that {x'} is open. Choose p E P r F such that F
^
p
~(F) =
~Xl
9
By Proposition 4.2.7.18, p e p C F and so p e p is a Gelfand C*-algebra. Put K :: p(H),
define for each u E s
~" K
) K,
~ : >pu~
and
Then by Proposition 5.3.2.24, (K, ~) is an irreducible representation of pEp. Since pEp is a Gelfand C*-algebra, K is one-dimensional (Corollary 6.3.6.7). Hence p C ~ ( H ) . By Proposition 5.5.6.9,
6. Selected Chapters of C*-Algebras
310
K:(H) C ~ ( E ) . -1
Since p is faithful, p(K:(H)) is a closed ideal of E . E being simple, K : ( H ) - g)(E), I
i.e. E is isometric to K:(H). Remark.
b) does not hold for the real C*-algebras llJ and IH.
P r o p o s i t i o n 6.3.6.13
Assume IK = C . Let (H,r
be a non-degenerate re-
presentation of E such that cp(E) C K~(H). Then (H, ~) is equivalent to the Hilbert sum of a family ((H~, cp~))~ei of irreducible representations such that ~ ( E ) : r(H~)
for every ~ E I and (II~XlI)LE, E Co(I) for every x E E . Let (H~)LEI be a maximal family of pairwise orthogonal invariant closed subspaces of H such that the compression (H~,~) of (H, ~) to H~ is irreducible for every ~ E I . By Proposition 5.5.6.9, ~(E)-K~(H~) for every ~ E I . Put
LEI
We must show that K = {0}. Assume the contrary. Then K is ~a(E)-invariant (Proposition 5.3.2.9) and there is an irreducible representation of the compression of p(E) to K (Corollary 6.3.6.4). By Proposition 6.3.6.10 and Theorem 5.5.1.24, there is a closed p(E)-invariant subspace K0 of K such that the compression of (H, ~) to K0 is irreducible. This contradicts the maximality of ((H~, ~))~ET 9 Take x E E
and assume that
(]]p~xii)~e, r
Co(I). Then there is an
orthonormal family (~)~ET in H such that ~ E H~ for every ~ E I and (II(~x)~II)~ET r Co(I) and this contradicts Theorem 5.5.5.1 a ~ d. T h e o r e m 6.3.6.14 a)
I
The following are equivalent:
There is a faithful representation (H, p) of E such that ~p(E) C ~ ( H ) .
6.3 Von Neumann Algebras
b)
311
If IK = ([J then there is a family (Ht)~E I d tiilbert spaces such that E is isometric to the C*-direct sum of (~(H~))~EI. /f IK = IR then there are families of sets (R~)aeL, (S,),EM, and (T,),EN such that E is isomorphic to F~ x F~ x F ~ ,
where Fia, Fc, and F~ are the C*-direct sums of the families
(~(e~(n~,na)))~, (r~(e~(&,e))).~., (~(e~(r.,~)))~, respectively. If these conditions are fulfilled then E is C-order complete. Case 1
IK = C
a =~ b. By Proposition 6.3.6.13, (H, ~) is equivalent to the Hilbert sum of a family ((H~, p~))~cj of irreducible representations such that
~ ( E ) = ~(H~) for every ~E J and (}tp~x[l)~cj E Co(J) for every x E E . Let I be a maximal subset of I such that if ~, n E I are distinct, then the representations (H~, pc), (H~, ~ )
are not equivalent. Let F denote the C*-direct sum of the
family (K:(H~))~EI and put ~:E
>F,
z,
>(~z)~i.
Since p is faithful, ~ is injective. By Corollary 5.5.6.13, ~ is surjective. b => a follows from Proposition 5.3.2.27. Case 2
IK = IR
a => b. By Case 1 a =~ b, Proposition 4.3.5.6, and Corollary 5.5.1.13 b), E is the C*-direct sum of a family (E~)~eI of C*-algebras such that for every o
E I there is a complex Hilbert space K~ such that either E~ is isomorphic to ~(K~) or E~ is isomorphic to the real C*-algebra underlying K;(K~). By Corollary 5.5.7.13, in the first case E~ is isomorphic either to F ( ~ ( K ~ ) ) or to ~ ( H ) for some real Hilbert space H . By Proposition 5.6.4.18 b), F(1C(K~)) is isomorphic to
for some set T . It is easy to see (and follows from Corollary 6.3.9.8) that
312
6. Selected Chapters o] C*-Algebras
C~(e2 (T, ~) ) n lC(e2(T, ~) ) = ~:~(e2 (T, IH) ) . The assertion follows. b =~ a. By Corollary 5.5.7.13, there is a family (K~)~el of complex Hilbert spaces such that /~ is isomorphic to the C*-direct sum of the family o
(K:(K~))~el. By Case 1 b =:~ a, E has a faithful representation (K, ~) with o
~o(E) c /C(H). If H denotes the underlying real Hilbert space of K (Proposition 5.6.2.5 a),e)), then (H, q01E) is a faithful representation of E with ~o(E) C K:(H). The final assertion follows from b) and Corollary 5.6.5.4 b). m Remark.
This theorem generalizes Corollary 6.3.6.5.
T h e o r e m 6.3.6.15
Every reflexive C*-algebra is finite-dimensional.
We may assume ]K = IIJ. Let E be reflexive and M the set of minimal elements of Pr E \ { 0 } . Step 1
[ Every commutative, linearly independent ( subset of R e E is finite.
Let A be a commutative, linearly independent subset of R e E and let F be the C*-subalgebra of E generated by A. Then F is a Gelfand C*-algebra (Corollary 4.1.2.3) and so it is isometric to Co(a(F)). Hence Co(a(F)) is reflexive (Proposition 1.3.8.5 b)), a(F) is finite, F is finite-dimensional, and A is finite. Step2
pCPrE\{0}=~3qCM,
q
Let A be a totally ordered set of Pr E \ { 0 } . Since A is commutative and linearly independent (Corollary 4.2.7.6 a =~ b), it is finite by Step 1. The assertion now follows by Zorn's Lemma.
Ep-1 Step 3
There is an A E ~ I ( M ) such that
peA p, q E A =~ pq = (hpqp.
E is a W*-algebra, so it is unital (Theorem 4.4.1.8 h)). Put 92 := {A e 9~(M) I P, q e A ::::, pq = 5pqp}. By Zorn's Lemma, 92 has a maximal element A. Since A is commutative and linearly independent, it is finite by Step 1. Assume that
6.3 Von Neumann Algebras
313
pEA
Then 1- Ep
E PrE\{0}
pEA
and by Step 2, there is a q E M , q< l-Ep. pEA
We have A U {q} E 9.1, which contradicts the maximality of A. Hence
E p-
1.
pEA
Step 4
(H, p) irreducible representation of E =:> Dim H < ec
~(E) is isometric to E / K e r p (Theorem 4.2.6.6), so it is reflexive (Proposition 1.3.8.6 b)). By Step 2 (and Theorem 4.4.1.8 h)), M is nonempty. By Corollary 5.5.6.10 (and Theorem 4.4.1.8 c)), /C(H) C ~ ( E ) . Hence /E(H) is reflexive (Proposition 1.3.8.5 b)) and therefore H is finitedimensional (Corollary 6.1.6.9 c ==~ a). Step 5
E is finite-dimensional
We may assume IK = C. By Step 3, there is a finite subset A of M such that Ep-
1,
pEA
p,q E A ~
pq = 5pqp.
For p E A , let xp' be the unique element of T0(E) with Xp'(p) = 1
(Corollary 4.3.4.13 a), Theorem 4.4.1.8 c)) and (Hp, ~p) the representation of E associated to Xp' . By Theorem 6.3.6.2 a ::~ b for every p E A the representation (Hp, ~p) is irreducible. Thus by Step 4, Hp is finite-dimensional.
6. Selected Chapters of C*-Algebras
314
Let (H, q0) be the representation of E associated to A, i.e. the Hilbert sum of the representations ((Hp, qop))peA. Take x E E , and p E A. Then px* x p = x~p(px* x p ) p = xp' (~*x)p
(Corollary 4.3.4.13), so II~pxll 2 ~
X'p(X*X)= IIx~o(x*x)pll- Ilpx*xpll =
Ilxpll 2
(Theorem 5.4.1.2 g)). Hence if qox = 0, then qapx = 0 and x p = 0 for every p E A. Since x = Exp, pEA
x-
0 and thus q0 is injective. Since H is finite-dimensional, E is finite-
dimensional as well. C o r o l l a r y 6.3.6.16
I Let E be an i n f i n i t e - d i m e n s i o n a l
a)
There is a strictly increasing sequence in Pr E .
b)
E is not separable.
W*-algebra.
a) By Theorem 6.3.6.15, E is not reflexive. Hence the predual of E and the dual of E are distinct (Proposition 1.3.8.4). By Theorem 4.4.4.2 c =:~ a, there is a strictly increasing sequence in Pr E . b) By a), there is a strictly increasing sequence (Pn)ne~ in Pr E . For every A c IN, denote by PA the supremum in Pr E of {E(Pn+l-
pn) l B E q3s(A)}
nEB
(Theorem 4.4.1.8 i)). If A and B are distinct subsets of IN, then (PA -- PB ) + -- PA -- PAnB E Pr E ,
(PA -- PB ) - -- PB -- PAnB C Pr E
(Corollary 4.2.7.6 a ::v g), so that IIPA -- PB
II = sup( IIPA -- P A n B II, IIPB -- P A n B II} = 1
(Theorem 4.2.2.9 b), Proposition 4.1.2.21). Since {PA [ A C IN} is uncountable, E is not separable. P r o p o s i t i o n 6.3.6.17
I I f IK - C
then the following are equivalent:
6.3 Von Neumann Algebras
a)
E is finite-dimensional.
b)
There is an a E IR+ such that
LEI
315
tEI
for every finite family (x~)~ei in E+. c)
a(x) is finite for every x E E .
d)
a(x) is finite for every x E E+.
e)
Every Gelfand C*-subalgebra of E is finite-dimensional.
a ==~ b follows from the Corollaries 4.2.1.8 and 1.7.1.7. a =v c follows from Proposition 2.2.1.17. c =v d is trivial. d :=> e follows from the isomorphism of the Gelfand transform. b =v e. Let F be an infinite-dimensional Gelfand C*-subalgebra of E . Then a ( F ) is infinite. Hence there is a sequence (xn)ne~ in F+ such that
x~=o
I1~11 = 1,
for any dinstict m, n E IN. Then Card M nEM
nE
for every M E ~s(]N) which contradicts b). e =v a Let A be an upward directed commutative subset of E # By Corollary 4.1.4.2 c), the C*-subalgebra F of E generated by A is Gelfand. By e), F is finite-dimensional. It follows that the upper section filter of A converges to an x E F . Let (H, ~) be a faithful representation of E . By Proposition 1.7.2.2, ~x is the supremum of ~(A) in E ( H ) . By Theorem 6.3.4.6, E is a W*-algebra. If E is infinite-dimensional then by Corollary 6.3.6.16 a), there is a strictly increasing sequence (Pn)nEIN in P r E . By Corollary 4.2.7.6 a => b, (Pn)ne~ is commutative. Hence the C*-subalgebra G of E generated by (Pn)ne~ is Gelfand (Corollary 4.1.4.2 c)). G has to be infinite-dimensional and this contradicts e).
Proposition 6.3.6.18 T:
I
The following are equivalent for every nonempty set
6. Selected Chapters of C*-Algebras
316
a)
E is reflexive.
b)
~2(T,E) is reflexive (Example 5.6.~.2 a)). a =~ b. By Theorem 6.3.6.15, E is finite-dimensional. Let (x~)~eI be an
algebraic basis of E with [[x~[] = 1 for every ~ e I and (x'~)~ei an algebraic basis of E' such that
for all ~, A E I (Corollary 1.3.3.2 a)). For every t E I put H~ := {~ E g2(T, E)lt E T ==v ~t e IKx~}, ~ "s
> H~,
a:
> ax,
and denote by 7r, the natural projection e2(T, E)
>H~.
For every (x',~) E E ' x g2(T, E) put (x',~) " g2(T,E)
~ IK,
<<~I~>,~'> ;
77,
(x', ~) is linear and continuous. Let ~' E (t~2(T, E)) ' . For ~ E I , the map ~.2(T)
> IK,
a,
>~'(ax~)
is linear and continuous. By the Theorem of Fr6chet-Riesz (Theorem 5.2.5.2), there is an a, E t~2(T) for every ~ E I such that ~'(~)
=
<~I~,>
for every a E t2(T). For every 77 E t~2(T, E) and ~ c I , (x'~~a~l)(r/)- E
(x'~, a , " ~ ) ( ~ r ~ r / ) - E((Tr~r/ia~l>, x : ) =
AEI
AEI
= (@,r/ia, l>,m',) = ~'(Tr,r/)
so that
~EI
~EI
6.3 Von Neumann Algebras
317
By Proposition 5.6.4.6 d) (and Proposition 1.3.6.27), the predual of E2(T, E) is reflexive so that g2(T,E) is also reflexive (Proposition 1.3.8.4). b =~ a. Take t 6 T and put ~'g2(T,E) r
~E,
~, ~2(T,E) ,
~, x,
>~t, ) xet.
Then ~o%b is the identity map E -+ E and so ~"o~" is the identity map E" --+ E" (Corollaries 1.3.4.4, 1.3.4.5). Thus ~" is surjective. Since g2(T, E) is reflexive, j~2(T,E ) and jEO~ -- 99"oj~2(T,E)
are surjective (Proposition 1.3.6.16). Hence jE is surjective, i.e. E is reflexive. m P r o p o s i t i o n 6.3.6.19 Let E be a finite-dimennsional C*-algebra and n,p C IN. The following are equivalent:
b)
En,~ is isomorphic to a unital C*- subalgebra of Ep,p.
c)
There is an involutive algebra homomorphism En,n --+ Ep,p.
a ~ b =v c is trivial. c ==> a. We may assume ]K = C (Corollary 5.6.6.8 c)). By Corollary 6.3.6.5, there is a finite family (mk)keI in IN such that E is isomorphic to I f Cmk,mk k6I By Corollary 5.6.6.8 b), E~,~ Ep,p are isomorphic to H(~nmk,nmk and H(~pmk,pmk, kEI kEI respectively. Take k 6 I . Since ~u,~,~-~k is simple (Theorem 5.6.6.7 b)), there is exactly one k' C I and a unital algebra homomorphism q~nm~,~mk --+ CPmk,,Pmk, 9 Moreover I = {k'lk e I } .
By Corollary 2.1.4.13 c),
6. Selected Chapters of C*-Algebras
318
pink nmk,
EIN
for every k E IN. Hence
(P---')CardI= I I (Pink ~ It kEI nmk, ]
E ]N.
It follows
I
P E]N. n
Proposition
6.3.6.20
Let E be a C*-algebra and p E Pr E .
The following
are equivalent:
a)
pEE c .
b)
p E is an ideal of E .
c)
E p is an ideal of E .
d)
p E p is an ideal of E .
e)
For every irreducible representation (H, (p) of E ,
a ~
~p e {0, 1}.
b & c is trivial.
b ==v c. Take x , y E E . By b), therc is a z E E such that
y*px* -- pz. It follows x p y = z*p G E p
so t h a t E p is an ideal of E . c =v b. Take x , y G E . By c), there is a z G E such t h a t x*py* = z p .
Then ypx = pz* E p E ,
i.e. p E is an ideal of E . b & c =:v d. Take x , y E E . By b) (by c)), there is a z E E such that xpy = pz
(xpy = zp) .
6.3 Von N e u m a n n
Algebras
319
Then xpyp = pzp C pEp
and p e p
(pxpy = pzp 6 pEp)
is an ideal of E .
d =~ e. Let x E E . By d), there is a y E E such that xp = xp 3 :
pyp.
It follows
( ~ ) ( ~ p ) = (~p)(~(yp)). Hence Im (qop) is qD(E)-invariant. By Proposition 5.3.2.9, Im (qDp) reduces 9~(E). Since (H, qD)is irreducible, ~p E {0,1}. e => a. Take x c E . By e), for every irreducible representation (H, ~) of E,
~(zp
-
p~) = o.
By Corollary 6.3.6.3, x p - p x = O.
Hence xp = px
and p E E c .
I
6. Selected Chapters of C*-Algebras
320
6.3.7 C o m m u t a t i v e Proposition
von Neumann
6.3.7.1
subalgebra of s
c
Algebras
( 0 ) Let lK = r
(IK = IR) and E a Gelfand C*-
Then the following are equivalent:
a)
E=E
(E=Re(EC)).
b)
E is a maximal Gelfand C*-subalgebra of s
c)
E is a maximal Gelfand yon Neumann algebra on H .
If these conditions are fulfilled, then:
d)
E acts non-degenerately on H .
e)
If H is separable then E has a cyclic vector.
a r br
b follows from Proposition 4.2.2.14. c follows from Corollary 4.4.4.12 b).
a =v d. Put K "= {ux I ( u , x ) e E x H} •177
Then K is the closed vector subspace of H generated by {ux I (u, x) e E • H} (Corollary 5.2.3.9) and so K is E-invariant. By Proposition 5.3.2.9, 7( g E E c , so that 1 - 71-g E E c . By a), 1 - r g C E . Take x C K • . Then x = (1 - ~K)X C K , x-0,
K~:{0}, K=H.
Thus E acts non-degenerately on H . a & d =v e. By Proposition 5.3.2.23, E has a separating vector. By a) and d), this vector is cyclic for E (Proposition 5.3.2.22 b)). E x a m p l e 6.3.7.2
( 0 ) Let T be a locally compact space and p a positive
Radon measure on T . Given x E L~ ~" L2(#) and
I
and jc C L ~ ( p ) , define
~ L2(#),
y,
) xy
6.3 Von Neumann Algebras
a)
321
L+(#) is a maximal Gelfand yon Neumann algebra on L2(#) (Example 2.3.2.32). It is called the yon N e u m a n n algebra on L2(#) of multiplication operators.
b)
~(T) is strongly dense in Lf..~(#) and L ~ ( # ) is the yon Neumann algebra on L2(#) generated by K ( T ) .
c)
If T is compact then eT is cyclic for C(T) and separating for L ~ ( # ) .
a) By Example 2.2.2.22 d), L ~ ( # ) -and the assertion now follows from Proposition 6.3.7.1 a =~ c (in the real case L~r C Re s b) By Example 2.2.2.22 e), c=
c _ L~(#),
so that (K~(T))r = L~~ and the assertion now follows from Proposition 6.3.5.1 and Corollary 6.3.1.12. c) is easy to see.
Example 6.3.7.3
m Let A be an orthonormal basis oif H . Given x C ~ ( A ) ,
define H __.+ H ,
~ ~ > E x(ri)({lr])~" ~TEA
Then
is a maximal Gelfand von Neumann algebra on H algebra on g
and the yon Neumannn
generated by {~ I x e co ( d ) } .
The assertion follows immediately from Example 6.3.7.2 a),b), by identifying H with t~2(A), m
6. Selected Chapters of C*-Algebras
322
/
Theorem
6.3.7.4
( 0 )
Let E be a Gelfand unital von Neumann algebra
on a Hilbert space H . Let E admit a cyclic vector. Then there are a compact space T , a positive Radon measure p on T , and an isometry o] Hilbert spaces
u" H -+ L2(#) such that u E u -1 is the yon Neumann algebra of multiplication operators on L2(#) (Example 6.3. 7.2). If g
is separable then we may take T
metrizable.
Let F be a unital C*-subalgebra of E dense in E H . By Theorem 2.4.1.3 c), T := a ( F ) is compact. Let ~v be the Gelfand transform of F and ~ a cyclic vector for E . The map C(T)
>IK,
x,
>((~-lz)~l~ )
is a positive linear form (Corollary 5.3.3.7) and so there is a positive Radon measure # on T such that xd# = ((~v-lx)~r
for every x E C ( T ) . Given x E L~162 and .T C L~162 define YL2(#)
>L2(#),
y,
>xy,
and
.r:= {Ylx e 7} We have
= ((~-'~(v*~))r162
= (~*vr162 = (vr162
= I1~r ~
for every v E F . Put
K : = { v r 1 4 9 F}, and
~-K
>C(T), ~ ,
>~(v).
Then u is linear and preserves norms. Since ~r is cyclic for E and F is dense in EH, K is dense in H . C(T) being dense in L2(#), u may be extended to an isometry H
> L2(#). Define
6.3 Von Neumann Algebras
r
>s
v,
323
> u o v o u -1
Take v E F and x E C ( T ) . There is a w E F such that = ~(~).
We get (~)V)X = ~ t V U - - I ~ ( w ) = l t V W ~ - -
= ~(~)~(~)
~9(VW) =
= ~(v)~,
so that N
Cv = ~(v) e C(T), N
r
= C(T).
Since F is strongly dense in E and C(T) is strongly dense in L ~ ( # ) (Example 6.3.7.2 b)), r
- L~(#).
Now assume that H is separable. By Corollary 6.3.5.3, s is separable and metrizable. Hence EH# is separable and metrizable. Let A be a countable dense set of EH# and F the unital C*-subalgebra of E generated by A. Then F is separable and dense in E H . By Theorem 2.4.1.3 d), T = a ( F ) is metrizable. C o r o l l a r y 6.3.7.5
I ( 0 )
Let E be a Gelfand yon N e u m a n n algebra on a
separable Hilbert space H . Then there is a compact space T , a positive Radon measure tt on T , and a bijective involutive algebra homomorphism of E onto the von N e u m a n n algebra on L2(#) of multiplication operators.
The unit of E being an orthogonal projection in H (Corollary 4.4.4.9), we may assume that 1H C E . Then E acts non-degenerately on H and by Proposition 5.3.2.23, E has a separating vector x. Put
and for each u C E , define ~" K
>K ,
y~
>uy
324
6. Selected Chapters of C*-Algebras
( K is E-invariant). The map E
>s
u,
>
is an involutive unital algebra homomorphism and it is injective, since x is separating for E . The map
Es
)
> f-,(g)s
U,
> "U
is obviously continuous. Since
Es
) = EE
>s
u,
(Proposition 6.3.4.2), the map E
>
is a W*-homomorphism (Proposition 4.4.4.6), so {glu E E} is a v o n Neumann algebra on K (Corollary 4.4.4.8 b)). x is obviously a cyclic vector for { g [ u c E } . Replacing E by { g i u E E} if necessary, we may assume that E has a cyclic vector. The assertion now follows from Theorem 6.3.7.4.
I
6.3 Von Neumann Algebras
325
6.3.8 Representations of W*-Algebras P r o p o s i t i o n 6.3.8.1 ( 0 ) Assume E unital and IK = ~ (IK = IR). Take x' E ~-(E), let (H,~) be the representation of E associated to x' and A a downward directed set of E+ such that inf x' (x* yx) = 0
yEA
for every unitary (invertible) element x of E . Then 0 is the infimum of ~(A) in s
~(A) is downward directed (Corollary 4.2.1.4). Let v be the infimum of ~p(A) in s (Corollary 6.1.7.14 b), Theorem 4.4.1.8 c)) and x a unitary (invertible) element of E . Put r := {y e E I ~'(y*y) = 0},
and let w denote the quotient map E -+ E / F . Then ~~
-
~ ~ l ~
~
-
( ~ 1 ~ ) -
= i n f { ( g y ) w x l w x } = inf(w(yx) l w x } = inf x ' ( x * y x ) = 0 yEA
yEA
yEA
(Proposition 6.3.1.4 a =~ b), so that 1
wx E Kerv~ - Kerv
(Proposition 5.3.3.10 a)). Since every element of E is a linear combination of four unitary (of two invertible) elements of E (Corollary 4.1.3.7), v vanishes on Im w. Since Imw is dense in H , v is identically 0. I C o r o l l a r y 6.3.8.2
( 0 )
Assume E unital and IK--ffJ ( I K -
lit). Let
0 # A c ~-(E), let (H, 9) be the representation of E associated to A , and let B be a downward directed set of E+ such that
inf x'(x* yx) -- 0
yEB
for every x ~ E A and for every unitary (invertible) element x of E . Then 0 is the infimum of (p(B) in s
6. Selected Chapters of C*-Algebras
326
For each x' E A, let ~xr denote the algebra homomorphism associated to X I "
Take ~ E H . Then
xrEA
for every y E B . By Proposition 6.3.8.1, inf ((~y)~[~) = 0.
yEB
Since ~ is arbitrary, 0 is the infimum of ~(B) in s
(Proposition 6.3.1.4
b~a).
I
C o r o l l a r y 6.3.8.3 ( 0 ) Assume E unital and IK = r (IK = ]R). Let 0 ~ A C T(E), (U, ~) the representation of E associated to A , and ~ a set of upward directed subsets of E+ such that: 1)
every B E ~ possesses a supremum
V x in E+. xEB
:
3) x*Bx E ~ for every B E ~ and .for every unitary (invertible) element x o/E.
Th~n for ~e,'y B e ~ , ~(~V x) is the s~p,'em~m oI ~(g) in f.(H). Given B E ~ , defne
B:-{VY-xlx~B} yEB
and put ~:={BIBE~}. Take B E ~ . Then B is a downward directed subset of E+, inf x'(x) = 0 zEB for every x' E A, and x*Bx = x*Bx E ~ for every unitary (invertible) element x of E (Proposition 4.2.2.21). By Corollary 6.3.8.2, 0 is the infimum of ~(B) in s Hence ~ ( V x) is the supremum of ~ ( B ) i n s 1 \xEB
l
6.3 Von Neumann Algebras
327
T h e o r e m 6.3.8.4 ( 0 ) Assume E unital, IK =(F, and 91 a set of upward directed subsets of E+ such that: 1)
Every A E 91 has a supremum in E+.
2) x*Ax E 91 for every A E 91 and every unitary element x of E .
3)
Every commutative well-ordered set of E+, which has a supremum in E+, belongs to 91.
Let B be a nonempty dense set of i
: xEA
xEA
Let (H, ~) be the representation of E associated to B and F the closed vector subspace of E' generated by {yx'z l y, z E E , x' E B } . Then the following are equivalent:
a)
E is a W*-algebra.
b)
E is order-complete and EF is Hausdorff.
c) EF is Hausdorff and every commutative well-ordered set of E#+ has a supremum in E . d)
~ is injective and every commutative well-ordered set of E#+ has a supremum in E .
e)
~ is injective and ~(E) is a yon Neumann algebra on H .
f) E is a W*-algebra and ~ is an injective W*-homomorphism. a ::~ b follows from Theorem 4.4.1.8 c). b =~ c is trivial. c =~ d follows from Proposition 5.4.2.7. d ~ e. By Theorem 4.2.6.6, p(E) is a C*-subalgebra of s
and the
map E~~(E),
x:
;~x
is an isometry of C*-algebras. Let A be a commutative well-ordered set of ~(E)+# and let v be its supremum in / : ( g ) (Corollary 6.1.7.14 b), Theorem
6. Selected Chapters of C*-Algebras
328
-1
4.4.1.8 c)). Then ~v(A) is a commutative well-ordered subset of E+# . Let x be -1
the supremum of ~ (A) in E . By Corollary 6.3.8.3, = ~
e v(E).
By Theorem 6.3.4.6, ~ ( E ) is a von Neumann algebra on H . e ==>f. By Corollary 4.4.4.9 a =~ b, ~(E) is a W*-algebra and the inclusion map p ( E ) -+ s
is a W*-homomorphism. By Theorem 4.2.6.6, the map E
~v(E),
x,
)~vx
is an isometry of C*-algebras. Thus E is a W*-algebra and the above map is a W*-homomorphism. Hence ~ is a W*-homomorphism. f =v a is trivial.
II
C o r o l l a r y 6.3.8.5
( 0 ) If E is a W'-algebra, then there is a Hilbert space
U and an injective W*-homomorphism ~ : E -+ s
such that (p(E) is a
von Neumann algebra on H . Put A := T(E) N and denote by (H, ~) the representation of E associated to A. Step 1
~v is injective
Let x E E . By Corollary 4.4.1.5, there is an x' E A such that
(xx*xx*, z') # 0. Denote by q : E --+ E/Nx, the quotient map and by (Hx,, ~ , ) the representation of E associated to z'. Then xx* r N,, ,so q(xx*) ~ O. It follows
(~,z)qx* = q(~z*) # o,
~,,x # o.
Thus T is injective. Step 2
~ is a W*-homomorphism
By Corollary 4.4.1.9, A is dense in T(E) and by Theorem 4.4.4.2 a =v b, x' is order continuous for every x' C A. By Theorem 6.3.8.4 a =v f, ~ is a W*-homomorphism in the complex case. Assume now ]K - ]R and put
6.3 Von Neumann Algebras
329
B:-{(x',0) lx'cA}. By Proposition 5.4.2.14,
is equivalent to the representation of E associated to B. By Theorem 4.3.6.4 ..
(and Corollary 4.4.4.4), B C / ~ . By Corollary 6.3.8.3 (and Step 1 and Theorem o
4.4.4.2 a ==~b), the supremum of every upward directed subset of ~(E)+ belongs to ~(E) so that by Theorem 6.3.4.6, ~(/~) is a W*-subalgebra of s Step 1 and Theorem 4.2.6.6, the map
~. ~
By
~ ~(~)
is an isomorphism of C*-algebras, so it is a W*-homomorphism. It follows that ~ is a W*-homomorphism. Thus ~p is a W*-homomorphism (Proposition
4.4.4.7). Step 3
~(E) is avon Neumann algebra on H
The assertion follews from Step 2 and Corollary 4.4.4.8 b).
m
C o r o l l a r y 6.3.8.6 ( 0 ) Let E be a complex l/V*-algebra and F a C*subalgebra of E such that the supremum in E of any commutative well-ordered set of F#+ belongs to F . Then F is a W*-subalgebra of E . By Corollary 6.3.8.5, there is a Hilbert space H and an injective W*homomorphism p : E -+ s such that p(E) is avon Neumann algebra on H . Then the map
E
~(E),
z~-~z
is an isometry of C*-algebras (Theorem 4.2.6.6) and ~(E) is an order faithful C*-subalgebra of s (Theorem 4.4.1.8 d)). Hence p(F) is a C*-subalgebra of s Let A be a commutative well-ordered set of ~(F)+# and v its su-1
premum in s ~ (A) is a commutative well-ordered set of F+# and x, its supremum in E , belongs to F . Since v belongs to ~ ( E ) , we get v = ~
c ~(F).
Hence ~(F) is a W*-subalgebra of s (Theorem 6.3.4.6) and so a W*subalgebra of p(E) (Corollary 4.4.4.10). Thus F is a W*-subalgebra of E . m
6. Selected Chapters of C*-Algebras
330
C o r o l l a r y 6.3.8.7
( 0 )
Let E be a W*-algebra, F a C*-subalgebra of
E , and G the W*-subalgebra of E generated by F . Then F # is dense in G#. E
By Corollary 6.3.8.5, we may assume E is a v o n Neumann algebra on some Hilbert space H . By Corollary 6.3.1.12 (and Corollary 4.4.4.10), G is the closure of F in s
Let x E G # . By Kaplanski's Theorem (Theorem
6.3.1.10), there is a filter ~ on s
converging strongly to x such that
F # E ~. Then ~ converges to x in s163
) . Since E may be identified
with a quotient space of E l ( H ) (Corollary 4.4.4.9), the restriction of ~ to G # converges to x in G # . Hence F # is dense in G#.. E
P r o p o s i t i o n 6.3.8.8
1
E
If R is the Hilbert dimension of H , then there is a
faithful unital representation (K, ~) of f_,(H)" (Theorem 6.3.2.1 b)) such that the Hilbert dimension of K is at most 2 ~ and ~ is a W*-homomorphism. Let (L, r
be the complex universal representation of s
By Corollary
5.5.2.2 and Proposition 5.5.2.18 b), the Hilbert dimension of L is at most 2~ . By Theorem 6.3.2.1 b), the strong closure r 1 6 3
of s
is isometric to s
a, s
Thus by Proposition 6.3.4.2 c ~
identified with a unital W*-subalgebra of s
may be
The assertion now follows
from Corollary 4.4.4.9 a =v b. P r o p o s i t i o n 6.3.8.9
in s
I
Let E be a W*-subalgebra of a W*-algebra F and
a E R e E . TRhen there is a b E F+ such that
a < blE. By Corollary 4.4.4.9, there is a c E Re F such that
a = clE. Then b --
c+
I
possesses the desired property. P r o p o s i t i o n 6.3.8.10
Let E be a yon Neumann algebra on a Hilbert space
H . For every a E E + , there is a n E l l ( H ) + oo
~IE-
a.
such that
6.3 Von Neumann Algebras
By Proposition 6.3.8.9, there is a v E s
331
such that
a <_ ~ l E . Let A be a set of pairwise orthogonal elements of H \ { 0 } such that
~EA
and
{EA We denote by (K, ~o,X), the GNS-triple associated to ~]E. By Proposition 5.4.1.6 b), there is a w e s N ~(E) c such that
(x, a) = ((qox)wXlX) for every x E E . Put 1
~:=w~X. Then (x, a) = ((px)(~lC) for every x C E . In the sequel we use the notation of Example 5.4.3.3. Then
QeH~CH for every { c A and
~EA
Define u :: E
{]Q)Q c s
{EA
For x e E ,
~6A
so that a = ~IE.
I
6. Selected Chapters of C*-Algebras
332
Corollary 6.3.8.11
Let E be a W*-subalgebra of a W*-algebra F . Then
for every a E E+ , there is a b E tT+ such that b I E - - a.
By Corollary 6.3.8.5, we may assume that F is a von Neumann algebra on a Hilbert space H . Then E is also a v o n Neumann algebra on H (Corollary 4.4.4.10). By Proposition 6.3.8.10, there is a u E I : I ( H ) + such that oo
a = ~IE.
If we put oo
b:= ~I F
then b E F + and biE = a.
I
6.3 Von Neumann Algebras
333
S~
6. Selected Chapters of C*-Algebras
6.3.9 F i n i t e - d i m e n s i o n a l
C*-algebras
Throughout this subsection we use the following notation: a) b)
E is a finite-dimensional C*-algebra.
(F)Fe~ is a finite family in {IR,q~,IH} and n E IN~ such that E is the C*-direct product of the family (Fn(f),n(F))Fel d of C*-algebras (Corollary 6.3.6.5).
c) For every F E ~'"
Cl) rn(F),n(F ) is identified with the corresponding closed ideal of -E.
C2) 1F denotes the unit of Fn(F),n(F ) .
c3) B(F) :=
{1F}
if F = I K
{1F, 1fi}
ifIK=IR
{1f, lFi, lfj, lfk}
if I K = I R and F = H I .
and F =
C4) w(F) := n(F)CardB(F).
C5) ~/JF" F--+ 1K,
a:
~~ a ( rea
if IK = r if I K = ] R .
c6) For r, s E INn(F), E Fn(F),n(F).
ers : =
d)
P r m E "= the set of minimal elements of Pr E \ { 0 } . n(F)
e) ~ - E - + ~K, ~ H. ~ ~ r F6~ r--1
f) For every Hilbert right E-module H , H denotes the Hilbert space obtained by endowing the underlying vector space of H with the scalar product H x H
) 1K,
(,t, ~),
> ~<,11,7>
(Proposition 5.6.2.5 a),e),g),h), Proposition 5.6.6.17 a)). g) G, H, I are Hilbert right E-modules.
6.3 Von Neumann Algebras
335
In this subsection we present some aspects of the theory of Hilbert right C*modules over finite-dimensional C*- algebras, which is not very far from the classical theory of Hilbert spaces. In particular, the main results of the theory of /:P-spaces (presented in Section 6.1) still hold in this case.
Proposition 6.3.9.1 ( 8 )
b)
A n element x of E belongs to P r m E
iff there are F C ~ and ~ C F n(F)
with
Moreover, F is determined by x .
c)
xEPrmE=~x=l.
a) follows from Proposition 6.1.5.19. b) First assume x E P r m E . Take F c ~. Since 1F E ECn P r E , XlF -- 1FX E Pr E
XlF ~ X,
XlF < IF
(Corollary 4.2.7.6 c ~ a). Since x is a minimal element of Pr E \ { 0 } , it follows XlF C {O,x}.
Since x - ~ XlF, FC~
it follows that there is exactly one F C ~ with x-
XlF ~ 1F.
Hence x is a minimal element of Pr Fn(F),n(F)\{O } . By Proposition 5.6.5.11 b), there is a ~ E F n(F) with
The reverse implication follows from Proposition 5.6.6.18 a). c) follows from b) and Theorem 5.6.6.1 k).
I
336
6. Selected Chapters of C*-Algebras
P r o p o s i t i o n 6.3.9.2
If we put "d" E
~ IK,
x,
) ~o(ax)
for every a E E then ~ E/2(E, IK) for every a E E and the map
E
)Z:(E, IK),
a:
;
is linear and bijective.
It is obvious that the map is linear. Take a E E with ~ - 0. Then 0 = ~ ( a ' ) = (p(aa*) and this implies a -- 0. Hence the map is injective. Since E and f~(E, ]K) have the same finite dimension the map is bijective. I
Proposition 6.3.9.3 ( 8 ) a) b)
H is self-dual and reflexive and the norms of H and f t are equivalent. EE(G, H) = f ~ ( G , H) .
c) C.E(H) is a unital yon Neumann algebra on H . d)
ICE(H) is a closed ideal and a hereditary C*-subalgebra of s so a r complete C*-algebra.
e)
Let p E Pr )~E(H) and let A be a Fourier basis of I m p . Then A is finite and
and
s
a) follows from Proposition 5.6.2.5 a),d). b) follows from a) and Proposition 5.6.2.4. c) By Proposition 5.6.2.5 c),i), s By b),
is a unital C*-subalgebra o f / : ( H ) .
f-.E(H) = T~E(H) = {u E Z:(H)I (~, x) 9 H x E ===::vu({x) = (u{)x},
so that f~E(H) is a strongly closed set of s By Proposition 6.3.4.2 d a, •E(H) is a unital von Neumann algebra on H .
6.3 Von Neumann Algebras
337
d) By Corollary 5.6.5.4 b), ICE(H) is a closed ideal and a hereditary C*subalgebra of s By c) and Theorem 4.4.1.8 c), s is C-order complete so that ICE(H) is also G-order complete. e) By Proposition 5.6.5.11 a) (and Corollary 4.2.7.6 a ~ c & d), the set
{~(1~) I ~ E A} 0 {p} is commutative. Since it is contained in Re ICE(H), the C*-subalgebra J of ICE(H) generated by it is Gelfand (Corollary 4.1.4.2 c)). By Corollary 5.6.5.4 d), ICE(H) C IC(H) so that by Proposition 5.3.1.17, a(J) is discrete. Since p is the unit of J , a(J) is finite. It follows that A is finite. By Proposition 5.6.5.11 c), . - Z ~()~:) tiEA
Definition 6.3.9.4
m
( 8 ) A Fourier set A of H is called extreme if
{<~l~> I ~ ~ A} C Prm E . T h e o r e m 6.3.9.5
( 8 ) Let u be a compact self-normal element of f~E(H).
a) u E ICE(H) and there is an extreme Fourier basis A of H and an f E co(A) such that
u = E f(~)~('[~) tiEA
(in f--,E(H)),
f(A) C aL~(H)(U) C f ( A ) U {0}. Moreover, for every g E C (oLE(H)(U)) with o 9 ~.(.)(~)
g(u) =
~
~ g(f(~))~(-I0 ~EA
g(o) = o,
(in ICE(H)).
b) If we put B := {f ~ 0} then
~EB
(Theorem 5.6.3.13 d)) is the carrier of u in s
338
6. Selected Chapters of C*-Algebras
a) By Theorem 4.1.2.12 and Proposition 5.6.2.5 c),
o ~ ( . ) ( ~ ) = o~(~)(u). Put C : - ~ ac(B)(u)\{0}
if H is infinite-dimensional
L as
if H is finite-dimensional.
By Theorem 3.1.5.1 a),c), C is discrete and Ker(al - u) is finite-dimensional for every a E IK\{0}. By Theorem 4.1.4.6 a =~ b, there is a family (Ps)sec in Prff.E(H) such that: 1) u = E a p s ; sEC
2) a, fl E C =~ PsPz = 5s~Ps ; 3) H is finite-dimensional ~
~ Ps - 1. sEC
Take a E C and ~ E Imps. Then
for every ~ E C. It follows successively u~ = ~
Zp~ =
a~,
f~Ec
(~1 - u)~ = 0,
E Ker(al - u),
Imps C K e r ( a l - u). Hence Imps is a finite-dimensional Hilbert right E-module (Corollary 4.1.2.6). By Proposition 5.6.3.16 (and Proposition 5.6.3.13 a)), there is a finite extreme Fourier basis As of Imps. By Proposition 5.6.5.11 c),
Take distinct elements a, fl of C and ~ E As, r]E Aa. Then
6.3 Von Neumann Algebras
339
so that U A~ is an extreme Fourier set of H. By Proposition 5.6.3.16, there aEC
is an extreme Fourier basis A of H containing U A~,. aEC
If we define f'A
>IR,
3'
>
a
i f ~ E A ~ for some a E C
0
if~EA\
U A~ r~EC
then f(A) c ac~(u)(U) C f ( A ) U {0} and by Proposition 5.6.3.14 b) and Theorem 4.1.4.6 a =v e (and Proposition 6.3.9.3 a)), u= E
ap~ = E
aEC
a E
aEC
g(u)
-
~(']~)= E
~EA~
f(~)~('l~)
(in ICE(H)),
~EA
~
g(f(5))5(-15) ~EA
(in
s
b) follows from a) and Proposition 5.6.3.15 b).
I
P r o p o s i t i o n 6.3.9.6 ( 8 ) Let u E f-.E(G, H), A an extreme Fourier basis of G, and f E co(A)+ such that u*ou - ~ f(~)~(lO ~EA
(in ~E(G)).
Put B:={f#0},
g := v / f i B E co(B)+,
and
~.= g-~u~ for every ~ E B . . v
b)
~ = E g(~)~('l~) ~EB
(in 1CE(G, H)), I1~11 = Ilgll~.
6. Selected Chapters of C*-Algebras
~0
a)
(~"1~--~ --
1
1
~(f(~),~177)--
(,~1,7) 9
b) For ~ E A \ B , (~:lu,1) = (~*~:1,1) = (f (e),tle) = o
so that u~ = 0. Take r/E G. By Theorem 5.6.3.13 d), f), ,7 = Z
~(,71~>
~eA
so that
~eA
~eB
By a), Proposition 5.6.3.14 b), and Proposition 6.3.9.3 a),b), = ~--~g(5)~(.l~) ~eB
(in 1CE(G,H))
and
Itull- IIgll~ T h e o r e m 6.3.9.7
m
( 8 ) Let u be a compact element of s
a) u E ICE(G, H) and there are an extreme Fourier set A of G, a family (()~em in H such that
for all ~, rl E A , and an f E co(A)+ such that
,+>>
=
(in ICE(G, H ) ) .
~en Moreover, for every such representation of u,
Ilull- IISIl~.
6.3 Von Neumann Algebras
b)
34~
Take g E g~176 and put fI v" g
>G,
rl,
> ~g(~)~
(Proposition 5.6.3.8 a), Theorem 5.6.3.12 b), Proposition 6.3.9.3 a)). Then
you - ~
f(Slg(5)5
(in ICE(G)).
5EA
a) By Theorem 6.3.9.5 a), there is an extreme Fourier basis B of G and a g E co(B) such that u*ou = ~--]~g(~C)~c(.Isc)
(in ICE(G)),
~EB
g(B) c o~.(~)(~*o~)u {0} c
Ia+
(Corollary 5.6.1.12). Now the assertion follows from Proposition 6.3.9.6. b) Take ( E G. Then
for every ~ E A so that
~EA
By Proposition 5.6.3.14 (and Proposition 6.3.9.3 you - ~
f(~)g(~)~<'l~>
a)),
(in ICE(G)).
I
~EA
Corollary 6.3.9.8
( 8 ) ICE(G, H) = s
H) M IC(G, g ) .
The inclusion from the left to the right follows from Corollary 5.6.5.4 d). The reverse inclusion follows from Theorem 6.3.9.7 a). I Definition 6.3.9.9
( 8 ) Let ~ E H such that <(1(> E P r m E . By Propo-
sition 6.3.9.1 b), there is a unique F E ~ such that
We define
w(~) := w(F).
6. Selected Chapters of C*-Algebras
342
Theorem 6.3.9.10 a)
( 8 )
Let ~ 9 H such that
(t~J~) 9 PrmE.
There is an F 9 ~ and a ~ 9 F n(F) such that
(r162 = 1F,
(~]~) = ~(.Jr 9 PrFn(F),n(F).
Put
~r :~ ~Xr
.for every r 9 ~n(F) and Ar := {~ryJr 9 INn(f), Y 9 B(F)}.
b)
r, s E IN.(F) =~ x~x*~ = 5~s(~J~).
c) r, s e INn(F), y, z e B(F) =~ ~(~ryJ~z) = 5~5y,. d) CardAr = w((). e) rl 9 A~ (in H)=~ (~J~)= O.
f) For every u 9 s ~ ( ~ ~ (u~.~)) = ~(~)~(u~,~) .
a) follows from Proposition 6.3.9.1 b). b) By Proposition 5.6.5.2 a),f), ~;
= (r176162
(;(~]~))(.]~)
= 6-(~1~).
c) By b),
= ~r~((~l~)(~l~)) = ~
(Proposition 6.3.9.1 a),c)). It follows
343
6.3 Von Neumann Algebras
- 5~(yz*)
-- ~ ( ( ~ l ~ ) y z * )
- 5r~5~z
(Proposition 6.3.9.1 a)). d) follows from c). e) For r E ]N,(F) and y E B ( F ) , 0 = qo(~xrylrl} = qo((~lrl)xry) =
~((~I~)o(C(I~>)y) =
(Proposition 5.6.5.2 d), Theorem 5.6.6.1 k)). It follows successively ((~1,7}r
= o,
<~1~>r = o ,
(~:l,~) = (~:(~1~:}1,7} = (~ ,7)(~:1~:} = ( ~ 1 , 7 ) o ( r 1 6 2
((~l~>C)< Ir
=
o
(Proposition 5.6.3.9 a =~ b, Proposition 5.6.5.2 d)). f) For r C IN~(r) and y C B(F), by b), ~(U(~ry) l~ry ) : (~)(y*x; (U~l~}Xry) -- (tg((U~I~}XrX; ) :
(Proposition 6.3.9.1 a), Proposition 6.3.9.3 b), Proposition 5.6.3.9 a =~ b). Hence by d), (p (~e~A (u~,r/))= w(~)(Z(u~,~) ( 8 ) ForpC{O}U[1,c~]
Definition 6.3.9.11
s
define
H) = f-.E(G, H) A f_P(C, U) ,
s
= f_YE(H,H )
6. Selected Chapters of C*-Algebras
344
s
is an involutive unital f_.E(H)-module (Corollary 6.1.3.13 b), Proposition 5.6.2.5 b)).
Corollary 6.3.9.12
( 8 ) L~t A b~ a~ ~t~m~ r o ~ i ~ b a ~ oS ~ .
U A~ is an orthonormal basis of H , where A~ was defined in Theorem
a)
~EA
6.3.9.10 a).
c) ~ E A, r/E H =~ tr(r/(-I~)) = w(~)~(r/l~ ) . a) By Theorem 6.3.9.10 c), U Ar is an orthonormal set of H . Take 7/E ~EA A~
. By Theorem 6.3.9.10 e),
(wlO =
o
for every ~ E A. By Theorem 5.6.3.13 b), 7/= 0. Hence U A~ is an orthonor~EA mal basis of H . b) By a) and Theorem 6.3.9.10 f),
tr u = E
E
~EA nEAr
~(uTllrl)= E w(~)~(u~[~). ~EA
C) By b),
(Proposition 6.3.9.1 a), Proposition 5.6.3.9 a :=~ c). Theorem 6.3.9.13
I
( 8 ) Let 91 be the set of Fourier sets A of H such
that
{<~10 I ~ e A} C {eF~ IF 9 iY}. a)
Every maximal element of 91 {91 ordered by inclusion } is a Fourier basis oyH.
345
6.3 Von Neumann Algebras
b)
There is a unique family (~F)Fe~ of cardinal numbers such that H is isomorphic to
(9 ~(s~, ~E).
FE~ C) If we identify H with
9 ~(~,~, ~ E)
F Etd
using the isomorphism of b) then { ,~F AgRF
,.,,,.t y I F e ~,
t
e ~F, Y ~
B(F)}
is an orthonormal basis of H .
Let A be a maximal element of 91. For every F C ~ put AF "-- { ~ C A I (,11,9 = e ~ } .
By Proposition 5.6.4.12 b), AF is a Fourier basis of H1F for every F . Since H is isomorphic to (J) H1F (Proposition 5.6.4.13 a),b)) and A - U AF, it FE~
FG~:
follows that A is a Fourier basis of H . b) follows from Proposition 5.6.4.12 c) and Proposition 5.6.4.10 c), and c) follows from Corollary 6.3.9.12 a). I Remark. The family (~F)Fe;~ may be considered as a kind of Hilbert dimension of H .
T h e o r e m 6.3.9.14 ( 8 ) Let A be an extreme Fourier set of G, (~)~EA a family in H such that
/or all ~, ~ C A, f c co(A), p E [1, c<)[, and
u := E f(~)~('l~) ~A Then u C s
(in ICE(C, H ) ) .
iff f C lP(A) and in this case
Ilull~ - ~ ~(~) If(5)I
p ,
,~A
u= ~ ~eA
f(5)~(15))
(in s
H)).
6. Selected Chapters o.f C*-Algebras
3~6
By Proposition 6.3.9.1 b), for every ~ 9 A, there are F 9 ~ and r] 9 F n(F) with -
= ly
-
and the assertion follows from Proposition 5.6.6.19 b),d),e) and Corollary 6.1.2.12. I
(8)
Definition 6.3.9.15
We denote for every u 9 s n(F)
u'a -~
>g ,
~,
>E
w(F)l
T h e o r e m 6.3.9.16 a)
u(~eFy) f 9 etsY "
e~B(F E
FE~
y
) s,t=l
( 8 ) Let p ~ {0} u [1, co].
u 9 fifE(G, H) for every u 9 s .
.
t-I) and the map
.
.
s
_.+
>s
is a projection of s
u:
H) onto s
; u
H) .
b) u e s
v E s
c) If u e s
~ E H , F E ~ , and r C INn(f) such that
=~ uov = u o v .
F
then 1
where A~ was defined in Theorem 6.3.9.10 a).
d) u C s
=~ tr u = tr u .
a) Take ~ E G and x C E .
For F C ~
n(F) E
F
F
F
s,t=l
s,t=l
n(F)
n(F) =
t:l
~ and
n(F) F
=
yEB(F)
=
s,t=l
s', t v cINn(F),
6.3 Von Neumann Algebras
347
n(F)
--- ~
u(~)
~ ~
etses, t, 9
s,t=l
Since u(F) yEB(F) s,t-1
n(F) yeB(F) s,t=l
for F r IK it follows ~(~z)
-~ . (~)z
=
By Proposition 6.3.9.3 b),
-~ C s
H) -- f~E(G, H).
Since the construction of ~ from u involves a finite number of steps and every step is linear and continuous, it follows u C / 2 ~ ( a , H) and the map
(~ , ~I)
s
>s
H) ,
u,
~ u
is linear and continuous. Now assume u C s
and take ~ c G . Then for every F E ~ and
y e B(F), n(F)
n(F)
Ett
eF
F
s,t=l
(u~)estets =
,= s,t=l
n(F) =
n(F) E (u~)e~s =n(F)(u~)lF s=I
so that n(F)
E E yEB(F) s,t=l
F
F,
6. Selected Chapters of C*-Algebras
348
It follows
~ = u~, --+ ~--U.
Hence the map s
~s
is a projection of s b) For ~ 9
u,
onto s
H).
n(F)
~ov~= ~ ~ ( F1 ) ~ FE~
~"v~I~eF~,~~e~F , 9 =
yEB(F) s,t--1
n(F)
1 FE~d
>u
%y yEB(F) s,t--1
(Proposition 6.3.9.3 b)) so that _.....+
_+
U O V -~ U o V .
c) By Proposition 5.6.3.9 a => b, = ~(~1~) = ~ r ~
so that A~ = {~ertY l e ]Nn(f),y 9 B ( F ) } . By Proposition 6.3.9.1 a), n(F) 7?EA~
yEB(F)
t=l
n(F) y (~(~y)
u~B(F) t=l n(F) y e B ( F ) t=l
I ~)) =
-~(~)
6.3 Von Neumann Algebras
=~
I y
Fy, ~ I E~B(Fn(F) ~
F'E~
I
F'E~
F'e;}
,
) t=l
~<~I~>=~~(F,)
1
~
yEB(F')
~ s,t=l
~'
~'*
n(F')
f'y.
1 ( 1
I y
)
)
yeB(F')s,t=l
~(F) ~
349
Fy,
E~B(Fn(F) ~
~,~
) t=l
1 ~(~) ,~, ~<~,lI,J>
I~) ) =
9
d) By Theorem 6.3.9.13 a), there is a Fourier basis A of H such that
{<~1~>I ~ e A} C {elF~ I F E ~}. By a),c), and Corollary 6.3.9.12 a),b), =
~EA ~EA~
P r o p o s i t i o n 6.3.9.17 I1~11 ~
( 8 ) For every u E s
: sup{v<~(l~:> I( e c, 3 F E ~, ~ r E ~n(.), <~I~>=
F ~,-,-} 9
Put a - = sup{~(u~[u~)I~ E G, q F E ~, 3r Take ~ E C. For all F E ~ and r
E
]Nn(F) put
E INn(F) , ( ~ l ~ ) -
erF} 9
350
6. Selected Chapters of C*-Algebras
1
~]F :=
((r/Ir/)F)r-~r/%F
if ((~/Ir/)F)~ # 0
0
if (<~lr/)F)~ = 0
(Theorem 5.6.6.1 g)). If F E ~ and r E INn(F) such that
((,71,7),~)~ # o then
Il
I
~
\--|[I
I
~
\
F'
F
so that
= ~(u(,~)
I ~(,~))
=
= ( ( o l o ) ~ ) ~ < u ~ r ~ I ~ C ) --< ~((~1~) ~ ) ~ . Hence n(F) FE~ r=l
n(F) FE~ r=l
Ilul : < (Proposition 6.3.9.3 c)). The reverse inequality is trivial. T h e o r e m 6.3.9.18
I
( 8 ) Let p E {0} U [1, oc], q the conjugate exponent of
p, and P
0"I:qE(H,G)
~ (s
a) 0 preserves the norms. b) If p # cr then 0 is an isometry.
u,
>~I/:~(G,H).
6.3 Von Neumann Algebras
351
c) Assume p # co, denote by i the inclusion map N
/:~(G, H) --+ LY(G, H ) ,
and identify s G) and s G) with (fffE(G,H))' and (s respectively (b) and Theorem 6.1.6.6). Then i'u-- u for every u C ff-fl(H, G). d)
If G = H then
0
is an involutive homomorphism of s
e) If G = H and if we identify s and f_,l(H) with the preduals of EE(H) and s respectively (b),d), and Corollary 6.1.7.14 b)) then .~I(H) ---+ f_..1E(H),
u"
; u
is the pretranspose of the inclusion map s a)
Case 1
>s
q # c~
Take u ~ s G). By Theorem 6.3.9.7 a) and Theorem 6.3.9.14, there are an extreme Fourier set A of H , a family (~)~eA in G such that N
N
(<~ln)- (<~ln---) for all ~, r/C A, and an f c gq(A) such that (in s
G)) ,
~EA
I1~11~=
~
~~
~A
Take B E ~ f ( A ) and put
~B
f(~)
"
S(O#o
If q # 1 then by Theorem 6.3.9.14, llvll~= ~ ~B
~
pq-p = ~ ~B
c~
<
llullq 9
352
6. Selected Chapters of C*-Algebras
If q -
1 then by Theorem 6.3.9.7 a),
I1~11, _< *.
By Theorem 6.3.9.7 b),
uov - ~ If(~)lq~(.lO ~EB
so that by Corollary 6.3.9.12 b) (and Proposition 6.3.9.1 c)), p
~(v) : tr (uov) - ~
w(,~)lf(~)l a .
~eB
Since B is arbitrary, p
P II~IZTE(C, H)II Ilull,,~ >_ I~(v)l >_ Ilullg, P
P
q-~
Ilull~ = I1~11 _> II~IZ:~:(G, H)II ___ Ilull~
= Ilull~
if q :/= 1 and p
p
p
Ilull~ = I1~11 >_ II~IZ:~.(G, H)II _> I~(v)l _> Ilull~ if q = 1 (Theorem 6.1.6.6). Hence p
II~'I~PE(G, H)II = Ilullq and 0 preserves the norms. Case 2
q = c~
Let u E f~E(H, G). Take ~ E H , F E ~, and r E INn(F) with F
and put V :---~('J?_t~) 9 ~ l ( e ,
H).
For w E ICE(H, G) # , 0
I~(~)1 = It~ (vo~)l = Itr (~(.Iw*uO)l = Itr((~*~)('lOI
=
6.3 Von Neumann Algebras
353
~(~)ll<~lw~>~l ~ ~(~)1 <~ w~>ll ~ ~(~)1 ~111 w~l ~ ~(~)llu I (Corollary 6.1.5.3, Proposition 5.6.5.2 a),e), Corollary 6.3.9.12 c), Proposition 5.6.3.9 a =, b, Proposition 6.3.9.3 b), Theorem 5.6.6.1 a)). By Case 1, 0
so that 1
1
ll~lz2~(a, n)ll~(~)llull ~ I~(v)l = Itr(vou)l- Itr(~<'lu*u~>)l-
(Proposition 5.6.5.2 a),e), Corollary 6.1.5.3, Corollary 6.3.9.12 c)). By Proposition 6.3.9.17, 1
F sup{~(u~l~)l~ ~ H, 3 F E ~, 3 r < INn(F), <~-Is c) = err}
-
-
= LI~II2
so that 1
-
1
~11 ~ I~lz2~(a,H)ll ~ I~ I,
(Theorem 6.1.6.6, Proposition 6.3.9.3 b)), i
11~11=
II
H)I .
Hence 0 preserves the norms. b) Take x' C ( s By the Hahn-Banach Theorem, there is a y' E (Z;P(G, H))' with y' c ) ( a ,
H) = .'.
By Theorem 6.1.6.6, there is a u C IF-..q(H,(7) such that P
= y'.
By Theorem 6.3.9.16 a),b),d), ~ C/2q(H, G) and
6. Selected Chapters of C*-Algebras
354
P P
__+
x ' ( v ) = y ' ( v ) = ~ ( v ) - tr ( u o v ) = t r u o v
for any v C s
-- t r (-~ov) = -u(v)
H ) so that P X t
--
-~ ,~
o
Hence 0 is surjective. c) For v 9 s H), P
(i~, u) = t~ ( ~ o v ) = t~ u o v = t~(-~o~) = (v,-~)
(Theorem 6.3.9.16 a), b), d)), so that iu ~ = u .
d) follows from Corollary 6.1.7.9. e) Denote by i the inclusion map s and v 9 s
-~ s
Then for u 9 s
1
(u, iv) = t ~ ( u o v )
(Theorem 6.3.9.16 a),b),d)),
= t~ u o ~ = t ~ ( , , o v )
= ( ~ , v)
m
6.3 Von Neumann Algebras
355
6.3.10 A g e n e r a l i z a t i o n
Let E be a finite-dimensional C*-algebra, H a Hilbert right E-module, /4 an upward directed upper bounded set of ICE(H)+, and the upper section filter of Lt. Then
P r o p o s i t i o n 6.3.10.1
V:--
VU uEU
exists and ~ converges to v in ICE(H). Let w be an upper bound o f / 4 and let :~1, :~2 be ultrafilters on /4 finer than ~. By Proposition 5.6.5.13 b), ~
and ~2 converge in (ICE(H))ff to v~
and v2, respectively. By Proposition 5.6.5.13 c),
u <_vl <_w for every u E / 4 , so that V2 ~___Vl 9
By symmetry, Vl - - V2 -
Hence ~ converges in (ICE(H))ii to the supremum v o f / 4 . By Theorem 6.3.9.5 a), there is an extreme Fourier basis A of H and an
f E co(H)+ such that
v - ~f(~)~(.l~)
(in ICE(H)).
~EA
Take c > 0 and put B := {~ E A I f ( ~ ) >_ e} E q3s(A ),
~EB
p E PricE(H) (Proposition 5.6.5.11 a)) and by Proposition 4.2.2.21 (and Proposition 6.3.9.3 d)),
V (pup) -- pvp. uEU
356
6. Selected Chapters of C*-Algebras
Since I m p is finite-dimensional, lim pup = pvp u,~
in /CE(H). Hence there is a u0 E H such that I l p u p - pvpll < c
for every u E H , u >_ u0. Take u E H , u _> u0, and r/E H # and put W:---V--U
~1 : =
pr/,
r/2 := (1 - p)r/.
By Theorem 5.6.1.11 c), II<w~ll~)ll = II<wprllp~)[I = II(pwp~l~)ll ~ Ilpwpll < c.
By Theorem 5.6.1.11 c) and Proposition 5.6.3.12 b), 0 ~ (w~21~72) ~ (vz~2 ~72)= ~
f(~)(~(z~21~)l~2)-
~EA
~EA\B
~EA\B
so that
(Corollary 4.2.1.18). By Proposition 5.6.1.2 b), (wy/Ir/) = (w(ra + r/2)lr/a + 7]2) ~ 2((wr/1l~71) + (wy/2ly/2)), so that
II(WT]IT])II ~ 2(II(w~711~I)II-Jr II(WT]21T]2)]I) < 4c. Since r/ is arbitrary it follows
I1~- ~11 = il~ll <_ 4~ (Theorem 5.6.1.11 c)). Hence ~ converges to v in /CE(H).
I
6.3 Von Neumann Algebras
357
P r o p o s i t i o n 6.3.10.2 Let (E~)~et be a family of C*-algebras and let E be its C*- direct sum. For every t C I , we identify E, with the corresponding closed ideal of E and denote by qd, an approximate unit of E , . Let F, G, H be Hilbert right E-modules and for every t c I put F~ := {~ E F [ ~ = lim~x} x,~
G~ := {~ c a l~ = lim~x}
~
x,~
H~ : - {~ E H I ~ - lim~x}. X~ t
a)
For every t C I ,
H~ is (in a natural way) a Hilbert right E~-module.
Define
:= {~ e ]-I g~ I (ll,~ll)~, e c0(I)}. b) H endowed with the right multiplication N
g x E
>H,
(~,x),
>((~x~)~et
and with the inner-product
g x g
>E,
(~,r/),
> ((~[rh))~eI
is a Hilbert right E-module such that
I1~11- sup I1~11 tEI
for every ~ C H .
c) ~ ~ C H for every ~ C H and the map
~H,
~~~ tEI
is an isomorphism of Hilbert right E-modules. We identify H and H using this isomorphism.
d)
Take u E f~E(G,H) and t C I . Then u~ E H~ for every ~ E G~. Put
~H~, ~, Then u~ E f-.E~(G~, H~) and
Moreover
ll~ll = sup II~ll tel
~u~.
6. Selected Chapters of C*-Algebras
358
e)
If u E s
G) and v E LE(G, H ) , then, with the notation of d), VOU)t
:
VtO?2 t .
If (~, rl) E a • H , then, with the notation of d),
f)
(~(-I~))~ = ,7, (. I~) for every t E I and
e co(I).
(11~( Ir g)
Put
Ic := {u e 1-I ICE,(G~, H,) I (llu~ll)~e, e c0(1)} tEI
and endow tg with the norm K:
>R+,
u,
> supllu~ll .
Then (with notation of d)) (u,)~T E E for every u E E E ( G , H ) and the map EE(G,H)
>K,,
u,
> (u,),ez
is an isomorphism of Banach spaces. h)
/.f K: denotes the C*-direct sum of the family ICE,(H,))~ez then (with the
notation of d) ) (u,),~z E Y, for every u E ICE(H) and the map EE(H)
>E ,
u,
>(u~)~e,
is an isomorphism of C*-algebras. a), b), and c) are easy to see. d) We have u~ = lim u(~x) = lim(u~)x x,~
x,~,
so that u~ E H~. For (~, rl) E G~ • H , ,
( ~ l ~ ) = (u~lv)= (~lu'~)= (~l(~*)~), so that u~ E s
H~) and
6.3 Von Neumann Algebras
359
~: = (~*)~. The equality
]1~1] = sup I1~1] LEI
is obvious. e) is easy to see.
f) For ~ c G~,
x,~
,~L
so that
(~(.l~))~ = ~(-I~> 9 By Proposition 5.6.5.2 a),
II,~(-I~)lI < II~l II~lI, so that ([[rh('[~}l[)~eI e co(I). g) Take ~ > 0. There is a finite family ((~x, ~?X))XeL in G x H such that
AEL
By d) and f), for every t C I ,
AEL
so that u~ e ~EL (G~, H~) and
~ ~ Ir~~, ~,rl +~ ~ ~
< ~
I1,~,(.1~)11 +
AEL
Since c is arbitrary it follows from f)
~.
6. Selected Chapters of C*-Algebras
360
(ll~ll)~/e 6o(I), i.e.
(~tt)tE I C I C .
Take v C IC and ~ C G. Then
(llv, ll)~,,
~ Co(I)
(11(r162
and from e I ==~ IIv~r ~ IIv~ll I1r it follows (v~,),cl C H -
H.
We put u" G
~ H,
~,
> ('VL~t),E I .
If for every t E I , p~-G
~G,,
q~-H,----+H
denote the canonical projection and the canonical injection, respectively, then q, ov~op, c ICE(G, H)
for every t c I and u = E ( q ~ o v , op,)
(in ICE(G,H)).
tel
Since ?.Q - - VL
for every t C I , the map ICE(G,H)
>IC,
u,
) (u,)~ct
is surjective. By d), it is an isomorphism of Banach spaces. h) follows from d), e), and g). P r o p o s i t i o n 6.3.10.3
We use the notation of Proposition 6.3.10.2 and assu-
me in addition that E~ is finite-dimensional for every t C I .
6.3 Von Neumann Algebras
a)
361
Let bl be an upward directed, upper bounded set of ICE(H)+ and let ~ be its upper section filter. Then ~ converges in ICE(H) to the supremum of ld in ICE(H).
b)
Let 92 be the set of extreme Fourier bases of H and for every ~ E I let 92~ be the set of extreme Fourier bases of He. Then for every A E 92, (A M He)~c, E H 92e eel
and the map
>H~e,
A~
> (A A HL)eeI
rEI
is bijective.
c)
If A is an extreme Fourier basis of H then <,1r
-
E), ~r
t~EA
for every r}, ~ E H .
d)
If A is an infinite extreme Fourier basis of H and if ~ denotes the filter of cofinite sets of A then
lim (~I~) = 0
lim u~ - 0
for every 7? E H and u E ICE(H).
e)
/ f p E PricE(H) and
and if A is a Fourier basis of Imp then A is finite
P - ~--~<'1~) 9 ~EA
a) By Proposition 6.3.10.2 g), ICE(H) may be identified with the C*direct sum of the family ICE,(H,)),eI and the assertion follows from Proposition 6.3.10.1. b) Let p be a minimal element of Pr ICE(H)\{0}. Then, identifying ICE(H) with the C*-direct sum of the family (ICE,(H,)),z, (Proposition 6.3.10.2 g)) it
362,
6. Selected Chapters of C*-Algebras
follows p, < p for every c 9 I . Since p is a minimal element of Pr K:E(H)\{0} it follows t h a t there is exactly one ~ 9 I with p, - p and p~ - 0 for every A 9 I \ { c } . This proves the assertion. c) By b) and T h e o r e m 5.6.3.13 f),
~6ANHe
for every ~ 9 I . It follows
<~Ir = ~ <~Ir ~6A
Take c > 0. By Proposition 6.3.10.2, there is a J 9 ~/(I) such that 6
II~II< for every ~ 9 I\J. By b) and Theorem 5.6.3.13 d),f),
~=
~
~<~I~>
~6ANHL
for every ~ 9 I . Hence for every ~ 9 I , there is an A, 9 q3/(A fq H,) such t h a t
~6B
for every B 9 ~3I(A N S~)~ B D A~. Put
:= [_jA~ e Vs(A) e6J
and let B 9 ~3I(A), B D A. T h e n
BNH, 9
BNH~ D A~
and so
~-
~
~<~I~>
~6BMH~
for every ~ E J . Take t 9 I \ J . Then by Proposition 5.6.3.12 a ) , b),
(6BnH~
~<~1~>>=
6.3 Von Neumann Algebras
~EBMH~
and so (by Corollary 4.2.1.18)
~<,1~>]1 <- I1,,11, (EBnH,
s
s
(EBNH~
Hence by Proposition 6.3.10.2 b), c), I
r/-
E <~)=
_<e.
sup
[
{EB
tel
~EB
Since e is arbitrary
(EA
d) By c),
lim <~1~> (,a
- 0.
Let e > 0. There are finite families (r/x)XEL and (r [I~ - ~~+L ~<1r
By the above, lim E
r/a (~]4a) -- 0.
~'~ ,kEL
Since c is arbitrary, lim u~ = O. e) By d), A is finite. Put
(6A
Then q E Pr K,E(H) and
in H such that
363
6. Selected Chapters of C*-Algebras
364
pq=q, so that Imq C I m p . Since A is a Fourier basis of Imp it follows Im q = Imp and p = q .
I
T h e o r e m 6.3.10.4 We use the notation and assumptions of Proposition 6.3.10.3 and denote by u a self-normal element of ~ E ( H ) . a)
There is an extreme Fourier basis A of H and an f E co(A) such that u-
~f(~)~(.l~)
(in ]CE(HI) ,
~EA
f ( A ) c aL~(H)(U) C f ( A ) U {0}. Moreover, for every g E C(as
with
o e o=~(.)(u) ~
g(o)= o,
g(u) = E g ( f ( ~ ) ) ~ < . l ~ )
(in EE(H))
~EA
b)
/]" we put B := {f -7/:0} then H
>H,
77,
~E~(rll~) ~EB
is the carrier of u in ff~E(H). Let t E I. By Proposition 6.3.10.2 g), u~ is a self-normal element of )UE,(H~). By Theorem 6.3.9.5 a) and Corollary 6.3.9.8, there is an extreme Fourier basis A, of H~ and an f~ E c0(A,) such that u~ = E
f~(~)~('l~)
(in ICE,(H~)),
~EA,
f,(A,)
C
ar.E,(H,)(U,) C f,(A,) U {0},
6.3 Von Neumann Algebras
g(u~) - ~
9(L(~))~(-[~)
365
(in ~E~(H,)).
CEAt
Put
A := U A t , tEI
>K,
f'A
~,
~f~(~)if~EAt.
By Proposition 6.3.10.3 b), A is an extreme Fourier basis of H . Since
(11~*,11)~, e co(Z), I
f E co(A). The assertions follow.
T h e o r e m 6.3.10.5 We use the notation and assumptions of Proposition 6.3.10.3. Let G , H be Hilbert right E-modules and u E K:E(G,H). There are an extreme Fourier set A of G, a family (~)r162 in H such that
(ely) = (r for all ~,~ E A , and an f E co(A)+ such that u-
~f(r162162
(i~ t C E ( G , H ) ) .
~EA Moreover, for every such representation of u, II~ll = Ilfll~.
Let t E I and put ut'Gt
~ Ht,
~,
~u~.
By Proposition 6.3.10.2 d),f), ut is well-defined and belongs to K:Et(Gt, Ht). By Theorem 6.3.9.7 a) (and Corollary 6.3.9.8), there are an extreme Fourier set At of Gt, a family (~)r in Ht such that
for all ~, r] E At, and an ft E co(At)+ such that ut = ~ ~EAt
ft(~c)~(-I~)
(in K:E,(Gt, Ht)),
6. Selected Chapters of C*-Algebras
366
II~II = Ilf~ll~. Put
A "= U A , , t,EI
fA
>IK, ~,
~f,(~)if~EA,.
Since (llu~ll)~,E Co(I) and
II~ll = sup Ilu~ll tEI
(Proposition 6.3.10.2 g)), it follows f E co(A)+ and
llull- llflloo. Let E > 0. There is a J E ~3F(I) such that c
Ilu, II < for every ~ E I \ J . For every ~ E I there is a B~ E ~3s(A~) such that
for every C E ~3s(A~), C D B~. Put B := U B e E Vs(A) tEJ
and take C E ~ s ( A ) , C D B . For ~ E J
~EC
=Itu~- ~ ~,~,~~,11 ~eCnA,
For ~ E
I\J,
~EC
=llu, ~ ~,,~,~,11~ ~ECNA,
6.3 Von Neumann Algebras
f,(~)(~)((-I~)
<_ I1~11 +
c
367
6
_< I1~,11 + IIf~ll~ < 5 + ~ --
~ECnA~
(Theorem 6.3.9.7 a)). Hence (Proposition 6.3.10.2 g))
Ilu
_<~.
~EC
LEI
~EC
Since e is arbitrary,
The final assertion follows from I1~11 = sup I1~,11 = sup Ilfl(A n G)lloo - Ilflloo t, E I
I, E I
(Proposition 6.3.10.2. g), Theorem 6.3.9.7 a)).
I
7. C*-algebras Generated by Groups
Many examples of C*-algebras are constructed by projective representations of groups, which are presented in the first section of this chapter. The Clifford algebras are special cases of such representations and are treated in the second section.
7.1
Projective
Representations
of Groups
7.1.1 Schur f u n c t i o n s P r o p o s i t i o n 7.1.1.1
( -[ ) Let T be a group, f" T x T
, {oz C IKIIo~I : 1},
and p, q E [1, oc] such that 1
1
-+->1. p q We set x , y 9T
>IK ,
t
>E
f ( t s - l ' s)x(ts-1)y(s) - E
s6T
f(r, r - l t ) x ( r ) y ( r - l t )
r6T
for all (x, y) C (P(T) x (q(T) (Proposition 1.1.2.6 b), Example 1.2.2.3 a)) and denote by E the set of x E (~(T) such that x , y E (q(T) for every y C t~q(T) and such that the map
eq(T)
>
(T),
y,
is continuous. Moreover, we put
:: {~I~ e E}.
>x,y
7. C*-algebras Generated by Groups
370
a)
If x 9 g~(T), y 9 e q ( T ) , and s , t 9 T , then ( x , e t ) ( s ) - f ( s t -1, t ) x ( s t - 1 ) ,
(et,y)(s) = f(t, t-ls)y(t-ls). In particular, if x 9 E then x 9 ~q(T)
b)
s, t 9 T =~ es*et - f (s, t)est .
c)
The following are equivalent for all r, s, t 9 T " =
~,(~,~)
c~)
(~,~),~
c2)
f ( r , s ) f ( r s , t) -- f ( r , s t ) f ( s , t ) .
d)
x 9 t~l(T) =:v x 9 E , II~lI_ IIxll,.
e)
x 9 E , p >_ q ::~ I[Xllp <_ II--xll.
f)
Let ~ be a filter on E u 9 s
converging weakly to an Xo 9 gP(T) and let
such that
lim x = u x,~
in s
g)
(Definition 1.2.6.1). Then Xo 9 E and Xo = u .
If p > q then the map
is injective, the map E-----+E,
x,
>x
is continuous with norm 1, where E is endowed with the norm induced -+
by ~P(T), and E is closed in s
h)
A s s u m e p >_ q and 1 < p < co. If u c s in s
belongs to the closure
of a bounded set of E , then u 9 E .
i) If S is a subgroup of T , x e gP(T), and y e eq(T), then
(~),(y~s) -((x~)*(y~))~
9
7.1 Projective Representations of Groups
j)
If S is a subgroup of T , XeR, xes E E and
k)
If f is real then
371
R is a subgroup of S , and x E E , then
1) If r is infinite, q # oc, and x E E \ { 0 } , then C a r d T is the topological __+ cardinality of Im x . a) We have
(x,et)(s) = E f ( s r - l ' r)x(sr-1)et(r) = f ( s t - l ' t ) x ( s t - 1 ) , rET
(et*y) (s) -- ~
f(sr -1, r ) e t ( s f - 1 ) y ( r )
-- f(t,t-ls)y(t-ls)
.
rET b) By a),
(es*et)(r) = f ( r t -1, t)es(rt -1) = f ( s , t)Ss,rt-1 --
= f(s,t)5~t,~ = f ( s , t ) e , t ( r ) for every r E T , so that
c) By b),
(e~,e~),et - f (r, S)er,,et - f (r, s) f (rs, t)e~st ,
er*(es,et) = e r , f ( s , t)est - f ( s , t ) f ( r , st)e~,t. d) Take y E fq(T). For t E T ,
I(X*Y)(t)l--{E f ( t s - l ' S)X(tS-1)Y(S) I ~ sET First assume q -
oc. Then
E s6T
127(t8--1)1lY(S) I "
7. C*-algebras Generated by Groups
372
I(x,y)(t)l <__Ilyll~ ~ I~(t~-*)l = II~ll~llylloo sET for every t C T , so that x , y E s
and
IIx*ylJoo ~ IlYlloollXlll 9 Now assume q :/= c~. Then for every t E T ,
I(x,y)(t)l <_ ~ Ix(ts-1)l ~-~-~(Ix(ts-~)l~ly(sll) <_ sET
( E IX(tS-1)]) ~ql ( E Ix(ts-1)l lY(8)lq) ~ -sET sET q.-1 1 --" JlXl'lq ( E Ix(ts--1)l lY(8)lq) ~ sET (Example 1.2.2.3 a)). It follows that
~ : i(x,~)(,)l ~ <_ I1~11~-'~ ~ I~(~-')1 i~(o)1~= tET
tET sET
: Ilxllql-lE ly(s)Jq(E Ix(ts-1)l) : sET
tET
: Ilxllq-lllxlll]lyll q = (llXlllllYllq) q . Hence x , y C eq(T) and
II~,yll~ < Ilxl ~llyll~. Since y is arbitrary, it follows that x E E and
e) By a), I1~111~- llxl]~ > Ilxll~, (Proposition 1.1.2.6 b),e)) so that
7.1 Projective Representations of Groups
373
f) Take y E ~q(T) and t E T . Then
(xo*y)(t) = E
f ( t s - l ' 8 ) X o ( t S - 1 ) Y ( 8 ) ---
sET
= lim E f ( t s - 1 s ) x ( t s - 1 ) y ( s ) = X~
sET
= l i m ( ~ y ) ( t ) = (uy)(t). x,~
Hence
xo*y = uy C gq(T) , XO C E ,
-4 X 0 ~- 72.
g) By e), the map E
>s
-4
x,
>x
is injective and the map -4
E
ii
-4
>E,
xa
)x
is continuous with norm at most 1. By b), this norm is equal to 1. --+
Let u be an element of the closure of E in s
Then there is a
--+
seuqence (Xn)ne~ in E such that xn converges to u. By e), (xn)~e~ converges -4
--~
-4
to an x in t~P(T). By f), x C E and u = x C E . Hence E is closed in
s
. -4 h) Let A be a bounded set of E and ~ an ultrafilter on A converging to u Cs in the topology of pointwise convergence. By g), the map ~'E--+E,
x,
>x
is continuous. Hence ~(A) is a bounded set of ~ ( T ) . By Example 1.3.8.9, t~P(T) is reflexive, so that ~(A) is weakly relatively compact (Proposition 1.3.8.3). Let x0 be the weak limit of p(~) in gP(T). By f), x0 e E and -4
-+
U=xoEE. i) For t C T \ S ,
((xes),(yes))(t) = E sCT
f (ts-l' s)x(ts-1)es(ts-1)Y(s)es(s) ----
7. C*-algebras Generated by Groups
374
= E
f ( t s - l ' s ) x ( t s - ' ) e s ( t s - 1 ) y ( s ) = 0,
sES
so t h a t
(x~),(y~) = ((~r j) We set
s ,,~ t :r for all s , t E T .
~
st -1 E S
is an equivalence relation on T . Let L~ be the set of
equivalence classes with respect to Let y E tq(T). Take Q E ~
and t E Q . T h e n
((xes),y)(t) = E
f ( t s - l ' s)x(ts-1)es(ts-1)y(s) =
sET
= ~
f(t~-', ~)x(t~-i)~Q(~)y(~) = (~(y~Q))(t).
sET
First assume q < c~. T h e n for Q E L!,
I((xes)*Y)(t)l q - ~ tEQ
_<
~
I(-~(YeQ))(t)l q <_
tEQ
I(~(~Q))(t)l ~= II~(y~e)ll~ < II~ I~lly~Qll~.
tET
It follows t h a t
I((~)*y)(t)l ~ = ~ tET
~
I ( ( ~ ) 9 y)(t)l ~ <
QED. tEQ
QE~ Hence (xes)*y E fq(T) and
I (x~),yllq <_ I~11 lyllq. Now assume q = c~. T h e n for Q E 2 ,
sup I((~),y)(t)l - s~p I(~(y~Q)l(t)l tEQ
tEQ
___
7.1 Projective Representations of Groups
_< sup i(-x(yeQ))(t)i = iI-~(yeQ)iioo <_ il~i] IlyeQiloo . tET
We deduce t h a t (xes)*y E g~
and
il(~),yli~
< II~il ilyll~.
Thus xes C E and
IIx~ il ___ l,~ii in both cases. It follows t h a t XeR E E and
llx ll = ii
ll _< llx --;il.
k) Take y C t~q(T). For t E T ,
(~,y)(t) = ~
f (ts -1, s)x(ts-1)y(s) =
sET
=- E
f ( t s - l ' s)x(ts-])Y(S) -- (x*~)(t),
sET
so t h a t
x , y -- x , y . Hence 5 E E and
flail- I1~11. 1) Let l'I be the topological cardinality of Im ~ . By Example 1.1.2.5, R < Card T . There is an r E T with x(r) ~ O. Let s , t E T . By a),
I(~)(rt)l = I~(~)1, i(~)(~t)l = i~(~t~-~)l so t h a t
IIx~- x~ll >__Ix(~)l- I~(~ts-~)l 9 Since q ~ c~, there is a finite subset St of T such that
375
7. C*-algebras Generated by Groups
376
IIx~,- ~x~ll ~ llx(r)[ for all s e T \ S t . Let ~R be the set of subsets R of T such that -~ ~ 1 I I x e ~ - xe~ll >_ ~lx(r)l for all distinct s, t E R. Let ff~ be ordered by inclusion and let R be a maximal element of 9~ (Zorn's Lemma). By the above
Ust=
T.
t6R
Hence Card R = Card T . Since the cardinality of every dense set of Im ~ is at least equal to Card R it follows Card T _< R, Card T = R. D e f i n i t i o n 7.1.1.2
I
( "~ ) Let T be a group. We denote by ~ T , the set of
finite subgroups of T and call T locally finite if ~T i8 upward directed and US=T. S6GT
In this case, r
denotes the upper section filter of GT.
T is locally finite iff every finite subset of T is contained in a finite subgroup of T . P r o p o s i t i o n 7.1.1.3
( ~ )
We use the notations of Proposition 7.1.1.1 and
assume that T is locally finite.
a)
For ~w~y 9 E ~ ( T ) ,
x E E iy
{llx~ll I s
E ~ T } i~ bounded and in
this case
I1~11- sup IIx~ll- lim IIx~-~ll SE~T
S,~)S
-+
y E gq(T) ~
lim xesy = x y.
S,{~T
377
7.1 Projective Representations of Groups
b)
The following are equivalent for every x E E bl)
~ - lim x e s .
b2)
~ belongs to the closure of {~1 z e ] K (T)} in /:(gq(T)).
b3) ~ belongs to the closure of {~lz e t~I(T)} in ~(gq(T)). b4)
For every c > O, there is an SO C ~ T such that -----+
I1~-
~oll <
for every S E GT, So C S . a) If x e E , then by Proposition 7.1.1.1 j), {[Ixe--+slll S e |
is bounded.
Now assume --
sup II~il SE~T
< o0
If p -- 1, then x E E by Proposition 7.1.1.1 d), so t h a t we may assume p ~ 1. This implies q -~ co. Take y E gq(T). For S E | E t((Xes)*y)(t)lq ~tCT
]]Xe---sIIqllYllq-~ o~qllYllq"
Hence E
](x*y)(t)iq < lim inf E
Thus x C E
]((xes)*Y)(t)lq < o~qllYlq "
_ _
S,Q~T
tET
_ _
tET
and ]1511 _< ~ .
By Proposition 7.1.1.1 j),
flail-
sup Ixe--sli = lim Ilxe--s I .
S6GT
S,~T
Take y E t~q(T). If q = c~, then p = 1 and Jim ] 1 ~
S,@T
- xll, = 0 ,
so that, by Proposition 7.1.1.1 d), lim xes = x ,
S,~T
q
7. C*-algebras Generated by Groups
378
lim ( x e s ) , y = lim x e s y = x y S,~T S,~T Hence we m a y assume q-r
x,y.
By P r o p o s i t i o n 7.1.1.1 a), lim xeset = xet
S,I~T
for every t E T . It follows lim x e s ( y e A ) = ~ ( y e A )
S,{~T
for every A E g3f(T ) . Let c > 0. There is an A E g3f(T ) such t h a t IlYeT\Ailq < ~. By P r o p o s i t i o n 7.1.1.1 j),
IIxesy-
~yll < - I I x e s ( y e ~ ) -
-J(yeA)ll+
+llxes(yeT\A)ll + II~(yeT\~)ll IIxe~(yeA) - ~(yeA)ll + 211~llc. Hence lim sup I l x e s y -
xyll <_
S,I~T
211~11~.
being arbitrary, it follows --+
lim x e s y = x y . S,~T
bl => b2 => ba and bl => b4 are trivial. ba => b l . Let e > 0. There is a z E f l ( T ) such t h a t c
I l z - ~ l l < 5" There is an So E (5 such t h a t c
IIzeso- Zlll < ~. By Proposition 7.1.1.1 d),j), for every S E (hT, SO C S,
II~-
xll ___I ~ -
z~l + I I z ~ - ?'l + 117- ~ll < _.....+
c
c
c
< II(~ - z)~ll + IIz~ - z I1 + 5 <_ 117"~-zll + 5 + 5 < ~,
7.1 Projective Representations of Groups
379
Hence lim xes = x .
S,I~T
b4 =~ bl. Take e > 0. By b4), there is an So in ~T such that --+
~
E
I I = e s - Xesoll <
for every S E |
So C S . By a), x , y - lim (xes)*y S,I~T
Take S E ~T with So c S. For R E |
for every y E lr --+
-----+
~
-
~
C
S C R, C
- X~oll + I 1 ~ o - x ~ l l < ~ + ~ =
~.
Hence, by Proposition 1.2.1.7, II~' - xesll < lim inf l l x e R - xe~ll < ~. R,d,r
Since e is arbitrary
--+
x = lim x e s .
I
S,Or
P r o p o s i t i o n 7.1.1.4
( "7' )
We use the notations of Proposition 7.1.1.1 and
suppose that T is locally finite and that p >_ q. Let G be the set of a E co(T) such that
lim E
S,Q~T
a(t)x(t)
tES
exists for every x E E and such that the map
g" E
> IK
x,
~ lim E a ( t ) x ( t )
'
S,~ST tES
is continuous with respect to the norm II'IIE'E~~+,
x,
~11~11
(Proposition 7.1.1.1 g)).
a) G is a vector subspace of co(T), a E G =~d
E E ',
and
G
>IR+,
a,
>llgl]
is a norm. We endow G with this norm and denote the closure of IK (T) in G by H .
7. C*-algebras Generated by Groups
380
b)
a E eq(T) =:> a E G, I1~11~< Ilallq-
c) H is the closure of gq(T) in G . d)
a E G :=~ Ilall~ _< II~ll.
e)
If R , S
are subgroups of T such that R c S then for a E G ,
Ila~ll ~ Ila~sll.
aeR, aes E G ,
f) a E G ==~ I1~11= sup Ila~sll. SEe
g)
{ ~ l a E G} is a closed vector subspace of E ' . In particular, G and H are complete.
h)
For every x E E , the map
x'H
>IK,
a,
>~(x)
>H',
x~
>a:
is continuous and the map E is an isometry.
i) If u" H -+ co(T) denotes the inclusion map, then
u " e~(T) ----+ E is the inclusion map, where we identified E with H' using the isometry
oSh). j)
For every a E H , = l i m h-~s. S,{~T
k)
Assume p = q = 2. There is a unique operator
r 9s
~ co(T)
such that (r
=
(et * ~71~) = E f(t,t-ls)~7(t-ls)~(s) sET
for all ~,r/Ee2(T)
and t E T . r has norm 1, I m r
the map
7.1 Projective Representations of Groups
r163
y,
381
>r
is an operator of norm 1, and r
= x
for every x E E , where we identified E with H' using the isometry of h). a) is easy to see. b) For x C E ,
ax E el(T) and E
a(t)x(t) _< ilall~llxll. < Ilail~llzli~
tET
(Proposition 7.1.1.1 e)). Hence a e G and [l~l[ <_ [Jailq . c) follows from b). d) Take x E g l ( T ) # . Then x E E # (Proposition 7.1.1.1 d)), so t h a t
I(a,x}l-IEa(t)x(t)[--I~(x)l I
tET
_~ ilali 9
I
Thus Ilall~ = sup{lI i x 9 ~I(T)#} _~ II~II. e) For x E E , xeR, xes E E so t h a t aeR, aes E G and
ia~(x)l = la~(x~)l
_< lia~]l l i ~ i i ~
< Ila~ll ]lxl].
(Proposition 7.1.1.1 j)). Hence
f) We may assume ~ :/: 0. Take a e ]0, I1~11[. Then there is an x C E # with
< ~(x). Furthermore, there is an S c |
such t h a t
< la~(x)l 9 Hence
< Ita~ll,
7. C*-algebras Generated by Groups
382
< sup ll~--~ll, S6~T
Ilall _< sup Ila~ll. $66T
The reverse inequality follows from e). g) Let x ' E E' and let (a~)ne~ be a sequence in G with
lim n-.-).oo
an:
x'.
By d), (an)ne~ converges to an a 6 c0(T). For every c > 0, there is an m~ E IN such t h a t II~,,- %11 < for all n , p E IN, n > m~, p > m~. Take x 6 E . By e),
Iambs(x) - a ~ s ( x ) ] <_ II'(a. - a,)e~]l i]x]]E <_ I]~d. - ~d,] I ]Ix ]E for all n, p E IN and S 6 |
It follows t h a t
la~(~) for all e > 0, S 6 |
- a~s(~)l _< ~11~11~
and n 6 IN, n > m~. Since e is arbitrary, lim
E a(t)x(t)
S,~T t6S
exists, a e G , and lim ~ - ~. n--~oo
Hence x' = g. h) Take x E E . It is obvious that ~: is continuous and ___q__
Ilxll _< Ilxll~.
Let a' E H ' . By b) and c), there is an x 6 eq-1 (T) such that
a ' = (',~> on iq(T). For S E |
there is an a E H # such that
a = a~, Hence
~(~)
= llx~ll~.
7.1 Projective Representations of Groups
IxesllE- ~
a(t)x(t)
= ( a , x } = a'(a) <
383
Ila'll-
tCS
By Proposition 7.1.1.3 a), x E E and
il~ii~ = ~up i ~ i ,
_< iia' i.
SEOT
It follows t h a t ) - a ' ,
I l x l l - tla'll > lixil~, so t h a t
II/11- llxll~. Since the m a p E
'tH',
x , ~ ~2
is obviously injective, it is an isometry. i) Let v" e I ( T ) -+ E denote the inclusion map. For x E el(T) and t C T ,
(~)(t)
= ~(t) = ( ~ , , ~} = (~, ~'~} = (~'~)(t).
Hence VX
j) Take c > 0. There is a b
--
r
ltlX
~[(T)
~
l)
--
?.t t .
such t h a t g
I~-bll
< ~.
Let S c 0 such t h a t b = bes.
By e),
I I ~ s - besll = II(a- b)~sll _< I1~- bll < ~, so t h a t N
g"
C
II~c~s - ~11 <_ tl~c~ - b~slL + lib - ~lt < ~ + ~ = ~. Hence
7. C*-algebrasGeneratedby Groups
384
= lim a~s. S,~T
k) Let ~, rl E t~2(T). By Proposition 7.1.1.1 a),
(r
f(t,t-1s)71(t-1s)~(s)
: ~ sET
for every t E T . Take c > 0. There is an A E ~ I ( T )
with
C
I~(~11~ < sET\A
2(1 + 11771121"
For every s E A, there is a Bs E ~ I ( T ) such t h a t I~(t-ls)lCard A <
T\Bs.
for every t E
2(1 + 11~]12)
Put
B := U Bs E ~ I ( T ) . sEA
For t E
T\B, I(~, 9 ~1~}1 = EsETf(t, t-ls)~(t-ls)~ I <_
sEA
sET\A
g-
< - 2(1 + 01~]12) Hence <e.,r]l~} E Let y E s
co(T). and let
nEN
be a Schatten decomposition of y. Then for any t E T ,
le.I I(~,~1~)1 __ ~ nEN
nEN
le.I ~ sET
I~.(t-~)l I~.(~)1 <
7.1 Projective Representations of Groups
~
10~111~11=11~112=
nEN
385
10~1: Ilyll~.
~
n6N
Hence the m a p
T ---~ ~ ,
t~
On (et*T]nl~n) nEN
is well-defined, belongs to c0(T), and its norm is at most [lulls. We denne CU to be the above map. Then r is an operator with norm at most 1. Let 1 be the neutral element of T . Since I<el.ellel)l : 1, the norm of r is 1. Take s, t E T . Then (ct.elles)
:
f(t, t-Is)el(t-Is)
:
f(s, 1)6st
so that r
= f ( s , 1)es.
Hence ]K (T) C I m r Let ( , ~ E g2(T), S E |
and x E E . For t E
T\S,
(et*(ves)I@s) = ~ f(t, t-18)rl(t-18)es(t-18)~(8)es(8) sET so t h a t := ~(<-l~s>n~) c n< (T) c H .
Moreover,
tET
:
O,
7. C*-algebras Generated by Groups
386
= ~(x,(v~s))(s)~(~)~(~)=
(;(v~)l~)--
sET
(('l~)n~s, ;).
= It follows
I1~11= =
sup{l(~,x)l Ix 9 E,
sup{l(('l~es)~es, ~)1 Ix
II;11 ~
9 E,
1} =
II;11_
___II(-I~s)~sll~ (Theorem 6.1.6.6). Since lim ('l~s),7~s = ('1~>,7
S,~T
and since, by g), H is complete, we get
r
e H,
Let y e EI(g2(T)) and let
nEN
be a Schatten decomposition of y. Then by the above,
Cy- ~
0~r
~H
hEN
and
hEN
= Z On(('l~)'7n, ;) -- (Y, ;) nEN
for every x C E . Hence I m r C H and
1} __
7.1 Projective Representations of Groups
IIr
- sup{l(r
387
x>l Ix E E, II~ll ~ 1} =
= sup{l
I
for every x E E . Theorem 7.1.1.5
( -[ ) Let T be a group, 1 its neutral element, p E [1, 21,
and
f'TxT
>{aEIK]
]a[--1}
such that
f(1,1) - 1 and f ( r , s ) f ( r s , t) - f ( r , s t ) f ( s , t) fo;" all r, s, t E T . We set x,y" T
>IK,
t~
>~
f(ts -1,s)x(ts-1)y(s)
sET
for all x, y E gP(T) and denote by E the set of x E gP(T) such that x , y E gP(T) for every y E gP(T) and such that the map z " gP(T)
>
(T),
y.
>x , y
is continuous. Put x
E'-
xEE
.
a)
t E T ::> f ( t , 1) - f ( 1 , t ) - 1, f ( t , t -1) - f ( t - 1 , t ) .
b)
x E gP(T) =v e l , X
c)
~, v e E ,
--
X$el
z 9 e~(T) ~
--
(~,y) , z __.+
d)
X.
x, y E E ::v x , y E E , x , y - -
__+ ---->
xY .
x,(v,~)
388
e)
7. C*-algebras Generated by Groups
E endowed with the bilinear map
ExE
)E,
(x,y),
)x,y
is a unital algebra with el as unit.
f)
The following are equivalent: fl)
s , t E T =~ es*et = e t , e s .
f2)
T is commutative and
~,t e T ~
fa) x,y E eP(T) ~ f4)
.f(~, t) = S(t, ~).
x*y = y , x .
The algebra E is commutative.
-.+
g)
E is a closed unital subalgebra of algebra), the map E
s
)E,
(and therefore a unital Banach
x ~. ~ x
is an isomorphism of unital algebras, and its inverse is continuous with norm 1, where E is endowed with the norm induced by gV(T).
h)
If p=2
and T is infinite, then
N/C(e2(T)) = {0}. We denote by F the set of x E gP(T) such that y , x E gP(T) for every y E gP(T) and such that the map x .~(T)
~~(T),
y ~
y,x
is continuous.
il) /jr u E
{etlt E T } c and x "
uel then x E F and +X~U.
i2) {~lt E T}C= (~)c= {yly E F}. i3) {~tlt E i4)
If T
T}CC=
(~)cc = ~.
is commutative and f (s, t) -
E = F, x
f (t, s) for all s, t E T , then
- x for all x E E and (E) ~ = E .
7.1 Projective Representations of Groups
i5)
389
If T is finite then the map
~E
~,(E) ~,
__~ X
i
+___ > X
is linear and bijective and
~(x y) = (~y)(~-~) for all x , y E E .
j)
k)
The following are equivalent for every x C E "
jl)
X C Ec
j2)
t C T ~ x,et = et,x.
j3)
s, t C T =:> f ( t s -1, s)x(ts -1) = f ( s , s - l t ) x ( s - l t ) .
j4)
s, t C T ~ x ( s - l t s ) = f ( s , s - l t s ) f ( t , s ) x ( t ) .
j5)
y e gq(T) =* x , y = y , x .
Let 9~ be the set of conjugacy classes of T and 2
the set of finite Q c 9~
such that f(r, r - l t r ) f ( t , r) -- f ( s , s - l t s ) f ( t , s) for every t E Q and for all r, s E T with r-ltr_
s-its.
An element x C E belongs to E ~ iff x - O (s, t) E T x Q ~ for every Q E 2 .
ff 2
on T \
U
x ( s - l t 8 ) - f (s, s - l t s ) f (t, s)x(t)
is finite, then
Dim E c - Card 2 . If { t } c 9 1
and
for a t C T , then { t } c 2 f(s,t)-
iff f(t,s)
for all s E T . I f T is commutative and S "- {s E Tit E T ~
f(s,t) - f(t,s)}
then
Dim E c = Card S.
7. C*-algebras Generated by Groups
390
1) For t E T and m, n E lN,
f(tm, t n ) = f ( t n , t m ) . m)
If
is a group homomorphism then (~)
9 (yv) = (~ 9 y)v
for all x, y E gV(T). In particular x e E =:~ x ~ e E ,
I1~11- I1~11.
a) If we set s = 1 in the equality for f , then we obtain
f(r, 1)f(r, t) = f(r, t ) f ( 1 , t), so t h a t
f(r, 1 ) : f(1, t) for all r, t E T . It follows that
f(r, 1) : f(1, 1) : 1, f(1, t) = f(1, 1) = 1 for all r , t E T . P u t t i n g r = t, s = t -1 in the equality for f we deduce t h a t
f(t, t - 1 ) f ( 1 , t) = f(t, 1 ) f ( t -1, t). By the above,
I(t, t - ' ) = f ( t -~ , t). b) By a) and Proposition 7.1.1.1 a),
(el.x)(t) = f ( 1 , t ) x ( t ) = x(t), (x.el)(t) = f(t, 1)x(t) = x(t)
7.1 Projective Representations of Groups
for every t E T , so that el*X
--" X * e
I
---- X .
c) Let t E T . By Proposition 7.1.1.1 a), for s E T ,
((x,y),et)(s) = f (st -1, t)(x,y)(st -1) = = E
f ( s t - l ' t ) f ( s t - l r - l ' r)x(st-lr-1)Y(r) =
rET
-= E
f ( s t - l r - ] ' rt)f(r, t)x(st-]r-])y(r) =
rET : E f ( s q - l ' q ) f ( q t - l ' t)x(sq-1)y(qt-1) -qET
: E f(sq-l'q)x(sq-1)(y*et)(q) qET
-- (X * ( y * e t ) ) ( 8 ) .
Hence
(~,y),~ = ~,(y,~). This implies that
(~,y),(z~) = z,(y,(z~)) for every A E ~ / ( T ) . Let ~ be the upper section filter of ~ I ( T ) . Since
liz,(y,(z~))-
~,(y,z)l ~ = i i x , ( y , ( z ~ A - z))li~ <
< II~li Ily,(z~ - z)ll~ < ll~li ii~ii l l z ~ -
zll,
for every A E ~ I ( T ) , it follows that
limx,(y,(zeA)) -- x , ( y , z ) A,iY in t~P(T). Thus for t E T ,
((~,y),z)(t) = ~ / ( t ~ sET
-1, ~)(~ 9 y)(t~-l)z(~) =
391
7. C*-algebras Generated by Groups
392
= lim((x,y),(zeA))(t) = l i m ( x , ( y , ( z e A ) ) ) ( t ) = A,td
n,td
= (~,(y,z))(t).
Hence (~,y),z
= ~,(y,z).
d) follows from c). e) follows from b) and c). f~ =~ f2. Take s, t E T . By Proposition 7.1.1.1 b), f (s, t)est = es*et = et,e, = f (t, s)et, .
Hence T is commutative and f(s,t)=
f2~f3.
f(t,s).
For t E T , (x,y)(t) = E
f(ts-l' s)x(ts-t)Y(S) =
sET
=E f (r, r - l t ) x ( r ) y ( r - l t )
=
rET
= E
f ( t r - l ' r)y(tr-1)x(r) = (y,x)(t),
rET
so that x,y = y.x.
f3 ~ f4 =~ fl is trivial. g) By b),d), and e), the map E
",C(e~(T)),
x,
~
is a unital algebra homomorphism. By Proposition 7.1.1.1 g), this map is injective, E is closed in s and the map E
>E,
x,
>x
7.1 Projective Representations of Groups
393
is continuous with norm 1. In particular, E is a closed unital subalgebra of
~(e~(T)). h) follows from Proposition 3.1.2.18 b). il) For t c T , et*X
~
etX
---- e t h e l
-- ~tetel
-- uet.
If y E g2(T) and s C T , then
uy : u ( E y(t)et) : E y(t)uet = E y(t)et,x, tET
tET
tET
tCT
tET
-- E y(t)f(t, t-ls)x(t-ls) = E f(sr-l' tET
r)Y(ST-1)x(r)
:
(y'x)(8)
rET t---
(Proposition 7.1.1.1 a)). We deduce that uy - y , x , x C F, and x = u. i2) Take x E E and y E F . B y c ) , f o r tET, _._.~ ~---
__+
~y~-
x(~,y) = ~,(~,y) = (x,~),y =
= y(x,~)-
~
...4
y(x~)-
+---..+
y~r
so that --.44----
xy
"~"----+ =
yx,
y e ( E ) ~, {YlY e F} c (E)~ c {Ztlt e T} ~. By il), 4--
{~-~lt e T} ~ c {YlY e F } , so that
{~lt e T F = (/:)~ = (~ly e r } . ia) Let u E {~lt e T} co. Put
7. C*-algebras Generated by Groups
394
X
:=
?./,e I .
By b) and i2), for y E F , x,y = Yx = Yuel = UYel = u(el,y)
= uy.
Hence x E E and ~ = u. Thus { ~ l t E T } r162C E . By i2),
{~lt
e T } c~ = (E) -* co= E -*.
i4) By f2 =~ f3, E = F and x = ~ for every x E E , so that the assertion follows from i2). i5) By g) and i2), the map v is well-defined. It is obviously bijective and linear. By c), for z E g P ( T ) ,
9 ,yz = z , ( ~ , y ) =
( z , ~ ) = ~-~YXZ,
so t h a t by g), v
(~)
=v
(x~, y)
~--
=x,y=
~_~ (-~)(~) Yx = v Y v
.
j~ =v jl =~ j2 and j3 r j4 are trivial. j2 =~ ja follows from Proposition 7.1.1.1 a). j3 =~ js. For t E T ,
(x,y)(t) = ~
I(t~ -', ~)~(t~-')~(~) =
sET
= E f(s, s-lt)x(s-lt)y(s) = (y,x)(t). sET
k) Take Q E ~ and t E Q. Let x E E c. By jl =~ j4), x(s-lts)
: f (s, s - l t s ) f (t, s ) x ( t ) ,
for every s E T , so that
I~(~-~t~)l = Iz(t)l. If Q is infinite, then x E gP(T) implies x - 0 on Q. Assume Q finite and let r, s E T such that
7.1 Projective Representations of Groups
395
r-ltr = s-Its. We obtain
f (r, r-ltr) f (t, r)x(t) = x ( r - l t r ) = = x ( s - l t s ) = f(s, s - l t s ) f ( t , s)x(t). If Q ~ 2 , it follows t h a t x = 0 on Q . Hence Dim E ~ < C a r d 2 . m
If Q E L~, then the m a p
Q
~ IK,
s-its,
~ f(s,s-lts)f(t,s)
is well-defined. E x t e n d i n g it by 0 on T \ Q , we get, by j4 ==~ jl, an element of E ~ . Hence Dim E c >_ C a r d 2 ,
Dim E c = Card 2 , {t} E 9~ implies s - i t s = t for all s E T and the final assertions follow. 1) We may assume m < n . Since
f(t'~,tn-'~)f(tn,tm ) -- f ( t m , t n ) f ( t n - m , tm), the relation follows from a) by induction. m) For t E T ,
((x~) . (y~))(t) = E
f ( t s - l ' s)x(ts-1)cfl(ts-1)y(s)~9(s) =
sET
= )9(t) E
f ( t s - l ' s)x(ts-1)y(s) -- ((x * y)qo)(t).
sET
Since y E gP(T) = ~ y ~ E eP(T), Ily~llp = Ilyll, it follows
E E ~
x ~ E E , I1~11 = I1~11.
m
396
7. C*-algebras Generated by Groups
Proposition 7.1.1.6
( 7 ) Let T be a group. Take
f:T•
>{a 9
]a[=l},
g: T ----+ {a 9 IK[ lal = 1}, and p 9 [1, 2]. We define x.y" T
~ IK,
t,
~~
f(ts -~,s)x(ts-1)y(s),
sET
x*: T
~ ~,
t,
~ g ( t ) x ( t -1)
.for all x, y E eP(T).
a)
b)
The following are equivalent: ~1)
~, y c ~ , ( T ) , x , y e ~,(T) ~ (~,~)* : y**~*
a2)
s , t E T =~ (es*et)* = e ; , e : .
a3)
s , t E T =~ f ( s , t ) g ( s ) g ( t ) = f ( t - l , s - 1 ) g ( s t ) .
t E T =:~ e; -- g ( t - 1 ) e t - 1 .
c) The following are equivalent:
Cl)
X ~ ~ P ( T ) ::~ x** ~- x .
c2) t E T ~ e~* - et.
~)
t e T ~ g(t) = g ( t - ~ )
c4) gP(T) endowed with the map ~(T)
~ e'(T),
~,
~ ~*
is an involutive Banach space.
d) Assume the equivalent assertions of c) are fulfilled, define (xlY} - E
x(t)y(t)
tET
for all x , y C gP(T), and let F denote the set of x C gP(T) such that 9 , ~ e ~,(T) fo~ ~u ~ e ~.(T)
following are equivalent:
Th~
~* e F for ~ y
~ E F ~d
th~
7.1 Projective Representations of Groups
e)
f)
dl)
x C F , y , z 9 gP(T)=v (x,y[z) = (y[x**z).
d2)
r, s, t 9 T =v (er,e~[et) = (e~[e~,et).
d3)
s, t 9 T ~ f (s, t)g(s) = f (s -1, s t ) .
397
The following are equivalent: el)
x 9 gP(T) C) ]R T ~ x* --- x .
e2)
t9
e3)
t 9 T =~ t = t -1 , g(t) -- 1.
x , y 9 t~P(T) =v (x*[y*} = (yIx).
al ==~ a2 is trivial. a2 =V a3. By Proposition 7.1.1.1 b), for r C T , (es:~et)*(r - 1 ) -- g ( r - 1 ) ( e s , e t ) ( r )
--__
= g ( r - 1 ) f ( s , t) e~t(r) - g ( t - l s - 1 ) f ( s , t)5~,~t, 9
9
(e t ,e~)(r
-1
1
) = E
9
f ( r - l q - ' q)et ( r - l q -
1) , ( q )
e~
=
qET
= E
f(r-lq-l'
q)g(r-lq-1)et(qr)g(q)es(q-1)
--
qCT
= f(r-ls, s-1)g(r-ls)g(s-1)et(s-~r)
--
= f ( t -1, s-~)g(t-1)g(s-1)Sr,st, so t h a t
f ( t -1 , s-1)g(t-1)g(s -1) -- f ( s , t) g ( t - l s -1)
f ( s , t)g(s)g(t) = f ( t -1, s -1) g(st) . a3 =~ a l . For t C T ,
(y**x*)(t) -- E sET
f(ts-l's)y*(ts-1)x*(s)
--
7. C*-algebras Generated by Groups
398
= ~ f(ts-~,s)g(ts-')Y(st-l) g(s)x(s -1) = sET
= ~
f( s-l, st-1)g(t) y(st -1) x(s -1) =
sET
= g(t)Ef(t-lr-l,r)x(t-lr-1)y(r)
=
rET
= g(t)(=,y)(t-,)
= (=,y)*(t),
and so
( x , y ) * = y* *x* . b) Given s E T , e;(s)
-- g(s)et(s
-1)
= g(t-')~,-,,,
-- g(t-1)5,,s
= g(t-')~,-,
-, --
(,),
so t h a t
Cl =~ c2 is trivial. c2 =~ c3. By b),
~, = g ( t - i ) ~ , _ l = g ( t - i ) g ( t ) ~ , so t h a t
g(t) = g ( t - ~ ) . c3 =~ c~. For t E T ,
x**(t) = g(t)x*(t -1) = g(t) g(t -1) x(t) = x ( t ) . Hence X**
cl r
Ca is obvious.
--
X.
399
7.1 Projective Representations of Groups
dl :=> d2 is trivial. d2 => d3. By b) and Proposition 7.1.1.1 b), for r E T,
f (r, s)5,.s,t = (f (r, S)erslet) = (er,e~iet) = = (~1~;,~)= (~;,~)(~)=
-- E
f(sq-l,q)er(Sq-1)et(q) ----f ( s t - l , t ) er(St -1) =
qET = f ( s t - l , t ) g ( s t - 1 ) e r ( t s -1) = f(st-l,t)g(st-1)Srs,t, so that
f(ts -1, s) = f ( s t -1, t) g(st-1), and
f(s,t)g(s) = f ( s - l , st). d3 ==> dl. (y[x**z) -- E y(t) (x**z)(t) :
tET
:Ey(t)(Ef(ts-18)X*(tS-1)Z(S)" tET sET --"
Ey(t)( tET
---
E f(ts-1,8) g(ts-1)x(st -1) ~(8)) sET
: E . ( , > ( E :(.,-1.,>-(.,-1> tET sET
-
)
~(x,y)(s)z(~)
-
-~-
-
(~ * ylz).
sET e) follows from b). f) (x*lY*) = E x*(t)y*(t) = E g ( t ) x ( t
tET
-1) g(t)y(t -1) =
tET
= E
tET
x(t-1)y(t-1)-- E
tET
y(t)x(t) = (yix).
I
7. C*-algebras Generated by Groups
400
Definition 7.1.1.7
( ~ ) Let T be a group with neutral element denoted by 1. A Schur f u n c t i o n f o r T (also called normalized factor set in the literature) is a function
> {~ ~ ~1 I~1- i}
f" T x T such that
f(1,1) = 1 and f(r, s ) f ( r s , t) -- f(r, st)f(s, t) for all r, s, t 9 T (Schur, 1904). We put
f'T
} {a 9
I [a[=l},
t,
>f(t,t-t).
For all x, y 9 g2(T), define x,y'T
) IK,
t,
) Ef(ts
-1 s)x(ts-1)y(s)
sET
x* " T
} IK,
t,
~ f(t) x(t -1)
and denote by S ~ ( f ) (or simply Sw(f)) the set of x 9 g2(T) such that x , y 9 g2(T) for every y 9 g2(T) and such that the map g2
g2
is continuous. We put .-}
A "= {~[x e A} for every A C S w ( f ) . Sw(f) is a vector subspace of g2(T) which we endow with the norm
(Proposition 7.1.1.1 g)). If T is locally finite, then we denote by $ ~ ( f ) simply S o ( f ) ) the closure of IK (T) in S w ( f ) .
(or
7.1 Projective Representations of Groups
401
If T' is a group, f' is a Schur function for T', and u" T --+ T' is a group homomorphism then T x T
> {a 9 ]K Ilal- 1},
(s,t),
>f'(u(s),u(t))
is a Schur function for T. If T is finite, then t l ( T ) = 8c(f) = 8 w ( f ) = t 2 ( T ) .
If T is infinite (and locally finite for oct(f)), then the above four spaces are pairwise distinct (Proposition 7.1.2.4). P r o p o s i t i o n 7.1.1.8 Let T be a group, 1 its neutral element, .T the set of Schur functions for T , and A the set of
e {~ e ~1 I~1- 1} T with I(1) = 1 . For f 9 .T and l 9 A , put
f*'T•
>{a 9
A''T•
A*'T
a]=l},
(s,t),
>{aeIK[la]-l},
>{a 9
(s,t),
]a]=l},
s,
>f(t-l,s-1),
>
~(~)~(t) ~(~t)
'
>A(s-1).
a) .T is a subgroup of { a e IK I lal = 1} TxT such that f is the inverse of f for every f E iT.
b)
If f, g E F then f* C J:, f** = f , and (fg)* - f , g * .
c)
{A'II e A} is a subgroup of iT and (~,). - (~.),
for every A C A .
7. C*-algebras Generated by Groups
402
a) is easy to see. b) For r,s,t E T,
f*(r, s)f*(rs, t) : f(a -1, r-1)f(t -1, 8-1r -1) : = f(t-1, a-1)f(t-18-l,r -1) = f*(s,t)f*(r, st) so that f* E $'. The other assertions are easy to see. c) Take A E A. Then A,(1, 1) = )~(1))~(1) = 1, )~(1)
~'(~, ~)~'(,-~, t) =
~(r)~(~) ~(~)~(t) A(rs) )~(rst)
~'(~, ~t):~'(~, t) = ~(~)~(~t) ~(~)~(t) ~(~t) ~(~t)
~(r)~(s)~(t) A(rst)
:~(r)~(~)~(t)
~(~t)
=A'(r,s)A'(rs, t)
for all r, s, t E T so that )r E $'. Moreover,
(~*)'(~, t) =
~* (8)A* (t)
~*(,t)
/~(s- 1))~(t -1 )
~(t-,~-,)
= ~'(t-',, -') = (~')*(,, t)
for all s, t E T . It is easy to see that {A'I)~ E A} is a subgroup of $'. Proposition
7.1.1.9
( 7 ) Let f be a Schur function for a group T and
s, t E T . a)
f(t)---- f ( t - i ) .
b)
f ( s - l , s t ) f ( t , t - l s -1) = f(st, t-ls -1) = f(st).
c)
f(s, t)f(s) = f(s -1, st).
d)
f ( s , t ) f ( s ) f ( t ) = f ( t - l , s -1) f(st).
e) f(s, t-1)f(t -1) = f(st -1, t). a) follows from Theorem 7.1.1.5 a). b) By Theorem 7.1.1.5 a),
f(s-l, s t ) f ( t , t - l s -1) = f ( s - l , 1 ) f ( s t , t-ls -1) = f(st, t-ls-1). c) We have
II
7.1 Projective Representations of Groups
f(s, t ) f ( s t , t-is -1) - f(s, s-1)f(t, t - I s - l ) ,
so that, by b), N
f(s,t)f(s)-
f ( s , t ) f ( s , s -1) = f(st, t - l s -1) f ( t , t - l s -1) =
-- f ( s - l , s t ) f ( t , t - l s -1) f ( t , t - l s -1) -_ f ( s - l , st).
d) By a) and c), f ( t - l , s -1) f(st) -- f ( t - l , s -1) f ( t - l s - l , s t )
--
= f ( t -1, t) f ( s -1, st) = f(s, t ) f ( s ) f ( t ) .
e) By a),c), and d), f(st -1, t) -- f ( t s -1 , s ) f ( t s -1) = = f(s -1, st -1 ) f(s -1) f(t) = f ( s , t - 1 ) f ( t - 1 ) .
403
7. C*-algebras Generated by Groups
4O4
7.1.2 Projective Representations T h e o r e m 7.1.2.1
( 7 ) Let f be a Schur function for a group T .
a) s , t 6 T =:~ e*s,et = f ( s , s - l t ) e s - l t ,
e,,e; = f ( s t - l , t ) e s t - 1 .
b) t E T =~ e~ ,et = et,e; = el. c) S ~ ( f ) endowed with the bilinear map Sw(f) x Sw(f)
~Sw(f),
> x*y
(x,y) ,
(Theorem 7.1.1.5 e)) and with the involution
Sw(f)
>Sw(f),
x,
)x*
(Propositions 7.1.1.9 a),d), 7.1.1.6 aa =r al,
ca : : ~ C l )
i8 a unital W * -
algebra with el as unit and the map
>s
qO'Sw(f)
x,
is an injective unital W*-homomorphism.
)x
(e2(T), ~, el) is the G N S -
triple associated to the positive linear form
x"Sw(f)
,IK,
x,
~'(Xlel).
d) If f is real then the map o IR
S ; (f)
>~ ( f ) ,
(x,y) ,
) x + iy
is an isomorphism of involutive unital complex algebras. e)
If S is a subgroup of T ,
then {xes
Iz e Sw(f)}
is a unital W * -
subalgebra of S w ( f ) .
f) If T is infinite then x E S ~ ( f ) , u E r(E2(T)) ==~II-xll ~ II~ + u I .
g) If T is infinite, then the restriction of the quotient map
s
>s
___}
to S ~ ( f )
is injective but not surjective.
7.1 Projective Representations of Groups
405
h) el(T) is an involutive unital subalgebra of S o ( f ) . el(T) endowed with the induced algebraic structure and with its usual I1" Ill-norm is an involutive unital Banach algebra such that every algebra homomorphism g1(T) -+ IK is positive.
i) If f is real then the map o
gI (T) m
~, e I ( T ) C ,
(x, y),
~ x --[- i y
is an isomorphism of involutive unital complex algebras. ____+
j) S~(f) is the (unital) von Neumann algebra on g2(T) generated by {r
k) We denote by F the set of x C g2(T) such that y , x C g2(T) for every y c g2(T) and such that the map x . 62 (T)
~ 62 ( T ) ,
V~
~V*~
is continuous. Then F = S ~ ( f ) , ~ G
for every x e S o ( f ) ,
and the map ~
--+
~'S~(f)
~
, z,
+-_ ; x
is an isometry of involutive Banach spaces such that
~(xv) = (~)(~Y) for all x, y C S o ( f ) .
l) If IK - ( ~
and if S o ( f ) is a Gelfand C*-algebra, then S o ( f ) maximal Gelfand C*-subalgebra of s
m) If IK = ~ and if y C el(T) such that
E lET\{1}
lY(t)l <- y(1)
is a
7. C*-algebras Generated by Groups
406
then the linear form
~'$w(f)
~ IK,
x:
> Ex(t)y(t) tET
is positive. In particular, if T ~ {1}, the positive linear form Xt'Sw(f)
> ]K,
x l
> (Xlel)
from c) is not an extreme point of T ( S ~ ( f ) ) and the representations of $w(f) associated to it is not irreducible. n)
Let A be a set of generators of T such that
t2= 1,
f (t) = 1
for every t 9 A . If E is a real C*-algebra with Dim Re E - 1
and u" S ~ ( f ) -+ E is an involutive algebra homomorphism then Imu C ReE.
o) x, y 9 Sw(f) :=~ x,y(1) = y 9 x(1),
x 9 x*(1) =
I1=11~.
a) By Propositions 7.1.1.1 b), 7.1.1.6 b), and 7.1.1.9 c),e),
~,~
/(s-')~-l,~ = Y(~-~)f(~-~ t ) ~ - ~ = f(~, ~-'t)~-,~,
es,e~ = es*f(t-1)et-1 = f ( t - 1 ) f ( s , t - 1 ) e , t - ~ = f ( s t - l , t ) e,t-i. b) By a) and Theorem 7.1.1.5 a),
et*et = f ( t , 1) el = e l , et.e t : f(1, t) el : el. c) By Theorem 7.1.1.5 e),g), Sw(f) is a unital Banach algebra with el as unit and ~ is a norm preserving unital algebra homomorphism. By Proposition 7.1.1.6 a3 =~ al, c3 =~Cl, d3 =:> dl and Proposition 7.1.1.9 d),a),c), Sw(f) is an involutive algebra and ~ is involutive. By Theorem 7.1.1.5 i3) and Corollary 6.3.5.5, $ ~ ( f )
is a v o n Neumann algebra on t2(T), so that S~(f)
7.1 Projective Representations of Groups
r
is a unital W*-algebra and p is an injective W*-homomorphism (Corollary 4.4.4.9 a => b). For x 9 ,So(f), X'(X*) --
(x*iel)
-- X*(I)
Xt(X * $ :T,) = (X* *
X]el)
~-
f(1)x(1)
(XlX *
--
el)
=
x(1)
--
(XIX) -- []Xll 2
=
x'(x),
(Proposition 7.1.1.9 a),c), 7.1.1.6 Ca ~ cl, da ~ d l , Theorem 7.1.1.5 b)). Hence x' is positive, Nx, - {0}, and
$o(f)/N,,Since So(f) is dense in
&(f).
eU(T), (g=(T),~,el)
is the G N S - t r i p l e associated to
x' (Theorem 5.4.1.2 k)). d) Put u" c r For x,y e ,5r
x,
;g.
and t 9 T ,
(u(x*y))(t) -- (x,y)(t) -- E f ( t s - l ' s) x(ts -1) y(s) -sET = (~,~)(t)-
((~x),(~y))(t),
(ux*)(t) = x*(t) = f ( t ) x ( t -1) = g*(t) = (ux)*(t) (Proposition 7.1.1.9 a)), so that
~(x,y) -(~),(~y),
(~*) = (~)*
Hence u is a conjugate involution and the assertion follows from Proposition 2.3.1.43 a). e) By Proposition 7.1.1.1 i), {xeslx e So(f)} is a subalgebra of S o ( f ) and it is easy to see that it is unital and involutive. It remains to show, that {xe--~ix E S~(f)} is closed in {71 x c So(f)}e2(T) (c), Proposition 6.3.4.2 d r
a, and Corollary
4.4.4.10). Let x0 E S o ( f ) and ~" a filter on {xe---+six C S o ( f ) } converging to Xo in the topology of pointwise convergence. Take t E T \ S and x C S o ( f ) . Then
7. C*-algebras Generated by Groups
4o8
(xe--*s et-1)(1) = E f ( s - l '
8)X(8-1)eS(8-1)et-I(8)
--
sET
= f(t, t-1)x(t)es(t) = O. It follows that
f(t,t-1)xo(t) = (Xo~t-1)(1)
=
= hm(xeset-,)(1) = 0
9 o(t) - o. Hence x0 = xoes and {xe---~slx e S,~(f)} is closed in {~l x e Sw(f)}e~(T) 9 f) follows from Theorem 7.1.1.5 h) and Corollary 4.2.6.7. g) The injectivity of the map follows from Theorem 7.1.1.5 h). By Proposition 5.5.1.27, s
is not a W*-algebra. By c), ST(f) is a
W*-algebra, so that the map cannot be surjective. h) By Proposition 7.1.1.1 d), t~l(T) C S T ( f ) . Take x, y e el(T). For t e T ,
/(t~-', ~)~(t~-')~(~)
I~.y)(t)] = sET
__ ~
Iz(t~-')l ly(~)l,
sET
[x*(t)l = If (t)x(t-1)l = Ix(t-1)l, so that
I(~.y)(t)l < ~ tCT
Z
I~(t~-~)l ly(~), =
tET sET
Ix*(t)l = ~ tET
Hence x . y , x* C gl(T) and
tET
Ix(t-~)l = Ilxll~.
7.1 Projective Representations of Groups
Itx,yll~ ~ Ilxll~llyll~,
4o9
I1~*11~--Ilxlll.
Thus gl(T) is an involutive unital subalgebra of oct(f) and gl(T) endowed with the induced algebraic structure and with its usual I1" lit-norm is an involutive unital Banach algebra. Let x' : gt(T) --+ IK be an algebra homomorphism. By Proposition 2.2.4.19, x' is continuous with tcT.
IIx'll---
0 or
II~'ll
- 1. We may assume IIx'll = 1. Take
Byb), 1 = x'(1) = x ' ( e t , e t ) = x'(et)x'(et ) .
Since
I~'(~;)1 _< IIx'il I1~;11- 1, I~'(~+)1 ~ IIx'll Ile~ll = 1, it follows that
x'(+;) = ~'(~+). Hence x ~ is involutive . By Proposition 2.3.4.5, x ~ is positive. i) follows from h) and d). j) By Theorem 7.1.1.5 ia),
s ~ ( I ) - {~lt e T ) ~ and the assertion follows from Corollary 6.3.5.6. k) Take x e S w ( f ) . By c), x* e $ w ( f ) . By c), Proposition 7.1.1.9 d), and Proposition 7.1.1.6 a3 =~ al, for y C g2(T),
(y,~)* = z**y* e e~(T),
y , z c e2(T),
II~,zll~ = II(y*~)*ll~ = II~**y*ll~ <_ II~*ll Ily*ll~ = I1~111 yll~. +.._
Hence x is continuous and
I1~11 < I1~11.
7. C*-algebras Generated by Groups
41o
By symmetry, F = S o ( f )
and II~ll -
II~ll. By Theorem 7.1.1.5 i2), ~ 9
By the above, u is an isometry of Banach spaces. By Theorem 7.1.1.5 c) and symmetry, for x, y 9 S o ( f ) and z 9
e2(T),
t----
~,yz = z , ( ~ , y ) =
t---
t---- t---
t'-- t - -
y(~z)=
~z
(z,~),y = y(z,~)=
so that ~(zy)
= ~(~,y) = ~,y = y 9 = (u~)(~
Take x 9 S o ( f ) . By Propositions 7.1.1.9 a),c),d) and 7.1.1.6 f), a3 =val, c3 & d3 :=v cl & dl, for y,z 9 g2(T), (~*ylz) = ( y l ~ z ) =
(ylz, ~)=
(~*,z*ly*)= +.-
= (z* Ix,y') = (y,~* Iz) = (z*ylz).
Hence u ( ; ) = u~* : ~* = ~* = ( ~ ; ) *
and u is involutive. 1) By Theorem 7.1.1.5 f4 =v f3, x, y E g2(T) = ~ x , y - y , x . Hence --at X--
X
for every x e S o ( f ) . It follows by Theorem 7.1.1.5 i2),
s~(f)
=
and the assertion follows from Proposition 4.2.2.14 b ==> a. m) Take x E S o ( f ) . By Proposition 7.1.1.9 c) and Theorem 7.1.1.5 a),
sET
7.1 Projective Representations of Groups
- '~
f(8,1)lx(~)l
= -
I1~11~
9
sET
For t E T , I(:,x)(t)l
=
E
f (ts-l' s ) f ( t s -1)x(st -1)x(s)
sET
~ [f(ts-1,s)[l/(ts-1)lIx(st-I)[Ix(s)[ sET
_<
)1 Ix(~t-x)l ~
Ix(~)l ~
=
I1~11~.
It follows
g(x* *x) = E ( x * *x)(t)y(t) >_ tET
_> (x**x)(1)y(1) -
E
](x**x)(t)l ly(t)l _>
tET\{0)
O.
Hence g is positive. Assume now T # {1} and take to E T \ { 1 } . By the above, the maps &-',S~(f)
> IK,
x,
> x(1) 4-x(to)
are positive. They obviously belong to r ( $ ~ ( f ) ) (Corollary 2.3.4.7). Since
1 (g+ + g_)
x'
2 it follows that x' is not an extreme point of r ( $ w ( f ) ) . By Theorem 6.3.6.2 b =v a, the representation of Sw(f) associated to x' is not irreducible. n) For every t E A ,
e t -- f(t)et-1 = et, so that ue~ E Re E . It follows successively,
7. C*-algebras Generated by Groups
412
t E T ==~ uet E R e E , u(IR (T)) C Re E ,
Im u C Re E . o) By Theorem 7.1.1.5 a), x 9 y(1) = E
f (t-l' t)x(t-1)Y(t) = E
tET
f (t, t -1)x(t)y(t -1) -
tET
= Ef(t-l,t)y(t-1)x(t)
- y . x(1),
tET
x 9 x*(1) = E f ( t - l , t ) x ( t - 1 ) x * ( t )
=
tET
= Ef(t-l,t)x(t-1)f(t,t
-1) x(t -1) =
tET
= ~
Ix(t)l 2 = Ilxll~.
m
tET
P r o p o s i t i o n 7.1.2.2 ( 7 ) Assume ]K - IR. Let f be a Schur function for the group T and let Q be the set of finite conjugacy classes Q of T such that f(r, r - l t r ) f ( t , r) = f(s, s - l t s ) f ( t , s) ,for every t E Q and every r,s E T with r-ltr = s-its. Put
~1 :- {Q e ~ I Q - {t-~l t e Q}}, ~2 = {Q e ~IQ n {t-lit e Q} - 0 } and denote by Q' (by Q " ) the set of Q c Q1
(Q c Q2) such that
(s, t) e T • Q, s - i t s = t -1 ==~ f(s, s - l t - l s ) f ( t - 1 , s)f(t) = 1
((s,t)E
T x Q :::F f(s,s-lt-ls)f(t-l,s)f(t) : f(s,s-lt-ls)f(t-l,s)f(s-lt8)).
7.1 Projective Representations of Groups
a)
413
The following are equivalent for every x C ($w(f))c " al)
x is selfadjoint.
a2)
x=0
on T \
U
Q and
QED/u,Q" N
x(t) -- f (t)x(t -1) for every Q c Q" and t E Q. b)
If Q' U Q" is finite then DimRe (Sw(f)) c = C a r d Q ' + 1 C a r d Q " . Z
al =~ a2. By Theorem 7.1.1.5 k), x = 0 on T \ U Q - T a k e Q c Q \ ( Q ' u Qe~ ~ " ) . By Theorem 2.2.2.7 d5), ~ = Q1 U ~2First assume Q r ~ 1 . Since Q ~ ~ ' , there is an (s,t) e T • Q with s - l t s - t -1 and
f(s, s - l t s ) f ( t , s)f(t) r 1. By Theorem 7.1.1.5 jl =~ j4, N
z(t) = x*(t) - f ( t ) x ( t -1) - f ( t ) x ( s - l t s ) = = f ( t ) f ( s , s - l t s ) f ( t , s)x(t) so that x(t) = 0. Again, by Theorem 7.1.1.5 jl =~ j4, it follows x - 0 on Q. Now assume Q c Q2. Since Q r Q " , there is an (s, t) E r x Q such that
f(s, s - l t - l s ) f ( t -1, s)f(t) r f(s, s - l t s ) f ( t , s ) f ( s - l t s ) . By Theorem 7.1.1.5 jl ~ j4 (and Proposition 7.1.1.9 a)), x(t -1) - x * ( t -~) - f ( t ) x ( t ) -
f(t)f(s,s-lts)f(t,s)z(s-lts),
x ( s - l t s ) -- x*(s-lt8) = f ( s - l t s ) x ( s - l t - l s )
--
= f ( s - l t s ) f ( s , s - l t - l s ) f ( t -1, s)x(t -1) -= f ( s - l t s ) f ( s , s - l t - l s ) f ( t -1, s ) f ( t ) f ( s , s - l t s ) f ( t , s ) x ( s - l t s ) ) ,
414
7. C*-algebras Generated by Groups
so that x(s-lts)
= 0. By Theorem 7.1.1.5 jl =~j4, again, it follows x = 0 on
Q. a2 ~ a l .
Take Q 9 2 ' and t 9 Q . There is an s 9 T such t h a t s-its = t -1.
By T h e o r e m 7.1.1.5 jl =~ j4,
x*(t) = f(t)x(t -1) = f ( t ) x ( s - l t s ) : = f ( t ) f ( s , s-lts)f(t, s)x(t) = x(t). Take Q E t.~" and t E Q . Then, by a2),
x*(t) = f(t)x(t -1) = x(t). Hence x is selfadjoint. b) Take Q c 2 ' and t E Q . The function on T equal to 0 on T \ Q
and
equal to
s-its:
> f ( s , s - l t s ) f ( t , s)
on Q is well-defined. By Theorem 7.1.1.5 j4 ::~ jl, this function belongs to (Sw(f)) c and by a2 ~ al, it is selfadjoint. Take Q c L~" and t E Q . P u t Q , - = {s-l[s 9 Q } . The function x on T equal to 0 on T \ ( Q u Q') and equal to
s-its: (s-It-Is,
~, f(s, s-lts)f(t, s) >f ( s , s - l t - l s ) f ( t - l , s ) ' f ( t ) )
on Q (on Q ' ) is well-defined. By Theorem 7.1.1.5 j4 ~ j l ,
x belongs to
($w(f)) c . Moreover, for every s E T ,
x(s-lts) = f(s, s-lts)f(t, s) = = y(s-lts)f(s, s - l t - l s ) f ( t -1, s)f(t) = Y ( s - l t s ) x ( s - l t - l s ) , x(s-lt-ls) = f(s-lt-ls)x(s-lts) (Proposition 7.1.1.9 a)). By a2 =v a l , x is selfadjoint.
7.1 Projective Representations of Groups
r
By the above, 1 ~, Dim Re (S~(f)) ~ _> Card ~ ' + ~Card .
Since the reverse inequality follows from a2 ==~ al and Theorem 7.1.1.5 jl ==~ja & j4, we obtain 1 2" . D i m R e ( S w ( f ) ) ~ - C a r d ~ ' + ~Card P r o p o s i t i o n 7.1.2.3
m
( -[ ) Let f be a Schur function for the locally finite
group T . a)
IK (T) is an involutive unital subalgebra of S c ( f ) , which will be denoted by S ~ ( I ) (or simply, by S ( f ) ).
b)
So(f) is the closure of gl(T) in $~(f) C*-hull of ~I(T) (Theorem 7.1.2.1 h)).
c)
If f is real then the map
and, in the complex case, the
o
IR
S~ (f)
>~(f),
(x,y),
>x§
is an isomorphism of unital complex C*-algebras. d)
If f is real then the map o
Sn<(f)
>~(f),
(x,y),
) x + iy
is an isomorphism of involutive unital complex algebras. e)
The predual of 8w(f) (Theorem 7.1.2.1 c)) is the Banach space H defined in Proposition 7.1.1.~ a) and the map r from Proposition 7.1.1.~ k) is the pretranspose of the W*-homomorphism &(f)
(Theorem 7.1.2.1 c)). r tET,
~s
~,
~
is surjective. For a C H , x C S w ( f ) , and
7. C*-algebras Generated by Groups
416
f)
If u and v denote the evaluations el(T) --+ el(T) " and Sw(f) --+ $~(f)", respectively, and w: el(T) --+ Sw(f) denotes the inclusion map, then (Im v) A (Im w") = Im (v o w) = Im (w" o u).
g)
If u : gl(T) --+ S~(f), v : S~(f) denote the inclusion maps, then: gl)
; g2(T), and w : el(T) --+ Sw(f)
u' and v' are injective.
g2) x' E (Sc(f))~+ =v u'x' E tl(T)~+, ' = g~) u'(S~(y))+
Ilu'x'll
= IIx'll.
e'(T)~_.
g4) If x E Sw(f), x' E ($w(f))', and t E T , then
I (w'(xx'))(t) = ~ f(t,t-'s)(w'x')(s)x(t-'s) sET
(w'(x'x))(t) = ~_, f(st-',t)(w'x')(s)x(st-1), sET
where t~ h)
and (tx(r))' are canonically identified.
Card T is the topological cardinality of Sc(f). In particular, S~(f) is
separable iff T is countable. i) Sw(f) is the W*-subalgebra of S~(f) generated by S~(f). a) follows from Theorem 7.1.:2.1 e). b) IN (w) is dense in Sc(f), so that el(T) is dense in $~(f). Let E be the C*-hull of el(T), q : e~(T) --~ E the canonical map, and u the factorization of the inclusion map tl(T) -~ Sc(f) through E (Proposition 4.1.1.22 f)). Then q is injective (Proposition 7.1.1.1 d),g)) and
e'(T) = u(q(t~(T))) C u(E). Since u(E) is closed (Theorem 4.:2.6.6) it is equal to $~(f). By Theorem 4.2.6.6, again, u preserves the norms. c) and d) follow from Theorem 7.1.2.1 d). e) The first assertion follows from Proposition 7.1.1.4 a) ,g) ,h) ,k) . By Corollary 4.4.4.9, r is surjective. Take s E T and y E Sw(f). Then (~,y)
= (5, x,y) = (z,y)(~) =
7.1 Projective Representations of Groups
417
= E f(st-l't)x(st-1)y(t)' tET
(ffes, Y} -- (es, y,x} -- (y*x)(s) = E f(t, t-ls)y(t)x(t-18), tCT
so that (Proposition 7.1.1.1 a)) es~x(t) - f ( s t - l , t ) x ( s t
-1) -- ( x , e t ) ( s ) - ( ~ , x * e t ) ,
zeus(t) -- f ( t , t - l s ) x ( t - l s )
-- ( e t , x ) ( s ) -- ( ~ , et*z}.
It follows
for every a C ]K (T) By continuity, these relations hold for every a C H . f) Let H be the predual of S w ( f ) and w 0 : H -+ co(T) the inclusion map. Then w w 0' and the assertion follows from Proposition 1.3.6.25. gl) follows from b) and Corollary 1.3.5.9. -
g2) follows from Proposition 2.3.4.20. ga) For S c | {xesl x E gl(T)} is a finite-dimensional unital involutive subalgebra of el(T) (Theorem 7.1.2.1 e)), so that the assertion follows from Proposition 2.3.4.21. g4) By Proposition 7.1.1.1 a),
(~'(~'))(t)
- (x~', ~,) = (~', ~**x) -
= E(w'x')(s)(et,x)(s)scT
scT
(~'(~'x))(t)
= E(w'x')(s)(x,et)(s) sCT
~(w'x')(s)f(t,t-ls)x(t-ls),
= (~'x, ~ ) =
(~', x , ~ ) -
- Z(w'x')(s)f(st-1
' t)x(st-1).
sET
h) By Example 1.1.2.5, there is a dense set A of ~I(T) with Card A - Card T . By b) and Proposition 7.1.1.1 d), A is a dense set of $ c ( f ) .
7. C*-algebras Generated by Groups
418
Let B be a dense set of $~(f). By Proposition 7.1.1.1 e), B is a dense set of g2(T). By Example 1.1.2.5, Card B > Card T . i) Since _._..ff
______ff
{~lt e T} cc C (S~(f)) cc C (S~(f)) ~ , it follows from Theorem 7.1.1.5 i3), ---+
(So(f)) cc = S w ( f ) . The assertion now follows from the Corollaries 6.3.5.6 and 4.4.4.10. P r o p o s i t i o n 7.1.2.4
m
Let f be a Schur function for the infinite group T .
a) Sw(f) :fl g2(T). b)
S~(f) :/: g~(T).
c)
If in addition T is locally finite, S~(f) # g~(r),
S~(f) # S ~ ( f ) ,
G # co(T),
H # g2(r),
where G and H are the sets defined in Proposition 7.1.1.4. a) Assume S ~ ( f ) - g2(T). Since the identity map Sw(f)
> g2(T)
is continuous (Proposition 7.1.1.1 e)), it is an isomorphism of Banach spaces (Principle of Inverse Operators). Since g2(T) is reflexive, Sw(f) is also reflexive (Proposition 1.3.8.8). By Theorem 6.3.6.15 (and Theorem 7.1.2.1 c)), Sw(f) is finite-dimensional, i.e. T is finite. b) By Corollary 6.3.6.16 a), there is a strictly increasing sequence (Pn)ne~N in P r S w ( f ) . For every n E IN, Pn+l -- Pn E Pr Sw(f)
7.1 Projective Representations of Groups
419
(Corollary 4.2.7.6 a =~ g), so that
Ilp~+~ - p~ll - 1 (Proposition 4.1.2.21). Beside (P,)neIN is weakly Cauchy (Corollary 4.1.2.7 d), Proposition 4.2.1.22, Theorem 4.2.2.1 a =~ d). Now assume &(f) : fl(T). Since the identity map
eX(T)
>S ~ ( f )
is continuous (Proposition 7.1.1.1 d)), it is an isomorphism of Banach spaces (Principle of Inverse Operators). Hence (Pn)ne~ is a weak Cauchy sequence in gX(T). By Theorem 1.3.6.11, (Pn)nClN is norm convergent in gl(T). Thus (P~)ne~ converges in norm in $ ~ ( f ) , which contradicts the above result. c) Let S be a countable infinite subgroup of T. Put E:={xes[xCS~(f)},
F:={xes[xeSc(f)}.
By Theorem 7.1.2.1 e), E is a W*-algebra. Since it is infinite-dimensional, it is not separabel (Corollary 6.3.6.16 b)). By Proposition 7.1.2.3 h), F is separable. Hence E -r F and & , ( f ) :fi So(f). Now assume
Sc(f) = f l(T). By Proposition 7.1.1.1 d) and the Principle of Inverse Operators, there is an a > 0 such that
IIXlll _< ~ll x I for every x E gX(T). Let H be the predual of 8w(f). By Theorem 7.1.2.3 i), 8c(f) generates
8~,(f) as W*-algebra. Let x c & ( f ) # . By Corollary 6.3.8.7, there is a filter on 8w(f) converging to x in ,S~(f)#H with So(f) # C ~. In particular, x(t) -- lim y(t) y,~i
420
7. C*-algebras Generated by Groups
for every t E T (Proposition 7.1.1.4 h)). By the above a t l ( T ) # E ~,. Hence
tET
Ix(t)l <_ lim inf E ly(t)l _< a y,~ tET
x 9 e'(T),
e'(T) = S ~ ( f ) . This contradicts b). Now assume G = co(T). By Proposition 7.1.1.4 d),g), and the Principle of Inverse Operators, the identity map G ~ co(T) is an isomorphism of Banach spaces. Since IK (T) is dense in co(T) it follows H = co(T). By Proposition 7.1.1.4 i) (and Corollary 1.3.4.7), el(T) = Sw(f) and this contradicts b). Finally assume t~2(T) = g . By Proposition 7.1.1.4 b),c),g) and the Principle of Inverse Operators the identity map t~2(T) -~ H is an isomorphism of Banach spaces. By Proposition 7.1.1.4 h) and Corollary 1.3.4.7, e~(T) = S~(f) which contradicts a). I P r o p o s i t i o n 7.1.2.5 ( 7 ) If f is a Schur function for the group T , then "tte following are equivalent: a) $w(f) is a Gelfand C*-algebra.
b) ~I(T) is an involutive Gelfand algebra. c) t~I(T) is a symmetric involutive Gelfand algebra. d) T is commutative, s, t E T==~ f ( s , t ) -- f ( t , s ) , and, in the real case, tET==~t=t
-1 f(t) = 1
If these conditions are fulfilled then the Gelfand transform of t~l(T) is injective, involutive, and with dense image. Moreover, if IK -qJ then Sw(f) is a maximal Gelfand C*-subalgebra and a faithful set o f / 2 ( f 2 ( T ) ) .
7.1 Projective Representations of Groups
421
a :=v b and c ==> b are trivial. b =~ c.
Case 1
IK = C
By Theorem 7.1.2.1 h), every character of ~I(T) is involutive so that by Proposition 2.4.2.3 d =v a, gl(T) is symmetric. Case 2
IK = IR
By Case 1 and Theorem 7.1.2.1 i), the complexification of gl(T) is a symmetric involutive Gelfand algebra. By Proposition 2.4.2.3, gl(T) is a symmetric involutive Gelfand algebra. c =v d ::v a follows from Theorem 7.1.1.5 f3 :=> f2 =~ f4 and Proposition 7.1.1.6 e2 =v e3 ==~ ei. We now prove the final assertion. By c) and Proposition 2.4.2.3 a ==> c, the Gelfand transform of gi(T) is involutive and by c) and Corollary 2.4.2.6, its image is dense. By Theorem 7.1.2.1 c), h) and Proposition 2.3.2.26), tl(T) is semi-simple, so that the Gelfand transform of ~I(T) is injective (Corollary _...+ 2.4.1.14). If ]K = r
then by Theorem 7.1.2.1 h),l) and Corollary 4.2.2.20, ,Sw(f)
is a maximal Gelfand C*-subalgebra and a faithful set of s P r o p o s i t i o n 7.1.2.6
m
Let f be the constant Schur function on the commuta-
tive locally finite group T such that, in the real case, t--t-i for all t c T . (Proposition 7.1.2.5 d =~ c) is a totally disconnected compact commutative group. Let )~ denote its Haar measure with A(a(gl(T))) = 1.
a)
a(gl(T))
b)
If u: gl(T) --+ So(f) denotes the inclusion map (Proposition 7.1.2.3 b)), then the map
o(s~(f))
~o(~I(T)),
~',
~'x'
(Proposition 7.1.2.5 d ~ a) is a homeomorphism. We identify a($c(f)) and a(g I(T)) using this homeomorphism. c)
(St)teT is an orthonormal basis of L2(A). Denote by v the isomorphism of Hilbert spaces v't2(T)
>L2(A),
x,
>Ex(t)'~t" tET
7. C*-algebras Generated by Groups
422
d) x E ,.,cc(f) => vx -- ~. e)
If H denotes the Banach space defined in Proposition 7.1.1.4 a), then there is a unique isometry w" LI(A) --+ H such that A Wet
--
et-1
for every t E T .
f)
For every z E S~( f ), vx e L~176
y e e~(T) ~
IIv~ll~ = I1~11,
v(ly)= (w)(~).
g) If we canonically identify H with the predual of S ~ ( f ) 7.1.2.3 e), Proposition 7.1.1.4 h)), then VX
(Proposition
--" W l X
for every x E ,S~(f), where w is the isometry of e).
h)
The map ~" S ~ ( f )
) L~(A),
x,
r vTc
is an isomorphism of W*-algebras and w is its pretranspose.
i) If Uo and vo denote the evaluations of S~(f) and S ~ ( f ) , respectively, and if wo " S~(I) -+ S w ( f ) denotes the inclusion map, then (Im Vo) N (Im w~') = Im (Vo o Wo) - Im (w~' o Uo).
a) follows from Theorem 7.1.2.1 e), Corollary 2.4.4.6, and Corollary 4.1.2.17 d) (and Theorem 2.4.4.4). b) By Proposition 2.4.1.17 a),c),d), the map is well-defined, continuous, and injective. Thus we have to prove only, that this map is surjective. Let y' E cr(gl(T)). By Proposition 2.4.2.2 c), y' E gi(T)~_. By Proposition 2.3.4.21 (and Theorem 7.1.2.1 e)), there is an x' E (Sc(f))~+ such that u t x ' = y'. Since gl(T) is dense in So(f), it follows that x ' E cr(Sc(f)). c) By a), Proposition 7.1.2.5, and Corollary 2.4.4.6, for s, t E T,
7.1 Projective Representations of Groups
423
= / ~,-~d~ = / V~,-,d~ = ~,, . Hence ('dr)tell is an orthonormal family in L2(A). By Proposition 7.1.2.5, the image of the Gelfand transform of gl(T) is dense, so that ('~t)teT is an orthonormal basis of L2()~). d) ]K (T) is dense in $~(f) coincide on IK (T) . Hence v x -
and v and the Gelfand transform of ,S~(f)
"~ for every x 9 $ ~ ( f ) .
e) P u t Wo" IK (T)
~ LI()~),
a,
~ Ea(t)~t-l" tcT
Take a 9 IK (r) and x 9 $ c ( f) . By d), for t 9 T ,
x(t) = (vxl6) -- . / ~ ( v x ) d A = f ~t-~'2d)~ =
so that
E
a(t)x(t) - E
tET
a(t)(~t-1, ~) = (woa, ~).
tET
By Proposition 7.1.1.4 h),j), Ilall = sup{l(a,x}[I x 9 s w ( f ) #} =
= sup{[(a,x}llx 9 8c(f) #} - sup{l(woa,'2}llx 9 No(f) #} = sup{l{woa, y)l Y 9
C(~(el(T)))#} =
Ilwoa[[,.
Wo preserves the norms, so that it extends to a linear m a p wl" H ---+ LI(A) preserving the norms. Since w0(IK (r)) is dense in LI(/~), w0 is an isometry. w = w{-1 has the required properties. f) For r, s E T , A
A
eset
Thus for t E T ,
A
-
es*et
i
--
est.
7. C*-algebras Generated by Groups
424
=
= E
x(rt-1)~r - E
r6T
(x*et)(r)~r = v ( X e t ) .
r6T
Hence
(~)(vy) = v(;y),
II(vx)(vy)ll~ = IIv(~y)ll2 = Illyll~ ~ I1~11 Ilyll~ for every y 9 IK (T) . Since IK (T) is dense in t~2(T), it follows
Ilvxll~ -I1~11,
vx 9 L~ and
(v~)(~y) = ~(~y) for every y e t?2(T). g) For t 6 T ,
<~,~'~> = < ~ , ~ >
= <~_,,~> = z ( t -~) =
= <~xl~> = <~, w > . It follows that w'x = v x .
h) For 9 9 & ( f ) , w* = ~
~(t)~_, =
tET
= E tET
x(t)"St = E
x(t)~t = v x .
tET
Hence v is involutive. By f), ~ is an algebra homomorphism and by g), the given map is an isometry of W*-algebras with w as its pretranspose. i) follows from d),h), and Proposition 1.3.6.24. Remark.
This proposition is a concrete realization of Theorem 6.3.7.4.
m
7.1 Projective Representations of Groups
Let f
Proposition 7.1.2.7
425
be the constant Schur function on the locally
finite group T . We put
E "= $~(f), and 1
Ps := C a r d s es for every S E ~T.
a) I f S is a subgroup of T , then {xes[x E gl(T)} is a closed involutive subalgebra of ~1 (T) and the map {XeSIX E el(s)}
~, ]Z,
x l
'~ E x(8) sES
is an involutive algebra homomorphism and therefore positive.
b) bl) b2)
R, S E |
R C S ~ pR, ps E P r E , ps <_ pR.
The map E ~
sES
J
is a homomorphism of unital C*-algebras for every S E |
C) If p E E c n Pr E , then: c~) pE(1 - p) - (1 - p)Ep - {0}. c2) E is the C*-direct sum of p E p and ( 1 - p)E(1 - p ) . c3) If T is finite, then Dim (pEp) = rang of [p(st -1)]s,tcT. d)
If S E GT is a normal subgroup of T then:
dl) ps E E c N P r E . d2) If T is finite, then Card T Dim (psEps) = Card S d3) I f T is the permutation group of a finite set containing at least two points, then E is isomorphic to the C*-direct product of IK 2 and of a C*-subalgebra of E .
7. C*-algebras Generated by Groups
426
e) If T is finite, then: el)
x 6 E=~X*pT:
(t~eTX(t))pT.
e2 ) For p 6 P r E , E p(t)=l
andpT<_p
tET or
EP(t)
= O and pT <_ 1-- p.
t6T
e3) PT is a minimal element of P r E \ { 0 } . e4)
Themap
tET
is a surjective homomorphism of unital C*-algebras. es ) pT E pT = IKpT . e6) If Card T :/: 1 then E is not simple. f)
If (S,)ne~ is a strictly increasing sequence in | then the infimum of (Ps.).eIN (of (PsL).eIN ) in ,-qw(f) (in Z.(e2(T)) ) is O.
a) By Theorem 7.1.2.1 e),h), {xeslx gebra of t~I(T). For x 6 t~I(T),
s6S
6
t?~(T)} is a closed involutive subal-
s6S
s6S
so that the map {xeslx 6 el(T)}
>]K,
x'~ ' E x ( s ) sES
is involutive. Take x, y
6
el(T). For t
6
S,
((z~),(y~))(t) = ~ ( ~ ) ( t ~ - i ) ( y ~ ) ( ~ ) = ~ ~(t~-l)~(~), s6T
s6S
so that
~((~), t6S
(y~))(t) = ~
~ x(t~-l)y(~) =
t6S s6S
7.1 Projective Representations of Groups
427
-- E y(s)( E x(ts-1)) -- ( E y(s)) ( E x(t)) . sES tES sES tES Hence the map
{xeslx E gl(T)}
,
IK,
x,
~~x(s)
sES is an algebra homomorphism. By Theorem 7.1.2.1 h), this map is positive. bl) We have p~=
1 , 1 E, Card S es = Card S et
z
tES
1 Z Card S
tES
---
et-1
i
= Card S 2~ et - Card s e s
Ps,
tES
1
PR*PS = (Card R)(CardS) ~
Es~s eT*es =
1
1
(Card R)(CardS) ~
e"~-(CardR)(CardS)~
~ 1
Card S
2 ~ et = Ps.
tES
Hence PR,PS E Pr,Sc(f) and by Corollary 4.2.7.6 d => a,
Ps <_ PR. b2) follows from a) and bl). c1) pE(1 - p) = Ep(1 - p) = {0}, (1 - p)Ep = (1 - p)pE = {0}. c2) follows from Cl) (and Corollary 4.1.1.21). Ca) For x E E
and s E T ,
(p 9 x)(s) - E p ( s t - 1 ) x ( t ) .
tET Hence Dim (1 - p)E(1 - p) - Dim {x E E [ p . x - 0} -
EtEs e t -
7. C*-algebras Generated by Groups
428
= Card T - rang of [p(st-1)]s,teT. By c2), Dim ( p E p ) = rang of [p(st - 1 ) ] s , t e T . dl) For t E T ,
sES
sES
sES
so that P s * et - - et * P s .
Hence P s E E c . By bl), p s E Pr E . d2) follows from ca). da) Let S be the normal subgroup of T of even permutations. By bl) and d 1), Ps , PT E E c M Pr E ,
PT _ PS .
By d2) Dim ( p s E p s ) = 2,
Dim (pTEpT) = 1,
SO that Dim ( ( P s - p T ) E ( p s
-- P T ) ) = 1.
By Proposition 4.2.7.19, E is the C*-direct product of pTEpT,
(Ps -- p T ) E ( p s
-- PT) , and (1 - p s ) E ( 1
- Ps) .
el) For t E T , (X*pT)(t)
-- ~--~X(tS-1)pT(8) sET
:
1 CardT
sET
so that
1)__ ~-~x(t8sET
1 carat ~ sET
x ( 8 ) -~
7.1 Projective Representations o] Groups
429
sCT
e2) By a),
E p(t) e {0, 1}, tET
and by
el), tCT
Hence
P * PT = 0 or p 9 PT = PT. By Corollary 4.2.7.6 c =~ a and d =~ a,
p <_ 1 - P T or PT <__P. e3) follows from e2) and bl). e4) follows from b2) and d2). %) follows from e4). %) follows from c2) and dl). f) By bl), (PSn)nEIN is a decreasing sequence in Pr E . By Theorem 7.1.2.1 c), Sw(f) is a W*-algebra and by Proposition 7.1.2.3 e), H is its predual. By
Theorem 4.4.1.8 b),c), (Ps,)ne~ has an infimum p in Sw(f) and (Ps.)ne~ converges to p in (S~(f))H. Let a r H and let c > 0. Put A := {lal > c}. For n E I N ,
I(a, ps,}l <_ ~
ps,(t)la(t)l + ~
tEA
tET\A
< Ila[l~Card A -t-c, Card Sn so that
lim sup](a, ps.)] <_c. n--~(x)
ps,(t)la(t)l <_
430
7. C*-algebras Generated by Groups
Since c is arbitrary, lim (a, Psi) = O .
/t-+oo
Since a is arbitrary, p - 0. By Theorem 4.4.1.8 d) (and Theorem 7.1.2.1 c)), the infimum of (P-~s~)ne~ in s 0. m
7.1
7.1.3 S u p p l e m e n t a r y
Projective Representations of Groups
43~
Results
7.1.3.1 ( 7 ) Let T1, T2 be two groups and let f l , f2 be Schur functions for T1 and T2, respectively. Take
Proposition
g: T~ x T~
; {~ c ]KI I,~1 = 1 } ,
h:TI•
>{~ 9
and put T := TI X T2,
f: T x T
> {c~ C IK[ [c~[ = 1},
((sl, s2), (tl,t2)),
, ~g(sl, s2) g(tl, s2t2) g(sltx,s2t2) h(tl,t2) h(tl,s2t2)fl(sl,tl)f2(s2,t2). Then the following are equivalent: a)
For (Sl, S2), (tl,t2) E T , g(tl, 1) = h(tl, 1) = g(1, t2) - h(1, t2) = 1 g(sl, t2)g(tl, t2) g(sltl, t2) - h(sl, t2)h(tl, t2) h(sltl, t2) g(tl, s2)g(tl, t2) g(tl, s2t2) - h(tl, s2)h(tl, t2) h(tx, s2t2).
b)
(resp. c)) f is the (unique) Schur function for T such that f((sl,1),(tl, 1)) = f l ( s l , t l ) ,
f((tl,1),(1,t2)) = g(tl, t2),
f((1, s2),(1,t2)) = f2(s2,t2),
f((1,t2),(tx,1)) = h(tl,t2),
for all (81, S2), (tl, t2) E T .
The above conditions imply: d)
For (tl, t2) C T , C(tl,1) * C(1,t2) --
g($1, t2)h(tl,
t2) e0,t2) * e(tl,x) 9
7. C*-algebras Generated by Groups
432
a =v b. Take (T1, T2), (81, 82), (tl, t2) e T. Then
f((r~,r2), (s~,s2))f((r~s~,r2s~), (tl, t2)) f((r~,r2), (s~t~,s2t2)) f((s~,s~), (t~,tg)) ---g(rl, r2) g(sl,r282) g(rlsl,r2s2) h(sl,S2) h(sl,r2s2) fl(rl,Sl) f2(r2, s2)• xg(rlsl, r2s2) g(tl,r2s2t2) g(rlsltl,r2s2t2) h(tl,t2) h(tl,r2s2t2)x Xfl(rlsl,tl) f2(r2s2, t2)g(rl,r2) g(sltl,r2s2t2) g(rlsltl,r2s~t2)x xh(sltl,s2t2) h(Sltl,r2s2t2) fl(rl,sltl) f2(r2, s2t2) g(sl,s2)x xg(tl,s2t2) g(sltl,s2t2) h(tl,t2) h(t,,s2t2) fl(sl,tl) f2(s2,t2)= = g(81, r2) h(81, r2) g(81, r282t2) h(Sl, r2s2t2) h(sl, s2t2) g(81, 82t2) - 1. Moreover f((1, 1), (1, 1)) = 1. Hence f is a Schur function for T. We have
f((Sl,1)(t~,l)) = g(81, 1)g(t~, 1) g(s,t~,l) h(t~, 1) h(t,, 1) f,(s~,tl) ]'2(1,1)= -- fl(Sl, tl) ,
f((1, s2),(1,t2)) = = g(1, s2) g(1, s2t2)g(1, s2t2) h(1, t2) h(1, s2t2) fl(1, 1) f2(s2, t2)= =/~(~,
t~) ,
f((tl,1),(1,t2)) = : g(tl, 1 ) g ( 1 , t 2 ) g ( t l , t2) h(1,t2)h(1, t2) fl(tl, 1) f 2 ( 1 , t 2 ) : g(tl,t2),
7.1 Projective Representations of Groups
333
f((1, t2)), (tl, 1)) = g(1, t2) g(tl, t2) g(tl, t2) h(tl, 1) h(tl,t2) - h(tl, t2) (Theorem 7.1.1.5 a)). b => c follows from the multiplicative property of Schur functions. c => a. By Theorem 7.1.1.5 a), g(tl,1)=f((tl,1),(1,1))=l,
g(1, t2)= f((1,1),(1,t2))= l,
h(tl,1)=f((1,
h(1,t2)=f((1, t2),(1,1))=l.
tl),(tl,1))=l,
We have f((1, s2), (8i, 1)) f((s~, s2), (tl, 1)) - f((1, s2), (sit1, 1)) f((81, 1), (tl, 1)) so that
h(sl, s2) f((sl, s2),(tl,1))- h(sltl, s2) fl(sl,tl), f((sl, s2), (tl, 1)) - h(Sl, S2) h(Sltl,s2) fl(sl,tl). Moreover,
f((sl,1),(tl,t2)) = g(tl,t2) g(sltl,t2) h(tl,t2) h(tl,t2) fl(sl,tl) f((sl, s2), (1, t2)) - g-771-,~) g(81, 82t2) f2(s2, t2) f((1, s2), (tl,t2))= g--~l~s--~2) g(tl,s2t2) -h-~1,~2) h(tl,s2t2) f2(s2,t2). From f((sl, 1), (1, t2)) f((sl, t2), (tl, 1)) = f((sl, 1), (tl, t2)) f((1, t2), (tl, 1)) f((1, s2), (tl, 1))f((tl, s2), (1, t2)) = f((1, s2), (tl,t2))f((tl, 1), (1, t2)), we deduce
g(sl,t2) h(Sl,t2) h(Sltl, t2) f l ( s l , t l ) - g(tl,t2) g(sltl, t2) fl(sl,tl) h(tl,t2), h(tl, 82) g(tl, s2) g(tl, s2t2) f2(s2, t2) = h(tl, t2) h(tl, s2t2) f2(s2, t2) g(tl, t2), { g(sl, t2) g(tx, t2) g(Sltl, t2) = h(Sl, t2) h(tl, t2) h(Sltl, t2) g(tl, s2) g(tl, t~.) g(tl, sgt~.) = h(tl, s2) h(tl, t2) h(tl, s2t2) . b => d. We have
e(tl,~) * e(~,t2) = f((t~, 1), (1, t2))e(tl,1)(1,t2) = = g(tl,t2)e(1,t2)(tl,1) = g(tl,t2) f((1,t2), (tl, 1)) e(1,t2) * e(tl,1) = = g(tl, t2) h(tl, t2) e(1,t2) * e(tl,1) 9
[]
7. C*-algebras Generated by Groups
434
Corollary 7.1.3.2 ( 7 ) Let T1, T2 be two groups and let f , , f2 be Schur functions for T1 and T2, respectively. For j E IN2, let T + be a subgroup of T3 such that
T; T; c T? ,
(T? T; ) u (T; Ti" ) c T; ,
where
Tj\r;. Put T := Tl x T2,
1 if tlET1 + e" T
> {~ ~ 1KI I~l = 1},
f" T x T ~
{a e ~1
I~1 =
(tl, t2):
;
-1
i f t~ ~ T1
or
t2CT +
and t2 E T~ ,
1}, ( ( S l , S 2 ) , ( t l , t 2 ) ) ~ e(tl,s2)fl(sl,tt)f2(s2,t2).
Then f is the unique Schur function for T such that f((sl,1),(tl,1))= fl(sl,tl),
f((tl, 1), (1, t2)) = 1,
f((1, s2),(1, t2))= f2(s2, t2),
f((1, t2), (tl, 1)) = r
t2)
for all (81, 82), (tl, t2) e T . Moreover, e(tl,1) * e(1,t2) -- c(tl, t2) e(1,t2) * e(h,1) for all (tl, t2) E T . Put g'=eT h:-c. g und h fufill the condition a) of Proposition 7.1.3.1. By the implication a =~ c of this proposition, it follows the assertion about f . The final relation follows from Proposition 7.1.3.1 d).
m
7.1 Projective Representations of Groups
435
Theorem 7.1.3.3
( 7 ) We use the notation of Corollary 7.1.3.2 and assume T1 is locally finite, T2 := ~ n for some n E IN, 2n
T+ : : { t e E 2 z 2 ~ l E
(mod2)},
t2(j)-O
j=l and f2 real. Put 0lN2n
>2Z~n,
j'
> e ~ 2~,
q "= Card {j e ]N2.lf2(O(j))- 1} and suppose f2(O(j), O ( k ) ) = - f 2 ( O ( k ) , O(j)) for all distinct j , k c IN2n. Let r : S(f2) ~ 1K2~2~ be an isomorphism of unital C*-algebras, r : S(f2) -+ (Sc(fl))2~,2,~ the map obtained from r by replacing the unit of IK with the unit of So(f1), and
IITI
> (Sc(fl))2n,2 n ,
tl '
>
{
in-q 1-I r j=l 1
i f tl e T f
if tl e T + .
~) For (t~, t~) e T,
(Theorem 5.6.6.1 f)),
H(tl) * ~(et2) - e(tl, t2)r
9 H(tl).
b) For sl, tl E T1, H(Sltl) - H(sl)$ II(tl),
(H(tl))*
-- i-I(tl) 9
c) There is a unique isomorphism of unital C*-algebras ~ 9 S~(f)
> (Sc (fl))2n,2,~
such that
~(~(~:,)) for all
-
~
9 I-[(t~)
(tl, t2) C T (Theorem 5.6.6.1 f)).
9
r
436
7. C*-algebras Generated by Groups
a) The relation
etl * r follows form the fact t h a t r
-- r
* etl
was defined by using ~bo. In order to verify the
second equality we may assume tl 9 T1- and t2 = O(k) for a k 9 IN2n. Then
E(tl) * r
- in-q ( ~
r
*r
-
\j=l 2n
= --in-q~b(e~
)*E r
-- e(tl,t2)r
* E(tl).
j=l
b) We may assume sl,t~ 9 T ~ . Then sltl 9 T + and by a),
1-i(~l), i-I(tl) = (_l)~-o ([if(~o(j))
=
\j=l
= ( - 1 ) ' ~ - q ( - 1 ) 2 ' ~ - q ( - 1 ) ( 2n-1)+(2'~-2)++2+1 =
= ( - 1 ) 3n-2q+n(2n-1) = 1 : l-I(sltl),
(H(t,)
).
--(-1)~-qi n-q
in
r
\j=l
)"
-
2n
= (--1)n-qin-q(--1)2n-q(--1)(2n-1)+(2n-2)+'"+2+lE r j=l
H(tl)= H(t~).
= (_l)an-2q+"(2n-1)
c) We define p first on ,5(f) by extending the map
{et[t 9 T} --+ ($c(fl))2n,2n,
e(~l,t2) '
> etl * H ( t l ) *
~/)(et2)
linearly. Take (sl, s2), (tl, t2) E T . By a) and b),
~(~(~1,~1* ~ l ~ , ~ )
= f((~l,~l
, (tl,t~ti~(~l~,~)l
-
7.1 Projective Representations of Groups
= c(tl,s2)fl(sl,tl)f2(s2,
= C(tl, 82)esl :
es 1 * 1 - I ( 8 1 )
t2)eslt~ * I I ( s l t l ) * r
* et I * H ( s 1 )
* ff2(es2) * et 1 *
(~( C*( t l , t 2 ) ) -
*
l-I(tl) * r f((tl
, t2))~(e(t
= c(tl,t2)f~(t1)~(t2)et71 = fl(tl)~(t2)r
II(tl) 9 r --
437
--
9r
-
~(e(sl,s2))
* P(e(tl,t2)),
T1 ,t2))-
* ~I(tl)
*r
--
* H ( t l ) * etT1 --
tl
Hence p is an involutive unital algebra homomorphism. It is easy to see that is injective. Case 1
T1 finite
Since D i m S ( f ) = C a r d T = 4nCardT1 = Dim ($(fl))2n,2n, is bijective. Hence p is an isomorphism of C*-algebras. Case 2
T1 infinite
Take S c GT. Let fl,s
be the restriction of fl to S x S and f s
the
restriction of f to (S • T2) • (S • T2). By Case 1, the map s(f~)
>(s(f,,~))~ ~
defined by ~ is an isometry of unital C*-algebras. Since S is arbitrary, the map S(f)
> (S(/1))2n,2~
defined by ~ is an isometry. It extends therefore to an isometry of unital C*algebras s~(f) (Theorem 5.6.6.1 a)).
> (s~(f~))~,~o m
7. C*-algebras Generated by Groups
43s
Corollary 7.1.3.4 ( 7 )
We use the notation and assumptions of Theorem
7.1.3.3 with n = 1. Put TI
CI "-- (?.7"11+-- eT~
and 6 : = 1
a)
(resp. 6 : = - 1
and 1K = r ).
The function f2 defined by
(1,1)
(0,1)
(1,0) (1,0) (0,1) (1,1) is a Schur function for 2Z2 such that
f2(O(j), O(k)) = - f2(O(k), O(j)) for all distinct j , k E IN2 and q =
b)
There is a unique isomorphism of unital C*-algebras r
[o1] '
such that
r176176 = -1 c)
1+5 2 9
0
" S(f2) --+ IK2,2
r176176 = i~-4 [ O1 01]"
For tl E T 1 ,
H(tl)=(-1)~-~ [ 10 -10 ] . d)
There is a unique isomorphism of unital C*-algebras
~ . s~(:)
~ (s~(f,)),~,:
such that qo(e(tl,t2))
etlO
= (-1) (1-5)(-le(lt1)I4
for every (tl, t2) E T .
o
el(tl)eh
] 9r J
7.1 Projective Representations o] Groups
e)
Assume
IK = ~ , 5 = - 1 ,
s
439
is a set, a n d ,So(f1) is identified with
S
via an i s o m o r p h i s m of unital C*-algebras.
Take ~- C {-1, +1}
and t l C T1 such that etl
---- T e t l
( D e f i n i t i o n 5.5. 7.1). T h e n
~(~(,,,o)) = ~ ( r and
~(e(o,(1,o)))--
(/9(e(o,(1,o))) ,
(/9(e(o,(o,1))) -- -~(e(o,(o,1))) .
a) is a straightforward verification. b) The uniqueness of r follows from the fact that e(1,o) and e(o,1) generate
s(A). Put a'-
[ _0
,
1
b:=i-v
0
c "= a b == i l :~-~" [01
10j
-10 I "
Then a2 = - 1 ,
b2 = 5 1 ,
c2 = 5 1 ,
a* = - a ,
b* = 5b,
c* = 5c,
ac = - c a = - b ,
First define r
and then extend r the above,
bc = - c b - - S a ,
ba - - c .
on {et21t2 e 2E~} by putting "t/20(C(1,O)) "'-- a,
r
b,
r
Co(e(1,1))"--
C,
"= 1,
linearly to 8(f2). It is easy to see that r
r
, et2) = r
r
= r
, r
,
is bijective. By
7. C*-algebras Generated by Groups
44o
for all s2, t2 E 2Z~. Hence r c) By a) and b), ii(tl)=il_~2_~ I 0 -1
is an isomorphism of unital C*-algebras.
11 ,i~@ [0 0
1 ] = (_l)t~ -~ [ 1
1 0
0
0 ] . -1
d) Take (tl,t2) E T. If tl E T1+ then B
0
4
et,
o
0
r
If tl E T 1- then by c),
Hence the assertion follows from Theorem 7.1.3.3 c). e) follows immediately from d). Proposition
7.1.3.5
Let T be an uncountable group, f a Schur function for
T , and W, W' cardinal numbers such that:
1) R ' < C a r d T . 2) W' <_ CardT. 3)
The sum of any countable family of cardinal numbers strictly smaller than W' is strictly smaller than P,".
Denote by .T the set of x E s
for which the Hilbert dimension of Imx
is at most equal to N' (is strictly smaller than R" ).
a) .T is a closed proper ideal of/2(g2(T)). ----4
----4
b) $" + Sw(f) is the C*-subalgebra of/2(g2(T)) generated by .TU S w ( f ) . ._._.4
~)
7 n &(f)
=
{o}. -----4
d)
a) b) c) d)
(~, y) e 7 • S~(/) ~ Ilvl _< I1~ + yJl.
was proved in Proposition 5.5.3.13 a). follows from Proposition 5.5.3.13 b) (and Theorem 7.1.2.1 j)). follows from Proposition 7.1.1.1 l) and Theorem 5.5.2.5. follows from c) and Proposition 5.5.3.13 c).
7.1 Projective Representations of Groups
Proposition
7.1.3.6
Let T , T '
be groups, f , f '
Schur functions for T and
T ~, respectively, 9~:T --+ T ~ an injective group homomorphism, and
A: T
> {a 9 lK] ]a] = 1}.
Then the following are equivalent:
~(s).x(t)
a)
s , t 9 T ~ ff(~a(s),~(t)) =
b)
For all x, y 9 g2(T') with y = ye~(T) ,
~(st) f ( s , t ) .
(~(xo~)),(~(yo~)) c)
= A((z,~)o~)
s , t 9 T => (~(e~(s)o~)),()~(e~(t)o~)) = )~((e~(s),e~(t))o~).
If these conditions are fulfilled then:
d)
A ( 1 ) = 1.
e) x e & ( f ' ) f)
==~ A(xo99) e 8 ~ ( f ) , IlA(xo~)[I <_ [[xl[.
If ~a is bijective then the map ,Sw(ff) --+,S~(f),
x.
> )~(xo~)
is an isometry of Banach algebras.
g)
The following are equivalent:
gl)
X 6 ~2(T')::~ (l(xo99))* : l(x*o99);
g2) t 6 T :::> ()~(e~(t)o~))* - l(e*~(t)%p); g3) t E T => ,~(t)2l(t-1) 2 = 1. h)
If g is a Schur function for T such that the map
,-q~(g)
>Sw(f),
x:
; Ax
is an isometry of Banach algebras then
g(~, t) = ~,(~),~(t) ~(~t----~f(~, t) for all s , t E T .
441
7. C*-algebras Generated by Groups
442
For t E T ,
a~b.
((~ ( z o : ) ) , ( ~ (yo:)))(t) = E
s)A(ts-1)x(:(ts-1))A(s)Y(q~
f(ts-l'
=
sET
f'(qD(t)~(s)-l' ~(s))x(q~
= A(t) E
=
sET
= A(t) E
f'(~(t)s'-l's')x(cp(t)s'-l)y(s')=
A(t)(x,y)(cp(t)).
s ~E T ~
Hence
(~(xo:)),(~(yo~)) b ~
= ~((~,y)o~)
c is trivial.
c=~a.
For r E T , ((A(~(~)o~)),(~(~(~)o:)))(~)
E
= ((~),(~))(~)-
f(rq-l'q)A(rq-1)es(rq-1)A(q)et(q)
-
qET
= f ( r t - ' , t)A(rt-i)es(rt-1)A(t) = f ( s , t)A(s)A(t)Sr,st, (A((~(s),~(~))o~))(~) = A(~)(~(~),~(~))(:(~))-
f'(q~176
= A(r) E q~ E T ~
= A(r)f'(~s(r)~(t) -1, qs(t))e:(sl(qo(r)qo(t) -1) = = A(st)f'(~s(s), qo(t))rSr,st. P u t t i n g r := st, we obtain f'(w(s), (:(t)) = A(s)A(t)
A(~t) f(~' t).
a~d.Take
s=t-1.
b ::v e. Take z E g2(T). There is a y C g2(T') such that
y - ye,(T),
A(yo(p) -- z.
By b), (~(xo:)),z
= (~(xo~)),(~(yo~))
-
7.1 Projective Representations of Groups
443
e~(T),
= A((z,y)o~)e
< llx,yll~ < I1~11 Ilyll~ = Ilxll Ilzll~. Hence A(xo~) e &~(f),
[IA(xo~)[[ < I[xl[.
b & e =~ f is obvious. g) For every x E t~2(T') and t C T,
(~(xo~))*(t) -
f(t)
(A(xo~))(t -~)
=
f(t,t-1)A(t -1) x ( ~ ( t - 1 ) ) ,
(A(x*o:))(t) = A(t)x*(~(t)) - A(t)f'(~(t), ~ ( t ) - l ) x ( ~ ( t ) -1) =
-
-
A(t)/~(t)A(t-1) A(1) f(t't-])x(9(t-1)) "
g3 =V gl follows by d). gl =a g2 is trivial. g2 =~ ga follows from the above and from d) by putting x := ev(t-1). h) Put
, g "TxT
>{c~eIKi[al-1},
(s,t),
~(~)~(t) ~(~, t). > A(st) "
By d) and Proposition 7.1.1.8. a),c), g' is a Schur function for T . By a =>f, the map
is an isometry of Banach algebras. Hence the Banach algebras Sw(g) and Sw(g') coincide and this implies g = g~. m P r o p o s i t i o n 7.1.3.7 Let T be a finite group, f a Schur function for T , and (R, 9~, #) a measure space. For every p e [1, c~] denote by Hp the set of
~'TxR
>IK
such that ~(t, .) e LP(#) for every t C T , and for every ~ E H1 put J
J
7. C*-algebras Generated by Groups
444
a) Let p, q, r e [1,:xD] with 1
1
1
p
q
r
and (~,71) 9 Hp x Hq. Then ~ , ~ E Ha and
(~*v)* = v**C, wh e re
~,V" T • R
-," IK,
(t, r),
>~
f ( t s - : , s)~(ts -1, r)rl(s , r)
sET
and ~* " T • R
b)
If ~ c
) IK ,
(t, r) ,
) f (t)~(t -1, r) .
H1 and x C , S ( f ) , then
f (x,~)d~=~, f ~d~,
where x,~" T • R
> IK,
(t,r) ,
>Ef(ts-"s)x(ts-:)~(s'r)' sET
~,x" T • R
~ IK,
(t,r),
~Ef(ts-'
s)~(ts-' ~r)x(r)
9
sET
c)
Let ~ E H1 such that ~(.,r) C 8(f)+ for It-almost all f ~d~ e S(I)§ and
f ~d~ - 0 implies
:(t,r) --0 for all t C T and for It-almost all r C R .
r C R.
Then
Z1 Projective Representations of Groups
r162
d) H2 endowed with the right and left multiplications Ha x S ( f )
>Hu,
(~,x) ,
>~ , x
S ( f ) X Hu
>H2,
(x,~) ,
>x*~
and with the inner-product H2 x H2 ~
S(f),
,)
. f (,*
is a unital Hilbert 8(f)-module. e)
For every t E T and k E L2(#), ket 9 H2, [[ketl[ = Ilk[12.
f) For every ~,~ 9 H2, (~l~?)" T
>IK,
t,
> ~-~(~(s,.)lf(st-l,t)~(st-1,.)), sET
IIr <_Z lr tET
g) If (k,),ei is an orthonormal basis of L2(p) then for every ~ 9 H2,
LEI
h)
For every ~ 9 H ~
define ....+
~c "//2
>/-/2,
r],
>~,r/.
Then ~ E gs(:>(H2) and
,~ -,~*,
sup ll(C ,O(t,-)l ~ _< II,~ll < ~ II,~(t,-)tlor tET
tET
for every ~ 9 Hoo and the map H a -----+s
~'
>
is an injective involutive unital algebra homomorphism. Moreover, for every k 9 L ~ ( p ) and t 9 T , ,...a
ket 9 Hor
llketl]= ]Iki]~.
7. C*-algebras Generated by Groups
~6
i) Assume there is an a E IR+ such that
sup II~(t,-)il~ < ~iI~li tET
for every ~ E Hoo and that # is a-finite or it is a Radon measure on a locally compact space. Put ~'H1
>IK,
77' >E f ~(t")~(t")dP tET J
for every ~ E Hoo. Then Ho~ endowed with the norm
H~ ----+ ]R+,
~'
>iit:ii
is a W*-algebra and H1 endowed with the norm H~
~§
,7,
> sup It:(~)]
is its predual and
sup ii,7(t, .),l~ < .,~1, _< ~ ~ tET
ii,~(t, )li,
tET
for every 77E H1. a) For every t E T , (~,rl)(t,-)
=
E f(ts-l,s)~(ts-1, .)y(s,.) E Lr(#), sET
(~.,7)*(t, .) = ~*(t, .).~*(t, .) = (~*.~*)(t, .). b) For every t E T ,
/ ( x , ~ ) ( t , r)d#(r) = / E f(ts-l' s)x(ts-1)~(s' r)d#(r) sET
---- E f(ts-l' S)x(ts-]) / ~(s, r)d#(r) -- (x, / ~dl.t)(t) , sET
/
(~,x)(t, r)d#(r) = / E f(ts-l' s)~(ts-l' r)x(s)d#(r) = sET
7.1 Projective Representations of Groups
= s~Tf(ts-l,s ) (f ~(ts-l,r)dp(r)) x(s) = ((f ~d,) ,x) (t), ~*(t, r)d#(r) =
= f(t)
f
f(t)~(t -1, r)dp(r) =
~(t -x,r)d#(r) --
(f)* ~dp
(t).
c) For every x ' e ( S ( f ) ) ~ ,
(~(., ~), x') 9 ~ + for p - a l m o s t all r C R . It follows
I / ' ~ d p , x') = /(~(',r),x')dp(r) C IR+ so that
f ~dp C S(f)+. Now suppose
/
~d#=O.
Then for every x' e (S(f))~_, 0=
(f ~d~,x')=f (~(.,r),x')d~(~)
so that
(~(., ~), ~') = o for p-almost all r E JR. It follows
~(t,r) --0 for all t C T and for p - a l m o s t all r E R . d) Take x, y E S ( f ) and ~, r] E H2. The relations
~,(~,y) = (~,~),y,
(x,~),y -- x,(~,y),
(x,y),~ = x,(u,~),
1,~ -- ~,1 --
447
7. C*-algebras Generated by Groups
448
are easy to see. By a) and b),
(~.~1~> = f(,**(+.x))e, = (f(,*.+)e,).x <x,+l.> = f(~*.(~.+))~,
/
= f((,*<).~)e,
=
= (+1,>.~. = f((r
=
((x* ,7"/)**~)d/z - (r
(el,>= f (,*.e)d,= f (c.,)*~, = =
(/)" (~*,r/)d#
=
(r/l,{)*
By c),
(,'I,')~ s(:)+ and (~1r = 0 ~
r = o.
By Example 5.6.1.5 (and Theorem 7.1.2.1 c)), o
((,'(.,,)I,:(.,~)) + (,7(.,,-)1,7(-, ,-)),(,7(.,,)I,:(.,,+))- (,:(-,,)1,7(-,r))) e (s(f))+
(x,:(.,~)ix,:(.,r))_< ilxil'+(,'(-, r)I,'(.,r)) for /z-almost all r E R. By c), o
((~Is:)+ (rllrl),(rll~)-(,:Irl))e (S(:))+ (xEiix~:) < I1~1]~(r162 e) For every s E T ,
7.1 Projective Representations of Groups (ket)(s, .)= 5stk e L2(#) so that ket E H2. By Theorem 7.1.2.1 b),
,[ket[[2= ,[(ket[ket)[' = llf -kketetd# =
= (flkl2d~)el
-[ikil2.
f) For every t E T, (rl* *~)(t, .) = E f ( t s - l ' s)rl*(ts-x' -)~(s, -) = sET
= E f(ts-l' s)'f(ts-1)zl(st-l' .)~(s, .) = sET -= E f (st-l' t) ~ ( s t -1, -)~(s,-) sET
(Proposition 7.1.1.9 c)), so that (~]~)(t) - f (~**~)(t,-)d~ = E ( ~ ( S , ' ) l f ( s t -1, .)r/(st -1 , .)). sET
By Proposition 7.1.1.1 d),
I1~11~ - I1(~1~)11< ~ I(~I~)(t)l < tET
~ i (~(s,-) If(st -~, .)~(st -~, -)J s,tCT
_< ~
ll~(~, .)ll~ I~(~t-~,-)ll~ -
s,tET
II~ll < Z Ilk(t,-)ll2. tET
II~(t,-)ll~
449
450
7. C*-algebras Generated by Groups
g) For every t E T and ~ E I , ((~]k~l))(t) = f k~(r)~(t,r)dp(r)= (~(t,-)Ik~) J so that
~-~(5(t,-)lk,)k~ = ~ k,((5lk, l))(t).
5(t, . ) =
tel
tel
Since T is finite
tEI
(Proposition 7.1.1.1 d)). h) Take r/, { E //2. By a), (~r/l(~) = (~,r/I(:)=
=
((~*,(~,r/))d# =
((~',~)**r/)dp = (r/I~',r
(((~**~),r/)d# =
= (r/l~*(:).
Hence ~ E Es(/)(H2) and ~ = ~*. Thus the map
Hoo
>
Es(/)(H2),
~,
>
is involutive. It is obviously an algebra homomorphism. Now we prove the inequalities. By Proposition 7.1.1.1 d),
tET
tET
for p-almost all r E R. Take 77 E/-/2. Then
o < ~**C,~,v <
11~l12v**~ <
II~(t, )11~
v**~
(Corollary 4.2.2.3), so that by c),
o_ By Corollary 4.2.1.18,
(~,~ ~,~) <
II~(t,.)
I~
(~1~).
#-a.e.
7.1 Projective Representations of Groups
I1~*,112 ~
II~(t,-)ll~
117112
It follows
II~'ll _< ~
Ilk(t, .)ll~.
tET
Take t E T and c~ E]0, 1[. There is a k E L2(#) # such that /
Ikl2(~ * 9 ~)(t, .)d# > -I1(~ 9 ~)(t, .)ilo~.
Since Ikl2(~ * 9 ~)(t,-)d# = {k~lk~}(t )
/ it follows
I(k~lk~)(t)l >__ali(~* 9 ~)(t, ")II~. By e), kel E H2# so that
ii,~ ii~ > ll,,(k~)il~ = II,~, (k~)ll~ = ilk,~it~ = = I]ii _> I(t)] _> ~ll(~* * ~)(t, ")il~ (Theorem 7.1.1.1 e)). Since c~ and t are arbitrary,
li,' II > sup il (r 9 ~)(t, .)il~. tET
In particular,
=0~r We prove now the last assertion. We have
for every s E T so that ket E H~ and
451
7. C*-algebras Generated by Groups
452
I1~';~11 < Z
II(k~)(~, .)11oo = Ilkll~.
sET
In order to prove the reverse inequality take a e]0, Ilklloo[. There is an h e L2(#) # such that Ilkh]12 > a . By e),
her-1 e H2,
Ilhet-lll = Ilhl[2 _< 1
so that, by e), again,
IIk~,ll >_ IIk~ll IIh~,-,ll _> IIk~,(h~-,)ll = = [](ket),(het-~)]= ][khel]] = ][kh[]2 > a . Since a is arbitrary, IIk~l > Ilkl oo.
i) By the inequality of h) and the supplementary hypothesis of i), Hoo endowed with the norm
is complete and thus a C*-algebra. It is obvious that H1
,~IR+,
77,
~ sup I~(~)1 ~en~
is a norm. For every r/ E H1 and { E H ~ ,
]~(T])] __~Z ]]~(t, ")l]C~]],(t,")l]l --~O/]]~J]E ]]?](t, ")1]1 toT
tET
so that
II,If _< ~ ~
ll,(t, .)ll, 9
tET
Let 77 E H1 and t E T . Take k C L ~ ( # ) # . By h),
k~, e H~,
IIk~,ll = Ilkll~ _< 1.
7.1 Projective Representations of Groups
Since
for every s E T ,
k~ (,) = ~ f(k~,)(,, .),(,, .)d, sET
=~ 5~f kv(~,.)d~-/ kv(t,.)d~. sET
It follows
f k,(t, .)d, Since k is arbitrary, we get
I1~11>_ Ilk(t,-)11~. Hence
Ilwll ~ sup Ilw(t,-)111. teT
It fellows that H1 is complete. Let k E L 1(#) and t C T . Then
(k~)(~, .) =5~k for every s C T so that ket E H1 and by the above,
sCT
For every ~ E H ~ ,
r
= ~s C T f ~(~, .)(k~,)(~, .)d, =
r162 scT
so that
453
7. C*-algebras Generated by Groups
454
ilketll -
sup I~(ket)l <_ c~llkl 1. ~EH~
Let ~' E H~. Take t E T . Then for every k E L 1(#), the map
LI(#)
> ]K,
k,
> ri'(ket)
is continuous. Hence there is an ht E L ~ ( # ) such that
htkd# = r/(ket) for every k E L 1(p). Put ~'TxR
>IK,
(t,r):
;ht(r).
Then ~ E H ~ and for every 77 E H1,
tET
tET
= ~,'(,(t,
.)e,) = q(,).
tET
Hence
and the map H~
>H~,
~:
;~
is surjective. Moreover,
I1~11 = sup I~(,)1 _< I ~11. ~6H1#
We want to show that H ~ is a closed set of ( H ~ ) H , . Let ~0 be a point of adherence of H ~# in (H~)Hx 9 There is a filter ~ on H ~ such that H ~ E and lim ~(r]) = ~0(r]) ~,i~ for every 77 E H1. Take k E L 1(#) and t E T . Then
/
_ = lim / ~(t, .)kd#. ~o(t,.)kd# = ~o(ket) = lim~(ket) ,;~ (,;~
455
7.1 Projective Representations of Groups
Take ~ 9 H2# and put
6 := ~or o H 2 . Then for every t 9 T ,
(~,(~o<))(t,
-~, ~)~(t~ -~, .)(~o<)(~,-)
.) = } ~ / ( t ~
=
sET
= E
f(ts-]' s)~}(ts-l' ")f(sr-l' r)~~
.)~(r, .).
r,s6T
Since ~(ts -1, -)~(r,-) 9 LI(#) for all r, s,t 9 T it follows
<~o,~l~o,r
f(r162162
=
.)d# =
= limr f ( { ~ , ( ~ , ~ ) ) ( t , .)d# = lim<~,4l~0,{} . r (t) By Theorem 7.1.1.1 d) and Proposition 5.6.1.2 d), I}~o~ll2 = II(~o*~l~o*~)ll = lim (,;~ [1(~*~1~o*~)11-< N
_< lim~,ainf I1r
I1~o,~1 ___ I1r
I1~oll = sup II~oCII ~ 1, ~6H2 #
,'o e g ~ Hence H ~# is closed in (H~)H1. Take {o (5 H~\H#~. Putting E := H1, F := goo, A' "- H ~ , and x' := {o in Proposition 1.3.1.8 we get an r / 9 H1 such that
< Co(n) _< li~oll linli,
iinll = sup Ir
~6H~
so that 1< Thus
II~olI.
456
7. C*-algebras Generated by Groups
I1,~oll _< I1,~oll , I1~011 = I1~011,
and the map Hoo---+H'l,
~,
>f
is an isometry. By Theorem 4.4.2.21, Hoo is a W*-algebra with HI as predual. II
P r o p o s i t i o n 7.1.3.8 Let T1, T2 be groups such that T1 is finite and T2 is locally finite and commutative and let fl, f2 be Schur functions for T1 and T2 , respectively. We assume f2 constant and, in the real case, t2 = t~ l .for all t2 E 7'2. Put T := TI • T2
and denote by f the Schur function/or T
f" T x T
> {~ e ~1 I1~11= 1},
((~,,~), (tl,t2)) ; > fl(Sl,tl).
We identify a(t~l(T2)) with a(S~(f2)) and denote it by R , by A the Haar measure of R with A(R) = 1 and by v the isomo~hism of Hilbert spaces v.e2(T2)
>L2(A),
x:
> E x(t2)~t2, t2ET2
(Proposition 7.1.2.6 a),b),c)). Put ~" TI • R---~ IK, 1o~ every x 9 e ~ ( T ) . For every p ~ maps
(tI,')'
>vx(tl,')
[1, cxD] denote
~" Tl X R - ~
by Hp (by Ho) the set of
IK
such that
~(tl,') E Lv(A) (~(tl,') ~_ C(R)) .for every tl C 7'1, endow H2 with the structure of a Hilbert $(fl)-module (Proposition 7.1.3. 7 d)), and put / fdA " T1 :for every ~ C H1.
> IK,
tl ,~ +
f ~(t~,.)d~
7.1 Projective Representations of Groups
a)
457
/f tl 9 T1, p 9 [1, c~], and k 9 I2(#) then ket~ 9 Hp and (ketl)(Sl, ") =
(~sl,tl k
for every sl 9 7'1.
b)
x 9 g2(T) ::> 5 9
c)
For every x , y 9 e2(T) and tl 9 T1,
(~ly-')(tl) = ~
(~(sl,')lf1(slt11,tl)y(slt11,')> 9
sl ET1
d) x 9 e ~ ( T ) ~ I1~11~ Ilxll~x/CardT~. e)
The map
g2(T)
>//2,
x~
>~
is an isomorphism of Banach spaces. In particular, there is an c~ 9 IR+ such that
for every x 9 g2(T).
f)
An element x of s
belongs to Sw(T) iff ~ e H a .
g) x 9 S~(T), y 9 g2(T) :=> ~ h)
= ~*y (Proposition 7.1.3.7 a)).
For every ~ C H ~ ,
sup II5(t,,-)11oo ~ ~11~ II
tl ET1
w h e re ,,,.a
{-H2
>H2,
,>{,r]
V,
and ~ E ff_,S(fl)(H2) (Proposition 7.1.3. 7 g)).
i)
We endow Hoo with the norm
Hoo put
> 1R+,
{,
> I1~11,
7. C*-algebras Generated by Groups
458
~ " H1
~ ]K,
rim tl ETI
for every ~ E Hoo, and endow H1 with the norm
H1
~]R+,
r/,
~ sup [~(~7)]. ~EH~
Then Ho~ is a W*-algebra with predual H1 and
IIk~,l II = Ilklloo for every tl E T1 and k E L~176
j)
Let H ( f ) , H ( f l ) , and H(f2) be the Banach spaces defined in Proposition 7.1.1.~ a) with respect to f, fl , and f2 , respectively, and :for every ~7E H1 put
77.T
~IK,
(tl,t2)')(Wrl(tl,'))(t2),
where w is the isometry L~(A),
~ H(f2) such that A
wet2 =
et"~ 1
for every t2 E T2 (Proposition 7.1.2.6 e)). Then the map ,-%(:)---+Hoo,
x,
~
is an isometry of W*-algebras with pretranspose
H1
~H(f),
r/,
~rl.
k) An element x of t~2(T) belongs to So(f) iff ~ E Ho. In particular, H~ is the W*-subalgebra of Hoo generated by Ho. a) and b) are easy to see. c) and d) follow from Proposition 7.1.3.7 f) (and Parseval's Equation). e) By b),c), and the Principle of Inverse Operators, it is sufficient to show that the map is bijective. Take x E e2(T) with 5 = 0. For every tl E T1, ~(tl,
so that
.) = ~(tl, .) = o
7.1 Projective Representations of Groups
x(tl, ") = 0 . Hence x - 0 and the map is injective. Take ~ C / / 2 and put
x" T
>IK,
(tl,t2),
> (v-l~(tl,'))(t2).
For every tl C T1,
~(t~, .) = vx(t~, .) = ~(t~, .) so that ~ = ~ and the map is surjective. f) Take tl C r l . By Proposition 7.1.2.6 h),
x(tl, .) e 8~(f2) ~
~(t,, .) e L~(A).
Since T1 is finite, x e $w(f) ~
x C Hoo.
g) Take (tl,t2) E T . Then ( x * y ) ( t l , t2) = E fl (t1811, 81)x(t1811, t2821)y(81, 82) -sET
= E
fl(tls~l'sl) E
sl ET1
z(tls~l't2s21)y(sl'su)=
s2ET2
= E fl(tlS-11'sl)(x(tls~1")*Y(sl"))(t2) sl ET1
so that (x*y)(tl,')
-- E f 1 ( t 1 8 ~ l ' s l ) x ( t 1 8 1 1 ' sl ET1
") * y ( s l ' ' ) "
By Proposition 7.1.2.6 f),
~(tl,
") -- v ( ( x * y ) ( t l , ")) --
-- E f1(t1811'81)(vx(t1811''))(vy(81''))--sl ET1
459
7. C*-algebras Generated by Groups
460
= ~
f1(t1811 , ~)~(tis~ -~, .)y(~l, .) = ( ~ . ~ ( t , , .).
sl ETI Hence
x,y = x,y. h) By Proposition 7.1.2.6 h), for every tl E T1 there is a z E $~(f2) such that
vz=~(t,,.). Hence there is an x E S~(f) with 5 = ~. Take tl E T1 and
D e 10, [15(tl, ")1[~[. There is a k E L2(A) # such that
115(t~, ")kl12 > ~. Put y - T - - - + IK,
(sl,82):
~"5~,,tl(k]~).
Then y E e2(T) and
y(sl, ") = 5sl,tl v - l k ,
~(~,.) =vy(~,,.) =~,,,,,k for every Sl E T1 so that Y - - ketl 9
By g),
For (s,, s~) ~ T, (x*y)(Sl, S2)
-
-
E
fl(slr[l'rl)X(Slrll's2r21)y(rl'r2)--
(rl ,r2)ET
-- E E f1(81r11'T1)x(81r11'82r21)(~rl,tl(v-lk)(r2)r2ET2 rl ET1
7.1 Projective Representations o] Groups
461
= E fl(slt-1]'rl)x(sltll's2r21)(v-1k)(r2)= r26T~
= f~(s~t~l,r~)(x(stt~l,.),(v-lk))(s2).
By Proposition 7.1.2.6 c),f), IIx*yll~ =
~
I(x,y)(sx, s~)l ~ =
(Sl,S2)6T
= E E IX(Slt~1")*(v-1k)(82)[2= s167"1 s26T2 -- E liv(x(81t11")*(v-1k))l122 = E iivx(slt11")ki12 = Sl 67"I sl 6T1
- ~
]l~(81t~-1, ")kl[2 ~ Ilk(t1,-)kll~ > ~=.
sl ET1
By Proposition 7.1.3.7 e),
so that by e) and g), ll,~ii > ]l~ll ]ik~,il > l i , ' ( k ~ ) i l
= ]l,',(k~)li
oz
=
oz
Since /3 and tl are arbitrary, it follows II~(t~,-)lt~ _< ~11~11, ....+
sup II~(t~,-)ll~ _ ~11~11.
t l 6T1
i) follows from h) and Proposition 7.1.3.7 i). j) By Theorem 7.1.2.1 c) and Proposition 7.1.2.3 e), Sw(f) is a W*-algebra with predual H ( f ) . By f) and g), the map
7. C*-algebras Generated by Groups
462
is an isomorphism of algebras. This isomorphism is obviously involutive. Take ~ E H1. By Proposition 7.1.2.6 e),
r/(tl,-)----wrl(tl, .)E H(f2) for every tl E T1 such that r/ E H ( f ) . For every x E S w ( f ) , by Proposition 7.1.2.6 g),
(rhX) = E f x(tl, ")vl(tl,")d)~: tl ET~
: E
f vx(tl, .)r/(tl,-)d)~ : E (rl(tl' ")' vx(tl, .)) :
tl ETI
tl ETI
= E (wq(t,,'),x(tl,'))= E tleT1
E ~(tl,t2)x(t,,t2)=
tleTl
t2eT2
= (~, ~ ) .
Hence the map Sw(f)
) Hoo,
x,
~~
is an isometry of W*-algebras with pretranspose H1
>H(f),
r/,
~ ~'.
k) x E 3c(f) iff x(tl, .) E 8c(f2) for every tl E T1 so that by Proposition 7.1.2.6 d), x E So(f) iff 5 E H0. By j) and Proposition 7.1.2.3 i), Hoo is the W*-subalgebra of H ~ generated by H0. I P r o p o s i t i o n 7.1.3.9 Let S , T be groups, f and g Schur functions for S and T , respectively, and n E IN. We assume that S ( f ) is isomorphic to IKn,n and T is locally finite. Put h ' ( S x T) x (S x T)
((81, tl), (82, t2))'
.~ {c~ 6 IK I Ic~I = 1},
) f(81, 82)g(t1, t2).
T h ~ h i~ ~ Schu~ Su~t~on So~ S • T ~nd S~(h) ~ ~omo~ph~ to
(8~(g))~,~.
7.1 Projective Representations o] Groups
It is easy to see t h a t
463
h is a Schur function for S • T . Let ~a be the
composition of an isomorphism of C*-algebras S(f)
> lK,~,n
with the canonical embedding
IK~,n
> (s~(g))~,~.
Put
for every (s, t) E S • T and extend u linearly to a map
It is easy to see t h a t u is a unital, involutive algebra homomorphism. Since S • T is locally finite, it follows that u has norm at most 1 with respect to the norm induced on IK (s•
by Sc(h) (Corollary 4.1.1.20). Hence u may be
extended to a unital involutive algebra homomorphism $c(h)
~ (Sc(g))n,n.
We want to show t h a t u is injective. Let x E So(h) with ux=O.
Put ys
lim 2 . . x ( s , t ) e t
:=
R,OT
tER
for every s E S (Proposition 7.1.1.3 b2 =3> bl). Then x -- ~ e s y s , sES
) ~ , ( ~ e s ) y s -- u x -- O. sES
For every x' E (So(g))',
'~(~)~'(y~) sES
=
0
7. C*-algebras Generated by Groups
464
so t h a t
9 '(y~) = o for every s E S . Since x' is arbitrary, Ys = 0 for every s E S , i.e. x = 0 and u is injective. Let p , q E INn and t E T . There is an x E S ( f )
such t h a t
~ X "--[~ip~jq]i,jEINn E ]Kn,n.
Put
sES
Then
sES
Hence ($(g))n,n C I m u . Since I m u is closed ( T h e o r e m 4.2.6.6),
(Sc(g))n,n C Im u and u is surjective. Hence u is an isomorphism of C*-algebras.
Remark.
For every n E IN, there is a Schur function f on ~
I
such that S r
is isomorphic to ~n,n (Proposition 7.1.4.9 d)). Proposition
7.1.3.10
Assume IK = IR and let f
be the Schur function for
a finite group T . Put p " - Card {t E TI t2 - 1, f ( t ) = 1}, q := Card {t E r l t2 = 1, f ( t ) = - 1 ) , r : - Card T . a)
DimReS(f)
b)
If n is an odd natural number, n ~ 1, then S ( f ) ~ n ~n 9
= p-q+r 2
is not isomorphic to
7.1 Projective Representations of Groups
465
a) Take t 9 T with t 2 = 1. T h e n
e t = f(t)e, so t h a t et is selfadjoint iff f ( t ) = 1. Take t 9 T with t 2 :fi 1. For c~,13 9 IR,
(OL~ t -Jr ]~e_.t-1 )* -- ozf(t)et-1 + 3 f ( t ) e t = f(t)(13et + ae,-~ ) such t h a t a e t + ~et-1 is selfadjoint iff 13 = a f ( t ) . By the above Dim Re&'(f) = p + b) A s s u m e $ ( f )
r-p-q
p-q+r
2
2
is isomorphic to IR . . . . Since n 2 is odd,
{t 9 T I t2 = 1} = {1}. Hence p----l, By
q=0.
a), l+n 2 n(n + 1) 2 = Dim Re S ( f ) = D i m Re IR,,, = - - - - 7 - - ,
which is a contradiction.
Remark.
Let n C IN. If there is a p C IN U {0} with n = 2p , then by T h e o r e m
7.2.2.7 k), there is a Schur function f on a group such t h a t S ( ] ) is isomorphic to IRn,n. I do not know if the converse of this implication also holds.
7. C*-algebras Generated by Groups
466
7.1.4 E x a m p l e s P r o p o s i t i o n 7.1.4.1
Let f be a Schur function for the finite group T and let L~ be the set of conjugacy classes Q of T such that f(r, r-ltr) f(t, r) = f(s, s-its) f(t, s)
for every t
E
Q and for all r, s
E
T with
r - l t r = s-Xts.
a)
There is a unique qo E (IN U {0}) (~) such that 6r is isomorphic to the C*-direct product of the family ((~p(n)~ Of complex C*-algebras ~, n,n ] nEIN (where ff~~ n = {0} ). We have
Dim(Se (f))c = E
qo(n) = Card L~,
nEIN
E
n2q~
= Card T.
nEIN
If T is commutative and f(s,t)= f(t,s) for all s, t E T , then Sin(f) is isomorphic to
b)
(I] C a r d T
.
There are uniquely A,#,v E (IN U {0}) (~) such that v vanishes at the odd natural numbers,
A+2p+v=qo, and S~t(f) is isomorphic to the C*-direct product of the family
(IR~(') ~'(")xIH~("~) --n,n
X
n,n
nEIN
of real C*-algebras. We have
Dim Re S ~t (f) = = Card {t E T i t = t - l , f ( t ) = 1} + ~Card {t E T i t # t -1} =
= E nEIN
( n ( n + 1)A(n)+ n2#(n)+ n(2n - 1)v(2n)/ 2
7.1 Projective Representations of Groups
Dim Re ( S ~ ( f ) ) c = E
467
(/k(n) + #(n) + ~,(n)).
nEIN
If T is commutative and f ( s , t) = f ( t , s) for all s , t C T , then there are p,q E IN U {0} such that Sn~(f) isomorphic to ]Rp • r
is
and p + 2q = Card T .
q = 0 iff t -- t -1 and f ( t ) = 1 for every t C T . a) ~ ( f )
is finite-dimensional. By Corollary 6.3.6.5, there is a unique ~ C
(IN U {0}) (~) such that $r
is isomorphic to the C*-direct product of the
family \ n,n ] of complex C*-algebras. We have E
n2~(n) = Dim'SO(f) = C a r d T .
nE]N
(,So(f)) c is a Gelfand C*-algebra. By Corollary 4.1.2.5, Proposition 4.2.7.20. and Theorem 7.1.1.5 k), E
~(n) = D i m ( S ( / ) ) c = C a r d ~ .
nEIN
Now assume T commutative and f(s,t)=
f(t,s)
for all s, t c T . By Theorem 7.1.1.5 k), Card T = Dim S r (f) = Card L~. By the above Y
~(n) = ~ Card T i f n = 1
t Hence ,Sr
is isomorphic to
o
if ~r
([~CardT.
b) By Theorem 7.1.2.1 d), ,So(f) is isomorphic to the complexification of $~(f).
By a) and Corollary 5.6.6.10, there are uniquely A, #, v with the given
properties. Let t E T such that t -
t -1 . Then
7. C*-algebras Generated by Groups
468
:r
~ = f (t)~, ,
so that et is selfadjoint iff f(t) - 1. Hence the intersection of R e S t ( f )
with
the vector subspace of Sn~(f) generated by {et ] t E T , t -- t -1} has dimension Card{t E T i t
- t -~, f(t) - 1}.
Let t E T such that t ~ t -1 and let a , ~ E IR. Then (aet + Zet-~ )* = ~e; + Zet-~ * = y(t)(c~et-, + ~et) ,
so that oLet + flet-~ is selfadjoint iff Z = c~/(t), i.e.
{ ~ +/3~-, I ~,
~ e
~} n ReS"~(/)
is one-dimensional. Hence the intersection of R e S ~ ( / )
with the vector sub-
space of S ~ ( f ) generated by {et I t E T , t ~ t -1} has dimension 1Card{t E Tit 2
r t-l}.
Putting together the above results, one obtains
DimaeS~(f,g)
1
= Card {t E T [ t = t - l , f ( t ) = 1} + ~Card {t E T i t ~ t - I } .
The relation Dim Re ( S ~ ( f ) ) c - ~ ( ~ ( n ) +
#(n) + u(n))
nEIN
is easy to see. The other equality follows from Corollary 5.6.6.9. The final assertion is easy to see.
Remark.
I
It may be a very difficult problem to find p, A,#, and u in con-
crete cases. Even if they are known it may be difficult to find a corresponding isomorphism. E x a m p l e 7.1.4.2
Let n E IN and f a Schur function for 2Zn.
a)
f(s, t ) = f(t, s) for all s, t E 7Zn and 5r
is isomorphic to (~n.
b)
If n is odd then S ~ ( I ) is isomorphic to ]R x ffjn~
7.1 Projective Representations of Groups
c)
If n is even then
8~(f)
isomorphic to IR • IR d)
is isomorphic to r
469
if f-(~) = - 1
and is
if f ( 2 ) - 1.
x r 9-1
If T is a group and f a Schur function for T then S ( f )
is not isomor-
phic to 1K2,2 • IK. a) By T h e o r e m 7.1.1.5 1), f ( s ,
t) =
f ( t , s) for all s, t e 2Zn. By Proposition
7.1.4.1 a), ,So(f) is isomorphic to C ~ . b) and c). By a) and Proposition 7.1.4.1 b), there are p, q E ]N t2 {0} such t h a t S i n ( f ) is isomorphic to IRp x Cq and such t h a t p + 2q = n ,
P + q = { 1 q n~l
if n is odd
1 + 51,)-(9) + ~-~ if n is even. It follows t h a t n-1 q--
n--1 2
n--1 2
if n is odd
n - 1 - a l , 7 , ~ ) j ~ - - -n -22 _-- n~
1 P =
51J(9) if n is even,
if n is odd
251,f(~) if n is even.
d) Since Dim (]K2a x IK) = 5 it follows Card T = 5
T - TZs . By a) and b), S ( f ) is not isomorphic to IK2,2 x IK. Remark.
Explicit isomorphisms for S ( f )
are presented in Example 7.1.4.10.
m
7. C*-algebras Generated by Groups
47o
E x a m p l e 7.1.4.3
Let .7z be the set of Schur ]unctions for 7Z~. Put
a:=(1,0),
b:=(0,1),
c:=(1,1),
A := {a er
= 1}
and for every 8 := (a, fl,~/,~) E A a x { - 1 , + 1 } defined by the following table:
denote by fo the function
Io
~o~
E')"
E~
a) feE.7 for every
O E A 3 x {-1,+l}
Aax{-1,+l}
and the map
~.T',
0:
;re,
is bijective.
b) For each 8 := (a, fl,"/,c) E A 3 x { - 1 , + 1 } , fo(a, b) re(b, a) = re(b, c) fo(c, b) = fo(c, a) fo(a, c) = c .
c) If f C ~ and f ( a , b ) = f ( b , a ) , then ~ ( f ) S ~ ( f ) is isomorphic to IR 2p-a x r where
is isomorphic to (~4 and
p := Card {t E 7/,,22] f(t) = 1} E {2, 4}. Moreover,
Ilxll = sup{Ix(0) + ~x(a) + Cx(b) + (pCx(c)] ] ~, r e {+1,-1}} for every x E S o ( f ) .
d) If f E .7z and f(a,b) ~ f(b,a) then , ~ ( f ) S ~ ( f ) is isomorphic to IR2,2 (to ]H ) iff Card{t s is equal to 3 (to 1).
2Z~ls ) :
is isomorphic to r
I}
and
7.1 Projective Representations of Groups
4~
a) is the result of a long calculation. b) follows from a). c) By b) (and Theorem 7.1.1.5 a)),
f(s,t) = f(t,s) for all s, t 9 72;,2 , so that by Proposition 7.1.2.5 d => a, ~qc(f) is isomorphic to
([j4. Since $r
is a Gelfand C*-algebra, I1~11 = I1~11
for every x 9 $r
and it is easy to see that
I]~[I -- sup{Ix(0 ) + cpx(a) + Cx(b) + qpCx(c)l I p, r 9 { + 1 , - 1 } } . Now suppose 1K = IR. By a) and b),
f (a) f (b) f (c) = 1, so that p C {2,4}. The system of equations { A+2tt--4 A+#-p has the unique solution )~=2p-4,
#=4-p.
By Proposition 7.1.4.1 b), ,ga(f) is isomorphic to IR2p-4 x i ~ 4 - p . d) By Proposition 7.1.2.5 a ~ d, ,~r is not isomorphic to ~4, so it is isomorphic to ([J2,2 (Proposition 7.1.4.1 a)). Now suppose IK = IR. By a) and b),
f(a)f(b)f(c) = - 1 , so that V := Card {t e 2Z~l f(t) = 1} e {1, 3}. The system of equations
7. C*-algebras Generated by Groups
472
A+u= 1 3A+u=p has the unique solution A=p-1 2
By Proposition 7.1.4.1 b), S ~ ( f ) i s p=l). Proposition 7.1.4.4
3-p '
v=--~.
isomorphic to 1R2,2 (to IH)iff p = 3 (iff m
Let X be a set, T a subgroup of the permutation groups
of X such that x, y E X =:=>(xy) E T , and f a real Schur function for T (we use here and in the following the usual notation of permutation groups: e.g. (xy) denotes the permutation of X , which interchanges x and y and keeps all other elements fixed).
a) Let (xj)je~2n+l be a family of distinct elements of Z .for some n E IN. Put a "-- (X2n_lX2n),
t :--
b
::
d := (X2,~--~X2nX2n+~),
(X2n_lX2n+l),
l-I (x23-1x2j) jEINn_~ 1
if n ~ 1 ifn=l.
Then f (ta) = f (tb) - f (ta, ta) , f(d) = f(ta, tb)f(tb, ta).
b) Let (Xj)ye~. be a family of distinct elements of X with n 9 IN\{1}. Put a : : (x,x2)(xax4),
b "-(XlX3)(x2x4),
t:: { j=3(x2j-lx2j) if n r 2 1
ifn=2.
Then f (at) - f (bt), f (a) = f (at, bt) f (bt, at).
7.1 Projective Representations of Groups
c) If (xj)je~2~, ( Y j ) j e ~ some n C IN, then
are two families of distinct elements of X for
kjEINn
d)
473
jEINn
Let (xj)j~4n be a family of distinct elements of X for some n E IN. Put n
2n
s :: I ] ( x ~ j _ ~ j ) ,
I I (x~j_~x~j).
t-:
j=l
j=n+l
Then f(st) - f(s, t)f(t, s). N
e) Assume X finite and f ( ( x y ) ) : C*-algebra with
1 for all x , y E X . If E is a real
Dim Re E - 1 and u" S ~ ( f ) --4 E is an involutivc algebra homomorphism then Imu C ReE. f) If X is infinite and
{~ e xlt(~) r x} is finite for every t C T then (Sw (f)) c - IKI. a) Put C ":
(X2nX2n+l),
e "--(X2n_lX2nH_lX2n).
By Proposition 7.1.1.9 a),d), f (c, ta) f (te, c) = f (c, td) f (ta, c) f (c, ta) f (c) f (ta) -- f (ta, c) f (te) f (c, td) f (c) f (td) = f (te, c) f (tb) N
f (td) = f (te),
7. C*-algebras Generated by Groups
474
since
te = (td) -1 It follows that
f (c) f (ta) f (te) = f (c, ta) f (ta, c) = = f(te, c)f(c, td) = f ( c ) f ( t d ) f ( t b ) , so that
f(ta, ta) = f(ta, (ta) -1) = f ( t a ) = f(tb), f (ta, tb) = f (ta, tb) f (ta) f (tb) = f (tb, ta) f (tbta) = = f(tb, ta)f(d) (Proposition 7.1.1.9 d)), so that
f (d) = f (e) = f (ta, tb) f (tb, ta) . b) Put C "---(X2X3) ,
d :--(XlX3X4X2),
e "= (XlX2X4X3).
By Proposition 7.1.1.9 a),d),
f (c, at) f (dt, c) = f (c, et) f (at, c)
f (c, at) f (c) f (at) = f (at, c) f (dt) f(dt, c)f(dt)f(c) = f(c, et)f(bt) f (dt) = f (et) , since
et = (dt) -1 It follows that
f (c) f (at) f (dt) - f (c, at) f (at, c) : f(dt, c)f(c, et) : f ( d t ) f ( c ) f ( b t ) ,
7.1 Projective Representations of Groups
475
so that
f (at) = f (bt) . We deduce
f (at, bt) = f (at, bt) f (at) f (bt) = f (bt, at) f (ab) (Proposition 7.1.1.9 d)),
f(a) = f((xlx4)(x2x3))-- f ( a b ) = f(at, bt)f(bt, at). c) follows from a) and b). d) By c) and Proposition 7.1.1.9 d),
f (s, t) = f (s, t) f (s) f (t) - f (t, s) f (st) , so that
f(st) = f(s, t)f(t, s). e) follows from Theorem 7.1.2.1 n). f) Take t E T\{1} and put
A "- {x e X[t(x) :/: x } . Let (An)neIN be a sequence of pairwise disjoint subsets of X \ A n C IN let ~ 9 A --+ An be a bijection. For every n C IN, put
sn " X
>X ,
x,
>
w~(x)
ifxcA
qp~l(x)
ifxcA~
x
if x E X \ ( A u
and for each
An)
Then (Snts~l)ne~ is an injective sequence in T , so that the conjugacy class of t is infinite. By Theorem 7.1.1.5 k),
( S ~ ( f ) ) ~ = IK1.
I
Let $3 be the permutation group of IN3, 3~ the set of Schur functions for $3, and f E ~ . Put
E x a m p l e 7.1.4.5
A'-{a 9 a "--(12),
b:=(13),
c:=(23),
d--(123),
and denote for every 0 "- (a, fl,~/,5, e) e A 5
by fo the function defined by the following table:
e'=(132)
r
7. C*-algebras Generated by Groups
fo
a
b
c
d
a
~5
~
~
a2~&
a275
b
7r
fl~
a
~27&
a~25
r
c~e
",/25
a72 &
~725
a275c
a/325r
#725r
a2f15
~275
a725
e
a2#23,252e
a2~272~2r
a) fo C . T for every O E A 5 and the map A5
>.T',
O,
>fo
is bijective.
b) ,5~(f) is isomorphic to r xq~ xr c) If f((12)) = 1, then ,S~(f) is isomorphic to IR x IR x lRz,a. N
d) /f f((1,2))= -1, then S~(f) is isomorphic
to IF_,x ]H.
a) is the result of a long calculation. b) The system
E ~(n) < 3 nEIN
n2~(n) = Card $3 nEIN
has a unique solution in # E (IN U {0})(~), namely
~(1)-2,
~(2)-1,
n>2==#cz(n)=O.
By Proposition 7.1.4.1 a) and Theorem 2.2.2.7 d4), So(f) is isomorphic to
(!:; xC xC2,2. c) and d). By Proposition 7.1.4.4 a), N
N
f((12)) = f ( ( 1 3 ) ) = f((23)).
7.1 Projective Representations of Groups
477
Hence, by Proposition 7.1.4.1 b),
d'-
5
if f ( ( 1 2 ) ) = 1
2
if f ( ( 1 2 ) ) = - 1 .
Dim Re ,S~ (f) -
Let A,#,u e (IN U {0}) 2 such that u(1) = 0 and A(1) + 2p(1) = 2
A(2) + 2 ~ ( 2 ) + ~(2) = 1 A(1) + p(1) + u(2) + 3A(2) + 4#(2) = d. Then p(2) = 0 and - # ( 1 ) + 2A(2) = d - 3. If #(1) - 0, then A(1)-2,
d-3
A(2)=
This implies
d=5, If # ( 1 ) r
~(2)=~,
~,(2)=0.
A(1)=0,
A(2)-
A(2)=O,
u(2)=1.
then if(l)=1,
d-2 2
This implies d=2, .....
By Proposition 7.1.4.1 b), if f((12)) = 1, then 8 a ( f )
is isomorphic to
IR x IR x IR2,2 and if f((1,2)) = - 1 , then ,Sa(f) is isomorphic to C x IH. m
E x a m p l e 7.1.4.6 Let A4 be the group of even permutations of IN4 and f a real Schur function for A4. Put a := (12)(34),
b "- (13)(24),
c "-(14)(23),
p := Card{t E A4lt -1 = t, f(t) = 1}. a)
{1, a, b, c} is a subgroup of A4 isomorphic to 2Z2 .
7. C*-algebras Generated by Groups
478
f(a,b)f(b,a)= f(b,c)f(c,b)= f(c,a)f(a,c).
b)
c) If f(a, b) = f(b, a), then
f (a) = f (b) = f (c) = 1, ~ ( f ) is isomorphic to r
p=4,
and S~t ( f ) is isomorphic to
Xr
]R X r X 11{3,3 .
If f (a, b) 7/: f (b, a) then
d)
f(a)f(b)f(c) = - 1 ,
p e {1,3},
~ ( f ) is isomorphic to r 2,2 and Sm(f) is isomorphic to IR2,2 xr p = 3 and isomorphic to r x IH if p = 1.
if
a) is easy to verify. b) follows from a) and Example 7.1.4.3 b). c) and d). A4 has 4 conjugacy classes. The equation
n2~(n) = Card A4 nEIN
has two solutions in ~ C (IN U {0}) (r~) with ~(n) < 4, nEIN
namely ~(1)=3,
r
~o(3)=1,
n>3:=~,,p(n)=O
~(1) = 0 ,
qo(2)=3,
qo(3)--0,
n>3~~(n)=0.
and
By Proposition 7.1.4.1 a), So(f) is isomorphic to r Nr 3 in the first case and it is isomorphic to r in the second case. Assume q0 is the first solution. The system of equations A1 q- 2#1 = 3 A3 = 1 A1 q- #1 q-6A3 = 4 + p has a solution in INU{O} only if p = 4 and in this case, there is a unqiue solution
7.1 Projective Representations of Groups
479
A1 =tq=A3=lSince p = 4,
f (a) = f (b) = f (c) -- 1 and by Example 7.1.4.3 c),d), f (a, b) = f (b, a). By Proposition 7.1.4.1 b), Sn~(f) is isomorphic to ]R x @ x IRa,3. Now suppose p is the second solution. Consider the system of equations
{ ,x+2#+~,= 3 3A + 4 # + u = 4 + p . It follows that 2#+2u--5-p. Hence p is odd and therefore
f(a)f(b)f(c) = -1. By Example 7.1.4.3 c),d),
f (a, b) ~- f (b, a). There is a t C {a, b, c} such that
f(t) = - 1 . For 8 :-- 1, 8--It8
_
_
t--i ,
N
f (s, s-lt-Xs) f (t -~, s) f (t) - - 1 . By Proposition 7.1.2.2 b), Dim Re (slR(f)) ~ _< 2 so that, by Proposition 7.1.4.1 b), A+p+u_<2.
7. C*-algebras Generated by Groups
480
If p = 1, then #+u=2, and A = 0, # = 1, v = 1 is the unique solution. Hence Sin(f) is isomorphic to r
x IH.
If p = 3, then
#+v--I and A = 1, # = 1, v = 0 is the unique solution. Hence Sin(f) is isomorphic to 1R2,2 x ~2,2. Example
II
7.1.4.7
Let $4 be the permutation group of
IN 4
and
f
a
real Schur
function for $4. Put a := (12)(34), a)
f(a,b)f(b,a)=
b := (13)(24),
c := (14)(23).
f(b,c)f(c,b)= f(c,a)f(a,c), f (a) = f (b) - f (c) ,
f((12)) - f((13)) - f((14)) = f ( ( 2 3 ) ) = f ( ( 2 4 ) ) = f ( ( 3 4 ) ) . b)
If f (a, b) = f (b, a) , then f (a) - f (b) = f (c) = 1, S o ( f ) is isomorphic to ~2 • ~2,2 x C a,3, 2 and S rt ( f ) is isomorphic to IR 2 x IR2,2 x
IR a,3 2
q~ x ]H xr c)
if f ( ( 1 2 ) ) = 1 if f ( ( 1 2 ) ) = - 1 .
If f (a, b) :/: f (b, a) , then f(a) = f ( b ) = f ( c ) -
-1,
Sr ( f ) is isomorphic to ~22,2 x ~4,4, and Sm ( f ) is isomorphic to r
x 1R4,4
IN 2 x 1H2,2
if f ( ( 1 2 ) ) = 1 if f ( ( 1 2 ) ) =
-1.
7.1 Projective Representations of Groups
481
a) follows from Example 7.1.4.6 b) and Proposition 7.1.4.4 c). b & c. 5'4 has five conjugacy classes (Theorem 2.2.2.7 d4)). The equation n2~(n) - Card $4
E nEIN
has two solutions in ~ e (IN t2 {0}) (~) with
~(~) < 5, nEIN
namely
~(1)--2,
~(2)=1,
~(3)=2,
n>3==~(n)--0
and ~(1)=~(3)-0,
~(2)-2,
~(4)=1,
n>4==~(n)=0.
By Proposition 7.1.4.1 a) , So(f) is isomorphic to ~2 • ~2,2 x q~2 3,3 in the first case and to ~22,2 x ~4,4 in the second case. Put p "- Card {t e S4 It -1 - t,
f ( t ) -- 1 }.
Assume ~ is the first solution. By Corollary 2.1.4.13 c) ~32,2 is not isomorphic to a subalgebra of ~2 x ~ , 2 x ~23,3
9
By Example 7.1.4.6 c),d)
f (a, b) = f (b, a),
f(a) = f(b) = f (a) - 1, since A4 is a subgroup of $4. By a), p C {4, 10}. Consider the system of equations
A1 -~ 2#1 = 2 A2 + //2 = 1 A3 + 2p3 = 2
A1 nt- ~1 -~- I/2 -~- 3A2 + 6A3 + 9#3 = 7 + p. We have Pl -9 2//2 -F 3#3 -- I 0 - - p . If p----- 10, then f ( ( 1 2 ) ) - - 1 and
7. C*-algebras Generated by Groups
482
~1
=
/]2 ~-~
~3
=
O,
and if p = 4, t h e n f ( ( 1 2 ) ) = - 1 #1
:
/]2
-"
P3
A1 =
A3 =
2,
A2 =
1
and 1,
:
A1 : A2 : A3 -- 0.
By P r o p o s i t i o n 7.1.4.1 b), $ ~ t ( f ) is isomorphic to ]R 2 x IR2,2 x ]R 3,3 2 in the first case and to ~ x IH x ~3,3 in the second case. Now suppose ~ is the second solution. By Corollary 2.1.4.13 c), C 3 x ~3,3 is not isomorphic to a subalgebra of ~22,2
X
~4,4- By E x a m p l e 7.1.4.6 c),d),
f (a, b) ~ f(b, a ) ,
f(a)f(b)f(c)
= -1,
since A4 is a s u b g r o u p of $4. By a ) ,
f (a) - f (b) - f (c) - - 1 and p E {1, 7}. Consider the system of equations
I A2+2p2+U2=2 A4 -+- u4 ---- 1 3Az + 10A4 + 4#2 + v2 + 6v4 = 7 + p . We have 2#2 + 2u2 + 4u4 -- 9 -- p . If p = 1, then f ( ( 1 2 ) ) = - 1
and
u2 = 2, and if p = 7, then f ( ( 1 2 ) ) -
f14
- -
u4 = 1,
A2 - A4 - #2 - 0
1 and either
#2
--
I,
A2
:
/22
---
/24
:
0
or
A2 -- A4 -- u2 -- 1,
#2 = u4 = O.
By Proposition 7.1.4.1 b), ,.q~t(f) is isomorphic to IH2 x IH2,2 in the first case while in the second case it can be isomorphic either to C2,2 • ]R4,4 or to IR2,2 • ]H • ]R4,4. But the last case cannot occur by a) and Proposition 7.1.4.4 e).
9
483
7.1 Projective Representations of Groups
Lemma
7.1.4.8
w =
Take n E IN and put
x :=
9
j,ke~
y := [51+j k]j k e ~
E ~,~ ~
E ~,~ n
where, in the definition of y , INn is canonically identified with ~ n .
a)
x , y e Un(r
b)
For p, q E 2~,
xPYq = [o)P(J-1) (~q+J,k] j,kEIN,~ ' yqx p -- 02PqxPyq "
c)
xn=yn=l.
d)
(xPyq)(p,q)EiN~ is an algebraic basis of the vector space ~n,n.
a) is easy to see. b) Obviously xP -- [(~Jkcdp(k-1)] j,kEINn
"
By induction yq = [Sq+j,k]j,kElNn 9
Hence
= [~P(J-1)5~+J, k] j,ke~
'
"-- [(~q+j'kO')p(k-1)] j,kCINn -- [(~qTj'kO')p(q+j-1)] j,kEINn ----
= ~
[~+~,~(J-~)]j,k~o
-- wPqx p yq
7. C*-algebras Generated by Groups
484
c) follows from b). d) Let
(O~pq)p,qEINn be
a family in q~ such that OZpqXPy q -- O . p,qEINn
By b), for j, k E INn, 0--"
~
OLpqO.)P(J-1)(~q+j,k ---- ~
p,qE]Nn
OLp,k_jO.)p(j-1)
pE]Nn
~-~ (Rp,qO.)p(j- 1) __ O . pEINn
Since Det [WP(J-1)]p,jeiN~ # 0 it follows OLp,q -- 0
for all p,q E INn. Hence (xPyq)p,qeiN~ is linearly independent, i.e. it is an algebraic basis for the vector space ~n,n. I P r o p o s i t i o n 7.1.4.9 Take n E IN and jl, j2, kl, k2, g E ~ such that njl, nj2, jl + kl , and j2 + k2 are even and put ~rz
9--
fjm "2En • 2En
e w
,
> {oz E(IJ[[a[-- 1},
(p,q),
>w 2jmpq
for m E {1,2}, X : : [(~jkw2(k-1)]j,kEiNn E(~n,n,
Y "= [51+j,k]j,ke~n E r where, in the definition of y, INn is canonically identified with 2En.
a) fjm is a Schur function for ~ n for every m E {1, 2}.
7.1 Projective Representations o] Groups
b)
There is a unique Schur function f for ~n • 2 ~ such that
f((p,O), (q,O)) = w 2j~pq, f((O,p), (O,q)) = w 2j2pq, f((p, 0), (0, q)) = w 2tpq ,
f((O, q), (p, 0)) = 032(g+l)pq
for all p, q e ~ n . For (p, q), (r, ,3) e 2Z~ • ~ , ,
f((p, q), (r, s)) = 022jlpr+2j2qsT2~ps+2(gT1)qr. c)
Fo~ (p, q) 9 2Z~ • 2z~, e (p ,q) - - w 2jl p2 + 2j2 q2 + 2(2t + 1)pq e (_p ,_q)
in 8 ( f ) .
d)
There is a unique isomorphism of C*-algebras
~.s(f)
~r
such that
~(~(~,o)) - ~ ' ~ ,
~(~(o,~)) = ~ y .
We have ~(e(p,q) ) -- W (jl +kl )p+(j2+k2)q-jlp2-j2q2-2~PqxPyq
for all p,q C ~ n .
a) For
r,s,t C ~n, fjm (r, 8)fjm (r + ,3, t) -- w2jmrso.) 2jm(r+s)t ~-
Moreover fj~ (o, o) = ~o = 1.
b) Define g-Z~n•
>{aer
(p,q),
>w 2epq,
485
7. C*-algebras Generated by Groups
486
h-2Z~ x 2Z~
> 4~ e r
1},
(p,q), >w 2(t+i)pq.
The claim follows from a) and Proposition 7.1.3.1 a =~ c. c) By b), for (r,s) e ~ n x ~ , ,
]'((r, s)) -- f ((r, s), ( - r , - s ) ) -- 022jlr2§
e(*p,q)((r, s)) = f ((r, s))e(p,q)((-r, - s ) ) = --__ 032jlr2+2j2s2+2(2~+l)rSS(p,q),(_r,_s)
=
so that * e(p,q)
~_
O.)2j~p2+2j2q2+2(2t+l)pqe(_p,_q)
d) The uniqueness is obvious. Define ~ as a linear map
such that
~o( e(p,q) ) -- w (jl +kl )p+(j2+k2)q-jlp2 -j2q2- 2tpqxP y q for all p, q C ~ n . By Lemma 7.1.4.8 c,d), ~ is well-defined and bijective. Take (p, q), (r, s) e 2En x ~ n . By b),c), and Lemma 7.1.4.8 a),b),
=
022jlpr+2j2qs+21ps+2(t+l)qr X
X O.)(jl +kl )(p+r)+(j2+k2)(q+s)-jl (p+r)2-j2(q+s)2-2~(p+r)(q+s) XP+r yq+S =
__ ~x)(jl A-kl )p-t-(j2+k2)q-jlp 2-j2q 2-2~pq X
~(e(*~.,)) = ~ J : + ~ J , e + W ~ + , ) ~ , ~ ( ~ ( _ ~ _ ~ ) )
=
7.1 Projective Representations of Groups
487
- - 03 2jlp2 T2j2q2 +2(2~T1)pq X
X 03 - (jl -k-kl ) p - (j2-k-k2)q-jlp 2 - j 2 q 2-2lpq x - p y - q
_.
._ O.)-(jl+kl )p-(j2"+-k2)q+jlp2+j2q2+2lPqy-q x-P : =
(~(~+~)~+(j~+~)~-J~-J~e-~~yq),
= ~(~(~,~)),.
Hence ~ is an involutive algebra homomorphism and therefore an isomorphism of C*-algebras. E x a m p l e 7.1.4.10
I Let n E IN and let .~ be the set of Schur functions f o r
2En. Denote A:={aEIKllal=l},
w:=e-
27r~
,
and for every a E A n-1 put
f~'~nX2~n
~A,
(p,q)'
>
~j
ak \
,
k=q
where 2Z~ and INn are canonically identified and a~ - 1.
a)
f~ E . T for every a E A n-1 and the map A n-1
) J~,
a,
~fa
is bijective. Fix a E A n-1 and take ~ E 9 with n--1
/3n = l-I aj. j--1
b)
For every p E INn,
el*el * . . . *el =
o/j
ep.
p - times
c)
For every k E ~ n
there is exactly one homomorphism of C*-algebras
with
For all k, p, q E 2En , p-1
f,~(p, q) = uk(ep)uk(eq)
j=l
u~(e,+q)
488
7. C*-algebras Generated by Groups
d) cr(Sc(f~))= {uklk e INn}. e)
If IK = ]R and n is odd then we may take 3 E ]Ft and >]R x r "~ ~ ,
Srt(f~)
~,
~ ( u~(~), ( ~ ( x ) ) ~ ~
)
is an isomorphism of real C*-algebras.
n--1
f) If ]K = ]R, n is even, and 1-I aj = - I
then
j=l
s~(fo)
..... >r
~,
~(u~_~(~))~
is an isomorphism of real C*-algebras.
n--1
g) If ]K = IR, n is even, and l-I aj = l the we may take 3 = 1
and
j=l
:(:o)
> ~ • ~ x c ~ -1 ,
~:
> (u~(~), ~ ( ~ ) ,
(~(~))~_,)
is an isomorphism of real C*-algebras.
h) For every x E r 21ri~k
llExJ,lJ, .nll- kEINn sup j = l e Xj "
a) For p , q , r C 7Zn, fa(p, q) f~(p + q, r) f~(p, q + r) f~(q, r) --
._ (~.~j) (P~loLk) (p~l.~j) (p-t-~-loLk) X \ k--q \ j=l \ k--r
-
l k)
Hence f~ C :-. Take f C :" and put o~j "- f(1,j) E A for every j E ~n\{0}. By Example 7.1.4.2 a),
1 ) = 1"
7.1 Projective Representations of Groups
f(j, 1)=c~j for all j 9 2Zn\{0}. Let p 9 2En\{0} and assume
for all q 9 2Zn\{0}. Then
f ( p + 1, q ) = f(p, 1 ) f ( p , q + 1)f(1, q ) =
---~p (~-~j) (kP+=Hq+lOLk)O~q: (j=~l-~j) (~=~OLk) for all q 9 ~ n \ { 0 } . Hence f = f~ and the map A ~-1
>$-,
~.~
~f,
is bijective. b) follows by complete induction. c) Uniqueness is obvious. Put p-1
j--I
for every p 9 ~
and extend Uk linearly to S ( f ) . For p, q 9 2En,
\ k--q
\ ~--1
= Uk(Cp*eq), uk(ep) = uk(f,~(p)e_p) =
~(p, 1l,-- p)'U,k(en_p)-- (PjH=IOLj)(1--I ~k) t~n-pWk(n-,)n--p--I H -~f"= \k=n-p e=l
489
490
7. C*-algebras Generated by Groups
--
O~j
~-PW -kp -- Uk(ep) .
Hence Uk is an involutive algebra homomorphism. We have
?Zk(eP)' lZk(e-'q=) (Pj~=I~ ) (qjN=l~ ) Uk(ep+q) (Pj~----1)(p~_~l ) -ff-] O~k \ k-"q
=
(p~~_K1) \ t=l O i l
=/(p,
--
q).
d) follows from c) (~nd nx~m01e 7.1.4.2 a)). e) Take x e S ~ ( f ~ )
with
Then for every k E I N . - x ,
2
n--1
p--1
p=O j=l n-1
p-1
p--O
j=l
n-1
p-1
p=O
j--1
=0.
It follows x - 0 and the map is bijective. By c), it is an isomorphism of real C*-algebras. f) Take x e S~t(f~) with
(u,_~(~)),~ Then for every k E IN~,
n--I
p-I
p:O
j:l
= 0.
7.1 Projective Representations of Groups
n-1
49~
p-1
EXpfl-Po3-(k-1)P H oIj p=O
:0.
j=l
It follows x = 0 and the m a p is bijective. By c), it is an isomorphism of real C*-algebras. g) Take x e S~t(fk) with
Then for every k C IN~_l, n-1
p--1
E x , II , =0, p=0
j=l
n--1
p--1
p=O
j=l
H n-1
= o,
p-1
E Xpwkp H (~j = 0 , p=0
j=l
n-1
p--1
F
=0.
p=0
j=l
It follows x - 0 and the map is bijective. By c), it is an isomorphism of real C*-algebras. h) Take c~j = 1 for every j C ~ n and ~ = 1. By Proposition 7.1.2.5 d =~ a, 5-r
is a Gelfand C*-algebra so t h a t the Gelfand transform is an isometry
of C*-algebra. By d), I l x J l - sup Juk(x)J = sup
kEINn
~--~WkJXj .
kEINn j = l
For every y CC n and j C INn,
(x 9 y)(j) - ~
x(j - k)y(k) .
k--1 Hence if we identify S ( f ) with the corresponding subalgebra of Cn,n then x is identified with the matrix
[Xj-k lj,kE2En . We get n llr
1
ii
[J[Xj_kJj,keTZ,~lJ = sup ~ e kEINn
j=l
- xj
I
492
7. C*-algebras Generated by Groups
7.2 Clifford Algebras Clifford algebras appear in many domains of mathematics and physics. We present them in this section only as examples of C*-algebras.
7.2.1 General Clifford Algebras
Throughout this subsection, T denotes a totally ordered set and p a map p: T x T
~ {-1,+1},
such that
p(s, t) = p(t, s) for all s, t C T . We define
t.
a:T--~{-1,+l},
w:~
>INU{0},
A,
>p(t,t),
(1 + Card A)Card A 2
and endow T with the composition
T x ~.
> T,
(A,B) ,
>A A B .
T is a locally finite commutative group with 0 as neutral element (Proposition 2.4.4.11 a)).
Definition 7.2.1.1 ( 7 ) A word (on the alphabet T ) is a family of the fo~m (tj)j~o i,~ T If a : = ( ~ j ) j ~ and b : = ( t j ) j ~ . a~ t~o ~o~d~, th~n we denote by ab (the product of a and b) the word (rj)je~m+n defined by
I sj
ifj<_m
r j "=
tj_~
iy j > m.
We add the empty word 1 to the set of words and define its products by
7.2 Clifford Algebras
493
la = al = a for any word a.
If we represent the above words a and b written as a:=sls2...sm,
b:=tlt2...tn,
then ab is represented by sls2...Sm tlt2...tn
which is obtained by concantenation, i.e. by writing the letters of a in order and following them by those of b in order. The empty word has no letters at all. If a, b, c are words, then (ab)c = a(bc) .
P r o p o s i t i o n 7.2.1.2
( 7 ) Let W be the set of words on the alphabet T .
We consider the vector space IK (W) , set
1"=1 and (tlt2...tn)*'= (Ha(tj))
t . . . . t2tl C ]K(W)
j--1
for every t i t 2 . . , tn C W , and endow IK (W) with the multiplication
IK (w)
x
]K (w)
>
IK(w) ,
( ooa
, bCW
a,bEW
and with the involution IK(W)
> IK(W) '
E
aaa J
>E
aCW
~-ga*,
aCW
where we identified ea with a for every a C W .
a)
lK (W) is an involutive unital algebra with 1 as unit.
b)
The ideal R of IK (W) generated by {st - p(s, t)ts l s, t G T, s ~ t} U {t 2 - a ( t ) l I t G T } is proper and involutive. We denote by Cg(p) (or more precisely, Cg~(p) ) the involutive unital algebra IK(W)/R and call it the Clifford algebra associated to p. If p is the constant - 1 ,
then we write Cg(T)
(or
Cg~ (T) ) instead of Cg(p) and call Cg(T) the Clifford algebra generated by T . In this case the elements of T are called the g e n e r a t o r s o f Cg(T) .
7. C*-algebras Generated by Groups
494
c)
We identify ~, with a subset of ce(p) via the map
~:
>c e ( p ) ,
0,
>1
{t~,t~,...,t,}.~ ,,q(tit=...t,), where tl < t2 < ... < tn and where q denotes the quotient map IK (W) --+ Cg(p). Then ~ is an algebraic basis of the underlying vector space of Cg(p). In particular,
DimCg(p) = 2CardT if T is finite.
d)
The map o
t e a ( p ) ---+ cee(p) ,
(x, y) ,
> x + iy
is an isomorphism of involutive unital complex algebras. e)
A e ffg=~ A * A = I .
f)
There are uniquely determined maps
f : ~2 x ff ---~ {-1, +1}, g:ff
> {-1,+1}
such that A B = f (A, B ) A A B ,
A* = g ( A ) A
for all A, B E ft.
g) f is a Schur function .for 9 (Definition 7.1.1.7), g = f , and Cg(p) = S(f).
h)
f is called the Schur Junction associated to p.
Given x, y E Cg(p) and A E 7~,
(xy)(A) = E f ( A A B , B ) x ( A A B ) y ( B ) , BET
x*(A) = g ( A ) x ( A ) .
7.2 Clifford Algebras
~95
i) If p(s,t) = - 1 for all s,t E T with s ~ t, then:
il) A , B e 9 =~ A B = (--1)Card(AxB)-Card(AnB)BA. i2)
Take A E f f . Put
n'=CardA,
a'=na(t). tEA
Then
A2=(-1)~a1,
A*=(-1)~c~A.
In particular, if p is constant equal to - 1 ,
A2=(--1)~(A)I, ia)
A*=(-1)~(A)A.
Take A C ~ , t E A , and /3 C IK with /~4 = 1 . Put a'-
na(s),
x = fltA.
sEA
Then
x 2 = - ( - 1)~(n)c~2a(t)1, x * = (-1)~(n)a~2a(t)x. a) ,b) ,c) ,d) ,e) , and f) are easy to see. g) The relation A r ff2 ~
f ( A , 0) = f(0, A) = 1
is obvious. Take A, B, C E ~. The relations f ( A , B ) f ( A A B , C) = f ( A , B A C ) f ( B , C), f (A, B)g(A)g(B) = f (B, A ) g ( A A B )
follow from Proposition 7.1.1.1. respectively. By e) and f),
Cl ~
C2
and Proposition 7.1.1.6 a2 ==~ aa,
7. C*-algebras Generated by Groups
496
f ( A , B ) g ( A ) A A B = g(A)AB = A*B = - A*f(A, A A B ) A ( A A B ) -
f(A, A A B ) A A B ,
so that
f (A, B)g(A) = f (A, A A B ) . We get
g(A) f (A, A) = f (A, O)= 1, so that
g(A) = f(A, A) = f ( A ) . h) By c) and f),
9~ = ( ~ . / ~ ) ~ ) CE'K
(~(,).)
-
BE~
= Z (Zx/~/~/-/~,)= BE~
= ~_. ~
CEg
x(AAB)y(B)(AAB)B =
B E T AEq~
= ~
~_. x ( A A B ) y ( B ) f ( A A B , B)A =
BET AEg
AEg
BEg
AEg
AEg
AEg
Thus
(xy)(A) - ~
f (AAB, B)x(AAB)y(B) ,
BEg
x*(A) = g(A) x(A).
7.2 Clifford Algebras
497
il) For every t E A, if t ~ B
tB = ~ (--1)C~rdBBt
(
(--1)CardB-1Bt
if t C B .
Hence AB
= ( - 1)Card (A\B)xCard B+Card(AVIB)x(Card B-l)BA =
= (--1)Card(AxB)-Card(AnB)BA "
i2) Take A = {tl, t 2 , . . . , tn} with tl < t2 < . . . tn. Then A 2 = tit2..,
tntlt2..,
tn =
__ ( _ 1) (n-1)+..-+2+1§247 2 = (--1) -~1~2...t~
A* = ( t i t 2 . . . tn)* - tn . . . t;t*l -- a t e . . ,
al,
t:tl =
= a(-1)(n-1)+"'+2+ltlt:...tn - (-1)~aA. i3) P u t n := Card A. By i2), X2-
~2tAtA
= (-1)n-lfl2t2A2=
(-1)n-l~2cr(t)(-1)~al-
= -- (-- 1)~(A)a/32a(t) 1,
--
--
~
x* - ~ A * t * = ~ ( - 1 ) ~ a a ( t ) A t
n(n--1)
~
= ~(-1) --~afla(t)
= --(--1)~(A)afl2a(t)x.
,,
.
(-1 )~-ltA = m
7. C*-algebras Generated by Groups
498
Definition 7.2.1.3
( ? ) Let f be the Schur function associated to p (Pro-
position 7.2.1.2 g)). We set Cg~w(p) :-- S~w ( f ) ,
Cg~(p) "= S ~ ( p ) ,
and call them the Clifford W*-algebra and the Clifford C*-algebra associated to p, respectively. In this context we denote the product by xy rather than by x , y and set Xs'T
~IK
A: '
; ~ x(A) (
0
if A c S ifA~_S
for all x, y E e2(T) and S C T . If p is the constant - 1 then we write Ce~(T) and Ce~(r) instead of Cg~(p), and Ce~(p), respectively, and call them the Clifford W*-algebra and the Clifford C*-algebra of T . The exponent IK will be omitted in general.
If T is finite, then ce(p) = ce~(p) = ce~(p) .
By Proposition 7.2.1.2 il), i2), we may drop the condition that T be totally ordered for Ce(T), Cs and Ce,~(T). Proposition 7.2.1.4
( ~ ) We identify e2(T) in a natural way with a vector
subspace of ~2(~). If p(s,t) = - 1 for all s,t E T , s ~ t, then: ~)
e~(T) c e e l ( p ) .
1 9 b) x, y e g2(T) =~ ~(xy + y*x) = {xly) l.
c)
z e e~(T) ~
II~ll < ~llzll~, ~ = E z(t)7. tET
d) If IK = IR and if a is constant, then x e e2(T) ==a x*x = I[xl121, [[x'll-
IIxII2.
7.2 Clifford Algebras
499
a),b), and c). Let x, y E IK (T) . Then
sET
tET
sET
tET
= 2 E x(t)a(t)y(t)t2+ tET
+ E
( x ( s ) a ( t ) y ( t ) st + x ( t ) a ( s ) y ( s ) t s + a ( s ) y ( s ) x ( t ) s t + a ( t ) y ( t ) x ( s ) t s ) =
s,tET s
= 2 ( x l y )1.
In particular,
l (xx* § x ' x ) - Ilxll~l 2
1
1
~11~11= - ~llx*xll-
1
1
I1~*=11 __< 116(xx* + x*~)il- Ilxll~
(Theorem 7.1.2.1 c), Corollary 4.2.1.18), so t h a t
I1~11 _< vSIIxll=. Let now x C t~2(T). Then
I1~ - ~11 <_ vSIIx~ - ~11~. for all R, S c ~s R c S . Hence if ~ denotes the upper section filter of %7, then u := l i m x s S,~ exists in
s
and
Ilull _< vSIIxll~. We have for every y C ~2(T) and A c r163
uy = lim x s y -- lim x sy , s,~ s,iy (uy)(A) = lim(xsy)(A) = (xy)(A)
s,~
500
7. C*-algebras Generated by Groups
xy = u y . __.),
Hence x E Cgw(p) and x = u, so
I1~11~ v~llxll~. By Proposition 7.1.1.3 b4 ~ b2, x E Cgc(p). b) and x=
E (x t)
t
tET
follow by continuity. d) For x E ~:~(T) ,
sET
= E
tET
x(t)2a(t)t2 + E
tET
(x(s)a(s)x(t)st + x(t)a(t)x(s)ts) -
s,tET s
= I1~11~1. By continuity, the above result holds for an arbitrary x E / 2 ( T ) . We get
I1~11~- IIx*~ll--I1=11~. P r o p o s i t i o n 7.2.1.5 a)
m
(~)
There are unital C*-algebras containing Cg(p) as an involutive unital subalgebra and all such C*-algebras induce the same norm on Cg(p) . We take Cg(p) with this norm.
b)
Let E be a unital C*-algebra and (xt)teT a family in E such that x~xt = p(s,
x2t - a ( t ) l ,
t)ztz~,
xt = a(t)xt
for all s, t E T , s ~ t. Then there are uniquely determined continuous homomorphisms of involutive unital algebras 9 c~(p)
~ E,
7.2 Clifford Algebras
v "CG(p)
501
>E ,
such that llull <_ 1 and ut z vt ~ Xt for every t C T . If, in addition, the set {XtlXt2 ...Xtn [ t l , t 2 , . . . t ~ E T , t~ < t2 < "'" < t~} is linearly independent, then u and v are injective.
c)
If e
is a unital C*-algebra containing C~.(p)
as a dense involutive
unital subalgebra, then there is a unique isomorphism of C*-algebras
ce~(p) -~ E which J ~ d)
~w~y d ~ m ~ t
of Ce(p).
/f p'" T x T
>{-1,+1}
is a map such that p'(s,t) = p ( s , t ) for all distinct s, t E T , then there is a unique isomorphism
~ . cer
> cer
')
(~;.
~ . ce~(p)
>c ~ ( p ' ) )
such that
~t = ~ t ( it
if p(t, t) = p'(t, t) if p(t, t) # p'(t, t)
for every t E T .
a) By Theorem 7.1.2.1 c), there are unital C*-algebras containing Cg(p) as an involutive unital subalgebra. The uniqueness of the norm is obvious if T is finite (Corollary 4.1.1.21). But every element of Cg(p) is contained in an involutive unital subalgebra of Cg(p) generated by a finite subset of T . b) Let W be the set of words on the alphabet T and w : IK (W) -+ E the homomorphism of unital algebras such that wt -- Xt
502
7. C*-algebras Generated by Groups
for every t C T . By the relations fulfilled by the family (Xt)teT, W factorizes to a homomorphism of involutive unital algebras u : Ce(p) ~ E . Since every element of Ce(p) is contained in an involutive unital subalgebra of Ce(p) generated by a finite subset of T , it follows from Corollary 4.1.1.20 that u is continuous and Ilull _ 1. Hence u may be extended to a continuous homomorphism of involutive unital algebras v : Ct~c(p) --4 E . The uniqueness is obvious. If the supplementary condition is fulfilled, then u is injective, so it preserves the norms (Theorem 4.2.6.6). It follows that v is injective. c) follows from b). d) Put / Xt
t
if p(t, t) = p'(t, t)
it
if p(t, t) # p'(t, t)
for all t E T and apply b) and c).
Corollary 7.2.1.6 are equivalent.
m
( -[ ) Let S be a totally ordered set. Then the following
a) Cg(S) and Ce(T) a~r isomorphic. b) Cg~(S)and Ce~(T) are isomorphic. c)
Cards = CardT.
These conditions imply: d) Cew(S) and Ce~(T) ~rr isomorphic. a ::, b follows from Proposition 7.2.1.5 b). b =~ c follows from Proposition 7.1.2.3 h). c =~ a & d follows from Proposition 7.2.1.2 il), i2).
m
C o r o l l a r y 7.2.1.7 ( "~ ) Suppose T finite. Take an element a not belonging to T and extend p to a function p': (TU {a}) x (TU {a}) .... > {-1, +1}
by defining p'(a,a)
/ +1
if T 2 = - 1
I
if T2 = 1
-1
7.2 Clifford Algebras
503
and pl (t,
a) - p' (a, t) = ~ + 1 l -
if t T = T t if t T = - T t
1
for every t c T . Then there is a unique isomorphism of real C*-algebras
u . c e * ( p ')
~ cer
such that ut=t for every t C T and ua : i T .
Put X t :--t
for every t C T and xa := i T . Then z~2 = p'(s, s ) l
ZrX~ = p' (r, s)z~Z,. ,
for all distinct r, s C T U {a}. By Proposition 7.2.1.5 b), there is a unique homomorphism of involutive unital real algebras
~ . c e ~ ( p ')
~ eel(p)
such that U8
--" X s
for every s C T U {a}. It is easy to see t h a t u is surjective. Since Cg~(p ') and Cgc (p) have the same real dimension, u is bijective. Hence u is an isomorphism
of real C*-algebras. C o r o l l a r y 7.2.1.8
I ( 1 ) Suppose T finite. Take z- E ]K with T2 E { - 1 , +1}
and a an element not belonging to T and extend p to a function
7. C*-algebras Generated by Groups
5o4
p ' : (T U {a}) • (T U {a})
> {-1,+1}
by defining p'(a,a) :=
T2
if T 2 = 1
--T 2
if T 2 = - I
and if tT = T t
p' (t, a) "= p' (a, t) "- ~ + 1
t -1
if tT = - T t
for every t C T . Then there is a unique isomorphism of C*-algebras ~ : c~(p')
~ C~(p) • ce(p)
such that
ut = (t, t) .for every t E T and ua = ( T T , - ~ - T ) . Put
~ := (t, t) for every t E T and x~ := ( r T , - z T ) .
Then x~x~ = p' (~, s ) x ~ x ~ ,
x~2 -
p'(s,s)l
for all distinct r, s E T U {a}. By Proposition 7.2.1.5 b), there is a unique homomorphism of involutive unital real algebras ~ : c~(p')
~ Ce(p) • c~(p)
such that U8
--
X s
for every s E T U {a}. It is easy to see t h a t u is injective. Since Cs
Cs
• Cs
and
have the same dimension, u is bijective. Hence u is an isomor-
phism of C*-algebras.
m
7. 2 Clifford Algebras
Proposition 7.2.1.9
(7)
505
Assume p(s,t) = - 1
for all distinct s, t E T . Denote by Cg+(p) (or more precisely, by cgIK'+(p)) the subalgebra of Cg(p) generated by {st l s , t 6 T} and put
~s := {A 6 ~ l C a r d A is even}. a) ~+ is an algebraic basis of the underlying vector space of Cg+(p) and Cg+(p) is an involutive unital subalgebra of Cg(p). In particular,
Dim Cg+ (p)
=
2 CardT-1
if T is finite.
b) For every t E T with a(t) - - 1 , there is a unique isomorphism of involutive unital algebras 9 Ce(p')
~ c e +(p)
such that us-
st
for every s E T \ { t } , where p' denotes the restriction of p to (T\{t}) 2
c) The map o
ce ~,+(p) ---+ ce r
(x, v),
~ x + iy
is an isomorphism if involutive unital complex algebras.
d)
The commutant of Cg+(p) in Cgw(p) is equal to IK1 if T is infinite and to
{al + fiT [ a, fl 6 IK} if T is finite. e)
The commutant of Cg(p) in Cgw(p) is equal to IK1 if Card T is not an odd natural number and equal to
{ a l + ~TIc~ ,t3 6 IK} otherwise.
7. C*-algebras Generated by Groups
506
a) is easy to see. b) We have
(rt)(st) = srt 2 = - ( s t ) ( r t ) ,
(st) 2 = s 2 = a ( s ) l ,
(~t)* = t*~* = - o ( ~ ) t ~
= o(~)(~t)
for all distinct r, s E T \ { t } . By Proposition 7.2.1.5 b), there is a unique homomorphism of involutive unital algebras
u "Cg(p')
~ Ct + (T)
such t h a t US
--
st
for all s E T \ { t } and u is injective. It is easy to see t h a t u is also surjective. c) is obvious. d) Let x be an element of Cgw(T) c o m m u t i n g with Ce+(T). Take A E ~:\{0, T } , s
E A, and t E T \ A . P u t B " - (A U {t})\{s} = A A { s , t } .
Then
0 = ( x s t - stx)(B) = = f(BA{s,t}, {s,t})x(BA{s,t})= (f(A, {s, t } ) -
f({s,t},BA{s,t})x(BA{s,t})=
f({s,t},A))x(A),
where f denotes the Schur function for ~s associated to p. By Proposition 7.2.1.2 il),
f ( A , { s , t } ) f ( { s , t } , A ) = ( - 1 ) Card(A• and so
x(A) = 0 . Thus x E IK1 if T is infinite and
: -1
7.2 Clifford Algebras
507
x e {al + 13Tla ,13 e ]K} if T is finite. On the other hand, any element of this form commutes with C~+(T) (Proposition 7.2.1.2 il)). e) By Proposition 7.2.1.2 il), T commutes with every t E T i f f CardT is an odd natural number. Thus the assertion follows from d). m Remark.
b) was proved by Stcrmer (1970) for CardT = R0.
P r o p o s i t i o n 7.2.1.10 ( ~ ) Assume T is infinite. Let t E T and let E be the subalgebra of Cg~(T) generated by {rs [ r,s e T\{t}}.
a) E is isomorphic to Cf~'+(T). b) t e e c)
c.
The map >C e ~ ( T ) ,
(~, y ) ,
~ ~ + ty
is an injective homomorphism of involutive unital real algebras.
d)
There is an involutive unital subalgebra of Cga(T) which is isomorphic to the involutive unital real algebra C t e ( T ) .
a) and b) are obvious. c) Let the map be u. u is obviously linear and injective. By b),
~((~, y)*)
- ~((~*,-y*))
- ~* - ty* - (~ + ty)* - ~ ( ( ~ , y))*
for all (x, y) C E and ?-t((Xl, Yl))U((X2, Y2)) ~- (Xl --[- tyl)(X2 + ty2) = (~x~ -
for
-
~11 ( z , , y l ) ,
- y~y~) + t ( ~ y ~
-
-
+ y~x~) -
U((XlX2 -- YlY2, XlY2 + ylX2)) ---- ~((Xl, Yl)(X2, Y2)) ( z : , y~) e E .
d) By Proposition 7.2.1.9 b),c) (and Corollary 7.2.1.6 c ::v a), CgC(T) is o
isomorphic to Ct~'+(p). By a) and c), Ct~'+(p) is isomorphic as involutive unital real algebra to an involutive unital subalgebra of Cg.~(T). m Remark.
d) was proved by G.P. Wene (1989).
7. C*-algebras Generated by Groups
508
Proposition 7.2.1.11
( ~ )
a) For every A 9 ~ and a 9 ]K with A* = a2A , ]
:-(1 + a A ) 9 PrCt~(p). 2 b) If IK = r (IK = IR) then the vector subspace of Cg~(p) generated by PrCgc(p) is dense in Cgc(p) 5n ReCgc(p)). c) Let A, B 9 ~ and a , ~ 9 IK such that A* = a 2 A , AB = -BA,
B* = fl2B , (resp. A B = B A ) .
Then the infimum of 89 + aA) and 89 + 13B) in Cry(p)+ is equal to 0 (resp. to 51(1 + aA)(1 + ~B)).
a) By Proposition 7.2.1.2, (aA) 2 = a2A 2 = a41 = 1,
(aA)* = -~A* = -~a2A = a A SO
(~ (1 -t- a d ) )2 = ~(1 1 + 2 a A + ( a d ) 2) = ~1(1 =t=2aA + 1 ) = 1~(1 + a A ) , 1 + (aA)*) = ~(1 1 • aA). ( 1(1 + aA) )" = ~(1 Hence
1
~(1 + aA) 9 PrCec(p). b) The complex case follows immediately from a). Let IK = ]R and x 9 Re Cgc(p). Then ( x ( A ) ) A e q ; - (x(A))*A~qZ = ( f ( A ) x ( A ) ) m e ~ ,
where f is the Schur function associated to p and we get
7.2 Clifford Algebras
f (A) :/: 1 ~
x(A)-
509
0
for every A e ~ . But if f ( A ) = 1 for an A E ~ , then by a),
1
2(1 + A) e PrCgc(p). 2 Hence x belongs to the closed vector subspace of Ctc(p) generated by Pr Cry(p). c) The case A B = B A follows immediately from a) and Corollary 4.2.7.4 a =v b, so assume A B = - B A .
Let x C Cgw(p)+ with
x_<~l(l+aA),
x _ ~ <1(1 + f i B ) .
By Proposition 4.2.7.1 d =v e,
l(l+aA)x_ -~z1 + g
z - -~
and so x = aAx.
It follows x = aAx = a/3ABx = -al3BAx = -/3Bx = -x,
x=O.
Proposition 7.2.1.12
( 1 )
I
Let S be a subset of T and (A~)~eI a family
of pairwise disjoint sets such that
UA~ = T\S LEI
and
Card A~ = 4 for every ~ C I . We set
1 p~ "- ~ ( 1 + A~) for every ~ C I
and denote by F , G ,
and H
the unital C*-subalgebras of
Cgc(T) generated by S , (p~)~eI, and S U {p~ ] ~ C I } , respectively.
7. C*-algebras Generated by Groups
510
a) {P, I ~ 9 I} C PrCe~(T),
F U {p~ 15 E I} C {p, 15 E I} c,
J E q3S(I)\{0 } ==V YIp
# 0,
LEJ
1,. tEJ
b) G is a Gelfand C*-algebra and the map
o(a)
~{o, 1} ~, ~': ,~(~'(p,)),~,
is a homeomorphism.
c) The unital C*-algebra C({0, 1}', Ce~(S)) of continuous maps of {0, 1}' into Cg.c(S) (Example 4.1.1.6) is isomorphic to H .
d) For every (c~,),e, E IK', the family (a,A,),et is summable in Cg~(T) iff (a,),e, E g~(I). If these conditions are fulfilled and if (a,),e, E ]Rt+, then
tel
LEI
a) follows from Proposition 7.2.1.2 c), il), i2), and Proposition 7.2.1.11 a). b),c), and d) follow from a) and Proposition 4.1.2.29, since F is isomorphic to Cry(S).
I
Corollary 7.2.1.13
( 7 ) Assume T infinite and let X be the Cantor set. Then the C*-algebra of continuous maps of X into Cec(T) is isomorphic to a unital C*-subalgebra of C~c(T).
The assertion follows from Proposition 7.2.1.12 b),c) (and Corollary 7.2.1.6 c =v b))by taking S and I such that CardS=CardT, Corollary 7.2.1.14
CardI=R0.
( 1 ) If T is infinite, then the C*-algebra of convergent sequences in tee(T) is isomorphic to a unital C*-subalgebra of ego(T).
7.2 Clifford Algebras
511
Let IN be the Alexandroff compactification of IN. The C*-algebra of convergent sequences in Cg~(T) is isomorphic to the C*-algebra C(]N, Cgc(T)) of continuous maps of IN into C ~ ( T ) . Since IN is homeomorphic to a quotient space of the Cantor set X , Cg(IN, Cfc(T)) is isomorphic to a unital C*subalgebra of C(X, Cg~(T)) and the assertion follows from Corollary 7.2.1.13. I
Remark. No C*-subalgebra of Ct~(1N) is isomorphic to g~, since Ct~(IN) is separable (Proposition 7.1.2.3 h)). C o r o l l a r y 7.2.1.15 ( 7 ) Assume T infinite and let ( T n ) n ~ be a sequence of subsets of T . Then the C*-direct sum of the family (Cec(Tn))n~ of C*algebras is isomorphic to a C*-subalgebra of Cg~(T). In fact, this direct sum is isomorphic to a C*-subalgebra of the C*-algebra of convergent sequences in Cgc(T), so the assertion follows from Corollary 7.2.1.14. I P r o p o s i t i o n 7.2.1.16
( 7 ) Assume p(~,t) = -1
for all s, t E T , s ~: t. Let 91 be a se~ of pairwise disjoint sets of q~ such that (mod4)
CardA = 0
for every A E 91 and let ~ E IK T such that ~2(t) E {--1, +1}
for every t E T . We set B:= U A , AE~
~
.-
l-I(o(t)~(t)) tEA
for every A E 91, a"T
> {-1,+1},
{ --~2(t)a(t)aA t~
~2(t)a(t)
if t E A E 9 1 if t E T \ B ,
512
7. C*-algebras Generated by Groups
>{-1,+1},
p"TxT
(s,t),
>/
t
-1
ifs~t
o'(t)
iS s =
t,
and denote for every t E T by t' the element of Cgc(p') defined by t' "= ~ ~ ( t ) t A
L ~(t)t
if t E n E 91 if t E T \ B ,
where the multiplications are considered in Cgc(p'). Then there is a unique isomorphism of C*-algebras u : ce~(p)
~ ce~(p')
such that ut = t ~ for every t E T .
Let A E 91. Take t E A. By Proposition 7.2.1.2 i3), t '2 = - ( - 1 ) w(A) l - I ( - - ~ 2 ( s ) a ( s ) a A ) ~ 2 ( t ) ( - - ~ 2 ( t ) a ( t ) a A ) l
= a(t)l,
sEA
t'* = a ( t ) t ' .
If s, t are distinct elements of A, then by Proposition 7.2.1.2 il), s't' = p ( s ) ~ ( t ) s A t A = - ~ ( s ) ~ ( t ) s t A 2 = - ~ ( s ) ~ ( t ) t A s A
If s E A and t E T \ B ,
= - t ' s' .
then
s't' = ~ ( s ) ~ ( t ) s A t = - ~ ( s ) ~ ( t ) t s A
= - t ' s' ,
t '2 = ~2(t)t2 = ~ e ( t ) a ' ( t ) l = a ( t ) l ,
t'* = ~(t)t* = ~ ( t ) a ' ( t ) t = p ( t ) p 2 ( t ) a ( t ) t = a(t)t'.
Let now A1,A2 be distinct sets of 91 and let tl E A1, t2 E A2. Then by Proposition 7.2.1.2 il), t~lt~ = ~(tl )p(t2)tl Al t2A2 = - ~ ( t l )~(t2)t2A2tl A1 = -t~t~l .
7.2 Clifford Algebras
513
If s, t are distinct elements of T \ B , then
s't r = ~(s)~(t)st = - ~ ( s ) ~ ( t ) t s = - t ' s' . By Proposition 7.2.1.5 b), there is a unique homomorphism of C*-algebras
such t h a t
ut = t r for every t C T . Let A c ~s and t E A. Then
I I ~ ' - II(~(s) ~A) sEA
sEA
= (H~(s))AA~(-1)
('+2++(~-1))= ( I I
sEA
p(s))A,
sEA
where n := C a r d A so
t' 1-I s ' - qg(t)tA( H qg(s))A - qg(t)( 1-I ~(s)) ( H ~ ( s ) ) t " sEA
sEA
sEA
sEA
This shows that u is surjective. From the definition of t p it can be seen that {t !l t I~ . . . t In l t l , t 2 , . . . , t n E T , t ~
...tn}
is linearly independent, so u is an isomorphism (Proposition 7.2.1.5 b).
I
( ~ ) Let A be a subset of T such that ~ is constant on A and such that either A is infinite or A is finite and
C o r o l l a r y 7.2.1.17
(mod4) .
CardA-0
Let further p' : T x T-----~ { - 1 , + I } such that
p(s, t)
=
j(s, t)
=
-1
for all s, t E T, s ~ t, and p'(t, t) = ~ - a ( t )
L ~(t)
iftEA if t E T \ A . I
7. C*-algebras Generated by Groups
514
/
C o r o l l a r y 7.2.1.18
( 7 )
A s s u m e IK = r
Let 91 be a set of pairwise
disjoint sets of ~, such that
Card A - 0
(mod4)
.for every A E 91. We set
B:=UA AE~i
and
xt :=
f
irA
if t E A E 9 1
iS t e T \ B ,
Lt
Then there is an isomorphism of C*-algebras
u .Cg~(T)
~ Ce~(T)
such that ut = Xt .for every t E T .
I
Proposition 7.2.1.19 ( 7 ) Let u .eg(p) -+ cg~(p) b~ a linear map such that:
1) ur = q~. ==V(01uA) = 0.
2)
A E r163
3)
A, B E ~s ::v u ( A * B ) = ( u A ) * ( u B ) .
4) u(Ce(p)) ~ d~,~ ~ e'(~). Then u is the restriction of a unitary operator on g2(q;).
Take A, B E ~Z and let f be the Schur function associated to p. By Propositions 7.1.1.9 c), 7.1.1.6 d3 :=v dl, and 7.2.1.2 e),f),g), ( u A l u B ) = (Ol(uA)*(uB)} = (OIu(A*B)} =
-
(0If(A, B ) f ( A ) u ( A A B ) )
= f(A, B)f(A){OIu(AAB)} -
7.2 Clifford Algebras
S 0
if A C B
1
if A - B .
515
Hence u(~) is an orthonormal set, so by 4) it is an orthonormal basis of g2(~2). Let x E Cg(p). By Pythagoras' Theorem,
Ilu( x
= ~
I~(A)I ~ = I1~11~
AE~
Hence u is the restriction of an isometry t~2(~2) --+ ~2(~2) and the assertion follows from Proposition 5.5.5.9 a =~ d. I P r o p o s i t i o n 7.2.1.20
( 7 ) The following are equivalent:
a) p(s,t) = 1 for all distinct s,t E T . b) A , B = BA for all A, B E ~. c) Cgw(p) is commutative.
If these conditions are fulfilled and T is infinite, then: d) C~c(p) is isomorphic to a unital C*-subalgebra of Cgc(T). a , , b , , c is easy to see. d) Let (A~)tET be a disjoint family in ~ such that Card A t - ~ 2 [ 4
if a ( t ) - - 1 if a ( t ) = l .
If we consider (At)tEr as a family in Cf.c(T), then given s, t E T, by Proposition 7.2.1.2 il), i2),
AsAt = AlAs, At2={ -1 1
ifa(t)=-i if a(t)= 1
{ -At At = At
if a ( t ) = - I if or(t)= 1.
By Proposition 7.2.1.5 b), Cgc(p) is isomorphic to the C*-subalgebra of generated by (At)tET.
Ce~(T) I
516
7. C*-algebras Generated by Groups
P r o p o s i t i o n 7.2.1.21
( 7 ) If C a r d T = No then Cgc(p) is isomorphic to
a unital C*-subalgebra of Cgr
We may assume T = IN. Using Proposition 7.2.1.2 il), it is possible to construct an injective sequence (tn)ne~ in T and a sequence (A~)ne~ in %" such that for all m, n E IN, m < n, t~ E A n , CardAr,=_2
(mod4) i f a ( n ) = - I
CardA._=4
(mod4) i f a ( n ) = l ,
Am M A , C {tk l k E INto}, AmAn = p(m, n ) A , A m ,
where the multiplication is taken in Cg(T). By Proposition 7.2.1.5 b) (and Proposition 7.2.1.2 i2)), Cgc(p) is isomorphic to a unital C*-subalgebra of Cgc(T). I
P r o p o s i t i o n 7.2.1.22 ( 7 ) Let a,b be two distinct elements not belonging to T . Take ~ := 1 (resp. 6 := - 1 and IK = ~ ) and 7 E {-1, +1}. Denote by p' the extension of p to (T U {a,b}) 2 defined by p'(a,a) = p'(a,b) = p'(b,a) = - 1 ,
p'(b,b) = 6
and p'(t, a) = p'(a, t) = p'(t, b) = p'(b, t) = 7 for all t E T . Then there is a unique isomorphism of unital C*-algebras ~ : Cg~(p')
~ (Cg~(p))~,~
such that
99t = ( - 1 ) '
(~-,)~-~) '4" "
It 0] 0
for every t E T and
~a =
[0 1] --1
0
where 1 denotes the unit of Cgc(p).
Tt
1-~[0 99b = i--w 1
1] 0
7.2 Clifford Algebras
517
Put Ti'=E,
T2:=2Z~,
T'-TlXT2.
T is isomorphic with the group ~3I(T U {a, b}). Let fi be the Schur function for E associated to p and f2 the Schur function for 2g2 defined by (1,0)
(0,1)
(1,0)
(I,I) -1
(0,1) (1,1) (Corollary 7.1.3.4 a)). Put if "r = 1 r l + :=
{A E E I CardA - 0 (mod2)}
if r = - l ,
T2+ := {t2 E 2Z~ { t2(1) + t2(2) - 0 ( m o d 2 ) ) ,
c:T
f:T•
- > { - 1 , + 1 } , (tl,t2),
[
1
t
-1
~{--1,+1}, ((Sl,S2),(tl,t2)),
if t l E T 1 + or t2 ET2 + if tl E Yl- and t2 G T2-,
}c(tl, s2)fl(sl,tl)f2(s2, t2).
By Corollary 7.1.3.2, f is a Schur function for T and it is easy to see that f is the Schur function associated to p'. By Corollary 7.1.3.4 b),d), there is a unique isomorphism of C*-algebras
having the required properties.
II
518
7. C*-algebras Generated by Groups
Cs
7.2.2
( 7 ) For p, q e
Definition 7.2.2.1
IN tJ (0}, define
Pp,q : INp-t'q X INp+q
>
{-1, +1}
by pp,q(S,t) = - 1 for all distinct s, t E INp+q and pp,q(t,t)
= ~ -1
(
+1
if t E iNp if t E INp+q\INp.
Denote
ce~,~.=ce~(p~,~),
ce~,~ ~'+ := Ce ~'+ (p~,~) ,
Ce~ = Ce~,o
and cat Ce~ the c~iIro.a algebra of degree p (Clifford, 1878). Cg~ and Cg~l are isomorphic to IK and IK • ]K, respectively. Cg~ and Cg~ are isomorphic to (I] and IH, respectively.
Proposition 7.2.2.2
( 7 ) Let p,q E IN U {0}.
a) Cg~q and Cgep+q are isomorphic. b) Cgp+4,q and Cgp,q+4 are isomorphic. c) Cgp+n,q+n and (Cgp,q)2,,2, are isomorphic for every n E IN. d) Cgp,q is isomorphic to a) b) c) d)
C~;+l, q .
follows from Proposition 7.2.1.5 d). follows from Corollary 7.2.1.17. follows from Proposition 7.2.1.22 by complete induction on n. follows from Proposition 7.2.1.9 b).
Proposition 7.2.2.3
( 7 ) Let p, q E i N U { O }
II
such that p + q is even.
a) /f ~ is even (odd), then the real C'-algebra Cepe,q is isomorphic to CeplRTl,q (to ceplR,q+i). b) If ~ is even (odd), then Cgv,q x Cgp,q is isomorphic to Cgp,q+l (to Ce~+ 1,q ).
7.2 Clifford Algebras
519
Put n:=p+q,
T:=IN,.
Then (with the notation of Proposition 7.2.1.2) tT = -Tt
for every t E T and T2= (-1)~(-1)Pl
= (-1)~-~1
(Proposition 7.2.1.2 il),i2)). a) follows from Corollary 7.2.1.7. b) follows from Corollary 7.2.1.8.
m
Corollary 7.2.2.4 ( -[ ) If p, q E IN U {O} such that p + q and p-q-! are odd, then Cgp+2,q and Cgp,q+2 are isomorphic. Put p':=p+l,
q':=q,
p":=p,
__
q":=q+l.
p. _qJt
Then p'+q', P'~q', and p"+q" are even and ~ is odd. By Proposition 7.2.2.3 a), C~,q, and Ct~,q,, are isomorphic to Ce~+l, q, and C~,,q,,+l , respectively, i.e. CgCp+l,q and Cg~,q+1 are isomorphic to CgplR+2,q and Ct~q+2 , respectively. By C IR Proposition 7.2.2.2 a), C~pc+l,q and C~p,q+ 1 are isomorphic, so that C~p+2, q and Ct~q+2 are isomorphic. Again, by Proposition 7.2.2.2 a), Cg~+2,q and Cg~q+2 are isomorphic, m P r o p o s i t i o n 7.2.2.5 ]or every n E IN :
( -[ )
There is a unique sequence (un)~e~ such that
1) u~ "Cg~n --+ C2~,2~ is an isomorphism of complex C*-algebras. 2) For every j C IN2~-2,
o 1
__ [ - - U n - l e j Unej
3) Un e 2 n - 1
[
0
U n - 1ej
[0 1] -1
where 1 denotes the unit of r
Un e 2n - ' - i
0
[01] 1
.
0
520
7. C*-algebras Generated by Groups
4) ~
= (-1)J+lu~ej :for every j 9 IN2..
Define the sequence (u~),e~ inductively, using 2) and 3). 1) follows from Proposition 7.2.1.22 (with 5 = ~- = - 1 ). 4) follows from 2) and 3) by complete induction. 9 Corollary 7.2.2.6
is isomorphic to r
( 7 ) Cgr
xr
for every
ne~u{o}. By Proposition 7.2.2.2 a) and 7.2.2.3 b), Cge2n x Cgr is isomorphic to Cg~,+I. Hence by Proposition 7.2.2.5, Cg2r is isomorphic to ([J2-,2- x ([J2-,2- 9 m T h e o r e m 7.2.2.7
( 7 ) Let n , p , q 9
such that p + q = un be the isomorphism defined in Proposition 7.2.2.5. Put
xj
f Unej
if j 9 INp
t
if j 9 IN2n\~p
iunej
.for every j 9 IN2n, P "= {j 9 IN2,~ I xj = x j } ,
A:={2j-1
Ij 9
1-I Y
D~
B'={2jlj 9
xj
if Card P is even
jeIN2n\P
1-I xj
/f Card P is odd,
jEP
and
T :: { n(n21)-i-P n(n+l)+p+l
a)
j C A nINp ~ 5j = x j , xj = - x j , .
j E A\INp :=v ~ = - x j , x3 - x j , j E B C?INp :=v x---~= - x j , x* = - x j , j C B\INp ~ ~-j = x j , x~ - x j .
if p is even if p is odd.
2n and let
7.2 Clifford Algebras
b)
Card (A C'lINp) = { ~p+l --5Card (B N ]Np) - { p-1 p 2
if p is even if p is o d d , if p is even if p is odd.
c) P = (A gl ]hip) U (B\INp). (
d) C a r d P = ~
t e)
n n+l
if p is even if p is odd.
y is unitary and ~ = y .
f) y*= (-1)~y. g) y*= y if T is even and y* = - y
if ~- is odd.
h) y-x-~jy* - xj for every j C ]N2n. i) There is a unique isomorphism of complex C*-algebras v " CgpC,q
~ ~2~,2~
such that v e j --- X j
for every j E IN2n.
j) v(Cg~q) - {x c r
l y~y* = x } .
k) If "i- is even, then Cgp,~q is isomorphic to lR2n,2n and Dim Re Ct~p,~- 2n-'(2 n + 1). 1) If 7- is odd, then Cg~q is isomorphic to IH2n-l,2~-i and DimReCg~q- 2n-1(2 n - 1). m) p - q is even and 7- even
~
(p - q 2
- 0 (mod4)) or
p-q
-- 3 (mod 4)).
521
7. C*-algebras Generated by Groups
522
a) b) c) d) e)
follows from Proposition 7.2.2.5 4). is easy to see. follows from a). follows from b) and c). For j e ]N2n, ejej *
=ejej * =
,
1
so that Unej and xs are unitary. Hence y is unitary. The relation ~ = y is easy to see. f) Case 1 p even By d), Card P -- n. By a),b),c), and Proposition 7.2.1.2 i2), y* Case 2
(--I)~( 1)~y (--I)"("-2')+" ~
~
y.
p odd
By d), Card P = n + 1. By a),b),c), and Proposition 7.2.1.2 i2), y*
(_1)~2-I~( 1)e~-~+~y m
~
( 1)"("+~ +p+' _
y.
g) follows from e) and f). h) If j e P (j r P ) , then by e), yx~ y , = yxsy* = x j y y 9 = xj
(y~jy* = - y x j y * = xsyy* = x j ) .
i) follows from j) follows from k) By g), y* = so that (Corollary
Proposition 7.2.1.5 d). h) and i). ~. By j) and Corollary 5.5.7.9, C,e~q is isomorphic to IR2n,2,, 5.6.6.9)
Dim Re Cg~q = Dim Re 1R2-,2- = 2 " ( 2 2" + 1 ) = 2 , _ 1 ( 2 , + 1 ) . 1) By g), y* - - ~ . By j) and Proposition 5.5.7.11 g), Cg~ is isomorphic to F(C2-,2-). By Corollary 5.6.6.3 a), F(~2-,2-) is isomorphic to IH2,-1,2,-1, so that (Corollary 5.6.6.9) D i m R e C g ~ q = Dim ReIH2,-:,2,-, = 2 - 2 2("-1) - 2 "-1 = 2"-1(2" - 1).
7. 2 Clifford Algebras
523
m) Since p + q is even, p - q is also even. P u t r "= a~-2q. T h e n
p+q=2n p-q=2r so t h a t
p=n+r.
Case1
r is even
n(n--1)+n+r T --
n2
r
=-~-+~
n(n+l)+n+r+l
=
ifp
(n+l) 2
~
+7
is even
r
ifp
is o d d
Since p even
r
even,
"r even
~
- even. 2
it follows
Case 2
r is o d d (n+l)2-2(n+l) r+l 2 -~-T
n(n-1)+n+r
2
r =
r
--
if
p
is e v e n
n(n+l)+n+r+l = n=+2n ~ + - ~r+l - if p is o d d 9
Since p even it follows -1- even
Corollary
7.2.2.8
( 7 )
~
~ n
odd,
r+l
even.
2
Let p, q , p ' , q' E IN U { 0 } , such that p + q = p' + q'
Ct ,q, are i omo phic i;] p-p'
m
even ==> ~
p - p' odd
==~
even
p-q' +l
2
even.
7. C*-algebras Generated by Groups
524
Put p + q 2 n(n-1)+p 2 T "--
if p is even
n(n-1)+p'
if p' is even
n(n+ l )+p' + l 2
if
T t .--
n(n+l)+p+l
if p is o d d
2
If
ff + ql 2
p' is odd
p - p' is even, t h e n P - P' 2
T -- T ! :
If p is even and p' odd, t h e n
n(n - 1) + p
~--r
n(n + 1) + p ' + 1 m
2
2
= ~1 ( n 2 - n + p -
n 2 -n-
1
p,
= ~(pIf p is odd and
p' - 1 + 2 p ' +
2n+
+if+l=
2) =
p-q' + 1 +1)=
2
p' even, t h e n
7 - - T ' - - q ' = n(n+ l ) + p + l _ n ( n - - 1 ) + p ' 2 2
1
= ~(n 2 + n + p + 1 - n 2 + n -
1
p,
=~(p+2n-
q, -
q, -
p,
_a
t _
- 2q') =
p-q'+l +1)=
2
Hence T -- T' is even iff
p-p' pBy T h e o r e m 7.2.2.7 k),l),
even = ~
p' o d d
==v
(C~p,q)16,16 .
7.2.2.9
even
p-q'+ l
2
even.
Cg~q a n d Cg~,q, are i s o m o r p h i c iff T - T' is even
a n d the assertion follows from the above, Corollary
~
m
( 7 ) If p , q E IN U {0}, then Cgp+S,q is isomorphic to
7.2 Clifford Algebras
525
Put T 1 :-
7[? +q -- ~ 3 ( I N p + q ) ,
T ; ( resp. T ~ ) : = {A C lNp+q I CardA is even (odd)},
T2 "= ~8 = ~(IN8), T+( resp. T ~ ) : = {A c INs [ C a r d A is even ( odd)}, T : - 7'1 • T2 - ~(INs+;+q). Let Pl,/92 be the functions on TI x T1 and T2 x T2 defining C~p,q and CQ,
respectively, and let fl and f2 be the Schur functions for T1 and T2, respectively, associated to the corresponding Pl and P2. Put
p " ]i8+p+q
c"
• INs+v+q ---+ { - 1 , + 1 } ,
T
(j,k)~
,~ / 1 if j - k E INs+p+q\lNs+p
(- 1 otherwise > {-1, +1},
f: T x T
(tl,t2)'
> {-1,+1},
>~
1
(
-1
((Sl, S2),(tl,t2)),
if
tI
e T1-b
iftl 9
or
t2
9 T2
and t 2 9
,
>e(tl, s2)fl(sl, tl)f2(s2,t2).
By Corollary 7.1.3.2, f is a Schur function for T and it is easy to see that f is the Schur function for T associated to p. By Proposition 7.2.4.5 and Theorem 7.2.2.7 k), CQ is isomorphic to 1K16,16 so that by Theorem 7.1.3.3 c), Cgp+s,q is isomorphic
to
9
(C~p,q)16,16 .
T h e o r e m 7.2.2.10
( ~ ) Let p,q 9 lN u {O} such that p + q is odd. Put n :--
p+q-1 2
a)
I f p-q+1 ---W--
b)
If p-q+1 is even, then Cgp,~ is isomorphic to
IR is isomorphic to ~2~,2~. is odd, then Cep,q
IR2~,2~ x IR2~,2~ i f p-q+X
i8 even and it is isomorphic to IH2n-l,2n-1 X IH2n-z,2n-1
if ~-q+l is odd. 4
526
7. C*-algebras Generated by Groups
Assume p even (odd) and put p' := p, q' := q -
1 (p' := p -
1, q' : - q ) .
p'
and q' are even, p' + q' = 2n,
p' - q' = p - q2
( p' - q' = p - 2q
and
= Cep,,q,+,
=Cev+~,r
a) ~ is odd (even). By Proposition 7.2.2.3 a) Cgp, c q, is isomorphic to Cg~q. By the Propositions 7.2.2.2 a) and 7.2.2.5, CgpC,,q, is isomorphic to ~2-,2n. 2
b) ~ 2 to
'
,
is even (odd), so that, by Proposition 7.2.2.3 b), Cg~q is isomorphic
m x C~,,q,. m C~,,q, Put
W :--
p-q+l 4
E 7Z,
7" :=
n(n-1)+p' 2
Then
q- 1 =p-4w,
n =
7"
--
p+q2
1
=
(p - 2w)(p - 2w - 1) + p' 2
p+p-4w 2
----
=p-
2w,
p2 _ 4pw + 4w 2 - p + 2w + p' 2
----
p2 _ p + p, = w + 2w(w
-
p) +
Since p2_~+p, is even, it follows that T is even iff w is even. By Theorem 7.2.2.7 k),l), Cf~,q, is isomorphic to ]R2n,2- if w is even and to ]H2,-1,2=-1 if w is odd.
m
Remark.
Since for all p,q E IN U {0}, Cep,a+4 and Cep+S,q are isomorphic
to Cfp+4,q and (Ct~p,a)lo,16, respectively (Proposition 7.2.2.2 b), and Corollary 7.2.2.9), the table below gives the complete list of isomorphic representations of Ce~q as product of matrix algebras.
527
7.2 Clifford Algebras
IR
IR2,2
ff32,2
IR2,2
]I::[2,2 X JR2, 2
]R4,4
H
1~2,2
]R4,4
IR4, 4 x ]R4,4
HxH
H2,2
([]4,4
1R8,8
]I-12,2
I[-I2,2 x ]H2,2
H4,4
~8,8
([]4,4
1H4,4
IH4,4 x ]H4,4
H8,8
]R8,8
(!]8,8
H8,8
]I-18,8 x ]H8, 8
JR8, 8 X JR8, 8
1F[16,16
(~16,16
~-I16,16
IRx~
This list may be calculated using Theorem 7.2.2.7 k),l), and Theorem 7.2.2.10. C o r o l l a r y 7.2.2.11
(7)
r
and
C~q,p+l are isomorphic
for all
v, q e ~ u { 0 } . By Proposition 7.2.2.2 a), we may assume IK = JR. Case1 p+q+l
p+q
is even, p - q
odd is odd, and p-(p+l)+l
=0.
By Corollary 7.2.2.8, CrOp,q+1 and Cgq,v+l are isomorphic. Case2 p+q+l
p+q
even
is odd and p - (q + 1) + 1 = p - q = - ( q -
so that the numbers
p) = - ( q -
( p + 1) + 1)
7. C*-algebras Generated by Groups
528
p-
(q + 1) + 1 2 '
q - ( p + 1) + 1 2
have the same parity. Moreover, if they are even, then the numbers p-
(q + 1) + 1 4 '
q - ( p + 1) + 1 4
have the same parity. By Theorem 7.2.2.10, CrOp,q+1 and Ct~q,p+l are isomorphic.
C o r o l l a r y 7.2.2.12
( -~ ) I f p , q , p ' , q ' E I N U { O }
such that p + q = p ' + q '
is odd, then the following are equivalent: a)
Cep,~q and Ce.p,,q, ~t are isomorphic.
b)
P-q-P'+q' E ~ and 4 p-q+ 4
l '
p - q - p' + q' p'-q' + l E 2~ :=:~ E~. 8 4
a ~ b. Put
p+q-1
E := IR2,,2n,
F "=r
G "= IH2,-1,2,-~.
By Corollary 5.6.6.9, E and G are not isomorphic and E , F , G
are simple.
It i011ows that F is not isomorphic to E • E or to G • G. By Proposition 4.3.5.5, E • E and G • G are not isomorphic. By the above and Theorem 7.2.2.10, p-q+l and p'-q'+l are either both odd or both even and in the last case ~
p- q + 1 2
and P'-q'4+1 have the same parity. Since
p' - q' + 1 2
p - q - p' + q' 2 '
this implies b). b =v a follows from Theorem 7.2.2.10. C o r o l l a r y 7.2.2.13
9
( 7 ) If p, q E IN U {0} such that p + q and
p-q+1 2
are
odd, then Cgp+4,q is isomorphic to (Cap,q)4,4. By Proposition 7.2.2.2 a),c), we may assume IK - IR. Put It :----
p+q-1
By Theorem 7.2.2.10 a), Cgp,q and
C~pT4,q
are
isomorphic to ~2~,2- and
C2~+2,2~+:, respectively, and the assertion follows from Corollary 5.6.6.8 a). m
7. 2 Clifford Algebras
P r o p o s i t i o n 7.2.2.14
a)
529
( ~ ) Let p,q C INU {0}.
C~q is simple iff p + q is even.
b) Cgp,~q is not simple iff p + q and p-q-1 are odd. c) Cgp,~q is not purely real iff p + q and p--q+l 2 are odd. d)
If C~p,q is simple (not simple), then Dim (Cfp,q) ~ = 1 (= 2).
a) By Proposition 7.2.2.2 a), C~q is isomorphic to C~pCTq. If p + q is even, then by Proposition 7.2.2.5 , Cgp+q c is isomorphic to q~2n,2~ for n ._ p+__~q, so it is simple (Corollary 5.6.6.9). If p + q is odd, then by Corollary 7.2.2.6, Cg~+q is not simple. b) and c). If p+q is even, then by Theorem 7.2.2.7 k),l), C~q is isomorphic to lR2m,2m or to 1H2~,2~ for some m, n E IN, hence it is simple and purely real (Corollary 5.6.6.9). So assume p + q odd. Case 1
p-q+l 2 is odd.
By Theorem 7.2.2.10 a), Cg~q is isomorphic to r that it is simple (Corollary 5.6.6.9) and not purely real. Case 2
for some n c IN, so
p-q+1 2 is even.
By Theorem 7.2.2.10 b), Cg~q is isomorphic either to 1R2~,2~ x 1R2-,2~ or to 1H2~-1,2~-1 x 1H2~-1,2~-1 for some n C IN, so that it is purely real and not simple (Corollary 5.6.6.9). d) If Cgp,q is simple, then it is isomorphic to some matrix algebra and therefore (Cgp,q)c is one-dimensional (Corollary 5.6.6.9). If Cgp,q is not simple, then it is isomorphic to the product of two matrix algebras (a),b), Corollary 7.2.2.6, and Theorem 7.2.2.10 b)) and (C~p,q)c is two-dimensional. 9 C o r o l l a r y 7.2.2.15
( ~ ) Let T be an infinite set and
p:T•
>{-1,+1}
such that p(s,t) = - 1
530
7. C*-algebras Generated by Groups
for all distinct s,t E T . For every finite subset S of T , denote by ps the restriction of p to S • S and identify Cg(ps) canonically with an involutive unital subalgebra of Cg(p). Put E:=
LJ
ct(ps).
a)
E is an involutive unital subalgebra of Ce(p) and ideal.
b)
Cgc(p) is simple.
c)
ctp(p) ~ p.~ly ~ l .
{0}
is its only proper
a) By Proposition 7.1.2.3 a), E is an involutive unital subalgebra of Cg(p). By Proposition 7.2.2.14 a),b), {0} is its only proper ideal. b) Let F be a proper closed ideal of Cgc(p). Since E is dense in Cg~(p), E r F . By a), ENF={0}. Let
u .ct~(p) ~
Ct~(p)/F
be the quotient map. Let S E ~ f ( T ) . By Corollary 4.2.6.6, u preserves the norms of the elements of Cg(ps). Since S is arbitrary, u preserves the norms of the elements of E . Hence u is an isometry, so that F = {0}. Thus Cgc(p) is simple. c) follows from b) and Proposition 4.3.5.3 a ~ b, since Cgf(p) is the complexification of Cgc~(p) (Proposition 7.1.2.3 c)). I P r o p o s i t i o n 7.2.2.16
Put
~~ ~ ~, n,~ (o)+ (:)§ (:)+ (~)+ (:)+ ~,-, ~(o)+ (1)+ (~)+ (~)§ (~)+ (;)+, and N
qD'IN
>IR,
n~
>
2n
- 2~(n)
29
'
531
7.2 Clifford Algebras
r
~IR,
2n - 2 r
n,
2~
Take n E IN and denote
T 9 INn U {0}
~ IN,
~v(n)=r ~(n)=2
a)
'~-1
~(n)+r
n
~ ( n ) = 2 n-1 b)
7-(q+4)-7-(q)
c)
For every q E INn U {0},
q,
IR
> Dim Re C~n_q, q .
if n -- O
(mod4)
if n = l
(mod4)
if n = 2
(mod4)
if n = 3
(mod 4).
f o r every q e i N n U { 0 }
"r(q) -- 2n-q E
with q + 4 < n .
\4k + 3] + p ( n - q ) E ( - 1 ) k
k=0
+r
- q) E ( - 1 )
k
k=O
d)
+
k---0
( ) q 2k+l
"
~(0) = ~ ( ~ ) , ~(1) = ~ ( n -
1) + r
~(2) = 2 ~-~ + 2 ~ ( ~ T(3) = 2 n-1 + 2 ( r
1), 2), -- 3) -- ~ ( n -- 3)),
7(4) = 5 . 2 n-3 - 4 p ( n - 4), 7-(6) = 9 . 2 n-4 - 8 r
- 6),
where n is greater than the a r g u m e n t of ~-.
e)
~(n) + ~ ( n + 4 ) = r
a) If n - - 0
r
=
(:)
+ r
= 0.
( m o d 4 ) then
+
+ n-1
+...+
+ n-4
n-5
(4)+(3)+(0)=
= ~(~). If n = 1 ( m o d 4 ) ,
~(n)=
then
(:)
+
n-3
+
(~ n-4
+""
+
+
'
7. C*-algebras Generated by Groups
532
so that ~(n) § ~(n) -- E
-- 2~'
k--O
~(n)
=
2 "-I .
If n = 2 (rood4) then n
r so that
n
n
( : ) + ( n - 1 ) + ( n - 4 ) + ( n - 5 ) +...+ (2) + (1)'
~(~) k=O
If n --- 3 (mod4), then n n n ( n - 1 ) + ( n - 4 ) + ( n - 5 ) +...+ (3) + (2)'
r so that
r247
n, k=O
r
=
2 ~-~
.
b) By Proposition 7.2.2.2 b), Cgp,n_p and Cgp+4,~-p-4 are isomorphic. c) By Proposition 7.1.4.1 b), 7(q) = Card {A C INn [e~ = eA in Cgn-q,q}. It follows
+(~) ( ( n o q ) + ( n l q ) +
§
((~oq) 9 (~
( n 4 q ) + (n5q) + . . . ) +
9 (~;q)9
)~
7.2 Clifford Algebras
--(qo)~(n-q)+ (ql)r
533
(q2)(2"-q-r
+(;) (~'-~-~(~ (q4)~(~ (~)r176 ~) + .... =2n-q((q2) + (~)+ (:)+ (q7)+'")+
+~(n-q) ((qo)-(q2) + (q4)-(q6)+'") + +r ((ql) - (;) + (qs) - (qT)+'") = =2"-qE\4k+3]
+~(n-q)
k=O
E(-1)
k
+r
E(-1)
k=O
k 2k+l
"
k=O
d) follows from c). e) By b) and d ) , ~ ( n ) = ~-(0) = ~-(4) -- 5 - 2 ~-3 - 4 ~ ( n - 4) if n _> 4,
2 n-2 -k- 2~)(n -- 2) -- 7"(2) -- T(6) = 9 . 2 "-4 -- 8 r
-- 6) if n > 6.
It follows
~(n) +
~(-'n - 4) = 2" - 2~(n) 2 n-4 - 2 ~ ( n - 4) 2~ + ,~-4 2-~
__ 4 - 2 ~ - 3 - p ( n ) + 2
~-3-4p(n-4)
=0
2n~--2 if n > 4 and ~(n-
2)+ r
=
2 n-2 - 2 g , ( n - 2)
2~ - 6 -
2r
6)
2~__ 2 n - 2
-- 2~)(n
- - 2 ) -~- 2 n - 4
-- 8~)(n
--
6)
=
0
2"g2 if n > 6 . Proposition
m 7.2.2.17
( 7 )
Take n E IN and let T be a set. Let
be the Hilbert Cgn-module defined in Example 5.6.4.2 c). Then
e~(T, ce~)
7. C*-algebrasGeneratedby Groups
534
E e2(T,Cen) n
tET
and the map g2(T)V(~-)
r e2(T, Ct.),
OA''A)
(O~A)ACINn' n
tET
is an isomorphism of Banach spaces. By Proposition 7.2.1.2 e),
ACINn
E
ACINn
aA(t)A ACINn
AcINn
AcINn
for every t E T. Hence
( E aA(t)A)teTEf2(T, Ce.) ACINn
and = IIEteT ( Ac~.E aA(t)A)*( Ac~.E aA(t)A)II_<
AcINn
Thus the map of the proposition is continuous. This map is obviously injective. Take ~ E g2(T, Ct~n). By Corollary 5.5.1.28,
E ] ~(t)l]2 = E [l~(t)*~(t)ll -< 2~
E~(t)*~(t)
tET
tET
Given t E T, let
tET
(aA(t))Ac~
be a family in IK such that
= 2-11ell
7.2 Clifford Algebras
r
= ~
535
~(t)A.
AcINn
By Proposition 7.1.1.1 e),
E
laA(t)12<-
AC]Nn
E
aA(t)A
= II~(t)ll2
ACINn
for every t E T , so that
E
E I~A(~)I~= E
ACINn t E T
E
I~A(~)l~ -< E tl~(~)tl~ <- ~~ ~
t E T ACINn
tET
Hence the map ~2(T)~(IN~)
(O~A)AcINnl ) (A~C~
>g2(T, Ct~),
aA(t)XA) n
tET
is surjective and so it is an isomorphism (Principle of Inverse Operator).
II
( -~ ) Let n E IN and let S , T be sets. We take our notation from Proposition 7.2.2.17 and identify t~2(S)~(r~) and t~2(T)~(~n) with ~2(S,C~n) and ~9(T, C~), respectively, via the isomorphism of this proposition. Given u := (UA)Ac~ E s g2(T))V(~), define
P r o p o s i t i o n 7.2.2.18
~ e~(s, ce~l-~e~(T, Ce~),
.,
> y~ (UAaB)AB. A,BcINn
Then Es
(g2(S, dr?n), t?2(T, CQ)),
Iv
{*
--
( ( _1 ) ~(A)"iI,A. ,)
-
ACIN ~
for every u := (UA)Ac~ E/2(t72(S), t~2(T)) q3(~)
and the map s
t?2(T)) v(~n)
>s
Cgn), g2(T, Cgn)),
u,
>
is an isomorphism of Banach spaces. The following are equivalent for every u := (UA)ACrCn E s
e2(T)) ~(~n) 9
7. C*-algebras Generated by Groups
536
a)
{?-/,AI A c IN~} c K:(e2(S), e2(T)).
b) ~ is compact. Let
U "= ('UA)AcINn 6 /:(e2(S), e2(T)) v ( ~ ) Take a 6 e2(S, Cen) and fl 6 e2(T, Cen). Then
(Ualfl>--(
E
=
:
~cC>=
(UAaB)AB
\ A,BcIN.
CCINn
~ <~AOIB I flc)C*AB = A,B,CCINn
~
(aB l U*A~c)(A*C)*B =
A,B,CCINn
n
A,CCINn
A,CCINn
(Proposition 7.2.1.2 i2)). Hence 6 s
Cen), e2(T, CQ)),
~, _ ((--1)~(A)u~)Ac~ , It is easy to see that the map E(e2(S),t2(T)) ~(~n)
>s
CQ), e2(T,C~n)),
is linear, injective, and continuous. Take v 6 s163163 s such that v(al)-
u,
>
There is a family u - ( U A ) A c ~ n in
E
(UAa)A
AcINn
7.2 Clifford Algebras
537
for every a 9 g2(S). We get ~(al) = v(al) for every a 9 ~2(S). By Proposition 5.6.1.8 d),
BCINn
BcINn
A,BCINn
AcIN.
BcINn
BCINn
for all
~
o~BB 6 ~2(S,C~n), so that ~ - v and the map
BCINn
is surjective. By the above considerations (and the Principle of Inverse Operator), it is an isomorphism of Banach spaces. a ==> b. Let
(~k)kC~N be a bounded sequence in I~2(S,Cgn). We set
ACINn
for every k 6 IN. Then (~k,A)k~r~ is a bounded sequence in ~2(S) for every A c INn (Proposition 7.2.2.17). By Proposition 3.1.1.19 a :=~ b, there is a
(UA~kj,A)je~ converges for every ('u~k3)jelN converges. By Proposition 3.1.1.19 b :=> a, ~ is
subsequence (~ck~)jer~ of (~ck)ker~ such that A C INn. Hence compact.
b ==> a. Assume there is an A C INn such that Proposition 3.1.1.19 b ~ such that
UA is not compact. By (rlk)k~ in ~2(S)
a, there is a bounded sequence
(UArlk)kc~ posseses no convergent subsequence. Then (~/kl)ker~ is a ~.2(S,C~.n) and from
bounded sequence in
k 9 1N ~
~(r/kl)= E
(USrlk)B
BcA
it follows that (~(rlkl))ker~ posseses no convergent subsequence. By Proposition 3.1.1.19 a =~ b, ~ is not compact.
1
7. C*-algebras Generated by Groups-
538
7.2.3 Of(IN)
Throughout this subsection we assume ]K - C
and use the following
notation. We put 7r0 := x0 :-- 1 E and define the sequence (x,)ne~ of matrices and the sequence ( ( x n , j ) j e ~ 2 , ) n e ~ of families of matrices inductively"
X n :---
[ 0 .] (-1)nxn_l
E [~a2n,2 n ,
0
and for every j E IN2~-2 and p E {0, 1},
Xn,j :=
E (~2n,2~ , 0
.__ Xn,2n-p
F
]
xn-t,j
0
[ il-PTrn_l
(--1)n-lit+PTr"-a0 I E C2-,2-
(matrices of matrices!), where 2n 7rn " : 1 1 xn,j " j=t
For every n C IN and A C INe~, define X n , A :-~ X n , j l
Xn,j2 ... Xn,j k ,
~(A) := (-1) j'+j2+jk , where A = { j l , j e , . . . , j k } , J t < j2 < " " < jk (xn,o " - 1 , ~ ( 0 ) ' - 1 ) . Finally, for every n C IN, ~ C ~n, and x C ([:n,n, put t"
_ ~-IN
[ ~j
>r
~~
j,
;e ~
>
/
~,
0
if j E INn if j r IN~,
>x(~l~)
539
7.2 Clifford Algebras
P r o p o s i t i o n 7.2.3.1
( 7 ) Let n c ] N .
7Fn - -
[
~'n2=(--1)n1,
j EIN2,~
--iTrn-1 0 0
iTrn_l
J
:,~*=:,~=(_l)n~-n,
Xn,jT~ n --- - - 7 ( n X n , j
x--~,3
( - 1) 89(J- sins j~) 2 Xn,j
P'qC{O'l}===~Xn'2n-pXn'2n-q-~--iP-q [ ]1 (--1)P-q1 0 0
b) j,k E IN2n ~ xn,jX~,k + Xn,kXn,3 = --25jkl,
X~,j = --X~,j.
C) For every A C IN2,~,
Xn,ATCn-- ~P(A)XnjN2n\A. d)
For every j E IN2n, Xn,jXn,j+(_l)J+l
--- ( X n , j X n , j + ( _ l ) j + l )* :
--Xn,jXn,j+(_l)j+l
9
e) Given subsets A, B of lN2n, there is an 5 E { 1 , - 1 } such that for every kcIN, Xn+k,A Xn+k,B ---~XnWk,A/kB.
f) xn is unitary, ~ x n* = (co~ -~
sin n_~) ~-~,
and X n X n , j X n ~ Xn, j
for every j C IN2n.
7. C*-algebras Generated by Groups
54o
g)
n e IN, A C IN2n => IlXn,AI] < 1, [l~n,all _< 1.
a) We prove the assertion by induction on n. So assume the assertion holds for n -
1. We have Xn,2n_pXn,2n_
-
q =
,-i,n-lil+p [0 n_l]l-qn_lO,-1,n-l l+q n_X]o 0 ]=ipq[i 0] 0
(-1)~iq-VTr~_,
(-1)P-q1
0
It follows t h a t
~'n
Xn,j
----
\
--~
=[.-1 o][_io] [_i._1= o], [2 o] [,_l,nl o ] 0
2 71-n
(Xn,2n-lX2,2n)
j=l
T'n_ 1
0
i
iTrn-1
0
(_l)nl
--7rn-1
-
_
0
2
--Trn-
7r~ =
(--1)nl
0 1
=
0
m
7[n
--iTrn_1
__
i~n-1
[
0
0
Xn,j7r n =
-(-1)~7r~,
(--1)niTrn_l
irr~_,
0
0
--iTrn_ 1
] [
l [xnlj 0 ][i ni 0]= __
0
--i'~n-1
0
,
._
0
1
Xn-l,j
__
0
iTrn-1
7[n ~
541
7.2 Clifford Algebras
ol[_i.lXn_i.j 0
iXn-l,jTrn-1
iTrn-lXn-l,j
0
o ][xn_l,j o 1 0
0
iTrn-1
0
:
--7rnXn, j .
Xn-l,j
If j C IN2~_ 2 , then
"Xn,j--
[xnl 01
-- (--1) 89
-Xn--l,j
0
0
j~) 0
( -- 1)89
j~2 )X n , j
Xn-l,j
.
By the above result,
Xn,2n-- 1 - -
~n--1
__ (--1) n-1 [
0 (--1)n~n-1 0 71-n_1
Xn,2 n - -
0
-- (-1)89
0
1 ,
[ o ,1,nl.1] -- i - ~ n - 1
= (_1) n
---2--'Xn,2n_
0
(--1)n-li 7rn-1 ]
i 7rn_1
= (_l)89
b) Assume the assertion holds for n -
9
1. Take j, k C IN2,~_2 and p, q C
{0, 1}. Then, by a),
Xn, j - -
2.~)2Xn,2n
0
[Xnlj 0 ] 0
Xn_l, j
7. C*-algebras Generated by Groups
542
=
F [ -Xn-l,j
L
0
__--Xn,j ,
0
--Xn-ld
Xn,jXn, k =
[Xn_l, o 0
o] 0
Xn-ld
[
Xn-l,jXn-l,k
o
0
Xn-l,jXn-l,k
Xn-l,k
Xn,jXnk nt- Xn,kXnj =
=[Xn--l,jXn--l, +Xn--i,kXn-lj 0
Xn_l,jXn_l,
0
]
= -25ykl ,
k Jr- X n _ l , k X n _ l , j
XXn,2n_p=[Xn-l'JO][O0 , Xn-l,j /1-Pn_l 0
__
(--1)n-lil+PXn_l,jTrn.. 1 ]
il-Pxn_l,jTrn_l
0
0
(--1)n-lil+PTrn-lXn-l'J 1
il-PTrn_lXx_l,j
0
Xn2n-pXn,j=[0 ,--i,n-liiPn_l][Xn_ij 0 il-PTrn_l
0
0
Xn-l,j
(-1)n-lil+PTrn_lXn_l,j ]
i l-P Trn_ l Xn_ l ,j
x~'2'~-P -
0
0
0 ( - 1) "-li-l-pTrn-* I
= -- Xn'j Xn'2n-p
iP-1~-1 I 0
-
=
7.2
= [
[
0
-i p-q
~.--Xn,2n_ --il-PTrn_l
p
0
[1 0 ] 0
c) Let A -
543
Algebras
-(-1)n-lil+P~'._l
Xn,2n_pXn,2n_q
=
Clifford
-
Jr- X n , 2 n _ q X n , 2 n _ p
i q-p
[1 0 ]
(-1)~-ql
0
=
- 25p,q I .
(-1)q-P1
where jl < j2 < " "
{jl,j2,...,jk},
=
< jk. Then by b),
X n , A Trn .-- X n , j l Xn,j2 9 . . X n , j k X n , l Xn,2 9 . . Xn,2n - -
-- (--1)Jk+Jk-l+'"+JlXn,INe,~\A
--
~(A)x,~,1N:,~\A.
d) For j E IN2n, put j' .= j + ( - 1 ) j+l By b), (Xn,jXn,j,)*
- - x n,j, 9 x n. ,j __ X n , j , X n , j __ - X n , j X n ,j' 9
We prove the other relation by induction. Assume it holds for n - 1. Then for j C IN2n_2
,
Xn,jXn,j,
- - X n , j X n , j, ---
__. 0
Xn_l,jXn_l,
-Xn-l,j
j,
0
[
--Xn_l,jXn_l, 0
0
0
1
Xn_l,jXn_l,
j,
-Xn--l,j'
j,
O --Xn_l,jXn_l,j,
1
7. C*-algebras Generated by Groups
544
= ix.ij o l[..lJ 0
Xn-l,j
o
0
=--Xn,jXn,j,
.
Xn--l,j'
Now let j E { 2 n - 1,2n} and put p'=2n-j,
q:=2n-j'.
By a), Xn,jXn,jl
--- X n , 2 n _ p X n , 2 n _ q =
--" - - X n , 2 n _ p X 2 n _ q - - - - X n , j X n , y "
e) By b), there is a c E { 1 , - 1 } such that Xn,AXn, B = CXn,AA B .
It follows that
Xn+I,AXn+I, B =
['na0 ][... 0
I
Xn, A
0
Xn, B
['.AX.. 0 ] [''.a..
0 ]
=
0
Xn,AXn, B
0]:
0
--" E X n + I , A / X B
9
CXn,ALXB
By induction, X n + k , A X n + k , B --- C X n + k , A A B .
f) We prove the assertion by induction on n. Assume the assertion holds for n - 1. Then
0 x.i] Xn_ 1
0
[
Xn_lXn_
-
0
(--1)nxn_l
1
0 X n*- 1 X n - 1
]
-
0
1,
7.2 Clifford Algebras
= (-1)" cos
9
o
Xn =
*
545
(1) -- ~* Xn_ 1 ] 0
Xn-1
(n- 1)~-
2
(--1) nXn_l
0
nTr = ( cos~-sin n_~) ~. For j E 1N2,-,-2, XnXn,jXn =
xn_l] [
=[ o (-1)~x~_~
[
0
o][o ][o
-Xn--l,j
0
o
Xn--l Xn--l,j
(-1)"x.-1N.-1,j
0
m
(--1)nXn--1
Xn--1
(-1)nx~-l]0
---
Xn--1
Xn-- lXn - 1,jXn_l
0
0
Xn--lZn-l,jXn_ 1
---- I Xn-l,Jo
0
]
0 __
Xn, j .
Xn-l,j
Take p C {0, 1}. Then XnXn,2n_pX n ---
V -
-
!
0
L (- 1)"x~,_,
xn-1 0
l[ ~
iP-l~n_ 1
(-1)~'-1i-'-P~n-10 ]
x
_
_
7. C*-algebras Generated by Groups
546
[
X
0
(-1)
Xn_ 1
=[P-Xxn-Xn-1 0
__E
n. ] Xn_ 1
0
][ 0
--i-l-PXn-l~n-1
o
X~_ 1
I
0
/-l/niP-lxn_ln_lx_x] 0
--i-l-PXn-l-~n-lXn_l
----
=
0
0
(--1) n-l"sp+ 171.n_1 ]
il-Prn_l
~
0
Xn,2n--p .
g) follows by induction.
m
P r o p o s i t i o n 7.2.3.2 ( 7 ) Take n 6 IN and let (aA)Ac~2~ be a family in ~. Given A c IN2n-2, define
]~A :'-- O~AU{2n-1},
'~'A "----O~AU{2n},
5A :----OlAU{2n-l,2n},
A := IN2n_2\A 9 a)
E OZAXn,A = A cIN 2n
~ _.
(aA -- iSA)Xn-l,m
(--1) n
Ac]N2n- 2
Y~
~(A)(/~A + iTA)Xn_l, ~
AcIN2n- 2
Y~ ~(A)(~A -- iTA)Xn_l,X ] ACIN2n-2 ~ (aA + iSA)Xn-I,A
AcIN2n- 2
b) IfAcIN2n-2 ~ ~AXnxAIl
AcIN2n- 2
AcIN2n-2
AcIN2n
AcIN2n
AcIN2n
AcIN2n
537
7.2 Clifford Algebras d)
There is a ~ E 9 2n-F1 such that II~II
= 1,
{eiN,,~ = O,
~,A)~II:II~ ~AX~,AII"
I1( ~
Ac]N2n
AcIN2n
a) Take A C IN2n-2. Then by Proposition 7.2.3.1 c), for p C {0, 1},
Xn,A --
Xn_l, A
0
]
0
Xn-l,A
Xn,AU{2n_p} ~ Xn,AXn,2n_p =
=[Xn_l,A o ][ o 0
Xn-I,A
[
0
(--1)n-lil+PrCn-lo
] ----
il-PTrn-1 (--1)n-lil+PXn-l'ATrn-lo
] --
il-P Xn_ l,A Trn_l
(--1)n-lil+Pxn-l'Ao
0
= ~(A)
]
il-Pxn-l,A
By Proposition 7.2.3.1 a),
Xn,AU{2n_l,2n} = Xn,AXn,2n_lXn,2n ---
[xnlA 0
[ Xn-I,A
0
i
E
OZAXn,A ---
Hence
AcIN2n
--iXn-l,A
0
0
iXn-l,A
548
7. C*-algebras Generated by Groups
~ ----
((~A-- i(~A)Xn-I,A
(--1)n
Ac]N2n-2
E
AcIN2n-2
~(d)(~a + i'TA)Xn_l,~
AC~2,~-2Y]~ ~(A)(flA -- i"/A)Xn-I'A ] Y]~ ((~A -k- ihA)Xn-I,A AcIN2n-2
b) By a) and Theorem 5.6.6.1 a),
Ac]N2n-2
AcIN2n
~(A)('TA • il3A)Xn_l,~l I <_ AcIN2n-2
Ac IN2n
It follows that
AC~2n-2
AcIN2n-2
ACIN2n
AC]N2n-2
The other three inequalities are proved similarly. c) For any j C IN,
[AXn+jIA 0 ]11 Ac]N2n
AcIN2,~
._
0
E OgAXn+j-l,A AcIN2n
A c IN2n
The assertion follows by induction. d) There is an r/eff~ 2n such that 117711= 1 and
Jl(A~cIN 2n oAxnA)~lJ-JlAZc IN2n oAxnAf[ Put
549
7.2 Clifford Algebras
~:--
0 ] 9 q~2~+1
Then
II~ll=l,
~e~:= = 0 ,
and
E
A cIN 2n IJ(AC~2nOLAXn-FI'A)~I]--]][
0
E OIAXn,A Ac IN2n
:1][( 0 l AcIN2n
0 ] [0]]1
O~AXn,A
?7
I( ~ ~ ~ ) . l l
O~A Xn,A ) ?7
Ac]N2n
IIAc]N2n z ~ll Proposition 7.2.3.3
I
(7)
a) If n , p 9 IN, n < p, A c IN2n, and ~ 9 62, then
b) For every n 9 IN and A c IN2n, the sequence ('Xn+k,A)kEIN converges pointwise to an X~,A 9 f~(~2). Given j 9 IN, put x~,j "- x~,{j} .
c) For all ~ 9 62 and A , B 9 lira "Xn,A'Xp,B~ - - X ~ , A Xoo,B~ . p-...+oo
d) If n 9 IN, (Jk)ke~n is a strictly increasing family in IN, and A "- {jkJk 9 INn} then n
X~176 --- H
k--1
Xc~
"
550
7. C*-algebras Generated by Groups
e) For all j , k E IN, Xoo,j -- --Xoo,j
--
2 Xoo,j ,
Xoo,jXoo,k + Xoo,kXooj = --2~j,kl 9 a) For every k C INp_~,
Xn+k,A ( (~elN2, ) i]N2(n+k)) =
=[Xn+kiA Xn+0 IA
0
and so
r
i
-- Xn+k-l,A ( (~e'lN2n ) liN2(n+k-1) i -- "Xn+k-l (~elN2n ) . Since k is arbitrary, we get
~.,.(r176
= ~,,,A(r176
b) Take ~ E g2 and p,q E IN, with n < p _< q. By a) (and Proposition 7.2.3.1 g)),
I1~,~-
~,,~ll = I I ~ , ~ ( ~ \ ~ )
- ~%,~(~\~,)
I<
oo
_~ ~lt~~,,I Hence
(Xn§
= ~( k=2p+ E 1 ,~l~) ~
converges. By Corollary 1.4.1.3, the pointwise limit of
(bn+k,A)ke~ belongs to E(g2).
7.2 Clifford Algebras
551
c) By Proposition 7.2.3.1 g), for sufficiently large natural numbers n , p ,
II~,A~p,B~- X~,A~,B~II _< I 1 ~ . ~ , . . ~
-
~.~x~..~ll
+ I~.~.-~
-
x~.~x~..~ll <_
Hence 1Lm ~ Xn,Axp,s~ -- Xo~,AX~,B~. p---~ o o
d) follows from c). e) The relations x~,jx~,k + x~,kx~,j =
--25jkl,
xc~,j - ( - 1) 89(j-sin2 -~-)xc~,j
follow immediately from c) and Proposition 7.2.3.1 a),b). Let (, 77 C t~2 . Then by Proposition 7.2.3.1 b), (x~,/~lr]> = lim (2n,j~lr]> = lim (~1- ~n,jr]> = n - - + (:x)
n--+(x)
= <~1- ~.Jo>.
so X cc ,j -- -- X cc ,j .
P r o p o s i t i o n 7.2.3.4 a)
i
(~)
There is a unique injective h o m o m o r p h i s m of unital C*-algebras
9ce~(r~)
~ L(e ~)
such that uej -- xoo,j f o r every j E IN.
b) u(Cgc(IN)) is the unital C*- -subalgebra of •(e 2) generated by {xcc,jlj e I N } .
c) ~(c~(~)) n ~(~) = {0}.
552
7. C*-algebras Generated by Groups
If
d)
q./:(g2)
> f_.,((~2)/K:(i2)
denotes the quotient map, then the restriction of q to u(Cgc(IN)) is injective but not surjective. a) follows from Proposition 7.2.3.3 d),e) and Proposition 7.2.1.5 b). b) follows from a). c) Take x e Cgc(]N)\{0} and let y E t~2(9l) such that (with the notation of .__}
Definition 7.2.1.3) x = y . By Proposition 7.1.1.3 b2 =~ b l , there is an no C IN such that -~
1
IlY- Y~oll < 511xll for all n E IN, with n > no. Take n E IN, with n > no. By a) (and Proposition 7.2.3.3 d)),
E
y(A)x~,A =
IlYr~,=ll ___Ilxll- Ix- Y~,.II > gllxll.
ACIN2n
By Proposition 7.2.3.2 d), there is a ~n E q~2~ such that
I1~11 =
1,
~ne~2=_ , -- 0,
and
AcIN2n
AcIN2(n_ 1)
y(A>x. ,,All
By Proposition 7.2.3.2 c),
y(A)x~,A ACIN2(n-1)
AcIN2(n-1)
so that 2 AcIN2n
We get (Proposition 7.2.3.3 a)) 2 ACIN2(n-1)
AcIN2(n-1)
II> ~llxli, 2
7.2 Clifford Algebras
~___[[X~nl[-Jr-II(X-
E
Y(m)x~176
553
~-- 'lx~n'l -t- 51'X'l,
AcIN2(n-1)
Ilx~.ll > 511~11. N
.---.
Hence (X(n)ne~N does not converge to 0. Since (~n)ne~ converges weakly to 0, x is not compact (Theorem 5.5.5.1 a =~ c). d) By c), the restriction of q to u(Cec(IN)) is injective. Since Cry(IN) is separable (Proposition 7.1.2.3 h)) and
c(e)/~c(e ~)
5.5.2.12 e)), it follows that the restriction of q to
Proposition
7.2.3.5 ( 7 )
is not separable (Proposition
~(ce~(~N)) is not
surjective. II
Let u be the representation of Cec(1N) defined
in Proposition 7.2.3.4 a) and put
E - - {x 9 ~(ce~(~))l~ = x}. a)
E is a real C*-subalgebra of u(Cgc(IN)) such that the map ~ ~(ce~(~)),
(x,y),
~ x + iy
is an isomorphism of complex C*-algebras. b)
E is isomorphic to Cry(IN). a) is easy to see. b) We denote B := {n e I N l n - 2 (mod4) or n = 3 (mod4)}
and denote by 91 a partition of B such that A is finite and Card A = 0 (mod 4) for every A C 91. By Proposition 7.2.1.18, there is an isomorphism of C*algebras v .ce~(~)
~ ~(ce~(~))
such that f
Yen
_ J ixoo,nXc~,A
if n C A C 91
X~,n
if n C IN\B.
7. C*-algebras Generated by Groups
554
By Proposition 7.2.3.3 e), u
Yen
= Yen
for every n E IN. Hence v(Ce~rt(IN)) = E and the map CG~(IN)
>E,
x,
> VX
is an isomorphism of real C*-algebras. Proposition
7.2.3.6
( 7 )
I
We define a map
r
>r~u {0}
in the following way: if n E IN and P
E 2kJ j=l
is the binary representation of n -
1 (where (kj)jEIN . is a strictly increasing
family in IN U {0} ), then we set r ]
2n
(\ j n= l
: - p.
x
)/ nEIN
,o
7rc~"
b)
* = ~oo
~o~ = too
,
2 = %0
1.
c) j ~ IN ~ xoo,yTro~ = -Trooxo~,j. d)
If u denotes the map defined in Proposition 7.2.3.~ a) and E denotes the real (complex) C*-subalgebra of/:(/?2) generated by
~(cey(~)) o {~oo} (~(c~(~))u {~}) then there is a unique isomorphism of C*-algebras
v : Ce~(IN)
>E
such that Vej
-- Xcx~,j-1
for all j c IN\{1} and vel - iTrc~.
7.2 Clifford Algebras
555
a) By Proposition 7.2.3.1 a),
0
-lrn-1
for every n E IN. We get by induction,
for every n C IN and ~ C 9 2n . The assertion follows from Proposition 7.2.3.3. b) follows from a) and Proposition 7.2.3.3 e). c) We have
Xco,j
in
Zoo,k
in
---- - -
k=i
Xco,k
Xc~,j
k=l
for every n E IN, j < 2n (Proposition 7.2.3.3 e)), so X c c , j T~oo =
- - 7~ o c X o c , j .
d) follows from b),c), and Proposition 7.2.1.5 b).
I
P r o p o s i t i o n 7.2.3.7 ( -[ ) Let 7r~ be the operator defined in Proposition 7.2.3.6 a) and u the representation of Cg~(IN) defined in Proposition 7.2.3.~
a). a)
The commutant of u(C~+(IN)) in E(~ 2) is equal to
b)
A closed vector subspace H of 62 is u(Ce+(iN))-invariant iff 71"H C {0, 1,~1 ( 1 + 7r~)}
c)
The commutant of u (Cgc(IN)) in s
d)
u(Cgc(IN)) acts irreducibly on 62 .
2) is equal to r
a) For n E IN, let Hn be the vector subspace of t~2 generated by (ej)je~N2n and set z~" g n
>g . ,
~,
". (z{)l~.
for every z C Z;(e2). Then for every n E IN, z C/2(62), and ~, ~ E Hn,
7. C*-algebras Generated by Groups
556
(z~r
= (zr
= (r
= (r
Thus
(z~)*=(z*)~. Let z be an element of s
commuting with Ct~+(lN). Take n E IN. Let
91 "= {A C IN2nlCard A - 0 ( m o d 2 ) } , and take ( a A ) A e ~ E r
Put y . - 2.., OlAXoo,A 9 AE~I
By Proposition 7.2.1.9 a), y E u(Cg+(IN)). By Proposition 7.2.3.3 a), Yn :
~
OLAXn,A.
AE~
Then for all ~, r] E Hn, (y~z~l~) = (z~l(~)*~)=
(z~l(~*)~)=
= (z~ly*~ = (yz~ ~ = ( z y ~ l ~ =
= (y~lz*~ = ( y ~ l ( z * ) ~ ) =
(y~ (z~)*~)=
= (z~y~l~),
so that zn yn = yn zn .
By Proposition 7.2.1.9 a),d), there are a,~, ~,~ E r
such that
zn = a ~ l + ~ i n ~ . By Proposition 7.2.3.1 a),
Z n + l -~ Oln+ 1
E10] 0
1
+ ~n+li n 0
01
-7~n
=
7.2 Clifford Algebras
=[
an+ll
--b
~n+l inTrn
0
0
]
an i - fin+ 1inTFn
J
557
Thus O[.n+ l -- an ,
fin+l--fin.
We get
for every ~ E t2(IN2n). Take ~ E t~2 . We set
& "= ,~IIN2. for every n E IN. T h e n Z~n = Zn~n -- Oll~n ~-
fllinTrn~n,
for every n C IN, so t h a t
z~
-
lim Z~n
-
ozl~ +
n---+oo
--
fll
lim
n-->oo
in j--1
Xcx~,j
-
OLI~ -~- f 1 1 7 1 - c ~ " .
Hence Z = a l l + f117roo 9 Since 7 ~ c o m m u t e s with u(Cg+(IN)) (Proposition 7.2.3.6 c)) it follows t h a t
is the c o m m u t a n t of u(Ce +(IN)). b) By a) and Proposition 5.3.2.9, H is u(Cg+(IN))-invariant iff there are a, fl C ff~ such t h a t 7l"H - -
a l + fl~-~.
7. C*-algebras Generated by Groups
558
By Proposition 7.2.3.6 b), a l +/37roo = (al +/37r~) 2 = (a 2 +/32)1 + 2a/37roo, ~1 + Z~= = (~ + Z~=)* = ~1 + Z ~ = ,
whence
We get a : / 3 : 0 or a : l u(Cg + (IN))-invariant iff
a=~=a2
+~2
and / 3 : 0
or a : ~
1
and /3 : : k ~ 1. H e n c e H is
1 ~ , e {1, 0, ~(1 :t: ~oo)}. c) follows from b) and Proposition 7.2.3.6 c). d) follows from c) and Proposition 5.3.2.9. Corollary 7.2.3.8
m
( 7 ) Let u be the representation of Cgc(IN) defined in
Proposition 7.2.3.4 a). Put p'=~ l(l+r~)
where ~
was defined in Proposition 7.2.3.6 a).
a) /:(g2) is the von Neumann algebra on g2 generated by u(Cgc(IN)). b) p E PrL(g 2) and ps
+ ( 1 - p)s
p) is the yon Neurnann
algebra on g2 generated by u(Cg +(IN)).
a) By Theorem 7.2.3.7 c), ~(ce~(~)) ~ - ~ 1 ,
so that ~(ce~(~))~ : ~(e~)
and the assertion follows from Corollary 6.3.5.6. b) By Theorem 7.2.3.7 a), u(Cg+(IN)) r
{ a p + ~(1 - p ) l a , 3 e r
By Proposition 7.2.3.6 b), p E Pr/:(g2) and by Proposition 2.1.3.25, u(Ce+(r~))
~ = p Z . ( e 2 ) p + (1 - p ) Z . ( e 2 ) ( 1
The assertion now follows from Corollary 6.3.5.6.
- p) .
m
559
Name Index
Name Index Alaoglu, L. 1.2.8.1 Arens, R.F. 1.5.2.10, 2.2.7.13 Arzel~, C. 1.1.2.16 Ascoli, G. 1.1.2.14, 1.1.2.16 Atkinson, F.V. 3.1.3.7, 3.1.3.11, 3.1.3.12, 3.1.3.21, 5.3.3.16 Autonne, L. Banach, S.
2.3.1.3 1.1.1.2, 1.2.8.2, 1.3.1.2, 1.3.2, 1.3.3.1, 1.3.4.1, 1.3.4.10, 1.4.1.2, 1.4.2.3, 1.4.2.19
Beurling, A. 2.2.5.4 Bourbaki, N. 1.2.8.1 Branges, L. de 1.3.5.14 Calkin, J.W. 5.4.3.5 Carleman, T. 3.1.3.1, 6.1.4.1 Cauchy, A. 1.3.10.6 Choquet, G. 5.4.3.5 Clifford, W.K. 7.2.2.1 Dedekind, R. 1.7.2.1 Dieudonn~, J. 1.2.8.2, 3.1.3.9 Dixmier, J. 4.4.4.4 Drewnowski, L. 4.3.2.13 Dworetzky, A. 1.1.6.14 Dye, H.A. 4.1.3.7 Eberlein, W.F. 1.3.7.15 Effros, E.G. 4.2.4.15 Enflo, P. 3.1.1.7 2.4.2.4 Ford, J.W.M. 2.4.6.2 Fourier, J.-B.-J. 1.1.1.2,1.1.2.13,5.2.5.2 Fr~chet, M. 3.1.6.23 Fredholm, E. 2.1.4.21 Frobenius, G.F. 4.1.4.1 Fuglede, B. 4.2.1.1 Fukamiya, M. 1.4.1.9,2.2.5.4,2.2.5.5,2.4.1.2,2.4.5.7,4.1.1.1,4.1.2.5 Gelfand, I.M. 4.1.3.1,4.2.6.6,5.4.1.2,5.4.2.5 3.1.3.12 Gohberg, I.
560
Goldstine, H.H. 1.3.6.8 Goodearl, K.R. 2.2.1.19, 4.1.1.1, 5.4.2.13 Gowers, W.T. 1.2.1.12 Gram, J.P. 5.5.1.18 Grothendieck, A. 1.6.1.1, 3.1.6.25, 4.2.8.13 Hahn, H. 1.1.1.2, 1.2.1.3, 1.3.3.1, 1.3.6.1, 1.3.6.3, 1.3.8.1, 1.4.1.4 Hamilton, W.R. 2.1.4.17 Hellinger, E. 5.2.5.4 Helly, E. 1.1.1.2, 1.3.3.13 Hilbert, D. 2.1.3.1 Hirschfeld, R.A. 2.2.5.6 Jacobson, N. 2.1.3.10 James, R.C. 1.3.8.1 Jordan, C. 5.1.1.6 Kadison, R.K. 4.3.3.20, 6.3.6.1 Kaplanski, I. 4.1.2.1, 4.2.6.5, 4.4.2.24, 5.6.1.1, 6.3.1.10 Kelley, J.L. 4.2.1.1 Kojima, ?? 1.2.3.11 Kolmogoroff, A. 1.1.1.2 Kottman, C.A. E 1.3.5 Krein, M.G. 1.3.1.10,1.3.7.3 Laguerre, E.N. 2.2.3.5 Laurent, P.A. 1.3.10.8 Lax, P.D. 5.3.1.3 Le Page, C. 2.2.3.8, 2.2.4.3, 2.2.5.6 Lindenstrauss, J. 1.2.5.13 Liouville, J. 1.3.10.6,6.2.2.5 Lomonsov, V.I. 3.1.5.10 LSwing, H. 5.1.1.1 Mackey, G.W. 1.3.7.2 Mazur, S. 2.2.5.5 Mercer, J. 6.1.9.4 Mihlin, S.G. 3.1.3.12 Milgram, A.N. 5.3.1.3 Milman, D.P. 1.3.1.10 Minkowski, H. 1.1.1.2,1.1.3.4 Murray, F.J. 1.2.5.8 Nagumo, M. 2.2.1.1
Name Index
4.1.1.1, 4.1.2.5, 4.1.3.1, 4.2.6.6, 5.4.1.2, 5.4.2.5, 5.5.1.24 Naimark, M.A. 2.2.3.5 Neumann, C. 1.1.1.2, 3.1.3.1, 5.1.1.1, 5.1.1.6, 5.2.4, 6.1.2.1, 6.3.5.5 Neumann, J. von 4.3.2.14, 5.2.1.2 Nikodym, O. 3.1.3.1 Noether, F. 4.1.1.1 Palmer, T.W. 5.6.1.1, 5.6.2.2, 5.6.2.6, 5.6.2.11, 5.6.3.3, 5.6.3.5 Paschke, W.L. 4.2.4.16, 6.3.4.3 Pedersen, G.K. Peter, F. 2.2.1.15 Pettis, P.J. 1.3.8.4, 1.3.8.5 Phillips, R.S. 1.2.5.14, E 1.3.3, 2.1.4.9 Pierce, B. 2.1.1.1, 2.1.3.6 Plancherel, M. 5.5.4.1 Putnam, I.F. 4.1.4.1 Rellich, F. 5.1.1.1 Rickart, C.E. 4.1.1.20, 4.1.2.12 Rieffel, M.A. 5.6.1.1 Riesz, F. 1.1.1.2, 1.1.4.4, 1.2.1.1, 1.2.2.5, 2.2.5.1 3.1.1.1, 3.1.3.8, 3.1.3.17, 3.1.5.1, 5.2.1.2, 5.2.5.2, 5.3.3.20 Rogers, C.A. 1.1.6.14 Rosenberg, A. 6.3.6.12 Rosenblum, M. 4.1.4.1 Russo, B. 4.1.3.7 Sakai, S. 4.4, 4.4.1.1, 4.4.3.5 Schatten, R. 6.1.2.1 Schauder, J.P. 3.1.1.22 Schmidt, E 1.1.1.2, 5.5.1.18 Schur, I. 1.2.3.11, 1.2.3.12, 1.3.6.11, 7.1.1.7 Schwartz, L. 2.4.6.5, 2.4.6.8 Segal, I.E. 4.2.6.2, 4.2.6.5, 4.2.8.2, 5.4.1.2, 6.3.6.2 Shirali, S. 2.4.2.4 Sierpi/~ski, W. 1.1.2.17 Silow, G. 2.2.4.27 ~;mulian, V. 1.3.7.3, 1.3.7.15 Steinhaus, H.D. 1.4.1.2 Stone, M.H. 1.3.4.10, 1.3.5.16, 2.3.3.12, 4.1.2.5 St~rmer, E. 4.2.6.3, 7.2.1.9 Sturm, C. 6.2.2.5
561
562
Takeda, Z. 6.3.2.1 Takesaki, M. 6.3.2.2 Toeplitz, O. 1.2.3.4, 2.3.1.3, 5.2.5.4 Vaught, R.L. 4.2.1.1 Vigier, J.P. 6.3.1.4 Vitushkin, A.G. 2.4.3.7 Volterra, V. 2.2.4.22 Wedderburn, J.M. 6.3.6.5 Weierstrass, K. 1.3.5.16 Wene, G.P. 7.2.1.10 Weyl, H. 2.2.1.15, 6.1.3.26, 6.1.7.23 Wielandt, H. 2.2.5.8 Wiener, N. 2.4.5.7 Yood, B. 3.1.3.11, 3.1.3.12, 4.1.1.13 Zelazko, W. 2.2.5.6
Subject Index
563
Subject Index NT means Notation and Terminology
(A, B, C)-multiplication
1.5.1.1
absolute value of a number 1.1.1.1 absolute value of a measure NT absolutely convex 1.2.7.1 absolutely convex closed hull 1.2.7.6 absolutely convex hull 1.2.7.4 absolutely summable family 1.1.6.9 acts irreducibly 5.3.2.19 acts non-degenerately 5.3.2.19 additive group NT adherence, point of NT adherent point NT adjoint 2.3.1.1, 5.6.1.8 adjoint differential operator 3.2.2.3 adjoint kernel 3.1.6.5 adjoint operator 5.3.1.4 adjoint sesquilinear form 2.3.3.1 adjoint sesquilinear map 2.3.3.1 adjointable 5.6.1.7 algebra 2.1.1.1 algebra, Calkin 3.1.1.13 algebra, complex 2.1.1.1 algebra, degenerate algebra, division
2.1.1.1 2.1.2.1
algebra, Gelfand
2.4.1.1
algebra, Gelfand unital
2.4.2.1
algebra, involutive 2.3.1.3 algebra, involutive Gelfand algebra, algebra, algebra, algebra, algebra,
2.4.2.1
involutive unital Gelfand 2.4.2.1 normed 2.2.1.1 real 2.1.1.1 semi-simple 2.1.3.18 strongly symmetric 2.3.1.26
564
algebra, symmetric 2.3.1.26 algebra, unital 2.1.1.3 algebra, unital Gelfand 2.4.1.1 algebra homomorphism 2.1.1.6 5.4.2.3 algebra homomorphism associated to A 5.4.1.2 algebra homomorphism associated to x' 5.4.2.2 algebra homomorphism associated to (xt~),el algebra homomorphism, unital 2.1.1.6 algebra isomorphism 2.1.1.6 algebra isomorphism, unital 2.1.1.6 algebraic dimension 1.1.2.18 algebraic dual 1.1.1.1 algebraic eigenspace 5.3.3.20 algebraic eigenvector 5.3.3.20 algebraic isomorphism, associated 1.2.4.6 algebraic multiplicity 5.3.3.20 algebras, isomorphism of involutive 2.3.1.3 analytic function 1.3.10.1 approximate unit 2.2.1.15 4.2.8.2 approximate unit of a C*-algebra, canonical Arens multiplication, left 1.5.2.10, 2.2.7.13 Arens mutliplication, right 1.5.2.10, 2.2.7.13 associated algebraic isomorphism 1.2.4.6 associated quadratic form 2.3.3.1 associated quadratic map 2.3.3.1 associated unital C*-algebra 4.1.1.13 atom 4.3.2.20 atomic 4.3.2.20 atomic part 6.3.3.7 atomless 4.3.2.20 atomless part 6.3.3.7 Baire function 1.7.2.12 Baire set 1.7.2.12 ball, unit 1.1.1.2 Banach algebra 2.2.1.1 Banach algebra, involutive 2.3.2.1 Banach algebra, quasiunital 2.2.1.15 Banach algebra, unital 2.2.1.1
Subject Index
565
Banach categories, functor of
1.5.2.1
Banach categories, functor of unital Banach category
Banach category, unital Banach space
1.5.1.1
1.1.1.2
Banach space, complex
1.1.1.2
Banach space, involutive Banach space, ordered
2.3.2.1 1.7.1.4
Banach space, real
1.1.1.2
Banach subalgebra generated by Banach system
1.5.1.9
Banach system, dual of a
1.5.1.9
Banach systems, isometric
1.5.2.1
1.7.2.1
basis, Fourier
5.6.3.11
basis, orthonormal basis of
2.2.1.9
1.5.1.1
Banach system, bidual of a
band
1.5.2.1
1.5.1.1
5.5.1.1
~.2(T), canonical orthonormal
Bergman kernel
5.2.5.9
Bessel's identity
5.5.1.7
Bessel's inequality bicommutant
5.5.1.8 2.1.1.16
bidual of a Banach system
1.5.1.9
bidual of a normed space bijective
1.2.9.1
binomial theorem
2.2.3.12
1.3.6.15
bound, lower
1.7.2.1
bound, upper
1.7.2.1
bounded map
1.1.1.2
bounded operator
1.2.1.3
bounded operator, lower bounded sequence bounded set
1.3.6.1
NT
bilinear map bitranspose
5.5.3.1
1.2.1.18
1.1.1.2
1.1.1.2
boundedness, principle of uniform
1.4.1.2
boundedness theorem, Nikodym's
4.3.2.14
C*-algebra
4.1.1.1
566
C*-algebra, canonical approximate unit of a C*-algebra, canonical order of a C*-algebra, complex
4.2.1.2
4.1.1.1
C*-algebra, complex unital C*-algebra, Gelfand
4.1.1.1
4.1.1.1
C*-algebra, purely real C*-algebra, real
4.2.8.2
4.1.1.8
4.1.1.1
C*-algebra, real unital
4.1.1.1
C*-algebra, simple
4.3.5.1
C*-algebra, unital
4.1.1.1
C*-algebra associated, unital Calkin algebra
4.1.1.13
3.1.1.13
Calkin category
3.1.1.12
canonical approximate unit of a C*-algebra canonical involution of E F
canonical metric of a normed space
1.1.1.2
canonical norm of a pre-Hilbert space canonical norm of /:(E, F)
5.1.1.2
1.2.1.9
canonical order of a C*-algebra
4.2.1.2
canonical orthonormal basis of g2(T)
5.5.3.1
canonical projection of the tridual of E canonical scalar product of IKn cardinal number
NT
4.3.3.1
carrier, left
4.3.3.1
carrier, right
4.3.3.1
carrier of a function
NT
carrier of a Radon measure category, Banach C*-direct sum
NT
1.5.1.1
C*-direct product character
1.3.6.19
5.1.2.4
NT
cardinality, topological carrier
4.2.8.2
2.3.1.1
4.1.1.6 4.1.1.6
2.4.1.1
characteristic family
6.1.2.1
characteristic family of eigenvalues characteristic function of a set characteristic number, n-th characteristic sequence
6.1.1.1 1.1.2.1
6.1.2.1 6.1.2.1
Subject Index
C*-hull
567
4.1.1.22
class NT Clifford algebra associated to p
7.2.1.2
Clifford algebra generated by T 7.2.1.2 Clifford algebra of degree p 7.2.2.1 Clifford C*-algebra associated to p 7.2.1.3 Clifford C*-algebra of T 7.2.1.3 Clifford W*-algebra associated to p 7.2.1.3 Clifford W*-algebra of T 7.2.1.3 closed graph theorem 1.4.2.19 closed involutive subalgebra generated by 2.3.2.14, 2.3.2.15 closed involutive unital subalgebra generated by 2.3.2.14, 2.3.2.15 closed subalgebra generated by 2.2.1.9 closed unital subalgebra generated by 2.2.1.9 closed vector subspace generated by 1.1.5.5 C*-module, Hilbert right 5.6.1.4 codimension 1.2.4.1 codomain NT cokernel of a linear map 1.2.4.5 commutant 2.1.1.16 commutative 2.1.1.1 commutative monoid E 2.1.1 compact, relatively 1.1.2.9 compact operator 3.1.1.1 compatible, simultaneously 1.5.1.1 compatible (left and right) multiplications 1.5.1.1 complement of a subspace 1.2.5.3 complemented subspace 1.2.5.3 complete, C-order 4.3.2.3 complete, order 1.7.2.1 complete norm 1.1.1.2 complete normed space 1.1.1.2 complete ordered set 1.7.2.1 completion of a normed algebra 2.2.1.13 completion of a normed space 1.3.9.1 completion of a pre-Hilbert space 5.1.1.7 complex algebra 2.1.1.1 complex Banach space 1.1.1.2
568
complex C*-algebra
4.1.1.1 4.1.1.1
complex C*-algebra, unital complex Hilbert space
5.1.1.2
complex normed algebra
2.2.1.1 1.1.1.2
complex normed space complex pre-Hilbert space
5.1.1.1
complex unital C*-algebra
4.1.1.1
complex universal representation
5.4.2.6
complex W*-algebra 4.4.1.1 complexification of algebras 2.1.5.7 complexification of Banach algebras 2.2.1.19 complexification complexification complexification complexification
of of of of
Hilbert spaces 5.3.1.8 involutive algebras 2.3.1.40 involutive vector spaces 2.3.1.38 right C*-modules 5.6.1.6
complexification of vector spaces component of x on A4~ composition of functors composition of maps
2.1.5.1
6.3.3.6 1.5.2.1 NT
compression of a representation
5.4.2.8
cone 1.3.7.4 cone, sharp 1.3.7.4 conjugacy class 2.2.2.7 conjugate exponent of 1.2.2.1 conjugate exponents
1.2.2.1
conjugate exponents, weakly 1.2.2.1 2.3.1.3 conjugate involution conjugate linear map
1.3.7.10
1.1.1.1 conjugate number 1.7.2.3 continuous, order 1.1.6.22 convergence, radius of convex
1.2.7.1
convex, absolutely
1.2.7.1
1.2.7.6 convex closed hull convex closed hull, absolutely convex hull 1.2.7.4 convex hull, absolutely 1.2.7.4 convolution 2.2.2.7, 2.2.2.10
1.2.7.6
Subject Index
569
~-order complete
4.3.2.3
~-order a-complete C*-subalgebra
4.3.2.3 4.1.1.1
C*-subalgebra, unital 4.1.1.1 C*-subalgebra generated by 4.1.1.1 4.3.4.1 C*-subalgebra generated by, hereditary C*-subalgebra generated by, unital 4.1.1.1 cyclic element
5.4.1.1
cyclic representation
5.4.1.1
cyclic vector 5.3.2.19, 5.4.1.1 cyclic vector associated to x' 5.4.1.2 decomposition, Schatten
6.1.3.4
decomposition, spectral 4.3.2.19, 5.3.4.7 degenerate algebra 2.1.1.1 derivative 1.1.6.24 diagonalization of u
5.5.6.1
differentiable 1.1.6.24 differential operator, adjoint differential operator, selfadjoint dimension, algebraic
3.2.2.3 3.2.2.3
1.1.2.18
dimension, Hilbert
5.5.2.2
Dirac measure 1.2.7.14 direct integral of E with respect to p direct sum
1.2.5.3
directed, downward
1.1.6.1
directed, upward
1.1.6.1
disjoint family of sets
1.2.3.9
distance of a point from a set division algebra domain
1.1.4.1
2.1.2.1
NT
downward directed dual, algebraic
1.1.6.1 1.1.1.1
dual of a Banach system
1.5.1.9
dual of a normed space dual space 1.3.1.11 E-algebra 2.2.7.1 E-algebra, involutive
1.2.1.3
2.3.6.1
E-algebra, involutive unital
2.3.6.1
5.5.2.19
570
E-algebra, unital 2.2.7.1 E-algebras, homomorphism of 2.2.7.1 E-algebras, homomorphism of involutive 2.3.6.1 E-algebras, homomorphism of involutive unital 2.3.6.1 E-algebras, homomorphism of unital 2.2.7.1 E-C*-algebra 5.6.1.10 E-C*-algebra, unital 5.6.1.10 E-C*-algebras, isomorphism of 5.6.1.10 eigenspace 3.1.4.1 eigenspace, algebraic 5.3.3.20 eigenvalue 3.1.4.1 eigenvalues, characteristic family of 6.1.1.1 eigenvector 3.1.4.1 eigenvector, algebraic 5.3.3.20 E-module 2.2.7.1 E-module, Hilbert 5.6.1.4 E-module, Hilbert right 5.6.1.4 E-module, inner-product right 5.6.1.1 E-module, involutive 2.3.6.1 E-module, involutive unital 2.3.6.1 E-module, semi-inner-product right 5.6.1.1 E-module, weak semi-inner-product right 5.6.1.1 E-module, unital 2.2.7.1 E-module, unital Hilbert 5.6.1.4 E-module, von Neumann (right) 5.6.3.2 E-modules, homomorphism of 2.2.7.1 E-modules, homomorphism of involutive 2.3.6.1 empty word 7.2.1.1 equicontinuous 1.1.2.14 equivalence class NT equivalence class of a point NT equivalence of GNS-triples 5.4.1.2 equivalence of representations 5.4.1.1 equivalence relation NT equivalent GNS-triples 5.4.1.2 equivalent norms 1.1.1.2 equivalent representations 5.4.1.1 essential spectrum 3.1.3.24
Subject Index
571
E-submodule
2.2.7.1
Euclidean norm evaluation
1.1.5.2 1.2.1.8
evaluation functor
1.5.2.1
evaluation operator of a normed space E-valued spectral measure exact set
4.3.2.16
1.7.2.12
expansion, Fourier
5.5.1.15
exponential function
2.2.3.5
exponents, conjugate
1.2.2.1
exponents, weakly conjugate extreme Fourier set
1.2.2.1
6.3.9.4
extreme point
1.2.7.9
face of a convex set
1.2.7.9
factorization of a linear map faithful, order
faithful representation family
1.2.4.6
4.2.2.18 5.4.1.1
NT
family, absolutely summable family, sum of a
1.1.6.9
1.1.6.2
family, summable
1.1.6.2
family of sets, disjoint Fatou's Lemma
1.2.3.9
6.1.3.15
filter, lower section
1.1.6.1
filter, upper section
1.1.6.1
filter of cofinite subsets
NT
finite-dimensional ~'-invariant
1.1.2.18
3.1.4.4
Fourier basis
5.6.3.11
Fourier expansion Fourier integral
5.5.1.15 2.4.6.2
Fourier-Plancherel operator Fourier set
5.5.4.1
5.6.3.11
Fourier set, extreme Fourier transform
6.3.9.4 2.4.6.2
Fr~chet-Riesz Theorem Fredholm alternative Fredholm operator
5.2.5.2 3.1.6.23 3.1.3.1
1.3.6.3
572
Fredholm operator, index of a NT free ultrafilter function NT function, Baire
3.1.3.1
1.7.2.12
NT function, step 4.1.3 functional calculus functor 1.5.2.1 1.5.2.1 functor, identity 1.5.2.16 functor, inclusion 1.5.2.1 functor, isometric 1.5.2.17 functor, quotient functor, transpose of a 1.5.2.3 1.5.2.1 functor of (unital) Banach categories 1.5.2.1 functor of (unital) A-categories 1.5.2.1 functor of (left, right) A-modules functors, composition of 1.5.2.1 Gelfand, Theorem of 2.2.5.4 Gelfand algebra 2.4.1.1 Gelfand algebra, involutive 2.4.2.1 Gelfand algebra, involutive unital 2.4.2.1 Gelfand algebra, spectrum of a 2.4.1.1 Gelfand algebra, unital 2.4.1.1 Gelfand C*-algebra 4.1.1.1 Gelfand-Mazur, Theorem of 2.2.5.5 Gelfand transform 2.4.1.2 generators of C~(T) 7.2.1.2 GNS-construction 5.4.1.2 5.4.1.2 GNS-triple of E associated to x'
GNS-triples, equivalence of 5.4.1.2 GNS-triples, equivalent 5.4.1.2 Gram-Schmidt orthonormalization graph
5.5.1.18
NT, 1.4.2.18
Green function 3.2.1.2 group, additive NT Hahn-Banach Theorem 1.3.3.1 hereditary 4.3.4.1 hereditary C*-subalgebra generated by Hermitian sesquilinear map 2.3.3.3
4.3.4.1
Subject Index
573
Hilbert dimension
5.5.2.2
Hilbert E - m o d u l e
5.6.1.4
Hilbert E-module, unital
5.6.1.4
Hilbert right C*-module
5.6.1.4
Hilbert right E-module
5.6.1.4
Hilbert right E-modules, isomorphic Hilbert-Schmidt operator Hilbert space
5.6.1.7
6.1.4.1
5.1.1.2
Hilbert space, complex
5.1.1.2
Hilbert space, complexification Hilbert space, involutive Hilbert space, real
5.3.1.8
5.5.7.1
5.1.1.2
Hilbert space associated to A
5.4.2.3
Hilbert space associated to x'
5.4.1.2
Hilbert space associated to (x'~)~i
5.4.2.2
Hilbert space of square summable sequences Hilbert sum of a family of Hilbert spaces
5.1.2.3 5.1.3.1
Hilbert sum of a family of representations HSlder inequality
5.4.2.1
1.2.2.5, 6.1.3.21, 6.1.5.11
homomogeneous W*-algebra
5.6.7.8
homomorphism of C*-algebras
4.1.1.20
homomorphism of E-algebras
2.2.7.1
homomorphism of E-modules
2.2.7.1
homomorphism of involutive E-algebras
2.3.6.1
homomorphism of involutive E-modules
2.3.6.1
homomorphism of involutive unital E-algebras homomorphism of unital E-algebras hyperstonian space ideal
2.2.7.1
1.7.2.12
2.1.1.1
ideal, left
2.1.1.1
ideal, maximal proper
2.1.1.1
ideal, maximal proper left ideal, maximal proper right ideal, proper ideal, proper left ideal, proper right
2.1.1.1 2.1.1.1
2.1.1.1 2.1.1.1 2.1.1.1
ideal, regular maximal proper ideal, regular maximal proper left
2.1.3.17 2.1.3.17
2.3.6.1
574
ideal, regular maximal proper right ideal, right 2.1.1.1 ideal generated by idempotent
2.1.3.17
2.1.1.2
2.1.3.6
identical representation 5.5.1.23 identity functor 1.5.2.1 identity map
NT
identity operator iff
1.2.1.3
NT
image of a linear map 1.2.4.5 imaginary part 1.1.1.1, 2.3.1.22 inclusion functor inclusion map
1.5.2.16 NT
index of a Fredholm operator index of U 3.1.3.21 induced norm infimum
1.1.1.2 1.7.2.1
infinite-dimensional infinite matrix
1.1.2.18 1.2.3.1
initial segment of lN injective
3.1.3.1
6.1.2.1
NT
inner multiplication 1.5.1.1 inner-product 5.6.1.1 inner-product right E-module
5.6.1.1
integral of E with respect to #, direct interior point NT invariant vector subspace inverse of a bijective map inverse of a morphism
3.1.4.4 NT 1.5.1.6
inverse of an element in a unital algebra inverse operators, principle of invertible
1.4.2.4
1.5.1.5, 2.1.2.1
invertible, left invertible, right
1.5.1.5 1.5.1.5
involution 2.3.1.1 involution, conjugate 2.3.1.3 involution of E F, canonical 2.3.1.1 involutive algebra
5.5.2.19
2.3.1.3
2.1.2.4
Subject Index
575
involutive algebra, complexification of an involutive algebra, strongly symmetric involutive algebra, symmetric
2.3.1.26
2.3.1.26
involutive algebras, isomorphism of involutive Banach algebra
2.3.1.40
2.3.1.3
2.3.2.1
involutive Banach space
2.3.2.1
involutive Banach unital algebra associated to involutive E-algebra
2.3.6.1
involutive E-module
2.3.6.1 2.4.2.1
involutive Gelfand algebra involutive Hilbert space involutive map
2.3.2.9
5.5.7.1
2.3.1.1
2.3.2.1 involutive normed algebra 2.3.2.1 involutive normed space involutive normed unital algebra associated to involutive set involutive space
2.3.2.9
2.3.1.1 2.3.1.1
involutive subalgebra generated by involutive unital algebra associated to
2.3.1.18 2.3.1.9
involutive unital E-algebra
2.3.6.1
involutive unital E-module
2.3.6.1
involutive unital Gelfand algebra
2.4.2.1 2.3.1.18
involutive unital subalgebra generated by involutive vector space
2.3.1.3
involutive vector spaces, isomorphism of
2.3.1.3
involutive vector subspace generated by
2.3.1.18
irreducible representation irreducibly, acts
5.3.2.19
isometric Banach systems isometric functor
5.4.1.1 1.5.2.1
1.5.2.1
isometric normed algebras isometric normed spaces
2.2.1.1 1.2.1.12
isometric normed unital algebras
2.2.1.1
isometry of Hilbert spaces, partial 5.3.2.25 isometry of normed algebras 2.2.1.1 isometry of W*-algebras 4.4.4.5 isometry of normed spaces 1.2.1.12 isometry of normed unital algebras 2.2.1.1
576
isomorphic algebras 2.1.1.6 isomorphic Hilbert right E-modules isomorphic normed algebras isomorphic normed spaces
5.6.1.7
2.2.1.1 1.2.1.12
isomorphic normed unital algebras 2.2.1.1 isomorphic unital algebras 2.1.1.6 isomorphism, algebra 2.1.1.6 isomorphism associated to a linear map, algebraic
1.2.4.6
isomorphism of E-C*-algebras 5.6.1.10 isomorphism of involutive algebras 2.3.1.3 isomorphism of involutive vector spaces 2.3.1.3 isomorphism of normed algebras 2.2.1.1 isomorphism of normed spaces 1.2.1.12 isomorphism of normed unital algebras 2.2.1.1 kernel, Bergman 5.2.5.9 kernel of a linear map 1.2.4.5 Kronecker's symbol 1.2.2.6 lattice 1.7.2.1 lattice, vector 1.7.2.1 Laurent series 1.3.10.8, 1.3.10.9 Lax-Milgram Theorem 5.3.1.3 6.1.3.20 Lebesgue's Dominated Convergence Theorem left Arens multiplication 1.5.2.10, 2.2.7.13 left left left left left
carrier 4.3.3.1 ideal 2.1.1.1 ideal, maximal proper 2.1.1.1 ideal, proper 2.1.1.1 ideal, regular maximal proper 2.1.3.17
left ideal generated by 2.1.1.2 left invertible 1.5.1.5 left multiplication 1.5.1.1, 5.6.1.4 left shift 1.2.2.9, E 1.2.11 left (unital) A-module 1.5.1.10 linear form 1.1.1.1 linear form, positive linear map, conjugate locally finite lower bound
7.1.1.2 1.7.2.1
1.7.1.9 1.3.7.10
Subject Index
577
lower bounded operator lower section filter
1.2.1.18 1.1.6.1
L2-distributions, in the sense of map
3.2.2.3
NT
map, bilinear
1.2.9.1
map, bounded
1.1.1.2
map, conjugate linear map, identity
1.3.7.10
NT
map, inclusion
NT
map, inverse of a bijective map, involutive map, nuclear
NT
2.3.1.1 1.6.1.1
map, quotient
1.2.4.1
maps, composition of matrix, infinite
NT
1.2.3.1
maximal proper ideal
2.1.1.1
maximal proper ideal, regular maximal proper left ideal
2.1.3.17 2.1.1.1
maximal proper left ideal, regular maxxmal proper right ideal
2.1.3.17
2.1.1.1
m a m m a l proper right ideal, regular mean ergodic theorem measure, Dirac
1.2.7.14
measure, E - v a l u e d spectral measure, Radon
4.3.2.19
NT
measure of x , spectral
4.3.2.19
measure space, a-finite
3.1.6.14
metric of a normed space, canonical module
module, involutive
2.3.6.1
module, unital
2.3.6.1
2.2.7.1
modules, homomorphism of
2.2.7.1
modules, homomorphism of involutive modulus monoid
1.1.1.2
2.2.7.1
module, involutive unital
modulo
2.1.3.17
5.2.4.3
NT 4.2.5.1, 4.4.3.5 E 2.1.1
monoid, commutative
E 2.1.1
2.3.6.1
578
morphism 1.5.1.1 morphism, inverse of a multipliable sequence
1.5.1.6 2.2.4.33
multiplication 2.1.1.1 multiplication, (.A, B, C) 1.5.1.1 multiplication, compatible (left and right) multiplication, inner multiplication, left
1.5.1.1
1.5.1.1 1.5.1.1, 5.6.1.4
multiplication, left (right) Arens 1.5.2.10, 2.2.7.13 multiplication, right 1.5.1.1, 5.6.1.1 multiplication operator 2.2.2.22 6.3.7.2 multiplication operators, von Neumann algebra of multiplicity 3.1.4.1 multiplicity, algebraic 5.3.3.20 negative 1.7.1.1 negative part 4.2.2.9, 4.2.8.13 Nikodym's boundedness theorem 4.3.2.14 nilpotent 2.1.1.1 non-degenerate representation 5.4.1.1 non-degenerately, acts 5.3.2.19 norm 1.1.1.2 norm, complete 1.1.1.2 norm, Euclidean 1.1.5.2 norm, induced 1.1.1.2 norm, quotient 1.2.4.2 norm, supremum 1.1.2.2, 1.1.5.2 norm of an operator 1.2.1.3 5.1.1.2 norm of a pre-Hilbert space, canonical norm of I:(E, F ) , canonical
1.2.1.9
norm topology 1.1.1.2 normal 2.3.1.3 normed algebra 2.2.1.1 normed algebra, completion of a normed algebra, complex normed algebra, involutive normed algebra, quasiunital
2.2.1.13
2.2.1.1 2.3.2.1 2.2.1.15
normed algebra, real 2.2.1.1 normed algebras, isometric 2.2.1.1
Subject Index
579
normed algebras, isometry of
2.2.1.1
normed algebras, isomorphic
2.2.1.1
normed algebras, isomorphism of normed space
2.2.1.1
1.1.1.2
normed space, bidual of a
1.3.6.1
normed space, complete
1.1.1.2
normed space, completion of a normed space, complex
1.3.9.1
1.1.1.2
normed space, involutive normed space, ordered
2.3.2.1 1.7.1.4
normed space, real
1.1.1.2
normed spaces, isometric
1.2.1.12
normed spaces, isometry of
1.2.1.12
normed spaces, isomorphic
1.2.1.12
normed spaces, isomorphism of normed unital algebra
1.2.1.12
2.2.1.1
normed unital algebras, isometric
2.2.1.1
normed unital algebras, isometry of
2.2.1.1
normed unital algebras, isomorphic
2.2.1.1
normed unital algebras, isomorphism of norms, equivalent
1.1.1.2
n-th characteristic number nuclear map
6.1.2.1
1.6.1.1
number, cardinal number, ordinal
NT NT
object of a Banach system onto
open mapping principle operator
1.5.1.1
NT 1.4.2.3
1.2.1.3
operator, adjoint
5.3.1.4
operator, adjoint differential
3.2.2.3
operator, bounded
1.2.1.3
operator, compact
3.1.1.1
operator, Fourier-Plancherel operator, Fredholm operator, identity
5.5.4.1
3.1.3.1 1.2.1.3
operator, index of a Fredholm operator, lower bounded
3.1.3.1 1.2.1.18
2.2.1.1
580
operator, multiplication
2.2.2.22
operator, order of an
3.1.3.18
operator, selfadjoint differential operators, principle of inverse order complete
3.2.2.3 1.4.2.4
1.7.2.1
order continuous order faithful
1.7.2.3 4.2.2.18
order of a pole
1.3.10.9
order relation of a C*-algebra, canonical order summable order a-complete
1.7.2.1
order a-continuous
1.7.2.3
order a-faithful
4.2.2.18
ordered Banach space
1.7.1.4
ordered normed space
1.7.1.4
ordered set, complete
1.7.2.1
ordered set, totally
NT
ordered set, a-complete
1.7.2.1
ordered vector space ordinal number orthogonal
1.7.1.1 NT
5.2.2.1
orthogonal projection
4.1.2.18, 5.2.3.2
orthogonal set of A orthogonal sets
5.2.2.1 5.2.2.1
orthogonal vectors
5.2.2.1
orthonormal basis
5.5.1.1
orthonormal basis of g2(T), canonical orthonormal family orthonormal set
5.5.1.1
parallelogram law Parseval's Equation
5.5.1.15 NT
1.1.2.5, 1.1.5.2, 6.1.2.1
point, adherent
NT
point, extreme
1.2.7.9
point, interior
5.5.1.18
2.3.3.2
partial isometry of Hilbert spaces partition of a set
5.5.3.1
5.5.1.1
orthonormalization, Gram-Schmidt
p-norm
4.2.1.2
1.7.2.10
NT
5.3.2.25
Subject Index
581
point of adherence point spectrum polar 1.3.5.1
NT 3.1.4.1
polar representation polarization identity pole (of order)
4.2.6.9, 4.4.3.1, 4.4.3.5 2.3.3.2
1.3.10.9
positive 1.7.1.1, 2.3.3.3, 2.3.4.1 positive linear form 1.7.1.9, 2.3.4.1 positive part power series precompact
4.2.2.9, 4.2.8.13 1.1.6.22 1.1.2.9
predual of a Banach space
1.3.1.11
predual of a W*-algebra 4.4.1.1, 4.4.4.4 pre-Hilbert space 5.1.1.1 pre-Hilbert space, canonical norm of a 5.1.1.2 pre-Hilbert space, completion of a 5.1.1.7 pre-Hilbert space, complex 5.1.1.1 pre-Hilbert space, real 5.1.1.1 prepolar 1.3.5.1 pretranspose of an operator 1.3.4.9, 4.4.4.8 principal part 1.3.10.8, 1.3.10.9 principle of inverse operators 1.4.2.4 principle of open mapping 1.4.2.3 principle of uniform boundedness product 2.1.1.1, 7.2.1.1 product, C*-direct 4.1.1.6 product of a family of sets NT product of a sequence 2.2.4.33 product, scalar 5.1.1.1 product associated to f , scalar projection 1.2.5.7
1.4.1.2
5.1.2.9
projection, orthogonal 4.1.2.18, 5.2.3.2 projection of the tridual of E, canonical proper ideal 2.1.1.1 proper ideal, maximal 2.1.1.1 proper ideal, regular maximal 2.1.3.17 proper left ideal 2.1.1.1 proper left ideal, maximal 2.1.1.1
1.3.6.19
582
proper left ideal, reguar maximal proper right ideal
2.1.3.17
2.1.1.1
proper right ideal, maximal
2.1.1.1
proper right ideal, regular maximal pure state
2.1.3.17
2.3.5.1
pure state space
2.3.5.1
purely real C*-algebra
4.1.1.8
Pythagoras' Theorem
5.2.2.3
quadratic form, associated
2.3.3.1
quadratic map, associated
2.3.3.1
quasinilpotent
2.2.4.20
quasiunital 2.2.1.15 quaternion 2.1.4.17 quotient functor 1.5.2.17 quotient map
NT, 1.2.4.1
quotient norm 1.2.4.2 quotient space 1.2.4.2 quotient A-category 1.5.2.17 quotient A-module
1.5.2.17
Raabe's ratio test radical
2.2.3.11
2.1.3.18
radius of convergence
1.1.6.22
Radon measure NT Radon-Nikodym Theorem range of values
4.4.3.15
NT
real algebra
2.1.1.1
real Banach space real C*-algebra
1.1.1.2 4.1.1.1
real C*-algebra, purely
4.1.1.8
real C*-algebra, unital
4.1.1.1
real Hilbert space
5.1.1.2
real normed space
1.1.1.2
real part
1.1.1.1, 2.3.1.3
real pre-Hilbert space 5.1.1.1 real W*-algebra 4.4.1.1 reduces u
5.2.3.11
reflexive 1.3.8.1 regular maximal proper ideal
2.1.3.17
Subject Index
regular maximal proper left ideal 2.1.3.17 regular maximal proper right ideal 2.1.3.17 relatively compact 1.1.2.9 representation 5.4.1.1 representation, associated to A 5.4.2.3 representation, associated to x' 5.4.1.2 representation, associated to (x~)~r 5.4.2.2 representation, complex universal 5.4.2.6 representation, compression of a 5.4.2.8 representation, cyclic 5.4.1.1 representation, faithful 5.4.1.1 representation, identical 5.5.1.23 representation, irreducible 5.4.1.1 representation, non-degenerate 5.4.1.1 representation, unital 5.4.1.1 representation, universal 5.4.2.3 representation, 05.4.1.1 representations, equivalence of 5.4.1.1 representations, equivalent 5.4.1.1 representations, Hilbert sum of 5.4.2.1 residue 1.3.10.8, 1.3.10.9 resolvent 2.1.3.1 resolvent equation 2.1.3.9 Riesz, theorem of 2.2.5.1 right Arens multiplication 1.5.2.10, 2.2.7.13 right carrier 4.3.3.1 right C*-module, Hilbert 5.6.1.4 right E-module, Hilbert 5.6.1.4 right E-module, inner-product 5.6.1.1 right E-module, semi-inner-product 5.6.1.1 right E-module, von Neumann 5.6.3.2 right E-module, weak semi-inner-product 5.6.1.1 right ideal 2.1.1.1 right ideal, maximal proper 2.1.1.1 right ideal, proper 2.1.1.1 right ideal, regular maximal proper 2.1.3.17 right ideal generated by 2.1.1.2 right invertible 1.5.1.5
583
584
right multiplication 1.5.1.1, 5.6.1.1 right shift 1.2.2.9, E 1.2.11 right (unital) A-module, 1.5.1.10 right W*-module, von Neumann 5.6.3.2 scalar 1.1.1.1 scalar product 5.1.1.1 scalar product associated to f scalar product of IKn , canonical
5.1.2.9 5.1.2.4
Schatten decomposition 6.1.3.4 Schur function 7.1.1.7 Schur function associated to p 7.2.1.2 Schwartz space of rapidly decreasing C~ 2.4.6.5 Schwarz inequality 2.3.3.9, 5.1.1.2 section filter, lower 1.1.6.1 section filter, upper 1.1.6.1 selfadjoint 2.3.1.1 selfadjoint differential operator 3.2.2.3 self-dual 5.6.2.2 self-normal 2.3.1.3 semi-inner-product right E-module 5.6.1.1 semi-inner-product right E-module, weak 5.6.1.1 seminorm 1.1.1.2 semi-simple algebra 2.1.3.18 separating vector 5.3.2.19, 5.4.4.1 sequence NT series, Laurent 1.3.10.8, 1.3.10.9 series, power 1.1.6.22 sesquilinear form 2.3.3.1 sesquilinear form, adjoint sesquilinear map 2.3.3.1 sesquilinear map, adjoint
2.3.3.1 2.3.3.1
sesquilinear map, Hermitian set, Baire 1.7.2.12 set, set, set, set, set,
bounded 1.1.1.2 complete ordered 1.7.2.1 exact 1.7.2.12 partition of a NT totally ordered NT
2.3.3.3
Subject Index
585
set, p-null
NT
set, a-complete ordered sharp cone shift, left
1.7.2.1
1.3.7.4 1.2.2.9
shift, right
1.2.2.9
simple C*-algebra
4.3.5.1
simultaneously compatible space, Banach
1.5.1.1
1.1.1.2
space, bidual of a normed
1.3.6.1
space, complete normed
1.1.1.2
space, completion of a normed
1.3.9.1
space, complex Banach
1.1.1.2
space, complex Hilbert
5.1.1.2
space, complex normed
1.1.1.2
space, complex pre-Hilbert space, dual
5.1.1.1
1.3.1.11
space, Hilbert
5.1.1.2
space, hyperstonian
1.7.2.12
space, involutive
2.3.1.1
space, involutive Banach
2.3.2.1
space, involutive normed
2.3.2.1
space, involutive vector space, normed
2.3.1.3
1.1.1.2
space, ordered Banach
1.7.1.4
space, ordered normed
1.7.1.4
space, ordered vector
1.7.1.1
space, pre-Hilbert
5.1.1.1
space, pure state
2.3.5.1
space, quotient
1.2.4.2
space, real Banach
1.1.1.2
space, real Hilbert
5.1.1.2
space, real normed
1.1.1.2
space, real pre-Hilbert space, state space, Stone
1.7.2.12
space, subspace of a normed space, vector space, a-Stone
5.1.1.1
2.3.5.1
1.1.1.1 1.7.2.12
1.1.1.2
586
5.1.2.3
space of square summable sequences, Hilbert spaces, isometric normed
1.2.1.12
spaces, isometry of normed
1.2.1.12
spaces, isomorphic normed
1.2.1.12 2.3.1.3
spaces, isomorphism of involutive vector spaces, isomorphism of normed spectral decomposition spectral measure, E-valued spectral measure of x spectral radius
1.2.1.12
4.3.2.19, 5.3.4.7 4.3.2.16
4.3.2.19
2.1.3.1
spectrum, essential spectrum, point
3.1.3.24 3.1.4.1
spectrum of an element
2.1.3.1
spectrum of a Gelfand algebra
2.4.1.1
square summable sequences, Hilbert space of state
5.1.2.3
2.3.5.1
state, pure
2.3.5.1
state space
2.3.5.1
state space, pure
2.3.5.1
step function
NT
Stone space
1.7.2.12
strong topology
6.3 2.3.1.26
strongly symmetric involutive algebra subalgebra
2.1.1.1
subalgebra, unital
2.1.1.3
subalgebra generated by
2.1.1.4 2.3.1.18
subalgebra generated by, involutive subspace, complemented
1.2.5.3 1.1.5.5
subspace generated by, closed vector subspace of a normed space sum, C*-direct sum, direct
1.1.1.2
4.1.1.6 1.2.5.3
sum of a family
1.1.6.2
sum of a family in FA
5.6.3.6
sum of representations, Hilbert summable, absolutely
1.1.6.9
summable, order
1.7.2.10
summable family
1.1.6.2
5.4.2.1
Subject Index
587
summable in FA
5.6.3.6
support of a function
NT
support of a Radon measure support of x
NT
6.3.3.4
supremum
1.7.2.1
supremum norm surjective
1.1.2.2, 1.1.5.2
NT
symbol, Kronecker's
1.2.2.6
symmetric involutive algebra
2.3.1.26
symmetric involutive algebra, strongly theorem, mean ergodic
5.2.4.3
Theorem of Alaoglu-Bourbaki Theorem of Banach
1.2.8.1
1.3.1.2
Theorem of Banach-Steinhaus Theorem of closed graph
1.4.1.2 1.4.2.19
Theorem of Fre!chet-Riesz Theorem of Gelfand
5.2.5.2
2.2.5.4
Theorem of Gelfand-Mazur
2.2.5.5
Theorem of Hahn-Banach Theorem of Laurent
1.3.3.1
1.3.10.8
Theorem of Lax-Milgram Theorem of Liouville
5.3.1.3 1.3.10.6
Theorem of Krein-Milman
1.3.1.10
Theorem of Krein-Smulian
1.3.7.3
Theorem of Minkowski
1.1.3.4
Theorem of Murray
1.2.5.8
Theorem of Pythagoras
5.2.2.3
Theorem of Radon-Nikodym Theorem of Riesz
4.4.3.15
2.2.5.1
Theorem of Weierstrass-Stone topological cardinality topological zero-divisor
2.2.4.24
topology, norm
1.1.1.2
topology, weak
1.3.6.9
totally ordered set trace
NT
6.1.5.1
trace operator transpose kernel
1.3.5.16
NT
6.1.5.1 3.1.6.5
2.3.1.26
588
transpose of a functor
1.5.2.3
transpose of an operator 1.3.4.1 transpose unital category of s 1.5.2.2 transposition functor of s 1.5.2.2 triangle inequality 1.1.1.2 tridual of a Banach system 1.5.1.9 tridual of a normed space 1.3.6.1 type I W*-algebra 5.6.7.11 u-invariant 3.1.4.4 ultrafilter, free NT ultraweak operator topology 6.3 uniform boundedness, principle of 1.4.1.2 unit 1.5.1.1, 1.5.1.4, 2.1.1.1 unit, approximate 2.2.1.15 unit ball 1.1.1.2 unit of an inner multiplication 1.5.1.1 unital algebra 2.1.1.3 unital algebra, normed 2.2.1.1 unital algebra associated to 2.1.1.8 unital algebra associated to, involutive 2.3.1.9 unital algebra homomorphism 2.1.1.6 unital algebra isomorphism 2.1.1.6 unital algebras, isometric normed 2.2.1.1 unital algebras, isometry of normed 2.2.1.1 unital algebras, isomorphic 2.1.1.6 unital algebras, isomorphic normed 2.2.1.1 unital algebras, isomorphism of normed 2.2.1.1 unital Banach algebra 2.2.1.1 unital unital unital unital
Banach Banach Banach Banach
algebra associated to 2.2.1.4 algebras, isomorphism of 2.2.1.1 category 1.5.1.1 subalgebra generated by 2.2.1.9 unital C*-algebra 4.1.1.1 unital unital unital unital unital
C*-algebra, complex 4.1.1.1 C*-algebra, real 4.1.1.1 C*-algebra, associated to a C*-algebra C*-subalgebra 4.1.1.1 C*-subalgebra generated by 4.1.1.1
4.1.1.13
Subject Index
589
unital E-algebra
2.2.7.1
unital E-algebra, involutive unital E-C*-algebra
2.3.6.1
5.6.1.10
unital E-module
2.2.7.1
unital E-module, involutive unital Gelfand algebra
2.3.6.1
2.4.1.1
unital Gelfand algebra, involutive unital Hilbert E-module
2.4.2.1
5.6.1.4
unital involutive algebra associated to
2.3.1.9
unital involutive Banach algebra associated to
2.3.2.9
unital involutive normed algebra associated to
2.3.2.9
unital left A-module 1.5.1.10 unital normed algebra associated to 2.2.1.4 unital normed algebras, isomorphism of 2.2.1.1 unital representation 5.4.1.1 unital right A-module
1.5.1.10
unital subalgebra 2.1.1.3 unital subalgebra generated by
2.1.1.4
unital subalgebra generated by, involutive unital von Neumann algebra 6.3.4.1 unital W*-subalgebra
4.4.4.5
unital W*-subalgebra generated by unital A-category 1.5.1.14 unital A-module
4.4.4.5
1.5.1.12
unital (A, A)-module unitary
2.3.1.18
1.5.1.12
2.3.1.3
universal representation
5.4.2.3
universal representation, complex upper bound
1.7.2.1
upper section filter upward directed vector, cyclic vector, separating
5.4.2.6
1.1.6.1 1.1.6.1
5.3.2.19, 5.4.1.1 5.3.2.19, 5.4.1.1
vector associated to x', cyclic vector lattice 1.7.2.1
5.4.1.2
vector space 1.1.1.1 vector space, involutive 2.3.1.3 vector spaces, isomorphisms of involutive
2.3.1.3
590
Volterra integral equation 2.2.4.22 von Neumann algebra 6.3.4.1 von Neumann algebra, unital 6.3.4.1 yon Neumann algebra of multiplication operators 6.3.7.2 von Neumann (right) E-module 5.6.3.2 yon Neumann (right) W*-module 5.6.3.2 W*-algebra 4.4.1.1 W*-algebra, complex 4.4.1.1 W*-algebra, homogeneous 5.6.7.8 W*-algebra, predual of a 4.4.1.1, 4.4.4.4 W*-algebra, real 4.4.1.1 W*-algebra, type I 5.6.7.11 W*-algebras, isometry of 4.4.4.5 W*-homomorphism 4.4.4.5 W*-module, von Neumann (right) 5.6.3.2 W*-subalgebra 4.4.4.5 W*-subalgebra, unital 4.4.4.5 W*-subalgebra generated by 4.4.4.5 W*-subalgebra generated by, unital 4.4.4.5 weak operator topology 6.3 weak semi-inner-product right E-module 5.6.1.1 weak topology 1.3.6.9 weakly conjugate exponents 1.2.2.1 word 7.2.1.1 word, empty 7.2.1.1 zero-divisor 2.1.1.1 zero-divisor, topological 2.2.4.24 A-categories, functor of (unital) 1.5.2.1 A-category 1.5.1.14 A-category, quotient 1.5.2.17 A-category, unital 1.5.1.14 A-module 1.5.1.12 A-module, left (right) 1.5.1.10 A-module, quotient 1.5.2.17 A-module, unital 1.5.1.12 A-module, unital left (right) 1.5.1.10 A-modules, functor of left (right) 1.5.2.1 A-subcategory 1.5.2.16
Subject Index
A-submodule (A, A)-module
591
1.5.2.16 1.5.1.12
(A, A)-module, unital
1.5.1.12
#-null set NT a-complete, C-order a-complete order
4.3.2.3 1.7.2.1
a-complete ordered set a-continuous, order
1.7.2.1 1.7.2.3
a-faithful, order 4.2.2.18 a-finite measure space 3.1.6.14 a-Stone space 0-representation
1.7.2.12 5.4.1.1
592
Symbol Index NT means Notation and Terminology la[
4.4.3.5
a*
2.3.1.30
A*
2.3.1.1
A•
5.2.2.1
A
NT
A
7.1.1.7
A
NT 1.3.5.1
N,A o
A c , A c~ , A ~
2.1.1.16
Jr', .A", .A"'
1.5.1.9
1.3.6.9 aa t , a ta 2.2.7.8 AA, ab
2.1.4.23, 7.2.1.1
ax
2.2.7.23
atx"
2.2.7.11
a"x'
1.5.2.8
(a,~)
5.6.3.2
(a, ~, r/)
5.6.3.2
.A/B
1.5.2.17
A + B
1.2.4.1 NT
A\B
NT
AAB
NT
A•
A + z B
1.2.4.1 1.1.2.4 NT
c Co c(T)
1.1.2.3, 2.1.4.3 1.1.2.3, 2.1.4.3 1.1.2.3, 2.1.4.3
co(T)
1.1.2.3, 2.1.4.3
C(T)
1.1.2.4, 2.1.4.4
C(T,E)
1.1.2.8
Symbol Index
Co(T)
593
1.2.2.10, 2.1.4.4
Card
NT
Cg~ Cg~q
7.2.2.1 7.2.2.1
Cg~,~+
7.2.2.1
Cg~(T)
7.2.1.2
Cey(T)
7.2.1.3
Cg~(T)
7.2.1.3
Cg~(p)
7.2.1.2
Cg~,+(p) Cg~(p)
7.2.1.9 7.2.1.3
Cg~(p)
7.2.1.3
Coker
1.2.4.5
dA
1.1.4.1 NT
Det Dim
1.1.2.18
D(k, p, u) Do(k, u)
E' E" E"' g.%
/~
eA et
3.1.6.15
3.1.6.1 1.2.1.3 1.3.6.1 1.3.6.1
2.1.5.1, 2.1.5.7, 2.3.1.38, 2.3.1.40, 5.3.1.8 1.1.2.1
e~ eT
1.1.2.1 1.1.2.1 1.1.2.1
ex
2.2.3.5
Ea(u) Eb(u) Em,n En,,~
ET E (T) E~ E~ E+ E#
3.1.3.18 3.1.3.18 2.1.4.23, 2.3.1.30 2.1.4.24, 2.3.1.31, 5.6.6.1 1.1.2.1 1.1.2.1 1.7.2.3 1.7.2.3 4.4.4.4 1.7.1.1, 4.2.1.1 1.1.1.2
594
E+# 1.7.1.4 E~, E~ 2.2.7.15 r 6.1.2.1 E(x) 2.3.2.15
E(x,1)
2.3.2.15
E--~F U! E-+ F
6.1.7.1 6.1.7.1
P
E -2+ F E _2+ F
1.5.1.1 1.5.1.1
A
E/F
1.2.4.1, 2.1.1.13, 2.3.1.42 4.3.1.7
F(E) 5.5.7.10 f' 1.1.6.24 f 2.3.3.1, 7.1.1.7 9~ A 1.2.6.1 .T(E) 3.1.3.1 .T(E, F) 3.1.3.1 ~I 1.1.6.1 flS NT f(a, .) NT f(., b) NT f(A) NT f(x) NT, 4.1.3.1, 4.1.3.2, 4.3.2.5, 6.3.3.1, 6.3.3.4 f-1 NT -1
f (B)
NT
-1
f (y) NT f "X - + Y NT f " X ~ Y , x ~ T(x)
F[~,t] F[t] F @G {f=g} {f ~: g} {f > a} gof ~T
NT
NT 1.2.5.3 NT NT NT NT, 1.5.2.1 7.1.1.2
NT
Symbol Index
595
IH
2.1.4.17, 2.3.1.46, 4.1.1.31
l~/
5.6.1.6
H
5.6.2.2
I//
5.6.3.2
H
5.6.3.2
Y)A im
1.7.2.3 1.1.1.1, 2.3.1.22
Im
1.2.4.5
Ind u
3.1.3.1
Ind U
3.1.3.21
jE jEF
1.3.6.3, 1.5.2.1
IK
1.5.2.1 1.1.1.1
IK1
2.1.1.3
IK[-], IK[., .] K:(E) K:(E, F)
1.1.1.1 3.1.1.1 3.1.1.1
1CE(G,H) 5.6.5.3 ICE(H) 5.6.5.3 kx 3.1.6.1, 3.1.6.15 k'
3.1.6.5
k* k ~x
3.1.6.5 3.1.6.1, 3.1.6.15
N
k
1.2.3.1
U
k Ker
1.2.3.1 1.2.4.5
s
1.2.1.3, 1.5.1.1, 2.1.4.6
s
1.5.1.1
s s
1.2.1.3 1.6.1.1, 1.6.1.3
Lb f3(E)
1.6.1.13
s176
s f_,,~ f--.E(G,H) f--.E(G,H)
6.1.2.1 6.1.2.1 6.1.2.1 6.1.2.1 5.6.1.7 5.6.2.2
596
f--,E(H) EYE(G, H )
5.6.1.7 6.3.9.12 6.3.9.12
/:~(H)
f~(H)H f--,(H)~:f(g)
6.3 6.3
Z:(H)L,r
6.3
gP 1.1.2.5 g2(I, F) 5.6.4.2 e2(T)
gP(T) g0
5.5.7.1 1.1.2.5 1.1.2.3
g~ goo
1.1.2.3
1.1.2.2, 2.1.4.3
g~(T)
1.1.2.2, 2.1.4.3
gP'q(S, T)
1.2.3.2
eO 'q(S, T) log A/Ib
1.2.3.2 2.2.3.9, 4.2.4.4 1.1.2.26
. ~ F,F
5.6.4.20
./~ F,G
5.6.4.19
IN
NT
INn Nx,
1.1.3.3 2.3.4.1
No
2.3.1.3
./V'F,F ,A/'F,G
N(u) Pr
5.6.4.16 5.6.4.15, 5.6.4.19 6.1.2.1 4.1.2.18 1.1.2.1
~f
1.1.2.1 NT
IR
NT
IR
NT
re Re
1.1.1.1, 2.3.1.3 2.3.1.1
Re E #
2.3.2.1
r(x), rE(x) Sn
2.3.1.3
2.1.3.1
Symbol Index
S(x)
6.3.3.4
8~(f)
7.1.2.3
S~(f)
7.1.1.7
$~(f)
7.1.1.7
s ( ~ ~) ~T
2.4.6.5 7.1.1.2
Supp f
NT
Supp #
NT
tr
6.1.5.1
T
2.4.4.1
u'
1.3.4.1
u'*
6.1.7.1
u"
1.3.6.15
u*
2.3.1.1, 3.2.2.3, 5.3.1.4, 5.6.1.8, 5.6.4.12, 5.6.4.14 6.1.6.1
P
6.1.6.1 o
u
2.1.5.11, 2.3.1.41
u
6.3.9.15
Ilulp IlUlo
6.1.6.1 6.1.6.1
u'w
6.1.7.1
Un
2.3.1.3
Us(t) uT(t)
1.1.1.2 1.1.1.2
vu
5.6.4.12, 5.6.4.14
vu ~
6.1.7.1
X~ ~
NT
xn
2.1.1.1
x~
4.2.4.1, 4.2.4.4
x -n x~
2.1.2.5 2.1.1.3
x -1 x*
1.5.1.6, 2.1.2.4 2.3.1.1, 7.1.1.7 5.5.7.1
_..+
x
7.1.1.7 E
~, ~ v
x
2.4.1.1 2.3.5.1
597
598
[x I x+,x -
4.2.5.1 4.2.2.9
x'+,x '-
4.2.8.13
Xs
7.2.1.3
(x~)~el NT xa 2.2.7.23 X~,A 7.2.2.3 x~,j 7.2.2.3 x'a" 1.5.2.8 x"a' 2.2.7.11 xx' , x' x 1.5.2.5 x~ 5.6.1.4 (x, x') , (x', x) 1.2.1.3 <-[x)y 5.3.2.12 xy 7.2.1.3 x 9y 2.2.2.7, 7.1.1.7 x" q y", x" F- y" 1.5.2.10, 2.2.7.13 x| y 3.1.1.25 (x', ~, r/)
5.6.3.1
{x ] F(x)}
NT
{x e X I F ( x ) } (.,x')y 1.3.3.3 2Z NT z+ A 1.2.4.1 1.1.1.1 [a[ 1.1.1.1 aA 1.2.4.1
NT
(n~) 2.2.3.10 ]a,/3[, ]a,/3], [a,/3[, [a,/3] A
NT
5,t 5(s,t)
1.2.2.6 1.2.2.6
5t O(u) On(u)
I#1 # 9 v,
1.2.7.14 6.1.2.1 6.1.2.1 NT 2.2.2.10
NT
Symbol Index
599
5.5.7.1 ~x
5.6.1.1
~(.[r/)
5.6.5.1
7rA
5.2.1.2
1-I Et
2.1.4.1, 2.3.1.4
tEE
l-I X~
NT
tel
I-I Xn, II xn nE A
nElN
PE
6.1.8.1
PE,F
6.1.8.1
ao(E)
a(E),
a(T) a(x),
2.2.4.33
2.4.1.1
2.4.4.1
aE(X)
2.1.3.1 3.1.3.24
ae(u) ap(u)
3.1.4.1
E x(t)
1.1.2.1
tET q
xn
1.1.6.2
n=p
~-~x~
1.1.6.2
LEI
~-'~(., x'~)y~
1.3.3.3
~EI oo
anxn
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