Number 424
Yasuo Watatani Index for C*-subalgebras
Memoirs of the American Mathematical Society Providence· Rhode Island • USA January 1990· Volume 83 . Number 424 (end of volume) . ISSN 0065-9266
TABLE OF CONTENTS
INTRODIJCTION ... CHAPTER
I.
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1. 7. 1.8. 1.9. 1 10 1.11. CHAPTER
II. 2 2 2 2 3 2.4. 2.5. 2.6. 2 7 2.8. 2.9. 2.10.
1
Purely algebraic preliminaries Assum tions .......................................... . Conditional expectations ............................. . Morita equivalence .................................... Uniqueness of conditional expectations ................ Index under direct sums and tensor products ........... Dual conditional expectations ..........•..•........... Composition of conditional expectations ... ............ Coring structure and separable extensions ... ......... Hattori-Stalling rank and index ............•.......... Generalized Casimir elements ...................... Modular automorphisms................................. Index for C*-algebras Reduced C*-basic construction
Unreduced C*-basjc construct jon Values of index....................................... Index for finite dimensional C*-algebras .............. Index for factors.. . .. . ... . . . ...... . ........ . ...... ... Some constants ..................... . ..... ....... ...... Finite index .......................................... Index, freeness of group actions and covering spaces . . Some other examples . . .. . . . ........ . ................... Radon-Nikodym derivatives .. . ........ . ..... . ....... . ... 2 11 Reduct jon ............ 2.12. Minimizing index ...... . . .. . . ..... . ........... .. . .... . . CHAPTER
III 3 1
3.2. 3.3.
6 6 11 14 16 17 20 21 23 25 27
39
43 52 56 59 63 65 68 74 77 84
87
Transfer Induced and co-jnduced modules . . . . . . . ....... 91 Transfer maps in Tor and Ext ............. ......... .... 93 Transfer maps in the K-theory of C*-algebras .......... l03
REFERENCES. . .•.....•.................. , . , " " " ' , .•. , .. , ............ 114
iii
ABSTRACT In this paper we study the index. denoted by Index E. for conditional expectations E on C* - algebras. which is a generalization of the index for subfactors by Jones and Kosaki. Under a certain condition we show that the spectrum of Index E is contained in {4cos2~/n ; n=3.4.5 •...}V [4. ~1. The notion of the index is related with coring structure. separable extension. generalized Casimir elements. G-Galois basis. covering spaces. modular automorphisms and Nakayama automorphisms on Frobenius algebras. We introduce transfer maps in Tor. Ext and K-theory. We show that the transfer map is related with the multiplication map by Index E up to the evaluation by a trace.
Key Words and Phrases C* -algebra. index. conditional expectation. basic construction. K- theory. transfer. modular automorphism. Galois theory. covering space. coring structure. separable extension. Casimir element.
iv
To my wife Toshiko for everything
INTRODUCTION
Jones' index theory[36j for subfactors of type III is an exciting and epochmaking work. but
also
He createS a new world, not only for operator algebras many
for
unexpected
branches
of
statement(38] concerning the geometric, combinatrial and discrete
of Ill-factors has been developed (16],
Goodman,
nature
Chodaf14l,(15J.
by Bian-Nadal [2J, M.
de la Harpe and Jones [29J,
His
rnathematics[39].
Ocneanu[53],
Pimsner
and
Popa[58] and Wenz'1[76J.
On for
the other hand, Kosaki (44] defined index, denoted by
a
(normal
semifinite faithful) conditional
onto
a subfactor.
He used
Index E,
expectation
arbitrary
factor
the
Connes[21J
and the theory of operator valued weights by
E
spatial
of
an
theory
of
Haagerup[31J.
The case for factors of type I I I with the ergodic theory has been studied by Hamachi and Kosaki [32] ) (33] ) [34] and Loi [45 J and Riai [35 J • In
this
E
expectation transfer
in
Index E. [58J.
note on
we a
shall
investigate
C1'-algebra.
Index E
We shall
establish
a a
conditional link
between
the K-theory of C)':-algebras and the multiplication
Our definition of index is inspired by the
map
Pimsner-Popa
by
basis
Kosaki's formula "Index E "" E-l(l)" -and the Casimir
elements
for
Roughly speaking, index [M:N]
measures
the
semi-simple
Lie algebras.
maximal number of (disjoint) copies of N in M. our
for
Hence it is evident
that
index should coincide with the number' of sheets of a covering
in the case of a commutative C1'-algebra. exactly
Hattori-Stalling
index
definition
tilge.braic
formulation immediately implies that
index
almost
equivalent
Jones index is
Hence we shall start from
algebraic
is
of
rank.
For Ill-factors
to
unify
many
to the union of
space
related
a
purely
notions.
Our
the property of finite two
things:
the
coring
structure (a certain generalization of Hopf algebra) by Sweedler[68]
and
the notion of separable extension. Let
M be
a factor of type III on a Hilbert space
Hand
NC M
a
Received by the editor January 24, 1989. Received in revised form May 1989.
YASUO WATATANI
2
We denote by
subfae tor.
di~(H)
the coupling constant of M on H [48J.
Then Jones index [M:N] is defined by Jones[36] as [M:N]
if N' is finite.
= dimN(H)/dimM(H)
Surprisingly, Jones determined that the set of possible
values of the index [M: NJ for subfactors is exactl, the set
n ~ 3,4,5, ... }
{4cos 2 w/n
Pimsner
and
V [4,w].
only if there exists a Pimsner-Popa basis {ro , ... l (Le. L j mjE(m/~X) and some mor.e condition on orthonormality. (II)
is
l:j mjeNmj* "" 1
calculated
condition
(U)
by
if
shows that the Jones index [M:NJ is finite
Popa[S~J
the
Pimsner-Popa
reminds
us
of
the
=
,ffi
+
n 1
and
satisfying
}C M
x for x0M) In this case the Jones
basis
notion
as
~
[M:NJ
of
E
j
G-Galois
index
*.
The
basis
(cf.
mjm
j
[23J, [46Jand [54]).
be
Kosaki[44]
extended Jones' index theory for arbitrary factors. Let
an
with a subfactor N.
factor
We assume the
existence
(semifinite faithful) conditional expectation E : M-7'N.
of
M
normal
The set of
all
normal semifinite faithful operator-valued weights is denoted by P{M.N). Haagerup peN' ,M').
constructed
an order-reversing bijection
between
P(M,N)
and
Based on the spatial theory by Connes[21]. the above bijection
(denoted by E ~ E -1 ) is given by the equation of spatial derivatives: d(¢oE) Id~ • d$/d(~oE-1),
where ¢ and ware normal semifinite faithful weights on Nand M'.
Kosaki
defined the index. depending on E, by Index E • E- 1 (1).
And
he shows that the set of values of indices is the same set as
obtained. bas~s
Jones
If Index E is finite. then there exists a Pimsner-Popa
{mI' ...
~mn}CM
satisfying (U) in this case too.
type
Furthermore we can
show that for x€:N'. Hence we recover that Index E "" E-I(l) "'" Ej mjmji'. In
our
approach, the Pimsner-Popa basis is generalized to a
guasi-basis {(u1.vl)' ...• (un'vn )} satisfying
(PU) (,i uiE(vix)
=
x
= Ei E(xui)v i
·
notion
of
INDEX FOR C*-SUBALGEBRAS
3
Here the coefficients over N are not necessarily unique, so M~Nn genera1l • 7 We shall define Index E analogously by
Index E
~
E. u.v. 1.
1.
1..
On the other hand, PiIDsner and Pop a shows that if M:::>N are factors of
III
type
and
N'n M ~
~,
then
the
Connes-St¢'rmer
coincides with Jones index in the sense that H(MIN) PiIDsner ,and Popa, Kawakami and Yoshida
after
several
interesting
succeeds in
III
in
formulae.
is
It
relative
= In(M:NJ.
[41J, [42J, [78J
surprising
that
entropy
Following developed
Ocneanu[53J
classifying subfactors of the hyper finite factor M
the case of (M:N]< 4.
But we shall not
discuss
of
these
type
matters
further here. Goodman, de la
Neumann
algebras
Harpe 'and Jones[29j extended the index theory to
with finite dimensional centers.
Their
work,
von
though
very interesting, differs somewhat from ours in its approach. While
this note was being prepared, a preprint[6] by Baillet, Denize·
au and.Havet appeared.
They consider the index of conditional
expectation
of von Neumann algebras and some of their result are similar to ours. In chapter I a purely algebraic definition of index is introduced and several coring
consequences structure,
G-Galois
are
noted. The notion of index
separable extension,
basis,
generalized
modular automorphisms
and
is
related
Casimir
Nakayama
with
elements,
automorphisms on
Frobenius algebras. In
chapter II we consider a conditional expectation E : B-)A
C*-algebra B onto a C*-subalgebra A.
of
a
We introduce the notion of C*-basic
construction C*
. The C*-algebra C* is exactly the algebra of "compact operators" on a Hilbert C*-module E, where 8. is the completion of
B with
the A-valued inner product
construction
of
Jones'
iterated
(xly)
basic
interesting result by Wenzl[tt1] to show that
=
E(x*y).
extensions,
By we
a
similar
can
use an
if Index E is contained in
A, then the spectrum of Index E is included in the set {4cos 2 Tf/n ; n"" 3,4,5, .. )
If any
V
(4,00[.
Index E is not contained in A, then the spectrum of index E can values
natural.
in [1,00[.
But to consider Index E as being in
A
is
take very
Because if the C'lLalgebra B is simple, then Index E is a scalar
YASUO WATATANI
4
and
hence
covering
in
A.
In the case
spaces.
of
commutative
C*-algebras,
Then that Index E is in A tells
that the
consider
conditional
expectation E is given by taking the mean on each fiber. If
C~Lalgebras
the
following
after
B and A are simple, then we can minimize
Riai's idea [35J.
Thus we can define [B:A],
independent of the choice of conditional expectations. pairs
B,:, A of simple
above
set
instead can
C*-alge~ras
of weak closure in
analyze
the
index
equivalence[63]. If the properties
which
is
Many examples
of
are constructed with the index
by considering AF-algebras obtained by
taking
Wenzl's construction [76].
owing to
index
Rieffel' s
work
is finite, then
two
in corrnnon, for example. Hyperfiniteness
index E
on
norm
in
the
closure
Furthermore (strong)
algebras [ZOJ)
we
Morita
share
many
fullness[58],
property T [Z].[ZZ],[40J. similarity problem[491. nuclearity, etc. In chapter III transfer maps are introduced in the theory of Tor, Ext
K.
and
We
multiplication
shall
show
that
the
transfer
map
is
related
to
the
map by Index E up to evaluation by a trace. Suppose under some condition, then Index E is an eigen-value
that of
a
certain integral matrix S which is constructed by using a transfer map. For
example
4cosZ~/5
=
let
e l .eZ.e 3 •... be Jones proj ections $Z, where $ is the golden number. Let
with
T
-1
C*(l,e Z ,e , ... ). Then there exists a conditional expectation 3 E : B-7 A such that E(xely) "" 1:xy for x,yE.A. If so. the above integral and
A
=
matrix KO(B)?i'-
corresponds to the multiplication by Index E
.2 on Z + $Z
c
IR
and Index E is the Perron-Frobenius root of S. This is an extended version of an o.nnouncement [75] in C. R.
Acknowledgement. I am greatly indebted to Prof. Vaughan Jones for many Prof.
suggestions
and new ideas on my research.
Special
thanks
go
Hideki Kosaki for his introducing me his theory and Y. Sekine
improving an inequality.
his to for
I would like to thank Y. Kato for the influence
of his singular methods on research.
I would like to thank Prof. H.
and
INDEX FOR C*-SUBALGEBRAS M.
Choda,
Suzuki,
M.
Enomoto, J. Fujii, M. Fujii, E. Kamei,
Masahiro
Kitamura,
I would
deep gratitude to Prof. Osamu Takenouchi for his
Finally it is my great pleasure
constant encouragement.
Prof.
K.
H. Takehana and T. Yoshida for their discussions.
to express my and
5
Nakamura,
a
pioneer
algebras in Japan, who predicted many
among
researchers
like
guidRnce to
of
H.
thank
operator
important tools, ideas and notions
at an early stage of the development of the research of operator algebras in Japan.
Added in Proof: After submission of this paper, I received (and was informed of) the following interesting preprints on the index:
(HaJ J.-F. Havet, Esperance conditionnelle minimale.
[HO] R. Herman and A. Ocneanu, Index theory and Galois theory for infinite index inclusions of factors. [Hi] F. Riai,. Minimum index for subfactors and entropy. [Jo] P. Jolissaint, Index for pairs of finite von Neumann algebras. [Lo] P. Loi, On the theory of index and type III factors. [Lo] R. Longo, Index of subfactors and statistics of quantum fields. [PP] M. Pimsner and S. Popa, Finite dimensional approximation of pairs of of algebras and obstructions for the index. [Po] S. Popa, Relative dimension, towers of projections and commuting squares of subfactors. [Tsl S. -K. Tsui, Index of faithful normal c'onditional
expectations.
CHAPTER I.
PURELY ALGEBRAIC PRELIMINARIES
1.1. Assumptions.
Let
k
Throughout
be
commuto...tive ring with an
a
identity
element
1.
this chapter, we consider a k-algebra B and a k-subalgebra
of B with a common identity element 1.
A
In this chapter, we consider only
purely algebraic aspects of conditional expectations E : B ~ A and
indices Index E as preliminaries for Chapter II
their
and III.
1.2. Conditional expectations
Definition 1.2.1. A conditional expectation E
is
B-1 A
an
onto
k-linear map satisfying E(ab) for
=
=
aE(b), E(ba)
a E A and b EB.
E(b)a
and E(a)
=
a
But when we consider operator algebras,
we
always
assume that E is positive, that is, E is a projection of norm one studied
J.
by
Tomiyama
[73J
We say E
E(bB)
o
implies b "'" 0, for
then
E is
non-degenerate
E(b*b) "'" 0
implies
inequality
'liE(x*y)1\
for
conditional
b:= 0,
bE B. if
is
non-degenerate If we consider
and only if
for
~ \iE(x*x)111/2I\E(Y*Y)lr/2 •
expectations
in operator
E(Bb) = 0
operator
E is
bE: B, because
if
we
or
algebras,
is.
faithful,
that
have
following
the
See also
algebras.
Umegaki[74j
Throughout
paper, we assume that conditional expectations are non-degenerate
this unless
stated otherwise.
Motivated
by Pimsner and Popa basis of subfactors in (58J. we
shall
introduce a notion of quasi-basis.
Definition
1.2.2.
A finite
family
{(u
l ' v l ), ... , (un' v n )} C B x.. B
called a guasi-basis for E if for x\:-B.
is
INDEX FOR C*-SUBALGEBRAS A
expectation
conditional
E : B-7A is of index-finite
exists a quasi-basis for E, Index E
Remark
Center
B
7
~
if
there
In this case we define the index of E by Li uiv i E: B
=
(1) If E is of index-finite type, then Index E is in fact
and
the
value
Index E does not
depend
on
the
choice
in
of
quasi-bases, cf. Proposition 1.2.8. (2)
Note
that
"coefficients" E(vix),
(or E(xu
i
»,
are
not necessarily
unique.
(3) If B and A 'are C*-algebras, then we can choose a quasi-basis {(ui,v ) i ; i
=
1, ... ,n} with vi = u i
(4) Sometimes advance.
we
But
do
not
the
non - degenera te.
*.
know whether E is
existence In
fact
cf. Lemma 2.1.7.
of a
non-degenerate
quasi-basis
guarantees
or
not
in
E
is
that
if
Similarly E(bB) = 0 implies that b "" O.
(5) We see that G-Galois basis (cf. [23J ,[46J,[54J) is also a quasi-basis for the trace map up to normalization
Example 1.2.3. group
algebras
B
Let G be a group and H a subgroup of G. ~
k[G]
{)"(h) ; h6 H} over k.
and A
~
k[H]
with
bases
{A(g)
Define a conditional expectation E :
E(Lg
(G:HJ
is finite, and then, Index E "" [G:Hl.l. Suppose that
G.
Then
Index E
; g cG} k[G]~
and
k[HJ by
" h
where
finite.
for g E G.
~
Consider the
Then E is of index-finite type if and only n"'" [G:H]
if is
.gn} be a set of (left) coset representatives of H in -I i=1, ... ,n} is a quasi-basis for E and {(A(gi),A(gi »)
Let {gl'"
[G,H]l.
In fact
Li A(gi)E(A(gi-l)(LgeG X(g)A(g»)) -1 "i A(gi)E(L gCG x(g)A(gi g» Li A(gi)E(Lg~GX(gig)A(g» Li A(gi)LhOH x(gih)A(h) "i "heH x(gih)A(gi h )
Lg"G x(g)A(g). Similarly we have that
YASUO WATATANI
8
_1
"i E«'g
-1
)A(hg i
-1
)
"gEG X(g)A(g).
Therefore E is of index-finite type and
Index E Conversely
~i
A(gi)A(gi
{(ui,v i )
quasi-basis
Z~X(g)A(g),
would
.
i
representatives
coset
-1
)
=
nl
~
[G:HJ1.
assume that [G:H] is infinite.
index-finite type.
of
not
=
of
Suppose on the contrary
1, ... ,n} for E . H in G.
Then fiR
Take
SISI\.J,
some
then
ACg ) ~ Ei uiE(vi,\(gS» S This is a contradiction. equation
Example
of G.
.
Let R is
that
there
(g" E G
For
~h'H x(h) A(h) .
a
be
x
c..
5UPP('\(gS»o\lJ(XE:;J gaR.
But
the
would imply that SUpp(,\(gs»cVaeJ gaH.
conditional
a
was
sUPP(u ) i
1, ...• n,
i
1.2.4. Let G be a countable discrete group and H a
exists
is
{g E G ; x g "" O}. There
subgroup C~\(H).
Consider the reduced group C"k-algebras B "" C*r(G) and A ""
There
E
a '" I}
infinite.
supp(x)
define a !lsupport" of x by
exist finite subset JeI such that for
Va~J gaR.
We shall show that
expectation It
will
be
E : B'-7 A shown
index-finite type if and only if [G : H] is finite
such
that
E
that is
of
in Proposition 2.2.16
and its Corollary 2.2.17.
Example over
Let B
M2(~)
be the algebra of two by two
and
Therefore positivity of a conditional expectation E is essential for
the
~ 1.
complex numbers IL and A = (£1.
Define a conditional expectation E
<.10)
E (" t
= ta
+
Take a scalar t with
matrices t ,\0
t
the
1.2.5.
t
: B-) A by
(l-t)d.
Then we see that
is a quasi-basis for E. Index E
If
°
t
•
Thus E t is of index-finite type, and
(l/t + 1/(l-t»I.
<1, then index E
t
~
[4,00).
But in general we have that
{Index EtE.~ ; tEC, t"'O and t""U • n{O}.
9
INDEX FOR C"-SUBALGEBRAS
restriction
of
the values of Index E to 2 {4cos 1f/n ; n"" 3,4, ... }V [4,ooJ.
Remark.
the
If N is a subfactor of a factor M of type III and
canonical
conditional expectation determined by
the
trace.
Jones index [M:NJ coincides with Index E by PilUsner and Papa generally,
faithful
suppose
that
M is an arbitrary factor
and
E :M -) N
Then More
[58J.
E : M..,N
is
normal conditional expectation. Then E is of index-finite
a
type
if and only if Index E is finite in the sense of H. Kosaki [44J, and
the
values of Index E coincide by Proposition 2.5.3.
Followi'fl5 [31J, we define a product (depending on E) on B
Then
(b 1
C3A
B by
for bl'bZ,b3,b4~B. We also consider a
B-bimodule
structure on B ®A B by
x(b1@bZ)Y
(xb1)@(bzY)
for x,y,b1,bZeB.
Below, we shall use a B-bimodule homomorphism F : B Q9 B -7 B defined A
F(x0y) • xy
by
for x,YEO B.
Proposition
1.2.6. Let E :
{(u ' v ) , ... , (un' v n )} l l
C.
B X B is
B~A
a
be a conditional
quasi-basis
expectation.
for E
if
and
Then
only
if
E·L u.L (B)v.L is an identity of B@.'AB.
Proof. Then
and
('i uil1Jvi)(x@y)
' i uiE(vix)® y
x®y
(x®y) ('i u i v i )
' i x@ECyui)vi
x®y
for x ,y IS: B.
an identity of B ®A B. ' i u i ®vi is
Thus
Conversely,
suppose that ' i ui@v i is an identity of
B®A B.
Then
for x in B we have that x@l "" Ei (uit$vi)(x~l) "'" £i u i E(v i x)G91. Hence
X""
F(x@l)
Let
F(E uiE(vix)®l) "" Ei uiE(ViX).
Similarly
x.
Q.E.D.
I be an identity of B 65;)A,I3. Then the
above
we
have
proposition
YASUO WATATANI
10
implies
,
This is similar
F(I) .
for
E : M-?'N of arbitrary factors M::>N -1 ,W-J>W. B~B -? B is replaced by E
Index E
F
Index E "
that
Corollary 1.2.7.
1.2.S.
index-finite type.
the
definition
by Kosaki
BG9A B has
Let E : B-7A be a
of
where
[44] ,
Let E : B-?A be a conditional expectation.
is of index-finite type if and only if
Proposition
to
Then
E
an identity.
conditional
expectation
of
Then we have the following:
(1) Index E does not depend on the choice of quasi-bases for E. (2) Index E is in Center B.
Proof. (1)
Let
quasi-bases for E. L:i ui~vi = E j
{(ui,v ) ; i"" 1 •. ,n}
and
i
{(Sj,t ) j
; j
1, . , m}
be
Then by Proposition 1.2.6. SjG9t
"" (the identity element of B@A B).
j
Therefore £:i uiv i "" F(E i ui®v i ) "" FO:j s/~)tj) "" l:j Sjt j . (2) We have that, forbG-B, ~
b(Index E)
b(6i uiv i )
~i,j UjE(Vjbui)v i = 6
6 i buiv i
E. J
Q.E.D.
E)b. j UjV j b "" (Index
Proposition
C (A) B
1.2.9.
Let E: ~
B-;.,A be
a
conditional
expectation
type and {(ui,v ) ; i ; 1, ... ,n} be a quasi-basis for i
index-finite Put
=
{bE:B;"ba=ab
forallaeA}.
For
XE;CB(A),
Then H is a well-defined map from C (A) B
and
does
not
depend
on the choice
of
quasi-bases.
If
let to
of
E. H(x)
Center B
Index E
is
invertible in B, then H is surjective. Proof.
For x €:.C (A) and bE- B, we have that B bH(x) "" 6 i buixv i ~ Ei 6 j UjE(vjbui)xV i Ej ujx 6i E(vjbUi)V i
Hence
H(x)
is
in Center B.
quasi-basis for E.
Let
"" I.: i uixvjb {(Sj ,tj)
0::
H(x)b.
; j '" 1, ... ,m}
Then for x f!:' CB (A) we have that
Ej Sjxt j "" 6jO':i uiE(visj)xt j ) Ei 6 j uixE(Visj)t j = 6 i
uixv i = H(x).
be
another
INDEX FOR G*-SUBALGEBRAS
that
Suppose
d • c(Index E)-lE- Center B. H(d)
Hence H
,
Luidv i
C (A) B
Corollary
assume G (x) E
.
~
d( Index E)
the same situation
In
Index E is invertible.
(Index E) -lb. l
UiXV i
put
. c.
Center B is a surjection.
1.2.10.
that
c E Center B,
any
Then d is in GB(A) and
dLi uiv i
=
For
is invertible in B.
Index E
11
·
Define GE
Q.E.D.
of
,
Proposition
GB(A)
-7
1. 2.9. ,
Center B
by
Then G is a conditional expectation. E
1.3. Morita equivalence Let Rand S be a ring.
We shall use the following usual
notations
describing the modules.
RM means M is a left R-module. MS means M is a right S-module. RMS means M is a left R- right S-bimodule. Let
A C B
expectation.
be
a pair of k-algebras and E : B -7 A
be
a
conditional
We have the canonical "left and right multiplications"
such that (A(b»(x) "" bx and (p(b»(x) "'" xb for b,XICB. We use e when we consider E as an element in Endk(B).
It is easy
to
see that the following hold, which is originally in Jones [36J.
Proposition 1.3.1. Let E : B-7 A be a cQnditional expectation. Then we have the following: For bE: B, (1) eA(b)e • A(E(b»e (2) eHb) • A(b)e
i f and only if b EA
(3) A7a I--YA(a)e&Endk(B) is an injection.
The
above
proposition 1.3.1. shows that we can define
a
k-algebra
homomorphism $ : B GA B ...-?Endk(B) by $(x~y) = ,\(x)e;\'(y) for x,y cB. example the multiplicativity of ¢ is seen by $(xE(yz)lilw) • A(xE(yz»eA(w)
,\(x)e,\(yz)e,\(w)
(Hx)eA(Y» ('(z)eA(w»)· ¢(x0Y)$(z@w).
For
YASUO WATATANI
12 We
also
that ¢ is also a B-bimodule homomorphism
note
if
we
give
a
B-bimodule structure on Endk(B) by
b·£·c
\(b)f\(c)
=
for b,cGB and fE:Endk(B).
for
f(ba) = f(b)a
a
Eucl (B ) is exactly the commutant of peA) in Endk(B). A A
Then
bGB}.
It .is easy to
see
that $ (B @A B) C End A (B A )·
Definition
1.3.2. In the above situation, the algebra
O.(x)eA(y) ; x,yIE;B}
construction by E.
generated
in En<1.c(B) is denoted by and called
Thus is exactly the image of
by
the basic
~.
The following proposition was shown to hold in the case of factors of type
III by Jones [361.
Proposition index-finite
1.3.3.
If
a conditional expectation
E : B -7 A
type, then the above map $ gives an isomorphism
onto EndA(BA) as k-algebras and B-bimodules.
is
of
B~ B
of
In particular we have
that
Moreover the inverse $-1 is given by
" EndA (B ) • A $-1(f)
:=
I.:
£(ui)@v
i
for f " End (B ),
i
A
A
where {(ui,v i ) ; i = 1, ...• n} is a quasi-basis for E.
Proof.
Define
for
All we have to show is that IjJ
for x,y,mGBind fE-EndA(B A ), ~$(x®y) " "i $(x@y)(ui)lwv i " "i (A(x)eA(y)(ui))@v i
L:i
XE(yui)~vi
=:
x ®(1: i E(YUi)V i ) = x@Y.
and ($¢(f))(m) = ($("i f(ui)@vi»)(m) = "i A(f(ui))eA(vi)(m) 1:
i f(ui)E(Viffi) = f(L: i uiE(viffi»
Remark.
The
above
decomposition f = 1:
Proposition
i
proposition
= f(m).
says that
Q.E.D.
any
fE- End (B ) A A
has
a
A(f(ui»eA(v ). i
1.3.4.
Let E
B~A
be a
conditional
expectation
of
INDEX FOR C*-SUBALGEBRAS
13
index-finite type, then we have the following: Left
(1)
(and
right) A-module B is a progenerator,
i.e.,
a
finitely
generated projective generator. (2) A and B ~ B (~
Proof.
fi
Define
system.
(1)
Let
{(ui,V )
i
~HomA (BA,AA)
This
A-module.
are Morita equivalent.
means
; i
"" l, ...• u} be a quasi-basis
by fi (x) ::::: E(vix) for x E: B.
that B is a finitely
Then
generated
the
Since A itself is a generator and E
E.
identity
projective
Als,o the identity l.:i E(xui)v i "" x shows that B is a
generated left A-module.
for
right
finitely B~
A
is
a A-bimodule onto homomorphism, B is a generator over A.
By
(2) Since BA is a progenerator, A and EndA(BA) are Morita equivalent. Proposition 1.3,3., EndA(B A ) and
Corollary
1.3.5.
index-finite type.
Let
BG9A
E : B-..yA be
a
conditional B~
If A is simple, then B &A
We can also consider
Q.E.D.
B are isomorphic.
expectation
of
is also simple.
a A-B-bimodu1e structure on HomA(BA,AA) defined
by, for fEHoffiA(BA,AA), aG-A and b,XEB,
(af)(x)
~
af(x)
and
(fb)(x)
f(bx) .
,We also define a B-A-bimodule structure on
(bO (x) for f
~
f(xb)
and
(fa)(x)
A(AB, AA) by
HOffi
f(x)a
HomA(AB'AA), a '--A and b,xEB.
The following is a kind of Radon-Nikodym theorem.
Proposition index-finite
1.3.6.
Let
t;,: I3-}A be
a
conditional
expectation
type and {(ui,v ) ; i"" 1, ... ,n} be a quasi-basis for i
of E.
Then we have the following: (1)
Define Ebf.HoffiA(BA,AA) by Eb(x) == E(bx) for b,xE;B.
A-B-bimodule isomorphism $ : B
~HomA(BA.AA)
Then we have
such that 4(b)
==
an
Eb , and the
inverse $-1 is given by $-l(f) == ~i fCui)v . i (2)
Define bE E:-Hom(AB'AA) by bE(x) "'" E(xb) for b,xG.-B.
Then we
have
B-A-bimodule isomorphism Jl : B -7Hom(AB'AA) such that \1(b) "" bE, and
a the
l4
YASUO WATATANI
inverse ~-l is given by V-I (f)
Proof.
By
symmetry,
a
we shall prove only
(l) .
aEA
For
and
b,c,XE:B we have that
Hence
E(abex)
~(abc)(x)
Eabc(x)
a¢(b)(cx)
(a¢(b)c) (x).
=
aE(bcx)
is an A-E--bimodule homomorphism.
Define W ,
= Ei f(ui)v · A (BA,AA) by W(O i
HOffi
for b,x€; B and f
=
=
W(E ) b
Then I/J
¢
-l
.
fact
In
we have that
Zi E(bui)v i
=
b
«W)(O) (x) = (¢("i f(ui)v i )) (x) = B( (Zi f(ui)vi)x) Q.E.D.
and
1.4. Uniqueness of conditional expectations Let
E : B~ A
a
be
conditional
expectation
and
CB(A)
be
the
centralizer
h
is
in
invertible
= {be. E
, ab
CE(A) ,
then h is invertible in B if
in CE(A) .
ba for all a e- A} • and
Suppose that h is in CE(A) and E(h)
only
if
= 1.
h
is
Then
hE
Eh : B -7 A are also conditional expectations, although they are not necessarily non-degenerate. Conversely we shall show that any other and
conditional expectation has this form.
Propositionl.4.l. index-finite another
(not
F : B --::; A. (1)
type
with
Let
E a
necessarily
:B~
A be a
quasi-basis
conditional {(ui,v i ) ; i
non-degenerate)
expectation =
1, ... n}.
conditional
of Take
expectation
Then we have the following:
There exists a unique hE, CE(A) with E(h)
1 such that F
Eh .
More
E F(ui)v i , (2) F is of index-finite type if and only if the above h is invertible.
precisely h
(3)
Suppose
{(ui,h
-l
~
that F is of index-finite type.
vi) ; i"" 1,,,.,n} and Index F "" Ei uih
Then F has -l
vi'
a
quasi-basis
INDEX FOR C*-SUBALGEBRAS
15
Proposition 1.2.9.
Proof.
Since F is in HomA (BA,AA)' there exists a unique
F "" Eh by Proposition 1.3.6, where h "" Ei F(ui)v i ,
that
hE B
such
Since for
aEA
and x E:.- B E(hax)
ha
Eh(",-Xl = F(ax) = aF(x) = aE(hx) = E(ahx),
= ah because E is non-degenerate.
that E(h)
Hence h is in CBCA).
= F(l) = 1.
Suppose
that F is of index-finite type.
exists k in CBCA) such that E
F , k
E
is
>::i uiF(h
for x
E(hkx)
non-degenerate,
invertible.
By the same argument
Similarly
hk = 1.
kh
Conversely suppose that h is invertible. -1 -1 1: i lii E(hh
vix)
there
Then
E(x) = Fk(x) = E(kx)
Since
It is trivial
vix)
&i uiE(vix)
=
E: ~
B.
1.
Thus
h
is
is
of
expectation
of
Then x
and -1
Ei F(xui)h vi Hence {(ui,h -1 Vi)
=
i = l , ... ,n}
is
aquasi-basisforFand
F
Now it is clear that Index F "" Li uih -1 vi
index-finite type. and HF(x)
h-1hx ::: x.
HE (xh- 1 )
Q.E.D.
By the symmetric argument we have the following.
1.4.2.
Proposition index-finite another
type
(not
Let E : B -7 A be a
with
a
conditional
quasi-basis{(ui,v ) ; i""l, ... ,n}. i
necessarily
non-degenerate)
Take
conditional
expectation
1 such that F
More
F : B ~ A. Then we have the fo llowing. 0) There exists a unique h G:CB(A) with E(h) precisely h
=
Ei uiF(V i )·
(2) F is of index-finite type if and only if the above h is invertible. (3)
Suppose
(.euih
-1
that F is of index-finite type.
,vi); i
=
1, ... ,n} and Index F
Moreover HF(x) = HE(h-lx)
Finally
we
shall
:=
Ei uih
Then F has -1
a
quasi-basis
vi'
for xE: CE(A).
give
a
criterion
for
the
uniqueness
of
the
16
YASUO WATATANI
conditional expectation.
Corollary index- fini te
type.
another
Take
centralizer CB(A)C A, then F
Proof.
By
such that F
~
conditional
Q..
expectation
expectation
F : B -) A.
Since. h is in CB(A)C A,
E(hx)
~
hE(x)
If
of the
E.
~
Proposition 1..4.1. there exists hE:.CB(A)
Eh'
F(x)
E : B-'7 A be.
Let
1.4.3.
~
E(h)E(x)
~
for
XE;
with
E(h)
1
B
Q.E.D.
E(x).
1.5. Index under direct sums and tensor products Let E.:B. -"7A. be conditional expectations for i 1.
1,2.
1.
1.
Consider the
direct sum E 1 (j)E 2 ' B1@B2--7A1$A2 defined by (E $E )(b (j)b ) ~ E1 (b 1 )@E 2 (b ) 1 Z 1 Z Z
for bieBi'
Then EItt> E Z is a conditional expectation.
1.5.1.
Proposition
index-finite type. El~E2
LetE i
: Bi -7 Ai be cond~tional expectations
of
Then the. direct sum
: Bl~B2--7Al@A2 is also of index-finite type and Index(E 1G7E ) ~ (Index E )$(Index E 2 )1i- B <:l)B ' 1 Z 1 Z
Proof
Let
quasi-bases
{(ui,v i )
for
{(uiGJO .v 6'O) i
E1 ;
i
and
;
i EZ '
l, ... ,n}
and
Then
{(Sj.t j )
E1 $E Z
1 •... ,n) Ij { (06)s j ,0$1: j )
has j
j
;
a
~
l, ... ,m}
be
quasi-basis
=1, .. ; ,m}.
Therefore El ~ E2 is of index-finite type and IndexCEi;)EZ) ~ (EiU v i $ O)-\- (O$'jsjt j ) ~ (Index E1)~(Inde" E Z )'
Q.E.D.
Next we consider the tensor product E @E 1 2 by
Notice that Al (i\ A2 may not be a subalgebra of Bl we
~
hence
we
cg\
B2 in general.
that the canonical map Al ~ A2-!1 B1 ~ B2 is may
consider
A1 GS)k A2 as
a
subalgebra
of
injective
Bl~ B2 .
So and Then
17
INDEX FOR C*-SUBALGEBRAS
B1 ~ B2 -)A
1
65k
Proposition 1.5.2.
index-finite
injective. El <0 E
type.
LetE i
: Bi
~Ai
be a conditional expectation
Assume that the canonical map Ai
4
AZ -) B1
Gk
B2
of
is
Then the tensor product
: B1 €kBz -liAl \?kA
Z
A turns out to be a conditional expectation. Z
Z
is also a conditional expectation of index-finite
type and Index(E @E ) = (Index E )Q9(Index E ).
1
Proof.
2
2
1
Let
I, .. ,m}
i
quasi-bases for El and E . 2
be
Then E @E 2 has a quasi-basis 1 "" l, .. ,n and j
{(ui@Sj,Vi@t j ) ; i
"" l •..• m}.
Therefore E1(g) E Z is of index-finite type and Index(E 1 @E 2 ) = "i,j
(ui@Sj)(vi0tj)
(L i u i v i )Q9(L j Sjtj) = (Index E1 )0(Index E 2 ). The existence of a quasi-basis guarantees that E1@ EZ is non-degenerate. Q.E.D.
1.6. Dual conditional expectations
Let
E
be
Proposition
a
1.3.3.
conditional expectation
of
index-finite
We shall sometimes identify
bEB
with l\(b)EEndA(BA). In this way, we can regard B as a of with the same unit 1.
Proposition 1.6.1. type.
E:
Let
B~
A be a
~B
subalgebra
expectation
with
Then
of
there
E-(A(x)eA(Y»
A(x)A«Index E)-l)A(Y) = A«Index E)-l xy )
Proof. F
conditional
Assume that Index E is invertible in B.
exists a unique conditional expectation E- : =
Then
shows
isomorphism as k-a1gebras and B-B-bimodu1es.
index-finite
type.
i
Let {(ui,v i )
E~
: -0B by E-(f)
(index
= =
E)
AF~-l(f)
-1
Define
Since Index E is in Center B,
Moreover
ui®v i ) Define
1, ... ,n} be a quasi-basis for E.
(Index E) -I xy .
BOA B-c>B by F(x@y)
F is a B-B-bimodu1e map.
=
O':i uiv i ) "'" 1. for fEo
Then
YASUO WATATANI
18
E-(A(x)eA(Y» E~
~ A(F(x@y»
is also a B-B-bimodule map and
=
A«Index E)-lxy)
=1
E~(l)
by Proposition 1.3.3.
The only matter that is non-trivial is to show that Take f "" I:k A(Xk)€A(Yk) in .
for x,yEB
Assume that
E~
is non-degenerate.
E~«B,e>f)
"" O.
Then
for
z If:: B
o
E-«A(l)eA(z»f)
= E-(E k A(E(zxk»eA(Yk»
A«Index E)-l)E
A(E(zxk)Yk)'
k
Hence l:k E(ZXk)Yk "" 0 for all zE-B.
Hf) (w)
Thus f
~
Lk (E i UiE(vi"k) )E(Ykw ) l:i u i I:k E(vixk)E(Ykw) = I: i uiE(E k E(vixk)Ykw) = O. O. Similarly E-(f
Remark.
It
"k "kE(Yk w )
Then for all w6B
is
=
necessary for the existence of E-
that
Index E
is
invertible, see Proposition 1.6.3.
Definition
1. 6.2.
We
the
call
above
conditional
expectation
E- : -) B the dual conditional expectation of E : B--7 A.
For any conditional expectation G : -)B,
such that G(\(x)e\(y» In
property.
"'" \(x)q\(y)
partiGular
G(e)
for x,y We
q.
~B.
there must exist qE:B
since G has
shall
single
conditional expectation E- among them by the property of q
Proposition index-finite q E.
=
1.6.3.
type.
Let E : B ~A be a
B-B-bimodule out =
the
dual
G(e).
conditional' expectation
Let G : --) B be a conditional expectation.
of If
G(e) is in Center B. then G is the dual conditional expectation E- of Furthermore, when this is the case, Index E is invertible.
Proof.
Let {(ui,v i ) ; i"'" 1, ... ,n} be a quasi-basis for E. Then A(q)A(Index E) ~ A(q)E i A(uiv i )
G(E i A(ui)eA(v i ») = G(l) = 1. Therefore Index E is invertible and q
=
(Index E)-I.
coincides with the dual conditional expectation E-.
This shows that
Q.E.D.
G
INDEX FOR C*-SUBALGEBRAS Corollary index-finite
1.6.4. type.
Let
E: B -? A be a
19
conditional
expectation
If B is commutative, then a conditional
of
expectation
G : ---:; B is uniquely determined if it exists.
1.6.5.
Lemma
Let
E ,
B-""7A
be
a
conditional
expectation
of
index-finite type and E-be the dual conditional expectation of E. Then -1 E-Cf) Ei ACCInde" E) fCui)v i ) for f~EndACBA)' where {(ui,v i )
1, ... ,n} is a quasi-basis fqr E.
=
By Proposition 1.3.3., f"" Ei A(f(ui)e>..(v i ). Hence -1 E-CE i ACfCui»eACv i ) ~ "i ACCIndex E) (fCui)v i )·
Proof. ~
E-Cf)
i
Q.E.D.
We shall identify A(E) with B.
Proposition
index-finite
type
conditional
Let E ; B ~ A be a
1.6.6.
for
which
expectation
E-
Index E is ~
i
i = uie(Index E) and qusi-basis for E- and
ti
8
=
conditional
invertible.
expectation
Then
the
B is also of index-finite
of
dual
type.
l, ... ,n} be a quasi-basis for E.
Put
= ev i ·
Index EMoreover,
if Index E is in A, then Index E-
(Center A)
f1
in
(Center B) (\ (Center
For X, y E- B, we have that
Proof. ~i
Index E and Index E is
siE-(tixey)
6 i uie(Index E)E-(evixey)
=
6 i uie(Index E)E-(E(vix)ey) Li uie(Index E) (Index E) Li uieE(vix)y Similarly {(si,t i )
;
-1
E(vix)y
xey. "i uiE(vix)ey that Li E-(xeysi)ti xey.
~
we
have
i
1, ... )n}
~
~
is
a
quasi-basis
for
This
E-.
shows Hence
that
E-
index-finite type and Index E-
= [.i siti
Li uie(Index E)ev i = [,i uiE(Index E)ev i · Moreover if Index E is in A, then Index E is in =
is
of
20
YASUO WATATANI
(Center B) (\ (Center A).
Therefore
E~= ~i
Index
uiE(Index E)ev i (Index E)L i uiev i = Index E.
Hence Index E is in (Center A) (\ (Center B) () (Center
Remark.
In
general,
Index E is not in A and
Q.E.D.
.
Index E-
does
not
coincide with Index E.
Example expectation
1.6.7.
B --, A
E
2.4.2, Index E
=~.
~(\)C
B·
by
(3,3/2)
is in Center M2
1.7.
Let
A'~'
Define
a
conditional
Then
by
Proposition
E(x,y) = (x + 2y)(3.
and it is not in A. Since ':d M , 2
Therefore Index
E-~
Index
E~
Index E.
Composition of conditional expectations Consider k-algebras C c B C A
Proposition expectations also
and
a
Index E
1.7.1.
Let E :B-) A
of index-finite type.
conditional is
with a common unit 1.
and
F,.; C -7 B
be
Then the composition
expectation of index-finite
conditional
E~F
type.
More
in Center C (in particular in the scalars),
then
C -7 A
is
over
if
Index
is
multiplicative, that is, Index (EQF)
Proof.
=
(Index E) (Index F).
Let
{(ui,v i )
quasi-bases
for
E
{(SjUi,vit j )
;
and
i
and
;
i
=
1, .. ,n}
F respectively.
. 1, .. ,n and j . 1, .. ,m}.
and
(Sj,t j )
Then
E'F
Ei,j SjUiE(viFCtjc»
EjSjCE i uiE(viF(tjc»
Zj sjF(tjc) •
similarly
E. . (EoF)(cs.u.)v.t. "'" c. 1.,
J
J
~
J
1.
C
1, ..
j
has
In fact for
Ei,j SjUi(EQF) (vitjC)
;
a
,m}
quasi-basis
&C we have that
C
Thus EoF is
of
index-finite
type and we have that Index{E"F) Therefore if Index E is in Center C, then Index(E F)
(Index E)(Zj Sjtj)
=
be
(Index E) (Index F).
Q.E.D.
INDEX FOR C*-SUBALGEBRAS
Proposition
1. 7.2,
E , B-?A
Let
and
21
F : C-:iI B
be
conditional
If EoF is of index-finite type, Then E is of
expectations.
index-finite
type. Proof.
Let {(Ui,v ) i
((F(u ) ,F(v i » i
i
; i
"'" l,o .. ,n} be a quasi-basis for
"" 1, ... ,n}
is a quasi-basis for E.
EolF.
Then
In fact for
bE B
we have that "i F(ui)E(F(vi)b) ~ ~i F(ui)E(F(vib»
Similarly 6 i E(bF(ui»F(v i )
=
~ F(b)~
F(LiuiE F(vib»
b
Q.E.D.
b.
1.8. Coring structure and separable extensions.
E : B -7 A
Let
Then
B
has
Furthermore extension
of
"compatible" E : B
~
be a conditional expectation of index-finite
a
A-coring
if
Index E
structure is
in
the
invertible in B.
sense
of
then
B
Sweedler is
a
[68J.
separable
Conversely if B is a separable extension of A
A.
type.
with
a
A-coring structure, then there is a conditional expectation
A of index-finite type.
Definition 1.8.1. (Sweedler A-coring
if
M
is
[b~])
Let A be a ring. LI: M -1 M @A M
an A-bimodule,
Then (M,LI,s) is a
and
e.:
: M -? A
are
A-bimodule maps satisfying the following three commutative diagrams:
!!.
M
1\ fe£
lHJ~
!!.l M~ M
M~M@AM
1 M &iA M
-:--:c7 M~ M~ A01
M
6 is called a comultiElication
M
and
M-L,M
~
~
M
1@1 M
€
is called a counit.
Let E : B-3 A be a conditional expectation of index-finite type. {(ui~vi)
6.(b) = 6 i
; i = l, ...• n} be a quasi-basis for E. bUi~vi'
Let
Define LI; B-)B G)A B by
YASUO WATATANI
22
1.8.2.
~
We have the following:
(1) For bf:B, ~i bui@v i
and
"" Zi ui@vib
(2) 6 is a B-bimodule homomorphism.
Proof.
(1)
For b {; B 1: i bui@v i == 2: i U: j
UjE(vjbu i »@ vi
uj&l.: i E(vjbui)V i "'" t.:j uj@Vjb.
Ej
(2) For x,Y,bE:B
Q.E.D.
M(b)y.
Proposition 1.8.3.
Under the above setting, (B,lI,E) is a A-coring.
Proof. For bEE, using Lemma 1.8.2.(1), we have that (I@/I)Il(b) = (I"'/I)(L bu.Gi)v.) = L
bu. &E.v.u.Gilv.)
1.1.J1.JJ
1.1.1.
6j I: i bUi~ViUj@Vj "" Lj Ei bUjlli®Vi®Vj (II®I)(E
j
bUjiJ)v ) = (II@I)Il(b). j
We shall show that E is a counit. (I@E)6(b)_ = (I@E)(E.1. bu.0v.) 1. 1. (I@E)(E i ui@vib) = ' i uiE(vi)b = b.
Similarly we have that (E<8I)6(b) "" L:i E(bui)v
1. 8.4. (cf. [57 J ) .
Definition
Define
k-subalgebra. "(x@y)
Then
"" xy.
B-bimodule
is a
homomorphism
Proposition
B-bimodule
a B
1.8.5.
Let
B
be
Let
k-algebra
a
,
homo~orphism
and
B
A its
B -; B
GilA " seearable extension of A i f there exists
B-"l B~ B such that
4>
Q.E.D.
b.
i
E
B-1 A be
of
""'
by
a
idB·
index-finite
type.
If
Index E is invertible in B, then B is a separable extension of A.
E.E.29.!. a
Let {(ui,v i ) ; i"" 1, ... ,n} be a quasi-basis for E.
B-bimodule
E)
~$(b)
map
by
Lemma 1.8.2. and Index E
-Ill is a B-bimodule homomorphism.
= Ei
~«Index
"" E) -1 bUi~Vi)
~
is
in
Center E,
For b b B,
(Index E) -1 btiuiv i
Define a
b.
$
INDEX FOR C*-SUBALGEB,RAS
23
,,
Hence B is a separable extension of A.
Q.E.D.
Conversely we shall show that If B is a separable extension of A
and
B has a compatible A-coring structure , then A has a finite index in B in a certain sense.
Proposition 1.8.6.
Assume that B is a separable extension of A hlk\i,,,~
a B-bimodule homomorphism ¢ : B-7B~ B with lJ¢ "'" idB' Then
A-coring. quasi-basis
E:
is of "index-finite type"
(although
Let (B,<jl,e:) be
in the sense that
~
A of index-finite type with e:(b)
=
has
£
a
if
Moreover
E(l) "" 1 does not necessary hold).
£(1) is in Center B and invertible in A, then there exists a expectation E : B
a
conditional
(Index E)E(b)
for
b "E.
Put
.(ll
Li 8 i
G9
t
i
.
Since ¢ is a
B-bimodule
homomor-
phism, "i bS i
Since
€
is a cauuit, for b b
~
type ll •
A.
oS.
B.
B
(IQE)$(b)" (l@E)("i bs i 0 t ) i
(I@c)O::i si Similarly
for b
• (Db
".(b)
b:::::
<e: ®
we
can
tib) "" L:i si£(tib).
£(bsi)t , Hence £ is of "index-finite i Furthermore suppose that E(l) is in Center B and invertible in
Then
E(b) "" E-(l)-l e (b).
i::::: 1, ... ,n} Since
"" I.:
I)¢(b)
define Put
i
a
conditional
expectation
E : B ---7 A
by
u
si and vi"'" e(l)t i · Then {(Ui'V. i ) i is a quasi-basis for E.Hence E is of index-finite type. implies
that Q.E.D.
Index E
1.9. Hattori-Stalling rank and index
We shall show that the Jones index for subfactors of type III can identified
with the Hattori-Stalling rank (cf., [7]).
with an identity and [A,A} its commutator.
We denote
be
Let A be a ring the abelian
group
YASUQ WATATANI
24
by
A/lA,AI
finitely
T(A)
generated projective left A-module.
a
Then p*
~
HOffiA(M,A)
is
a
Define a map a : p* Q
P-7 A by a.(f,x) "" f(x) for A Then the trace Tr (f) of an endomorphism f E Haffi (P ,F) is A A
isomorphism.
f E: P*,
be
(x,y E P, f
e(f®x)(y) = f(y)x an
P
Then the mapping e : P*®A P-7 HOffiA(P,P) defined by
right A-module.
is
Let
the natural map A-7Af[A,AI by q.
and
x E- P.
defined by
que
Definition 1.9.1.
-1 (f).
We define the Hattori-Stalling rank RA(P} of P by
RA (P) = Tr (id p ) ET(A). A
Let
N be
a
factor of type III and, be the
unique
trace
of
N.
(Warning: the trace T is not the trace defined in the above of Definition
1.9.1,
although we use the same word.
and de 1a Harpe [28] shows that ,(x) of
commutators
,* :
T(N):=
of N
N/[N~NJ41[
Proposition of
M
RN(M)
for x 6 N.
Do not confuse them.)
=
Hence we can identify T(N)
Then the
ro
(:
by
for xE:N.
sub factor
Hattori-Stalling
M as a finitely generated projective N-module
of
with
Let M be a factor of type III and N a
such that Jones index [M:Nl <
Fack
0 if and only if x is a finite sum
such that T*(q(X»:= ,(x)
1.9.2.
Then
rank
coincides
with
N be the canonical conditional expectation
with
Jones index [M:N] through the above identification T*.
Proof. respect for
E,
bimodule
Let E :
M~
to the trace T. (cf. Proposition
2.5.1.).
quasi-basis
By Proposition 1.3.6., we have a
isomorphism \l : M..J.jl:M* = HomN(M,N) defined
for b,XE::M.
by
\l(b)
l:i E(xui)u i * for T*(RN(M))
T*(q(2:
=
"(b)(x)z
XE<
= ,*(q(a(E
i
M,
=
i
)0
by
E(xb)z.
e-l(i~)
"(u
M-N
(x) "" E(xb)
Then the isomorphism e : Mi~
Since x
; i = 1,2,3, ... } be a
Let {(ui,U{)
= l.i iJ(ui)@u *.
u *)))
i
E(uii,U i ») "" ,(l:i E(uiu i *» i
i
Therefore
INDEX FOR C*-SUBALGEBRAS
25 Q.E.D.
More
generally
Index E
and Hattori-Stalling rank
are
related
as
follows:
Proposition k-algebras.
1.9.3.
Let
Let
k
be
commutative
ring
E:B-"7 A be a conditional expectation
type.
Assume
"[(xy)
T(YX) for x,yE-B and that tE =
the
a
that
canonical map.
there
exists a k-linear map
T
Let TIA*
T.
of B~
:
and
index-finite
k
T •
B:>A be
such
that
A/[A,AJ-> k be
Then the Hattori-Stalling rank and Index E
coincide
up to the evaluation by t, i.e.,
Proof. Just follow the proof of Proposition 1.8.2.
Q.E.D.
1.10. Generalized Casimir elements The
index
aim
of this section is to unify the two notions of
for factors of type III and the Casimir elements
the
of
Jones!
semi-simple
Lie algebras. Let
L
be
a Lie algebra over a field k and U(L)
algebra
envelopin~
invariant form (,). cL
of If {e
L. i
Assume
that
L
be
universal
possesses a. non-degenerate
iEI}CL is a basis in Land
is
a dual basis, i.e., (ei',e j ) "" 0ij (i,j &1), element c"'" l:i e i e i '6 'U(L) is in the center of 'U..(L). (25)
the
then
{e i
f
the
in such a way that a non-degenerate invariant form on L
needed.
Casimir
example,
both of them
the
Casimir
A.S.Dzhumadil'daev
constructs a generalized Casimir element to give a central
in 1A.(L)
iEI}
;
elements and Index E have something
in
element
is
connnon.
For
are defined as sums of (quasi-)basis and live
center of the algebras.
We shall extend Dzhumadi 1 ' daev' s
not
idea
in to
unify casimir elements and Index E. Let
k
be
k-subalgebra.
a commutative ring with a unit 1, B Let M be a B-A-bimodule.
a
k-algebra
Then M*
bimodule by (at)
(x)
=
af(x) and (fb) (x)
f(bx) for fE;M*, a<sA, bG;B, XE;M.
and
A
YASUO WATATAN1
26
Assume that M is finitely generated projective right A-module.
Then
there exists an additive-group isomorphism ~
such
, H QA
that
~
(6(",0f))(x)
B-B-bimodule
(cf.[7 ,1-8-3))
M* -7 EndAMA
form,x~M.
mf(x)
fE:W'~.
e
Moreover
homomorphism if we put B-B-bimodule structures
on
is
a
M GA N*
and EndA(MA) by b (mlllf)b ~ (b m)G)(fb ) and (b hb )(x) ~ b h(b x), 2 2 1 1 2 2 1 1 for mE;M, fE::-M*. b ,b ,b ,b t;:B, h GEndA(M ). Suppose that there exist A 1 2 3 4 B-A-bimodule homomorphisms F:M-"7B and A-B-bimodule homomorphism F'
Define a B-B-bimodule homomorphism F@F'
: W"'--7 B.
(F®F')(m@f)
Definition 1.10.1.
~
: M0
M*-"'? B
A
by
F(m)F'(f).
In the above setting, let ~-l(i<\r) '" 1i ui@f , i
M-gene~,9-J-J..~.
Then a generalized (more precisely
Casimir
element
c t::: B
is defined by c ~ (FI/i F') e-l(i~) ~ 1:
Proposition 1.10.2.
Proof.
F(ui)F' (f ). i
i
A generalized Casimir element c is in Center B.
For any bE:: B,
bc" (FQF,)(~-l(bi~))
cb.
Let M be a B-B-bimodule and
v.;:
Q.E.D.
M'i M -7 A be a non-degenerate A-valued
inner product such that tJ(ax,Y) "" aw(x,y) and w(x,ya) "" .v.(x,y)a for a x",M,
yc::;M.
~ ",(x, by)
Assume that
w is B-invariant in the sense
for b~B. Define a A-B-bimodule homomorphism Now
conditional
expectations
and
we unify
Casimir
the
elements
notion for
that
E::
A,
w(xb, y)
1w:
M---7M"'~
of
index
for
semi-simple
Lie
by
algebras.
Proposition 1.10.3. Assume that M is a finitely generated right A-module and the above map
~w:
M-) M* is an onto isomorphism.
Let
such that x =!.i uiW(vi,x)
for
l, ... ,n) be elements of M be
a
projective
B-B-bimodule
homomorphism.
Then
INDEX FOR C"-SUBALGEBRAS
C
27
Ei F(ui)F{v ) is a generalized Casimir element. i
M*-) B by F' Proof. Define a A-B-bimodule homomorphism F' Then e- 1 (i"l = E.~ u.". Hence -M ~w w (v·l. ~ 1 c = (F0 F 'le- (id ) = ' i F(ui)F'(¢(v » 1: F(ui)F(v ) M
i
i
i
Q.E.D.
is a generalized Casimir element.
Example
envelop;\"\~
the universal on
L.
embedding.
c "" I.: i
Let
{e
by
Then €i€i
A =
for
{(ui,v i )
;
be the Killing
)
Let
K.
1.10.3. ,
the
the
be
F
,
usual
form
canonical its
i'" I) Casimir
I, ... n}
F
be
a
;
B -> B
and
k- algebras
Put
of index-finite type. Let
x,yG:B.
i
L, w =
B:7Ao;>1 be
Let
expectation
= E(xy)
=
K(
and ')A(L)
~
dual
element
is a generalized Casimir element.
1.10.5.
conditional
M
Let
i '" I) be a bas is of L and {e i
;
i
~,
Proposition
'of: tU..(L)
Examele
algebra of L
B =I'A(L),
Put
basis.
Let L be a semi-simple Lie algebra over
1.10.4.
the
be
E
= B and
M
identity
quasi-basis for E.
B....,A
;
Then
w(x,y)
map.
by
a
Let
Proposition
1.10.3. , Index E = ' i uiv i is a generalb.ed Casimir elements. 1.11. Modular
automorphisms
Tomita-Takesaki
theory[69]
of
von
N~umann
automorphisms
was
unexpected.
automorphisms
on
structure
of
general
is one of the most important work on
algebras.
The
discovery
In this sec.tion we
k-algebras
under
.of
the
introduce
certain
the
modular "modular!!
conditions
like
analytic
Our "modular" automorphism is an analog of the E E generator a -i of the modular automorphism group (Ot)t",R for an
operator
valued weight E on a von Neumann algebra by Haagerup[311.
finiteness
index.
"modular" automorphisms also contain Nakayama automorphisms on
Our
Frobenius
algebras.
There exists a strong analogy between von Neumann algebras and
Frobenius
algebras.
correspond
to
For
example
semi-finite
von
Neumann
symmetric (Frobenius) algebras because both of
characterized
by
the
existence of traces, and both of
characterized
by innerness of modular (or Nakayama)
them
algebras them are
automorphisms.
are also An
YASUO WATATANI
28
example
of
outer
"modular"
automorphism is
given
by
the
principal
automorphism on a exterior algebra.
In
Warning:
section we do EQE
this
assume
that
A-bimodule
map
B -7 A satisfies that E (1) "'" 1.
E
Definition 1.11.1.
Let E : B-7A be a A-bimodule map.
Then a finite
family {(u1,vi), ... ,(un,vn)}CB:l
they
satisfies that for xEB.
Then
E
is
necessarily non-degenerate,
i.e.,
o or
E(xB)
E(Bz.)
o
implies x "" O.
Definition 1.11.2. Let e be an automorphism on the
centralizer CB(A)
of A in B, where {b G B ; ab "'" ba for all a
~
A} .
Then a A-bimodule map E : B -"J A is said to satisfy the modular
e if
for
the following condition hold:
E(xy) • E(ye(x»
The
Remark.
imported
for x f:.C (A) B
and y
original modular condition in
~
B.
operator
algebras
from theoritical quantum statistico.1mechanics by R. Haag,
Hugenholtz
and M. Winnink [30]
Schwinger)
condition.
been
independently
used
automorphisms putting
condition
A "" k.
on
by
Of
Nakayama
[52]
to
algebras in 1940's as in
Frobenius course
N.
and is also called the KMS (Rubo
But we also note that the modular
A. Cannes' monumental
was M.
M~rtin
condition
had
characterize
his
the
work
above
(t9]
form,
had
not
appeared yet I .
Theorem 1.11.3. for
E.
Let E
B-)A be a A-bimodule map with a quasi-basis
Then there exists a unique automorphism
e
on C (A) for B
which
E
satisfy the modular condition.
i
1, ... ,n} be a quasi-basis
for
E.
Put
INDEX FOR C"-SUBALGEBRAS
SeX) = Li uiE(xv i ) x E CB(A) and y E B,
and
~
E(yS(x»
Since
E
~
E(Y('i uiE(xv i »)
E(xya)
E(xy)a
is non-degenerate.
= E(xy).
E(ye(x»a
~
as (x)
)
a~A,
In fact for
~
""
for
' i E(yui)E(xv i )
= Ei E(xE(yui)v i
Moreover 8(x) is in CB(A). ~
Then
o:(x)
Ei E(E(yui)xv i ) E(ya8(x»
29
e (x)a,
~
E(yS(x)a).
so that e (x)
is
in
C (A). B
Similarly we have that E(o:(x)y) "'" E(yx) for x<=CB(A), YG-B and o:(x) is
C (A). B
Moreover a: "" e -1.
E(a(S(x»y) shows that as "" 80:
E(ys(xw»
~
In fact equations
E(ye(x»
~
=
~
E(xy) and E(yS(o(x»)
id on CB (A) .
E(xwy)
E(wyS(x»
~
E(o(x)y)
~
E(ye(x)S(w)).
S(xw) "" 9(X)SCw).
CBCA)
for which E satisfy the modular condition.
Thus we have shown that e is an automorphism on
Finally
E(xy)
~
E(yS(x»
for x G C
B
(A)
and
y
another
~
Then
B.
Q.E.D.
Hence e' "" e.
Definition
The
take
a' on CBCA) for which E satisfy the modular condition.
E(ye'(x»
called the
E(yx)
For x) wE- CB (A) and y f. B
Hence
automorphism
in
1.11.4. The above automorphism e on CE(A) given by
E
is E
"modular" automorphism associated with E and denoted by e .
uniqueness
of the "modular" automorphism yields
the
following
results at once:
Corollary
1.11.5.
A-bimodule.
Let
E :
Let E~
B
be an automorphism on B as
A be a A-bimodule map having a
k-algebra
and
quasi-b?-sis.
Then ES also have a quasi-basis and
SES ~ s-leEs.
Example basis V.
1.11.6.(exterior
algebra). Let k = IR and {e , ... ,en} l
for the vector space V "" !Rn.Let B=A(V) be the exterior algebra Let
be
Q
the set of all
J
finite
(increasingly)
with
e. e . . . . e ... 1.)
be
1.:0,.
1.
J
Then {e
J
; J€;-Q} forms a basis
ordered
a of
subsets
Put of the vector space
E,
YASUO WATATANI
30
where we put e~ "" lB'
Define a linear functional E
B -? A
~
IR by
E(L J xJe J ) "" xl,Z",. ,n that
is,
E(x) is the coefficient of the largest
e e , "en' For l 2 J~ n, we denote by JC the complement of J ordeyed increasingly, Next put signature
the
where
word
""fJ
"" 1Jel e 2 · .. en' Then tee J' v J) ; J En) is a quasi-basis for E. E "modular" automorphism is exactly the principal automorphism:
The
e
eECei) ~ C-1) Cu-lJ ei
for i ~ l, .... ,n
or
~
E CLJ xJe )
J
L
C-dn-1)1J1"JeJ'
J
where \J I is the cardinality of J. a
functional with E(xy)
then
B
does
not
automorphism
is
E(yx) for x,yeB) and
have outer.
non-degenerate trace ""C.
If n is odd, then E is a trace (i. e. ,
a
non-degenerate
eE ""
trace
id,
If n is
and
the
In fact on the contrary assume that
even,
"modular" B
has
a
Then
1:(e l e Z ···en ) "" "'C(enel···en _ l ) ~
e(C-l) n-l el···e n )
~
-,Cel.·.e n ).
Hence '""(;(e e ,. ,en) = 0, Therefore 1;:(xel" ,en) "" 0 for all xc::.A..(V) l 2 Since
is non-degenerate, el'" en "" 0,
1::
This is a
contradiction.
Thus
the exterior algebra J\.(V) is a symmetric algebra if and only if n is odd. By Corollary 1.11.17 below, if n is even, then the "modular" automorphism Therefore if n is even, the exterior algebra A(V) behave
is outer.
type III factors in the theory of von Neumann algebras, although
~(V)
like is
finite dimensional!
1.11.7.
Example
Let B' be a Frobenius algebra over a
there exists a non-degenerate functional Let ~ul"" a
dual
,un") be a basis of B.
basis ; i
~
k.
Then
PutA~kandE~q\.
Since 1> is non-degenerate, there exists
(V l "" ,vnl C B in the sense 1,_ •. n)
B-lk.
fi~ld
Then
that
is a quasi-basis for E.
Therefore the
"modular"
automorphism is exactly the Nakayama ,automorphism in [52.]
Example 1.11.8. over IR.
Let k "" A "" Rand B "" Mn (!R) be the
Let h be an invertible element in B,
n by n
Define E ; B-)A
matrices by
INDEX FOR C*-SUBALGEBRAS
E(x) "'" tr(xh) for xGB. where tr is the normalized trace of Mn(JR). the "modular" automorphism eE on C (B) = B is inner and given A 1 SE(x) "" hxh- . In fact, for x, yE-B.
Tr(xyh)
Example 1.11.9. Let k
~
A
~
IR, Bi'2MZ(R) and C n
Define an injection Cn--yCn+l by x~x\&)l.
of
Let
C , n
h
h®, .. fl)h E-e
n
Define an automorphism an on Cn x E:C , n
E(xy) .
n by
~
B €,)· .. 0Bn. 1
Let B be the inductive
for an invertible
element
Define E : B--')IR by Ele
on M 0R). 2
satisfy the modular condition for
n
0:.
M2 (k) be the 2 by 2 matrix algebra OVer k and D We fix the standard matrix units: =
0)
o Each
1)
ell~(O
xED is represented
quasi-bases.
=
B~M2'
e2\~(O
0'
0
I
Let tr be
"" tr&:> ... Q)tr The"
A-bimodule maps with M2
h €: M2 (lR
h xh -1 n n
an(x)
Then O:n can be extended to an automorphisl'YJ a on B.
normalized trace
1
by a matrix
1Z X ) " L e,; "X'·i x 22 The balanced map G ~ El ~E2 of E1 and E2 is defined to be the
x'"
map of
ee'i
(XU x 21
A-bimo.
D to A, given by
1.11.10.
Lemma for E t Then
1
(t
=
1,2).
Let {(ui(t),vi(t» By adding
° if
; i "" 1, ... ,n } be a quasi t
necessary, we assume that n
{(ui(t)@est,viCt)@ets) ; s =- 1,2,
t "" 1,2,
Let
x "" L:k,r x kr ®e kr E--D.
Then we have that
[s,t [i (Ui(t)!Sest)G«viCt)<6)est)o.:k,r xkr@e kr »
=
~S,t
I.."",'r
L:i (uiCt}~est)G(vi(t)xkr!Z)etsekr)
Zt,k,r Zi (ui(t)®ekt)G(vi(t)xkr<:,;;)etr)
"" n 2
i " ' l , ... ,n}
quasi-basis for G.
Proof.
1
b.
i:
32
YASUQ WATATANI
[k,T Si (u i (r)G0ekr)Er(vi(r)xkr) tk,r I.: i
(u i (r)Er(v i (r»
)XKV'® e.;;.y
l:k,r xkr®e kr "" x.
Similarly we have that
Q.E.D.
x.
Let
E : B-"7A
invertible
be
element
a
h
A-bimodule map with
in
CBCA).
Put
F(x)
quasi-b'asis.
a
E(xh)
for
Take
x EB.
a'
The
F : B--:)A is a A-bimodule map with a quasi-basis as in Proposition 1.4.2 A
certain
method
converse hold, because we can follow
in
1.19]
give Radon-Nikodym
to
Cannes!
derivatives.
2 by 2
matri'
also
fol10'
We
Araki[lJ.
Proposition quasi-bases. that
1.11.11.
Let E and F : B-j A be a A-bimodule maps
wit
Then there exists a unique invertible element h E-CB(A) sue' E(xh)
F(x)
for
x€:-B.
Moreover
the
"modular"
automorphis'
associated with F is given by OF (x) = hOE (x)
By
quasi-basis. "modular" X""
1.11.10.
Lemma
Hence
h- 1 the
>
balanced
map
Theorem 1.11.4. says that there
automorphism
G
on the
centralizer
G«lliJ e 11 )x) = G(xll@e 11
+ x12we12)
e
G "" E@F exists
a
has unique
CD(A) ;:: C (A)Q9M . B 2
Fo
Lij xij@eijE-D,
G(x
11
0e
G
Hence 8 (l®e
ll
)
11 =
+ x 21 0e 21 )
=
G(x00 e
11
E(x 11 )
))·
l
For xil E:. B,
G
e O@e
G
) = e « 1 ®e ) ('U0') e )( 10e )) U U U u (lQ; eu)&G(x ®e ) O@e )' u U U
Hence
ij
G
(xll®e
ll
) "" y(xl1)G;)°
11
for some y(xll)f.CB(A).
Since
E(yU y(x U )) = G«yu0e11)eG(xu0eu)) G«x U 0 e U )(YllGge U )) = G(X 11 YU 0e ll) = E(XllY11), we have that Y(x ) = eE(x ). Thus eG(xll
INDEX FOR C*-SUBALGEBRAS @e ) ~ e G«1
(xd
e G(x
we
Similarly
eG(x12®e12)" eE,F(x12)@e12
h" OF,E O ) and g ~ eE,FO)<;CB(A). (l@e we
33
have
12
that
)0(/)e
21
gh "" 1.
invertible in CB(A). ~
) " 16g e
ll
for some
Applying e G to
,
Similarly we have
that
hg
1.
Hence
h
is
For c<.;;CB(A) and b6. B.
) ~ G«c€;>e ) (b
G(cb
G(beF,E(c)1i\) ell) ~ E(beF,E(c». Putting c "" 1, F(b) = £(bh) h is unique.
for bE: B.
Since E is non-degenerate such
For be B.
eF(b) ~
aG(b®e 22 ) ~
eG(0<8)e21)(b@ell)00e12»
~ heE(b)g ~ heE(b)h- l .
Definition 1.11.12. and
Q.E.D.
We call the above h the Radon-Nikodym derivative
denote it by h "" dF/dE. Let {(ui,v ) ; i'" 1, .. ,n} i
for E.
a
be a
~uC\si-basis
Then dF/dE is explicitly given by
dF/dE ~ ' i uiF(v i )
as in Proposition 1.4.1.
Corollary 1.11.13. (Chain rule)
Let E, F and G : B-7"A be A-bimodule
maps with quasi-bases.Then dE/dF ~ (dF/dE)-l and (dG/dF) (dF/dE) ~ dG/dE.
Proof.
Since F(b) ~ E(b(dF/dE»), E(b) ~ F(b(dF/dE)-I)
Hence dE/dF '" (dF/dE)-l. E(x(dG/dE»
Proposition
~
G(x)
~
for b €OB.
The second equality follows from F(x(dG/dF»
~
E(x(dG/dF)(dF/dE»
1.11.14.(generalized chain rule)
.
Q.E.D.
Let E, F and G
B...... A
be A-bimodule maps with quasi-bases '''1Then eE,F(x)eF,G(y) ~ eE,G(xy)
..J i,j
1.2.3} be the matrix units of
34
YASUO WATATANI
M . 3
Define the balanced map H : D-? A by
E2 ~ F and E3 ~ G.
where proof
of
Lemma 1.11.11.
Then H has a quasi-basis
We can check that the
"modular"
as
in
the
automorphism
oH is given H
o O::ij xij®e ij ) "" Eij S
E' E' 0L' ~(xij)\t)'eij'
Applying e H to (x12@e12)(xZ3@e23) = x 12 x 23 @e,3' we have that eEl ,E~(x )e E"E3(x ) = eE , ,E,(x X , Q.E.D. lZ 23 n
>"
Lemma 1.11.15. Let E and F: B~ A be A-bimodule maps with quo..si-bases. Let
(X
be an automorphism of B.
Then
d(Fa-l)/d(Ea- l ) = a(dF/dE).
Proof.
It follows from that
= E(a-1(x)dF/dE)
F(u-l(x»
Q.E.D.
E(a-1(xu(dF/dE»).
The following is an anolog of a characterization of semi finiteness of von Neumann algebras by innerness of modular automorphisms.
Proposition
1.11.16.
Assume
E : B-4 A with a quasi basis. (1) All "modular"
that there
exist
a
A-bimodule
map
Then the following are equivalent:
automorphisms are inner.
(2) There exists a A-bimodule maps F : B --7 A with a quasi-has is such that
F(xy)
=
F(yx)
for x "CB(A) and yE-B.
(3) There exists a A-bimodule map F : B-'7A with a quasi-basis such the "modular" automorphism sF is inner.
Proof. (l)-=:; (2) :
Since the
I1
that
8
mo dular" automorphism SE is inner,
there
E -1 for Xc e (A) . exists an invertible hE: e B (A) such that 8 (x) "" hxh B y ~ B. quasi-basis.
Then F : B -) A is a A-bimodule
Then for xf:.CB(A) and yGB,
F(xy) = E(xyh- 1 ) = E(yh-1eE(x» E(yXh- 1 ) = F(yx). (2)=9(3):
Since e F "'" id, of is inner.
= E(yh-1hxh- 1 )
map
with
Put a
I INDEX FOR C*-SUBALGEBRAS
(3)-=9(1):
35
Then by
Take any A-bimodule map G : S-,A with a quasi-basis.
Proposition
1.11.11,
that eG(x)
veF(x)v- 1 .
=
there exists an invertible element
v €-CB(A)
such
Since e F is inner, e G is also inner. Q,\Z,D,
As a Corollary, we obtain the following known result:
Corollary
Let B be a Frobenius algebra over a field
1.11.17.
k.
Then the following are equivalent. (1) All Nakayama automorphisms are inner. (2) B
is a symmetric algebra, i.e. there exists a non-degenerate
functional
(3)There
t
on B with t(xy)
exists
a
for x,Yf-B.
l(YX)
¢
functional
linear
linear
such
that
the
Nakayama
automorphism e$ is inner.
be the group of all automorphisms of a k-algebra
Aut M
Let Int M be
group
the
of
all
inner
automorphisms
M,
of
M and
Let
£
,
Aut M-'> Out M = Aut Mllnt M be the canonical quotient map.
Assume
1.11.18,
ProEosition
E : B---;>A with a quasi-basis. (1)
A
"modular"
automorphism
that
A-bimodule
a
is
uniquely
determined
°:
up
the homomorphism l. -7 Out C (A) i. e. E n N (\ "'" £«0 ) ) does not depend on the choice of E,
(2) Suppose that A :: k.
map
Then we have the following:
automorphism. n
there exist
J
Then on €: Center(Out B) for n
~
The assertion (1) is an immediate corollary
to
inner
defined
by
t. of
Proposition
1.11.11,
= k. CB(A) =B, Hence on
Definition quasi-basis.
1.11.19.
Let S Eo- Aut B, -1 £(6) 0n£(6).
Let
E : B-7 A
Then by Corollary
be a
1.11.5, •
Q,E,D,
A-bimodule
Then the centralizer BE of E is defined by
BE = {b" B , E(yb) = E(by) for all Y" B),
map
with
a
36
YASUO WATATANI
Proposition
quasi-basis.
1.11.20.
Then
Let
the
E : B---1' A be a
A-bimodule
map
with
CBCA)
fixed point algebra (CB(A)G of
by
a
the
e E is exactly the centralizer BE of E.
"modular" automorphism e
Then for all y f. E,
E(by)
~
Hence bE BE'
E(y8(b»
~
E(yb).
Converse ly let b <;, BE'
Then for a
E«ab - ba)y) , E(aby) - E(bay) ~
a(E(by) - E(yb»
O.
Hence e(b) , t, so
The
following
that is
A and Y E- B,
E(aby) -E(ayb)
Hence ab
E(yb) , E(by) , E(yG(b»
6:
ba.
for all
Thus bE C (A). B
Moreover
y~B,
Q.E.D.
b E(CB(A»G
an analog of T set for von
Neumann
algebras
by
Connes [ 19 ] .
Definition A
k.
1.11.21. Let B be a Frobenius algebra over a field k
Hence CE(A) = B. Y(B) , {nE l
where 6 : Z --7 Out B
and
We put on
idl,
is the map defined in Proposition 1.11.18.
6 the module homomorphism.
By construction, Y(8)
We
call
is a subgroup of land
an invariant of Frobenil),s, algebras.
Example
1.11.22.
Let
B "" AORn) be
the
exterior
algebra.
Then
Y(B) "" l if n is odd and Y(B) = 2l if n is even by Example 1.11.6.
Example exterior the
1.11.23.
algebra.
universal
The
Let k
unital
following example is If.
Fix w~ If with Iwl
(;-algebra
Bn(w)
a
deformation
1 and
generated
nG-1N by
of
theil
Consider
n-element
{u l 'u 2 '··· ,un} with relations: ( l ) u 2 ,O
,
(ifj>i). As
in the exterior algebra 1\ (lR) in Example 1.11.6., let I be the set
all finite (increasing) ordered subset J "" {iI' i
2
, ... ,i } of {I, ... , n}. k
of
I
I
INDEX FOR C*-SUBALGEBRAS
Put u
J
"" u. u . . . . u.
L" .
~'~2
Then {u
37 Define
; J ,E-I} forms a basis of Bn(w).
J
a non-degenerate linear functional
~
: Bn(w)--)
~
by
Xl, 2, ' .. ,n that is,
Bn(w)
$(x) is the coefficient of the largest word
is
a
Frobenius
We
algebra.
can
show
Hence
u 1u 2 ' .. uu'
that
the
"modular,t
automorphism e4> is given by 2i-n-l e~(Ui) "'" w u.
,
Finally Blattner
we
and
can
use an analog of Takesaki's
Montgomery
[5] ,[47], to show a
duality
[71]
due
duality
certain
to
between
non-symmetric Frobenius algebras and symmetric algebras. It
corresponds to the duality between type III von Neumann algebras
and
semifinite von Neumann algebras by Connes[19] and Takesaki[711.
Lemma finite Ct
:
1.11.24.
Let B be a Frobenius algebra over a field k and G a
Let
group.
$
be
a
G......l,I Aut B be an action.
non-degenerate
Then the
functional
B.
on
Let
following hold:
(1) The dual functional ¢- on B ~ G given by $-(E
XgAg)
=
$(x )
1 is faithful and B)
(2) The "modular '1 automorphism 8 e$- (E
(3)
Take
= goG XgAg)
another
and
degenerate, that x
= O.
b G B.
goG
that
onB'4,G is given g
1jJ
$-(Yx) = 0 for all
Let
y~A
=
O.
$-
the
dual
B><& G.
Then
for
Since
is
be
non-
[g xg Ag = O. Similarly ¢(xA) = 0 implies Thus ¢- is non-degenerate, so that A is a Frobenius algebra. Xg
Thus x
g
on B.
$-((bA _' )x) = $(be _. (x » g g g
O.
by
e$(Xg) (d$/d$a - 1 )A .
functional
faithful
Proof. (l)Suppose g Eo G
[
$-
=
(2)Define a linear map L , A -7 A by L(x)
It is enough to show that $-(xy) = $-(Y
=
"g e$(xg)(d$/d$a
L<x»
for x,Y0A.
g•
)Ag
We may
assume
that x "'" aA g and y "'" bAh for a,b(; Band g,h (;G. Put Vg "" d
YASUO IIATATANI
38
iP~ (xy)
""
4->
~ (8A.gbAh)
6
"" <1)- (aO-g(bH gh )
g,
h-1,p(aa (b».
g
On the other hand, ,-(yL(x)) - 0-(bl h (U'(a)v
¢~(buh(&Q(a)Vg)Ahg)
g'g8
Og,h-l~(bQg-l(8$(a»ug-l(Vg» '5
g
,h: 1 $(a"g(b)) .
Hence 0 9
"" L.
Put h - dy/d,.
(3)
6, h-10(0. (b)e'(a)) g, g
Then ,(b) - ,(bh) for b "B.
A. Then ~J~(x) "" tjI(x ) 1 d4'-/d~,-
;jJ(x1h) "" ~-(xh).
=
Let x
Hence
Q.E.D.
- do/do·
Proposition 1,11.25. Let B be a Frobenius algebra over a field k 9 : B-) k
be
automorphism primitive A ><1 H
a non-degenerate functional.
e ~
has
u¢
and
a period n, i.e. en "" id
n-th root of 1.
the
Then B is isomorphic to the
k
"modular"
contains
crossed
a
product
of the symmetric algebra A by the cyclic abelian group H of
order
up to the matrix algebra MuCk).
n
Proof. Let G "'" 'l/nl and A "'" B ><19 G. Lemma
1.11. 24 .
invariant
-
Assume that
and
0'- (L'g
(II)
the
XgA g )
-,
where
f
g
=¢.
Since
A.
¢
is
Hence
e$(XglA . g T
on A by
T(X)
is a generator of G such that e·(b)
is a trace.
T
on
-k
(2) ,
a non-degenerate functional
Define
~-
dual functional
under the "modular" automorphism 8, ¢8
By Lemma 1.11 . 24
1.
that
Consider
Then A is a Frobenius algebra by
-
.-(XA
AfbA f
-1
f
- 1 ) for
We
x<EA,
shall
show
First we note that
~+~(o\,)
'" iI,'X.
~t
-\-o,..u
x'" A
by (#).
Then we have that T(XY) = .-(xYA - 1 ) '_(XAf-10 o - (y)) f .-(YXA - 1 ) - T(YX). f Therefore A is a symmetric algebra. Let H = G be the dual group of Then
by
an
analog
of
Takesaki's
duality
theorem
by
Blattner
G. and
Montgomery [S J, [4'1.], B@Mn(k) ~ AXlH, the crossed product of A by H with 2 the dual action, where the dual action SA satisfies = e 7=-i/n T .
'L.f
CHAPTER II. INDEX FOR C"-ALGEBRAS
In
this
chapter, we assume that B is a C*-algebra and A
algebra of B with a common unit.
most cases, means
a positive
a
C*-sub-
A conditional expectation E : B7A,
~aithful)conditional
in
expectation ,that
is,
a projection of norm one, cf. Tomiyama [73]. Recall that E is faithful if and only if E is non-degenerate, when in the case of C*-algebras.
2.1. reduced C*-basic construction Let E : B--?>A be a conditional expectation.
Put
to '"
B "" B , then [:0 A
is a pre-Hilbert module over A with a A-valued inner product E(x*y) for x,yE B. where we use the notation n(x)E.
be the completion of
C
Then
EO
by the norm n(x)
for xEB.
'"
C is
Since
£. -? E
we
injective.
E.
be the set of all (right) A-module homomorphism T : £~
adjoint A-module homomorphism S :
Let2
E(x*x) 1/2
is a Hilbert C*-module over A, see (4]. [55], [63J, [64].
assume that E is faithful, the canonical map n : B-)
tA(O
EO
Let
with
an
such that
for all s,so[. We
Then i..
denote S as T*.
norm
I\T\I ~ Supl\\T,\\ ;
operator the
operators".
span
~
B-)!A(2)
n(bx)
Let
K.
the
be the "rank
be the norm
algebra
operator
of
one"
closure the
of
"compact
for xE:B.
turns out to be an
;; \iE(x*x)\\= i\n(x)h A
~.
0",
let
usual
injective
Then lIeAII" 1.
*-homomorphism.
In fact
2
can be extended to a bounded linear operator on~.
projection in tA(t) , i.e. e A = e A 2 2 _ E ; E shows that €A - e , And A <eAl)(x),ll(X»
For
= I\n(E(x»\\ 2 = I\E(E(x)*E(x»1\ = IIE(x*)E(x)\i
\\e n (x)\12 A
e
!;,s
For
{e~,1;
xeB, put eAn(x) = n(E(x».
So
is a C*-algebra with the
For bE- B, define II (b)f {A (EJ by
A(b)n(x) II
of
(D
~ l).
defined by e~,1;(Y)
linear
Then
\\'il
A
=
2
= eA*~~A(t). E(E(x*)E(y) )
Then e
In fact,
E(x*)E(y)
A
is
a
L
YASUO WATATANI
40
E(x*E(y) )
The following is just a modification of Jones[36j
Lenuna
2.1.1. The *-homomorphism A: B-,'i,A(£) and the projection
e
A
satisfy the following "covariant relation": for b (2) Let bG B.
Proof (1) For x
(2) If b<' A,
b~A
Then
Go
B we have that
n(E(bx» "
A
nCb)
A(b)eACn(l»
E(b)
b, which shows that bE-A.
(),(x)eAA(y)f:.i..-AO:)
eAA(b)n(l)
for XE B.
A(b)eAn(x)
=
n(E(b».
Since
n
is
injective,
Q.E.D.
Let C*r be the closure of the linear span of x.y~B}.
;
n(bE(x» "
Conversely if bE:- Band )"(b)e A :::: EAA(b). then
Thus A(b} commutes with e ,
Definition 2.1.2.
B
if and only if eA),,(b)
then e , (b) n(x) A
~
~
Then we call the C*-algebra
C*r the
reduced C*-basic construction.
Lemma
For x,Y
2.1.3.
IS;.
B, )'(x)eAA(y*) is exactly
the
"rank
one"
operator 6 n (x) ,n(y)'
Proof.
For Z GB,
n(x)
Remark. construction
~
The
"
A(x)eAA(y*)n(z)
n(x)E(y*z)
n(xE(y*z»
Q.E.D.
8 (x),n(y)n(z). n
above
C*r
Lemma 2.1.3. shows that is
exactly
the
the
algebra
reduced
/!..( ()
of
operators".
Lemma 2.1.4.
We have the following:
0) eAC'\eA "" A(A)e A (2)w(a) = A(a)e gives a *-isomorphism 0/ of A onto eAC*re A A
C*-basic "compact
r INDEX FOR C*-SUBALGEBRAS
Proof.
(l)By Lemma 2.1.1., >..(a)e A
)"'(A)e A c eAC*re A ,
41
€AA,(a)eAleA for af;A,
so
that
The equality )(E(x)E(y»e
eAA (x) e A), (y) e A • e A ) (E(x) leA' «E(x» e A
A
shows that eAC*reA C >"(A)eA, (2)If .(a) · 0 , then o(a) • And~)
is injective.
o(aE(l»
• A(a)eAo(l) • • (a)o(l)
is surjective by (1).
0, Hence
~)
Q.E.D.
Proposition 2.1.5. Let E : B -7A be a conditional expectation,
Then
the following are equivalent: (1)E is of index-finite type. (2)C~\ E(x~'{x)
»
has an identity and there exists a constant c
f: cx*x
0 such
that
for xE-B.
Proof.(l)~(2):
Let {(ui,v i ) ; i = 1, .. ,n} be a quasi-basis for E.
Then
for
Li )..(ui)eAA(v i ) is an identity of C*r.
Define
Thus
x6 B.
u, v,
X(x) E- B
a; Mn
,-\"\- 'u, B, \'y
Cons ider
equality
a condi tional expectation F "" E ~Hd : B@Mn ---;7 A~Mn'
Li uiE(vix) "" 1-.
particular U~O and V~O.
means
that UF(VX(x»
By the Schwarz
~
X(x)*X(x) • F(X(x)*V*)U*UF(VX(x»
"" X(x)
for
Then
the
x o::-B.
In
inequality we have that l!u\l2F(X(X)"V*)F(VX(X»
{IIU1\2 nx (xT"V"VX(x) JIMI 21lV I\Z" (XI jiT*x(x)):· Put c •
lIulI- 2 I!VIl- 2
> 0.
F(X(x)*X(x»
Then we have that ~
cX(x)*X(x)
for xE B.
This means that
b.l~''')'
(.
0, .....
0
0) ">
o
Thus we have shown that E(x*x) ; ; cx*x
=
('A.~)" C.
0 for x G B.
0, , 0 ) 0
I
I
.I YASUO WATATANI
42
(2) ::=9(1):
~
cl/Zlixil
xE'B,
E(x*x) ~ cx~~x
Since there is a constant c > 0 such that
\\n(xi\i:llxll·
1: 0
Thus
itself
• BA
is
complete,
CO· C.
Since C*r
be
set
A
the
{8 e E:LA(cJ (", II
of
"finite
; ~,ll
""}(.C'U,
C*r A r. e
...
nCb) "" lnCb)
Thus
b
[1 uiE(vib).
ti E(b*ui)v i , quasi-basis for E. =
~A
x
~
B
span
of
C*-algebra
we have that
n(t i uiE(vib».
,
Since 1::::: lrr "" [ i 8 ne v t.:*),n(ui)' we also Taking adjoint, we also have that
shows
This
{(ui,v ) i
that
; i"" l, ... ,n}
have'
is
a'
Q.E.D.
There exist C'k-algebras B '":) A and a conditional
Remark. E : B
linear
J
l:i en(u~),n(v~*)n(b) "" 1.:1 n(ui)
=
=
b = [i vi*E(ui*b). b*
Then for
nCui)E(vib)
the
Let
Thus there exist u1 •... un,v1' ... ,vnE-B such that
.
1: i
i.e.,
Since ~ is a dense ideal of a unital
lEJ "" 1
n(u~).n(v~")
1.
ranko operators,
i.e.,
LA (£)·
A
for
expectation
such that there is a constant c > 0 with E(x*x) ;;: cx*x for x€:-B
but E is not of index finite type, see Example 2.8.4. or Example 2.9.1.
Lenuna
2.1.6.
index-finite {(Ui,V )
; i
i
Let
E : B-iA
type on a
C~"'-algebra
Let {(ui,v ) i
T -1
~
Put s1 is
~i
=
conditional
Then we can choose U
i
*
for i
+
and
invertible, 1
putting
n(s~),n(s~)
in the proof of Proposition 2.1.6.,
such that Pr,(si)
=
~
a
of:
quasi-basis
1, ... ,no
Moreover
we
ll(t i ).
that'
1*)/2,
(A(ui)eAA(u i *) + A(v i *)eAA(v i »/2 ~ O. u i + vi* for i = 1, ... ,no Then T = (~i
positive
expectation
; i"" l, ... ,n} be a quasi-basis for E, so
1 . T- I / 2 T1.- I / Z ·PTF • P(Z. 0
As
a
= 1.
(1
T
E,
"" It ... ,n} such that vi
have that Ei A(ui)eAA(u i *)
Put
be
Then
Co
on(sl;)
,q(s~)
p "'" y-l/2,
)/2. Since T we
have
)P/Z· (Zi 0Pn(s,.),Pn(s,»/Z. =
G.
Therefore
we have that
1'" Li 0 11 (t.:J,T1(t.:) "" Ei A(ti)eAA(t i *)·
there
exists
INDEX FOR C*-SUBALGEBRAS
Hence
neb) "" In(h)
have that b*
= l •...• n}
0::
=
n(l:i tiE(ti*b»
l:i E(b*ti)t i
*
for bE-B.
for b*E-B.
2.1.7.
Taking adjoint, we
This shows that
B::J A be C*-algebras.
Let
conditional expectation of index-finite type.
{(ti,t i *)
Let
E: B -7 A
2.1. 8.
Let
conditional
expectation
i " 1 •... ,n}
be
be C*-algebras.
of
UiXU i
*·
i
be
a
Q.E.D.
Let type
index-finite
E
,
B .....
and
A
be
a
{(ui,u i *)
GE , CB (A) -7 Center B Then GE is a conditional expectation.
a quasi-basis for E.
GE(x) " (Index E)-lL.L
Proof.
B ?A
;
Then Index E is positive.
Proof. By Lemma 2.1.6 .• Index E
Corollary
also
Q.E.D.
is a quasi-basis for E.
Corollary
43
Define
by
Use Corollary 1.2.10.
2.2. Unreduced C*-basic construction Let B ~ A be C*-algebras and E Following [38] we define a product
B--;:} A be a conditional
expectation.
on B ~ B by
(b Gb )·(b Q9b ) "b E(b b )®b 1 2 3 4 1 2 3 4
for b 1 ,b 2 ,b 3 ,b 4 lOB.
and now we define an involution * by (x®y)* "'" y*@x*. The
involution * is well defined by considering conjugate C*-algebras.
Thus B GJA B turns out to be a *-algebra . .,................. , Lernilfif·-- . 2--;·Z--;~r;'· ·D-e-f-iffe- -iV
injective
Thus
l.j;
'~c®-L··"· Thelf
... 1p.... 'is" .. ·an .
*-homomorphiS~.
Proof. For :x:. yEA,
and
·A'-=:':·:fB""@A:"·B·--·bY""--iji'On- ..='
we
have that
~(xy)
xy® 1
xE(y)(31" (x(i91)(y®1) "
~(x*)
x*@l
l@x* "
is a *-homomorphism.
~(x)*.
(x
Suppose that
map B
1j!(a) =
O.
~(x)~(y)
""
a&l "" O.
Applying
Q.E.D.
a
44
YASUO WATATANI
Lemma
Then
Let C*r be the reduced C*-basic const'n.,I,.(.t,Ovl. A
2.2.2.
there
is
*-homomorphism
a
¢
A
,(x)eAA(y) ~I(a)
defined by
=
:':-homomorphism
4>(a01) is injective, Moreover if
is of index-finite type, then ¢ is an onto ,r-isomorphism.
Proof.
Since
e
and (Ida) ; a
A
~A}
commute, rp is
a
well
defined
By Lemma 2.1.1.) ¢ is seen to be a *-homomorphism.
linear map.
In
fact
we have that 4(x@y)¢(zt9W) • 4(xE(yz)\i9w) •
A(x)eAA(y)A(z)eA),(w)
4«xliily)(z
and .«x@y)*) •
Since is
(A(x)eAA(Y»"
index-finite
LACE>
=
type.
Then by
= EndA(B )·
A
Proposition
•
~(x@y)*
Suppose that
A
of
2.1.6.,
~
Let
2.2.3.
~O
=
t.
E and
Therefore Proposition 1.3.3. shows that
* - isomorphi.sm.
is an onto
:
,(y*)eAA(x*) •
!pCa) "" A(a)e , !jJ is injective by Lemma 2.1.4.(2).
C~'r
11
4(y*0x*) •
rp
Q,E.D.
E: B -7 A be
a
conditional
Let
expectation.
B ®A B ......., L(H) be a *-representation of B A B on a Hilbert space H.
Consider
a
conditional expectation
there exists a *-representation
1T'
F "" E@id :B@M n
: (BQ9Mn )
---7 A6QM. n
Then u
)
such that n' «x ij )ijll9(Yjk)jk) (xij)ij'
(Yjk)jk~B@Mn' . (B~®M~)"·(B@M)··~t.(H)®M~~by
n
f«Xij)ij'(Yjk)jk) •
('j n(xij@Yjk»ik'
For
n
(ajk)jkE- A@Mu '
n
we
have
that f( ("ij )ij (a jk ) jk(Ykm)km) = f( ([j xija jk ) ik' (Ykm)km)
O:k 1TO:j xija jk Ykm»im "" O:k rr(E j x ij ajkYkm»im U j
Hence
there
TI(x ij
"k ajkYkm»im' f«xij)ij,(ajk)jk(Ykm)km)'
is a linear map
CB@M ) Q) ""M CBQ M ) -7 L(H)4)M such n ~~" n n that 1T'«X ij )ij (Yjk)jk) "'" O=j 1T(X ij Yjk»ik' We shall show that rr' is a
*-hOffiomorphisffi.
Tl'
:
INDEX FOR C*-SUBALGEBRAS
45
we have that TI'
«x ij ) ij @(Yjk)jk)TI' «ukr)kr@(Vrm)rm)
(E j TI(X ij Q9Yjk» ik(L r TI(u kr O\lvrro»kro (E k (tj TI(Xij@Yjk»(E r TI(~r@vrm»)im (E k tj Er TI(XijI1>Yjk)(Ukr@Vrm»)im O:k L. j
Cr. r TIl
L. r 1f(XijE(Yjkllkr)@Vrm»im
1f(L. j
xij(L.k E(YjkUkr)~Vrm)im
«L.j Xij(L.k E(YjkUkr»ir~(Vrm)rm)
TI' «(x ij ) ij (E( Ek YjkUkr» jr)
® (vrm)rm)
n' «Xij)ijF«Ek YjkUkr)jr»@(Vrm)rm) TI'
«X ij ) ijF«Yjk) jk(Ukr)kr) @(Vrm)rm)
1f' ( ( (x ij \j®(Y jk) j k) «ukr)kr@(vrm)rm) ) .
Thus
is multiplicative.
TI
And we have that
TI'«(Xij)ij@(Yjk)jk)*) ~ TI'«Yjk*)kj@(Xij')ji) (E j 1f(Yjk*<J> Xij*»
~ (E j TI«Xij@Yjk)*»
<"j 1f( x ij
Hence
Lemma 2.2.4.
For c "'" 1: i
!\c\\max ""' Sup{lhr(c)\I; hC\\nax
~
UEi xi®Yi1luax
Proof.
a
BGAB
~
"
:
~
xi@Yi €- B A B.
let
is a *-representation of B
IT
~ B}.
I\(E(Xi*Xj»ij)1/2«E(YiYj*)\j)1/2
Then
11.(+0<>
Define X, Y If. B@Mn by
Consider 'IT
Q.E.D.
is a *-representation.
TI'
conditional
L(H)
be
expectation
F
a*-representation.
E ©id : BI&)Mn -:l A~Mn'
Define
a
(B@M ) ~1i\IM (BQ9M )-) L(HI$ en) as in Lemma 2.2.3. n n
IITI (c) 1\
2
~
n
II Ei
1f
(xi Il> Yj )
1\
2
~ C''''~''
Q 0
Let
*-representation Then
YASUO WATATANI
46
1.
Ii ,' (X/l)y) \l 2 lIn' (X@Y)*"(X@y)\
nn ' (y*® X*)'"' (X f61y) II 1\ n' (Y*F(X*X)® y)l\
II n ' (Y*F(X"X) 1/ ~F(X*X) 1 / 2y) \1 = 1\ "
«Y*F(X*X)1/2® 1) (1/l)F(X*X)1/2 y »
II
II n ' Otl/F(X*X)1/2 Y )*n ' O®F(X;'X)1/2 Y )\1 = II n' (liS F(X*X) l/Zy),, ' ( l@ F(X*X) 1/2y)* II =
=
1\ '"' (O@F(X*X)1/2 Y )(Y"F(X*X)1/2®1» II II ,'( F(F(X*X) 1/ 2 YY *F(X*X) 1/2) (01) II
=
II
=
n ' (F(X*X)1/2 F (YY*)F(X*X)1/2® 1 )
~ \\ F(X"X) 1 /2 F (YY*)F(X*X)1/2
II
II
(u)
=
II F(X*X) 1/2 F (yy*) 1/2(F(X*X) 1/2 F (yy*) 1/2 )" II
=
~F(X*X)1/2F (yy*)1/2 1\2
,
~«E(Xi*Xj»ij)1/2«E(YiYj* »ij)1/21· where
the
i n equality
defined by $' «aij)ij) ""
Lemma 2 . 2.1.
holds because
(If) 11 '
«a i j )i j ~l)
the
2 C )
is a ~':-representation of Afl)Mn by
U C~ax <
Thus we have shown that
~!':::: A ~Mn -::J L(H(jJ
map
00 ,
Finally we shal l
show
that the equality
l~ti
xiti'J Ytl\max = \\ «E (X i *X j » ij)1/2«E(Yi Yj*»ij)1/2/1
holds. It is enough to find a i("-representation n : B ~A B ~ L(H) such ._-...-......__ •.•.•.•....•.•..__ .•.•......._-_.•__.......--..............--........-....- ---------.---.--.-. ----.--.----.-.--.. -----.-...•--.--....................-..•..--.. --:ti..--------.. ....•.--..--.....-......---.. _----_._-~---
that
the
above
because (#)
*-representation
Let
B A 11
'"
A®Mn---?L(H
is
injective.
C*r be the reduced C*-basic construction . Let A faithful representation. be a Let 1l : be a *-homomorphism such
poQ> : B ~A B---} L(1-l). '$ '
by $' (x ®y)
:
is the only ine q uality in the above
calculation.
Put
W'
,
q>(x Q y)
Define a *-homo ¥'--\
(B ®Mn) "2iM, (B @ Mn) --i C* r Mn , ep,QlM,'
,
"" )..( x)eA@MA(y)
C*r
that
for x,Y f.
with C-A-r®Mn ·
B~Mn'
We note that we
In fact,
let £.1
Co.l'\
identify
and ~ be
Hilbert
47
INDEX FOR C*-SUBALGEBRAS
C"-modu1es
\«t1) (9)(
then
£~
r.
K(t\) "
external
their
be
t: 2) ~ )(£1~ L2)'
(2" (Mn)M,' s inee F ""
1\ G\l (2
and
see
tensor
product.
for example [3 ;p.130] Put
C*r and K([2)
-=
Mn
and
[1
=
Then ~A
and
e A0M "" eA@I n ,
Therefore we can define a map
p' "" poidn : C*r --1 L(H)QMn ,
Then
is faithful and moreover
p
IT
t
""
P '0 cp'.
In fact
P'o
P(A(xij)eAA(Yjk»ik) " (L j
P'~(Xij/b)Yjk»ik "
P(A(xij)eAA(Yjk»)ik
(~j n( x ij 0Yjk»ik
n' «Xij)ij @'(Yjk)jk) for (xij)ij'
(Yjk)jk~B~Mn'
Therefore, combining with Lemma 2.2.2.,
we
have that ~'
, AIblMn ,,(aij\j Hn'«aij)ijItl1)
p' (.p' (aij)ijQj1»" L(H)®M n
is injective.
Remark.
Q.E.D.
The above Lemma 2.2.4. shows that
l\ Ilmax
is a C*-seminorm on
B~B.
Definition 2.2.5. The completion of B taking
by the ideal {C~ B®AB ,
quotient
G9A
B by the norm \1 IImax
lIe lhax
called the unreduced C*-basic construction.
:==
O}
if
(after
necesso..ry)
We denote this
Ci~-algebra
is
by
-"-homomorphiSm.
We shall show that C*max has a certain universality. it. need some notation to
Definition
2.2.6.
A
covariant
representation
of
a
First
we
conditional
expectation E : B-? A is a triple (p,e,H), where p is a representation of B on
a
Hilbert
space Hand e is a projection
on
H satisfying
the
following covariant relation: p(E(b»e
We
call
a
~
ep(b)e
forbE:B.
covariant representation (p,e,H) is
non-degenerate
if
C*-algebra generated by {p(x)ep(y) ; x,y<=-BJ is nondegenerate on H.
the
YASUO WATATANI
48 Proposition conditional
2.2.7.
Let (p,e,H) be a covariant representation
expectation
E
!B~
A.
Then there
C*max -7 L(H) such that A p-(i(x6l>y» " p(x)ep(y)
p
Moreover,
a
a
*-representation
for X,Y &B.
injective if and only
is
p
is
of
if
~
: A
~a~p
fL(H)
is
injective.
Proof.
By the definition of
is a *-representation
11
\\ l'max' it is enough to show that
of B Q)A B on H such that
TI(xl1)y) " p(x)ep(y)
This
o
O.
"IT
for aEA.
we have that max
This shows that
~
" \l(E(a*a)1/2 E (1)
is injective.
II"
\la11
As in the proof of Lemma if
2
Conversely suppose that ~
injective.
since
Therefore
Let l/J(a) "" p(a)e "" 0
injective.
also
f:
is injective.
[lTI(a)ell" \\i(a@l)ll
Hence a
fact
As in Lemma 2.2.2., 'rr is a ~'<"-homomorphism.
is a well-defined linear map. Next, suppose that p
In
for a.e A, e commutes with pCA),
p(a)e " p(E(a»e " eTI(a)e
is
forx,yGB.
is immediately followed by the covariant relation.
Then by Lemma 2.2.4.
there
® id. Then
l/J
W is
2.2.4.,
and
L(H)@Mn is injective.
Since
TI'«aij\/i)l)" (TI(aijl'i>l»ij" (p(aij)e)ij" ';'«aij)ij)'
we have that
Hence
p
Q.E.D.
is injective.
Lemma
2.2.9.
canonical
The
*-homomorphism
$
shows that *-homomorphism
$
C*max-7
C*r is an onto *-isomorphism. A Proof. $(i(~y» if>
Lemma
2.2.2.
= A(x)eAA(y).
defined
Lemma 2.l.4.(2) and Proposition 2.2.7 show
by
that
Q.E.D.
is isometric.
Definition
is
2.2.10.
In the following we shall
identify
C*max
INDEX FOR C*-SUBALGEBRAS
with
C*r.
49
We call it the C*-basic construction and denote
by
it
C*· C*-basic construction is characterized as follows:
Proposition
2.2.11.
Let E : B -7A be a conditional
expectation.
Suppose that B acts on a Hilbert space H faithfully and e is a projection on H such that ebe "" E(b)e for bE B.
If _Ill
A?a l-7ae E:oL(H)
is
injective, then the norm closure BeB
of
BeB
is
isomorphic to C* canonically.
r.u,-.o,
Proof. This is just a restatement of Proposition 2.2.7.
We may use the GNS construction to realize the C*-basic construction.
Let
E be a conditional expectation and w be a faithful state on
that w(E(b))
web)
w. so that n
B -7 H is a w
Let (TIw,Hw.n) be the GNS construction
for b €: B.
with
=
natural
embedding
b,xt':B.
Then we can define a projection eE-L(H ) byen(b) w
In fact.
B
w(y*x) and
nw(b}n(x} "'" n(bx}
such for
for
n(E(b}).
since U en(b)i\2
'w(E(b*b))
w(b*b)
=
= w(E(b)*E(b))
Un(b)h 2 ,
We have that e 2 "" e by E2
e is bounded.
<en(x),n(y» w(E(y*)E(x*))
Proposition.
=
In
E and e "'" e;" by
= w(y*E(X))
= w("(E:{~~>'.»"
2.2.12.
=
the
wE(y*E(x) )
<1l1'-) , e1(il)
above
situation
isomorphic to C* canonically. A
The
eow(b)en(x) eTIw(b)e
=
=
covariant
en(bE(x))
E(b)e.
=
relation
n(E(bE(x)))
For a€:: A, let TIw(a)e
is clear.
In
n(E(b)E(x)) O.
fact
for
b,x E B
= E(b)en(x),
Then Thus a
~TI
w
(a)e
and
50
YASUO WATATANI _ _ _ _ _ _ 11\\
is
injective.
Proposition 2.2.11
By
lr
w
(B)eTI (B) w
is
isomorphic
Q.E.D.
Proposition
2.2.13. There is a bijective correspondences
among
set of non-degenerate covariant representations {(p,e,H)} the set of non-degenerate *-representations {'n} of B non-degenero- te.
Proof. i(x@y) of \\ of
=
B and the set
*-representations hl~} of C*. A
Let
the
be
canonical
'i!~i
Then by the equation 'n =-
.>t(x)eAACy).
and
the
\\max on B @A B, we have a bijective correspondence between the (non-degenerate)
*-representations
(p,e,H)
be
a non-degenerate
of B @A B
{TI}
(non-degenerate) *-representaitons {TI-} of Let
GA
C1:
covariant
Proposition 2.2.7. there is a *-representation
and
the
set
>
A
representation. 11
:
C* -) L(H) A
that lT~
1f(X@Y) Conversely let
C;1'
11-
Recall that C* ~ A non-degenerate, 1l~
which extends
i(x@y) A
>-j
K(El
=
p(x)ep(y)
for x,y* B.
L(H) be a non-degenerate *-represent:ano!
as in Remark after Lemma 2.1.3.
there is exactly one ;1'-representation by DixVYlIe.'r
TIl :
[24;2.10.4.Propositionj.
1f (z)(,,-(x)S) = ,,-(zx), l
is
defined Then
by
for z<{A(iD, x
A(b)n(x)
=
n(bx)
e is a projection.
ep(b)e = "I (eA)'"l O(b»"l (e A ) "I (A(E(b»e A
4 (£) -7 L
We have
Define a *-representation p : B--'7 L(H) by pCb) "" TIl (A(b» A(b)c--{A(8
Since-rr
Put:
for b E-B, e
=
We shall show that
"I (eAA(b)e A )
(by Lemma 2.1.1.)
9(E(b»e. Hence (p,e,H) is a covariant representation.
Moreover
In fact p(x)e.(y)
=
'I(A(x»1f l (e A )"I(A(y»
"-O(x)eAA(Y»
=
1fO(x)eAI(y»
=
=
"1(I(x)eAA(Y)
"(x@y)
p
INDEX FOR Since
51
C"-SUBALGEBRAS
is non-degenerate, so is (o,e,H).
'IT
Finally we shall show that the correspondence is bijective. a
*-representation of B ~A Band (p,e,H) be a
,.
from
obtained (p, e ,R).
be a
n'(x®y)
Then
Conversely
,'
Let
(p,e,H)
let
*-representaion of B@A B
p(x)ep(y)
a
be
p(l)~
lH'
(p,e,H).
,.
TI'
be
a
Let (p I ,e I ,H ')
be
a
and
First we claim that p(l)
'IT.
from
obtained
,
representation
then p(B)ep(B) would not be non-degenerate,
contradiction.
representation
Therefore
,(x@y) .
from
covariant representation obtained from If
.
covariant
of B ~A B obtained
*-representation
covariant
Let n be
=
which
lH'
is
a
Then we have that e' •
,(l\l)l) •
p (l ) ep (1)
e.
And for bE-B, x/&lYEBQ»A Band t;:EH "1 (A(b»rr-O(x)eAA(Y»E;
p' (b),(x0y)E;
,-(l(b)A(x)eAl(y»E; p(bx)ep(y)E; •
=
Hence p
,-(l(bx)eAl(Y»E;
p(b)p(x)ep(y)i;
•
p(b),(x0y)F,.
We have shown that the correspondence is bijective.
p.
Q.E.D.
Corollary 2.2.14. If A is simple, then C* is simple. A
Let C·;I<-. A Then
J
a (closed) ideal of C* A
be
In
q(e A )"" O.
fact
suppose
that
for x, y (:- B. This
=
q(eA)~
q is injective.
Proposition Let
0:
J ~
O.
$(a)
=
Therefore
q(ae ). then ¢ is a A
Since A is simple, ker ¢ Hence J ""
2.2.15.
q(e ) = O. A
This shows that q = 0, that
is a contradiction. by
¢(l)
that
C* -7' C*/J be the canonical quoti~nt map. A
Let q
J
such
~
O.
q(e ) A
~
O.
Define
*-homomorphism
By Proposition
{o1
is,
and
2.2.7.,
Q.E.D.
Let E : B
~A
be a conditional
be an automorphism on B such that aE "" Eo:.
Then there exists an automorphiSM B on C* such that A
for x,yc=- B.
expectation.
52
YASUO WATATANI
Proof.
Use the universality of C'k
Ci~-basic
Finally we shall give an example of
Proposition subgroup
2.2.16.
of G.
Let B
~
>,
A
construction.
Let G be a countable discrete group C*r(G) and A
~
C*r(H).
Consider a
and
H
a
conditional
expectation E :B-?A such that
E('g
0:
:
G ---7 Aut COCC/H)
be the action induced by traslation
Then there is an *-isomorphism e :
Proof.
Define
C~~-?
COCC/H)
a characteristic function e
=
~
k'{\). . . left.
G,
XH E-CO(G/H).
We
may
suppose that Band C (G/H)-"1 G act on a Hilbert space ~. j.2(G/Hl 0~\G). O Then e is a projection on Hand ebe E(b)e for bE: B. In fact for g E:G,
It
is
easy
to
see
that
A=;;8t--,)8e E:.L(Ff)
is
injective.
Hence
by
--."
Proposition 2.2.11. there is an *-isomorphism e of C* onto BeE such A Hence XgHAs "" GCAgeAAg~15) that 00 g e AAh ) "" AgXHAh "" cdXH)A gh "'" XgHAgh' ~
'1re13.
This shows that CO(G/H)">(il G =-
Corollary 2.2.17. and only if (G : Hl <
BeE.
In the above setting, E is of index-finite type if 00
Assume that [G; Hl is finite. as
in Example 1.2.3.
C*~
have
an
COCG/H)
~
identity.
Q.E.D.
Then E is of index-finite type
Conversely assume that [G: HI is
infinite.
Then
G does not have an identity because CO(G/H) does not Hence E is not of index-finite type
Proposition
Q.E.D.
2.1. 5.
2.3. Values of index
We shall identify be B with A(b)e
by
iJ't...A ).
-- -.- ,,-- ----
INDEX FOR C*-SUBALGEBRAS
53
Lemma 2.3.1. Let B?A be C*-algebras and E ; B., A be
expectation
of index-finite type.
C*~
F :
B
such
homomorphism, F(I) Index E
defined
By
F
that F(xeAy) "" xy for x,yt" B.
2.2.2.
Lemma
linear
positive
Then there is a positive
linear
map
B-bimodule
is
Index E and Index E is positive, invertible (in fact
B ~ B~ C* A
:
conditional
1).
~
Proof.
=
a
there
exists
such that q;(x® y)
xeAy.
onto
*-isomorphism
Therefore F is
We shall show that F is positive.
map.
element
an
in C*. A
Then there exist
a
Let
xi'Yi €::B
(i
z
well be
a
1, ... n)
=
such that z F(z) z
~iLj
Yi*E{Xi*Xj)Yj
for
F(l) :::: &i uiu i Index E Thus
0
(since E is completely positive),
[~~;IV.Coro~lary
c.f.,Takesaki quasi-basis
~
*
Index E
E
as
in Lemma
Index E.
is
3.4.J.
Let {(ui,u i *) ; i =1, .. ,n} 2.1.7.
index-finite
positive
Hence
and invertible.
It is clear
that
F
is
a
Q.E.D.
2.3.2.
type.
a
Since F is positive,
B-bimodule homomorphism.
Proposition
be
Let
Then
E : B -7
there
is
A be a conditional a
conditional
expectation
expectation
of
EB
C* ~ B such that for x,y' B.
-,,~
----- ----Prooe ----
---B'y---~Lemma----Z-:T:T:--,-------Inaex-E-"Ts
--- -po s'rtf ve ----ana -'--Tnverti15Hf-:"--
Therefore EB is a well-defined positive linear map such that
Since
EB(l) = EB(&i uieAu i *) :::: (Index E)-lEi uiu i * : : 1. Index E is in Center B, EB is also a B-bimodule homomorphism.
Hence EB is a conditional expectation. as that in Proposition 1.6.1.
Definition C* - ) B
2.3.3. in
We
call
EB is faithful by the same
proof
Q.E.D.
the
conditional
expectation
Proposition 2.3.2. the dual conditiorlO..l
EB :
expectation
of
-54
YASUO WATATANI
B-7 A.
case.
This is exactly the same as that in the purely algebraic
Therefore,
we
can
use
properties
of
the
dual
conditional
expectations in 1.6.
The following is a C*-algebraic refinement of Proposition 1.6.6.
Proposition index-finite
type.
for E.
expectation
Then the dual expectation EB ; C*-) B is
of index-finite type. quasi-basis
Let EA : B~ A be a conditional
2.3.4.
More pre(isely let Put wi
{(Ui,U *)
;
i
E) 1/2.
Moreover, if Index E is in A, then Index EB
of also
i '" 1, ... , n} be a
Then
{(wi,w *) i
i
;
""
Index EA and Index EA is in
(Center A) f\ (Center 8)n (Center(C*
Proof.
For x,yE:B, we have that
Li wiEB(wi*xeAy) [i
uieA(Index EA )·1/2 EB«Index E 'A )1/2 eAu * xeAy ) i
ti uieA(Index EA )1/2 ES «Index EA)I/2EA(ui*x)eAY) [i uieA(lndex EA ) 1/2 (Index EA ) -1 (Index E ) 1/2 EA(Ui,kX)Y A Li uiEA(ui*x)€AY ~ xeAy, Hence
(wi,w i *)
index-finite
; i
type.
= 1, ... ,n} And
is
we
Index EB "" ';i wiwi*
=
a quasi-basis for EB and have
I: i
EB
is
of
that
uieA(Index EA)eAu
i
*
Li uiEA(Index EA)eAu "'. i Moreover if Index E is in A, then
"'" (Index EA)(L i uieAu i *) '" Index E , A Hence Index EA is in (Center A)
The
following
Proposition 3.4.1.
Lemma
Lemma
is
n (Center
B)f"\(Center C*
essentially
the
same
as
Jones
Q.E.D.
[36
J.
2.3.5. In the above Situation, suppose that Index E is
then we have the following:
in
A,
INDEX FOR C*-SUBALGEBRAS -1 ( Index E) e
(l}eB€A€B
(Index
(2)e Ae B€A Proof.
(1)By
eB€A€S
-1
E)
55
B
eA'
Proposition 2.3.2.,
= EB(eA)e B
=
-1
(Index E)
e A,
(2) We may suppose that €A and €B act on a Hilbert B-module with inner product
"" EB(S}~t)
Eg ""
for s,t<= C*'
C* A
Then we have
that for x,yl;;- B €A €B€A n (xeAy) "" €A eBn (eAx€AY) "" €A eBn (E A (x) €AY»
eAo(EB(EA(x)eAY)) • eA"«Index E) n(eA(Index E) -1 (Index E)
Now
-1
-1
EA(x)y)' "«Index E) -1
ll(eAx€AY)
(Index E)
Theorem
Q.E.D.
2.3.6.
Let
B"':JA:;'l
be
Put is
C1~-algebras
of
and
E: B--.yA
be
a
Suppose that Index E is
a
1 "" (Index E)-I. index-finite
V
(4,00).
Then"[ is a scalar with
type,
there
is
the
0 < 1 5- 1.
dual
1Xy for .-
Index E
is in
{4cos 2 'rr/n ; n"" 3,4,5, ... }
Proof.
when
The proof depends essentially on H. Wenzl's result[77].
Then Index E
E
eAEA(x)y)
€An(xeAy)
conditional expectation of index-finite type.
Since
-1
we shall. determine the possible values of Index E
are scalars.
scalar.
EA(x)y)
conditional
x,y
c..B.
by
Proposition 2.3.2. Then EB is also of index-finite type and Index EB __ ._------"" Index E "" ITt'- 1)-y--'Propos-rtLon--2:-:C4--:-----De-:f1ne-Tfie-Tn-c re as-fi1g- s equence----E~----
--- ._-- -----------_.
(n
0,1,2, ",) of C*-algebras with the same identity 1 by the
BO
B,
Let Boo
Bl ""C*,···,B n +1 ""C* =:
l:.1m. n
sequence
Bn
forn;;;l,where
be the inductive limit of C*algebras {Bn}n'
{en; n:= 0,1,2, ... }
of
infinitely
many
relations en=eBI'I~l'
Then
projections
in
the Boo
satisfies
(1) e i 8 t1 e i and
(2)
=
1e i
eie j '" eje i
li-j\;;; 2
by Lemma 2.1.1.(2) and Lemma 2.3.5.
Then we can apply a result by
Wenzl
56
YASUO WATATANI
Q.E.D.
1771.
Remark,
(1)
If B is a simple C'k-algebra, then Index E is
=
scalar, because Index E is in Center B
always
~.
(2) Even if Index E is not a scalar, the spectrum of Index E is in
a
included
the above set under the assumption that Index E is in A, see
Theorem
2.11.4
PropOSition
Let B:> A 91 be C*-algebras and E : B -7 A
2.3.7.
conditional expectation.
Proof.
Suppose that B
E and
for
Then Index E
=
Index E
Proposition
2.3.2
A.
1.
there
Then E
is
the
EB(e A ) '" 1. 1 - e
projection, that E
A
Hence
;;: O.
that
A.
=
Index E
conditional
(Index E)
EEO - e A )
-1
=
EB
xy
xy
for x,yf:B.
1 -1
O.
Since =
eA'
By
1.
expectation
By the faithfulness of E , 1 B
B-7 A is the identity mapping,
!
suppose
dual ':::
a
= id and {(l,l)} is a quasi-basis
Conversely
such that EB(xeAy) particular
1 if and only if B
;=
be
e
In is
A This
a
means
Q.E.D.
that is, B = A.
2.4. Index for finite dimensional C*-algebras
First we recall the path algebra construction due to Ocneanu [53J and Sunder [29J:
[67].
We follow the notation in Goodman, de la Harpe
IDi~\
-_.--" ·----·--·--ano----=
Mat(n)
is
Nj
.6:>,:,
qjN
define the ~-vector
s ei
with Nj
=
s E-
Let tr
the be
Mat(v j )
(tr(e ), ... ,tr(e l n
the
traCe tr
over
(:.
Let
II
be a faithful trace on M.
We
CD by
»,
is a minimal projection in Mi'
determining
~
the algebra of n by n matrices
(Aij)ij be an inclusion matrix.
where
Jones
Let 1 G N eM be a pair of finite dimensional C'''-algebras such that N •
where
and
=
Let
tE:- [n
triN restricted to N.
conditional
expectation
tr(E(x)y) = tr(xy) for x EcM and yG:: N.
be
the
row-vector ~
Then
determined
by
M
s/\N'
Let
tr
with
We shall identify the pair of path
INDEX FOR C*-SUBALGEBRAS
algebras
Let B be the Bratteli diagram for N C M and
AO C Al ·
Qu~mented
diagram.
starting
at
Let lIrJ be the set
of
monotone
and ending on the r-th floor.
increasing
starting
at
infinite
monotone increasing paths starting on the r-th floor of
Let lI[r
be
the
the set of monotone increasing paths starting on the and
ending
s-th
the
on
the
B~
We denote Q the set of monotone increasing paths
*
*
57
floor
r-th
on
paths
set
of
and
B~
floor
Given
(r < S).
set
(0 > r)
'[r.sJ • ('r+l' ···.'s) E Q(r.sJ
(-1 , r).
" [ r · ('r+1·· .... ) E Q[r
{(Cn)"'''rJ 2 Teo'" L(t (,,)) by Let
Rr
•
~Qr]
;
end(1;)
TCllw "" o(nrl·wr])~r)pw[r
end(n)) .
identify N
=
AO and M "" AI'
For
(t;", n) b Rr
define
(W '" Q)
The path algebra Ar is the linear span of {T
We
(-1 ~ r < s)
Cll
; (Cn) ERr} in LO,2(Q».
We shall regard sand
t
as functions
of
vertices on the O-th floor and l-st floors respectively: t(v.O) ~ t. and J J 1 ) ~ si' By [24;2.6.5.1.J, we have the following formula for E: S(V i Put
Proof.
First we shall show that, for (Ct,8), (Cn)C&R , l
Ta.SE(T$,aT(,n) • 6(a.()o(Sl,01)(S(S[1])/t(S[OJ)T(,0·
In fact, TCt,SE(TS,O?S:
,n)
6(a.()T a • SE(T S • ) o 6 (o.$)T".S(s($[ l j )/t(6[0)) )o(Sl' 01 lT 6 , . " ,
'.
~.:.".
58
YASUO WATATANI
6 (a, B) (s ( 8 [ I ) ) 1 t ( 8 [0) ) )0 ( S I ' n I) T" , n, S
° 0(0, B)6(8 1 , °1 ) (s(S[ I] )/t(S[O) IT,, O. 6 (a,S)6 (8 1 , 1 ) (s(S[I) )It(3[0] )T a , n
To complete the proof, it is enough to show that, for (Son) G;:R ,:{u
ex,
SE(u
0:.
o*T,
P
"',
n
)\(U,3)EcRI)~T,
<."
n
1
.
In fact, !(u
0. •
oE(u, S*Tc fJ
c, •
v; ,
n ) ; (a,S)eRI)
Z{ ( t (fl [ 0 ) ) 1v (S [ 0 ) ) s (S [ I ) ) ) T" , BE (T S ,,,T f" n)
(!
(Z
e-
j (a"l)
H(0(a,<)6(Sl'nl)/v«l[0)))Tf"n
(" , S) c RI )
Rj
}
e(SI'OI)/v(S[O}) ; BE-ill))T!;,o
°[0 ) ) T £ , °
(1/ v ( 0 [ 0 )) ; y" nO)' Y [ 0) ~
(v(o[O) )/v(o[O)) )T
Cn Q.E.D.
Tco'
Proposition
C*-algebras.
2.4.2. Let
Let
N
eM
be
a
pair
of
tr be a faithful trace on M.
finite
dimensional
Let E : M -7 N
be
conditional expectation determined by tr. Then E is of index-finite
and
. Index E "" E. c.p., where the scalar c. L
L 1
1
-I L. s.
=
J
1
and
A,,),t),
the type
p.
1
are
the minimal central projections of M. (A = (Aij) is a 0-1 inclusion matri~
Proof.
Lemma 2.4.1. shows that E is of index-finite type.
Then
Index E
-(a.-S"rt:K
~-,-----.---------~----- --~-=-r-{-u-~-;Bu~~·-i3,r--;
): {( t
1
}------
(S [ 0) 1 v (B [0] s (3 [ 1 ) ) T" , 8T B ," ; (a, 3) " RI }
' i ,(t(S[Ol/v(S[O))S(S[l))Ta,a ; ' i (,{t(S[O)/v(B[O])S(S[I) ' i (sc
-I.
#
')( (S"QI)
; 3[01
~,fI
}
S"ili] })('{Ta,a;a
0
I
and 3[1) ~ vi })(t(v
0 j
)/v(v)
0
)))Pi
Li Si-1Lj V(Vj°)Aijt(VjO)/V(VjO»Pi
' i (si
-I
Corollary
' j ,ijtj)Pi
2.4.3.
Let the situation be as above.
Q.E.D.
Then Index E
is
a
INDEX FOR C*-SUBALGEBRAS
if
scalar "s (I,i\ t )
~
&5
that
suppose that there exists a positive scalar P such
that
i f there exists a
and in this case Index E
~
positive
B
p,
e,
Suppose that Index E is a scalar
Proof.
number
such
only
and
59 '
Then by
Proposition 2,4,2, , we have si
-I
2: j
This shows that tAt
Conversely,
=
fot i
1" .. ,m.
~
,Since t , 81\, we
es.
~
t S(f\A ) • BS.· Then EAt Index E
B
Aij tj
have
Bs, that is, Zj Aijtj
3
,-
s
8s.
(I\!\ t)
Therefore
6si,
Li Si-10:j Aijtj)Pi "" 1.:1 Si-1GBiPi '" 6[1 Pi "'" in
is a scalar.
Q.E.D.
Corollary 2.4.4.
In the same situation as above, any
fE:EndN(~)
has
a decomposition
Proof.
Q,E,D,
Combine Proposition 1.3.3. and Lemma 2.4.1.
Example 2.4.5.
normalized trace.
Let M:= Mat(n) and N "" 11:.
Let {e
i,j
Put u ij = /l1e ij , Then {(Uij,uij"")
and Index E "" i.:i,j UijU ij *
""
M~N
be
the
1, ... ,n} be matrix units of Mat(n).
; i,j
ij
Let E "" tr :
=
1, ... n} is a quasi-basis for E
nl..
Example 2,4,6_ Let M "Mat(2)(\)Mat(5)E\lMat(3) and N
Mat(2)(j)Mat(3),
])"f i n
~-""_it>C:~\l"-i()Il.ll1atr!:~.Ab)'7C~C~_~;.7. ... ...... ... ... ... .... . . . .... .......................... .... ..... ...
Let
t
be a trace on M determined by "'s "" (1/6,1/15,1/9).
restricted
on
N is determined by
t
(7(30,8/45)_
Then the
Let E,M-i>N
canonical conditional expectation determied by the trace
1.
trace be
the
Then Index E
2.5. Index for factors We of Jones
shall [361
show that our definition of Index E for
factors of type III if we take
coincides as
the
with
that
conditional
.
60
YASUO WATATANI
expectat.io"theone d.etermined by the unique traCe; also, it generally our
coincides
with Kosaki!s one for arbitrary factors. As a matter
definition is motivated by Kosaki
work.
more
of
In what follows
we
fact shall
assume that von Neumann algebras are always a-finite.
Proposition 2.5.1. Let M"JN be factors of type III and E : M ~N
be
the conditional expectation determined by the unique normalized trace
tr
on H.
Then E is of index-finite type if and only if Jones index [M:NJ is And, in this case, we have Index E "" IM:NJ.
finite.
Suppose
Proof.
that [M:Nj is finite.
Then Pimsner and
M such that [j ffijeNffij* = 1
shows that there exl.sts a family {ffij}l:£j;i;n+l and
This means that for
Zj
mjm/~
of index-finite type and Index E "" l:j that E is of index-finite type. N-module by Proposition 1.3.4.
Kosaki[ 4ll] arbitrary
xfM,
(~J'
m.E(m.*x) "" x. J
J
; j - 1, ... ,n+l} is a quasi-basis for E.
Therefore
extended
factors
::0
[M:N].
Thus E
Conversely
Then M is finitely generated
Jones!
theory
on
index
for
based on Cannes' spatial theory [21J
subfactors and
on
a
normal
expectation. weights
Hand
E: M-) N
faithful
factors
conditional
The set of all normal faithful semifinite operator
form M to N is denoted by
P(M,N).
There is a
of
Haagerup's
(a-finite)
space
suppose
Then [M:NJ is finite by [58;1.4.RemarkJ
Let 11:; N be
Hilbert
is
projective
theory on operator valued weights [31]. a
[58]
Popa
valued
canonical
order
- ~;v~~;i~g-bijectio-;::;- -from P(M,N) to P(N" ,-M~--:--denotedbY--E ~ E--l-~--- such that
where
~')
and
respectively, E- 1 (1). J..
Let
I4i
are normal faithful semifinite
(c.f.[66j.)
* be
weights
Kosaki [44J defined Index E
a fixed normal faithful state on N.
is a faithful normal state on M.
the Hilbert space H "" L 2 (M,\jJ). R¢(O from L 2 (N,qJ) to H
=
we may
on by
Put ~
Nand the
scalar
cpoE.
assume that M and N
M"
act
Then on
For a vector r, E: H, we define the operator
L 2 (M,\j;) with
Domain(R(j)Ci;:)) "" llq)(N)
by
INDEX FOR C*-SUBALGEBRAS
61
xEN
is said to be ~-bounded if R¢(}) is bounded.
vector
J
the
case,
the
H
2
The
~
L Wl,
See
[2l]
extended
bounded
linear
operator
When this
is
from
2 L (N,
to
If
me M.
f)
4» is again denoted by Rf()) and we set ~~(J, J) ~ Rcj)R"'O)*EN i ~ .jfM,eN)J". and [66J.
JlfomJlJr E:- M/ eN',
In particular
Hence
etc ~(1), ~(l))
J4'mJIf~(l) is 4'-bounded
e(J~mJ~ i( 1) ,J~mJ~ ~(l)) (J.mJ'f)a1(~(l),~(l))(J"mJ"),,
eN'
then
and
Eo Nt
(J~mJ'f) eN(J'j> mJ'j>J *
J!f(meNm*)J\fI"'
Kosaki kindly let me know the following fact which will be needed
to
prove the equivalence of definitions of index when they are finite.
Proposition
2.5.2. (Kosak!)
Let M:>N be factors and E : M -1 N
faithful normal conditional expectation.
sense of Kosaki.
be
a
Suppose that Index E.(oo in the
Then the following hold:
(1) M is type I i f and only if N is type I (2) M is type II if and only if N is type
II
(3) M is type III if and only i f N is type III (4) M is finite if and only if N is finite.
Proposition
2.5.3.
Let M ::> N be factors and E : M -7 N
a
faithful
cond-.!~l.onal ~p_~~tatl.Qn. __ Then--Index-E-<.-oo -in-the-s-ens-e-o-r-KosalU -iT-----anaonly
E is of index-finite type.
if
And in this case.
the
values
of
Index E coincide.
Proof.
Suppose that E is of index-finite type.
l •... ,n)bea
~
{(u ' u *) i i Hence
E- 1 (l) 1 E- 0:i
Let
(J"uiJ~)eN(J4>uiJ.)*)
1 Li (J",u i J",JE- (e ) N
(J~UiJ4»*
"" 4i JttJ(uiui*>]o/ "" Jl\l(J"i u i u *)J4i
i
62
YASUO WATATANI
= J~(lndex
Therefore
= Index E.
E)J
Index E
<
in the sense of Kosaki and the value
if,'
of
Index E
coincides with that of Kosaki. Conversely
that Index E
suppose
<
<X)
in the sense of
Then
[(osaki.
there exists a finite set {u ,u , ... ,u n ;c M such that l 2
1
(I)
=
'i Ui€Nui*
by l44;Corollary 3.4. J.
Indeed, this is seen by a slightly more
refined
process as follows: We have to show that there exists a finite set {Pl,P2"" pro~ections
such
'Pn~ of
1
that
In fact if <M,e N> is type In' type III or Type m, then it that it exists. If <M,e N> is type I~ or Type Ir w ' then it
clear
enough to show that eN is an infinite projection in <M,e N>· that e N is a finite projection in the contrary, N ~
8
N
<M,e >e N N
Proposition
=
Ne
a finite factor.
is
N
This is
a
Then
This
is
finite.
1, ... nJ be a quasi-basis for E. £-1(z)
~ l~.; U.2U.'~ ~
L
by
shows
Q.E.D.
Index £
that
on
contradiction
that £ is of index-finite type.
Suppose
is
Suppose,
2.5.2. (4).
RemarJs.·
is
1
for
zf:-
In fact it: is sufficient to show i t for z
=
Let
{(ul,Ul~'()"
Then
N~. xeNy for x and y b in
We
M~.
have that
Proposition
Let
2.5.4.
conditional expectation. Put M
=
'17(B)"
conditional
If
and N
=
Let
-;:(A)".
exp~~ctation
£-
B -.:> A be C"'-algebras and 11
E
B -'>A
B'-7 L(H) be a faithful representatio\,\.
Suppose that there exists a fatthful. normal M--;lN such that E-(Ii(b)) "'" 'l(E(b»
E is of index-finite type, then E- is also of index-finite
Index E-
=
?J::oof.
for type
b~
B.
and
,,(Index £).
Let {(ui,u i "') ; i. "" 1,2,3, .. nJ be a quasi basis for E.
Then
INDEX FOR C"-SUBALGEBRAS
63
Hence E- is of index-finite type and :1
(Index E).
Q,E,D,
2.6. Some constants
Definition
2.6.1.
Let B JA 71 be C'/(-algebras and
conditional expectation.
E : B-7A
be
a
Define
c(E) "" SupleE:: lR+ ; E(b) ~ cb for all positive bE- B}
1.
would
like to thank Sekine for his improvement
on
the
original
estimation as follows:
Pr~ositi()n
f.:t;oof
2.6.2. If E
Let {(ui,u '>") i
""lernencs U and X in B@M
i
;
n
1) -)
1, ... ,ol be a quasi basis for E.
and
B G9Mn-7 A GDHn'
Define
by
U
for xl"" xn ~ B.
A is of index-finite eype, then
X
Consider a conditional expectation F Then X "" UF( U*X) .
F(X·X) - FIFIX·U)U*UPIU·X»
E <21 id
Hence we have eha t
- FIX.U)lIU.U)PIU.X)
F(X"UF(U"U) )FIU"X) - FIX;'U)FIU"X)
Cons idering
1-1 component,
Hence clE) .' [[Index
If on(~,
E (x1-,'''X ) ~ 1
II Index
E
Elrl,
n-lx1'kx1
for xl in E B.
Q,E,D,
the conditional expectation is the the dual expectation
of
some
then the above inequality turns out to be an equality,
And It gives an alternativ8 proof of the original estimation by and Popa [S8J in the case of factors of type III'
Pimsner
YASUO WATATANI
64
Proposition 2.6.3. Suppose that EA : B-7A is of index-finite type.
Let EB : C* -7 B be the dual expectation. A
e(E B) - ~Index Proof.
If Index EA is inA)
EBI- l
Since
COr.1binig with Proposition 2.6.2 .• <";8 have the desired equality.
Q.E.D. In general c(E) is
Example 2.6.4.
Index E "" 4
not equal to
-
1/2. In fact, for x If:: M2 there is a unitary
u(il.o O)IA" to •
and scalars a , b ;;:0 such that x*x
tr H~
~)u") - tr(~~)=
El~ ~l- tr\~~) -
and
2.6.5.
Example
te i
Let
t~
ui
;
-1/2
i
-
as follows:
Let E "" tr : M (C)-) IL be the normalized trace. 2
and c(E)
E(x~'<'x)
E\(-1
\\Index
1/2.
Therefore e(E)
Let B
1, ... , n}
(a + b)/2
= ~n
be
and A
the
e i · Then {(ui,u i *)
Index E
MZ
Ihe~
U(~~)U*/2
x-J'xj
Let t 1 , ... , tn ejR +
~.
basis
for
2
B
such
-
(n.
tha t
Put
1, .. ,n} is a quasi-basis for E
and c(E) '" min{t i 1 , ... ,l/t n ) 1'her-efore Index E "" C(E)-l if and only if ti '"' lin for i Then
U~
1/2.
canonical
i
;
=
~
Then
(l/t:
i""l, ... n}.
1, ... , n.
In
general c(E) -1 - \\Index E \\ .
~xample
2.6.6.
Let
B
expectation E "" tPA : M2 --:'1 C
Then
Index
Index 9;.. ""
in.
for
lfA + l!(l-A)
~A
+
may assume that 0
1/(1-,\)
< ,\ <
and
M 2 A
with 0
A
Define
a
conditional
< A <1 by
and
by Example 1,2.5.
1/2.
c.
In We shall compute
We will show that
C(;;;;..)'
fact, We
INDEX FOR C* - SUBALGEBRAS
ad-be
~
A.
Assume that
O.
We have that
because (l-A)d (l-A)d(!ca
(:.
~)
G
0, Aa + (l-2A)d
+
(l-2A)d) - A2 be
d( (l-ZA 2 )a
Hence e($,) ;;; A.
+
(l-A) (I-ZA)dJ
O.
<;;
Then a
~
,
0 and
+
,2(ad - be)
65
0, d;;; 0 and
?;
O.
Since
~) Therefore c(¢)..)
Example 2.6.7. Let B functions <X
on
a
~
compact
A.
Q.E.D.
C(X) be a commutative C*-algebra of continuous T2
G -7 Aut B be a free action.
:
,
space X. Put A
Let G :=
Ci
B
be
a
finite
group
and
be the fixed point algebra.
Define E : B -:> A by
E(b)
Then Index E
G'
for b~ B. g. G a g (b» and c(E) • I/G', (c.f. Proposition 2.8.1.). = (l/G
11 ) (E
2.7. Finite index Let M be a type III factor and N a subfactor of finite index. and N share many properties with each other.
M is hyperfinite if and only if N is. M is full if and only if N is full.
factor by
if and only if N is.
(22],
[40], [59].
Then
For example by Connes [20J
Pimsner and Pop a [58J showed They also proved that M is a
that McDuff
And M has property T if and only if N
has
More generally Let M ::>N be arbitrary factors
and
E :M -) N be a normal faithful conditional expectation with Index E < Then and
M
as in Proposition 2.5.2. M is type I
(resp. Type II, Type
only if N is type I (resp. Type II, Type III).
And M is
oJ
III)
if
finite
if
YASUO WATATANI
66
and
only if N is finite. For the property of being Type
studies are made by Hamachi and Kosaki [32], [331 Let B J A::;.1
Suppose
that
properties
~
111\,
detailed
[341 and Loi[45].
be C*-algebras and E : B -7 A a condit;Onv.l expectation.
E
with
is
of index-finite type.
each other as well.
Then
A and
B share
For example, Nagisa
and
many
Song[49]
showed that the similarity problem can be solved for B if and only if
can
be solved for A.
Here we shall add some such properties
which
it
are
easily proved.
Suppose that E : B ~ A is of index-finite type.
Lemma 2.7.1.
Then A is finite dimensional if and only if B is so. Suppose that A is finite dimensional.
Since A and
C;« A
is Morita equivalent by 1.3.4., C* is finite dimensional, so is B. Q,E,D,
The converse is trivial.
In
general a C*-subalgebra A of a nuclear C*-algebra B need
nuclear
as
noted
by Choi [101 and Blackaday
[3].
But
we
not have
be the
following:
Proposition 2.7.2.
Suppose that E : B4A is of index-finite type.
Then A is nuclear if and only if B is nuclear.
.Proof.
Recall that B is nuclear if and only if the identity map
on B can be approximated (in the topology of simple norm convergence) complete positive contractions $n of finite rank (cf. Choi-Effros [lOJ .) Suppose that B is nuclear.
Put
by complete positive contractions Conversely Q'ilAivalent
$n - EQ$n' Then iA can be ~n
of finite rank.
suppose that A is nuclear. and
C*~pMn(A)p
approximated
Thus A is nuclear.
Since A and C* A
by Proposition 1.3.4. or
above Lemma 3.3. l l., C* is nuclear (c.f. also [4). A
are the
Morita argument
Since there
is
the dual conditional expectation EB : C*-7 B by Proposition 2.3.2., A B is also nuclear. Q.E.D.
Remark.
The
above argument can be applied to
discuss
many
other
67
INDEX FOR C*-SUBALGEBRAS For example B has the metric approximation property if
properties.
only if A has it.
and
And B has a faithful finite trace if and only if A has
it.
Proposition 2.7.3.
Suppose that E
B -7 A is of index- fini te type.
Then the following are equivalent: (1) CB(A) is finite dimensional. (2) Center A is finite dimensional.
(3) Center B is finite dimensional. (4) C
,eA
>(B) is finite dimensional.
Proof.
finite
:
{l)"=9 (2):
It is trivial.
Take
dimensional.
Center A--.?!I:
Proposition
index-finite
faithful
trace
on
$
Center A.
is
Then
is a a conditional expectation of index-finite type
Hence
2.4.2.
a
(2)-=9 (l): Suppose that Center A
for type,
there
a
is
Lemma
xC:;.Center A
by
there
exists
also
constant
c1 > 0
2.1. 5.
a
constant
such
Since c
2
>
E
0
by
that is
such
of that
E(bi~b) ~ c b*b for b ~
F \jJ
B. Let F be the restriction of E to CB(A). Then 2 C (A)-4 Center A is a conditional expectation. Then the composition B <jlI'F: CB(A)-;l C is a faithful state on CB(A) such that there exists a
, o with
constant c
on
!J;(x*x)
~
cx*x
for x ';'CBCA).
Hence the operator norm
CBCA) and the norm II \\2 defined by ijx!2 = .Cx*x)I/2 are
Hence C CA) is complete with respect to B space
with
Since
any
a inner product <x,y> =
li Ii 2'
~(y*x)
equivalent.
so that CBCA) is a
Therefore
CB(A)**
Hilbert CBCA).
C* - a 1geb r!_ ~~wi th_~_--=- M*:_~~ f i~~--=--=- di~en~ona 1 ~_~~~~_ ~--=
finite dimensional. Since the dual conditional expectation EB : C*-)B is of
(3)::::il (4):
index-finite
argument,
type
Cl)
(1) ~ (4):
by
Proposition 2.3.4., we
can
von
the
previous
9 (2).
There
exists
a (not *) anti-isomorphism
Bailiet, Denizeau and Havet
p
of
CB(A)
onto
Q.E.D.
See Lemma 2.10.6.
Remark.
apply
[61 obtain the same result
Neumann algebras by a different argument.
In
particular
for
Jones(36]
__.___________
68
YASUO WATATANI
and
originally that if B and A are factors and
Kosaki [44] showed
index
is finite, then the relative commutant is finite dimensional.
2.8. Index, freeness of group actions and covering spaces.
Let X be a compact Hausdorff space. G
on
g~
is free if, for gEG,
X
B "" e(X)
be
Consider
the
Recall that an action of a group
1, and X0X,
one
gx~x.
has
Let
the commutative C""-algebra of continuous functions on corresponding action
0:
G"7 Aut B
:
"" f(g -I x) for f <'=C(X). g G.G and x E X.
by
defined
X.
ag(f) (x)
Throughout this section, we assume
Define a conditional expectation E : B-j sO: by
that G is a finite group.
for fc"B
~
C(X).
In this section, we shall show that, if an action of G on X is free, then E
is
index-finite type and
of
Index E
=
lie.
Consider
the
covering
Tf'/( : C(X/G) -? C(X) is injective and projection 'If : X .....-yX/G. Then a B ~ n*(C(X/G». Therefore Index E is exactly the number of sheets of
the covering projection
11
:
X ~ X/G.
Consider a C:I;-algebra B in general
First we need the following fact: and
a finite group G.
let
E
,
B-!A
Let
be E(x)
define unitaries u g "
<)
,
(IlliG)): "g(x) .
~
t.( t A) ~
ugn(x)
BQ
and
Since "gE " E for g.::. G, we
can
B be an action.
G -? Aut
Put
A
~
for gG G by forxc"B.
n(ug(x»
Then we have the covariant relation u
By
the
universality of the crossed product, there is a
(!I)
Put
g bu g ~" "" a g (b) *-homomorphis~
~("g.G bgAg) - Lg
p ~ (1/!l G )(! g
A).
g
eAn(b) ~ n(E(b»
w(p) ~ CA'
Then
(l/IIG)n(Lg"G "g(b»
In
fact,
~ (l/ilG)ugn(b).
Since lj.J(xPy) '" xeAy, we have 1j;(B~G) ")K.J~A) "" C*. Proposition Hausdorff
space
« : G-7 Aut S. E(x) ~ (l/IlG)!
2.8.1. X.
Put B "" C(X).
Define g
ag(x).
Assume that a finite group G acts on
a
Let a be
conditional
the
a
corresponding
expectation
compact action
E: B--)A ~ BQ
If G acts freely on X, then E is of
by
index-finite
INDEX FOR C*-SUBALGEBRAS type,
:
Index E "" #C and c(E) "" l/II G.
Ck
Proof.
~
69
Moreover there is
a
*-isomorphlsm
B1G such that $(X€AY) :::: (l/#GH geG xUg(Y)'\g'
Suppose
that G acts on X freely.
Then for
any
x t: X
there
exist open neighborhoods Vx ::> U of x and fx G..C(X) such that x
supp fx C.Vx ,fxlUx Then
if
Ctg(fx)O!h(f x ) "" 0
choose a finite set
~
g , h.
Since X
::=
~ ¢
i f g~h.
~)( Ux is
compact,
x(l) ,xv'2), ... ,x(n) } C X such that
i = 1, ... ,n ) is a quasi-basis for E.
Then {(ui,u i *) f"C(X)
1 and g(V x ) (\ h(V x )
we
can
0 UX(i)' c- ,
X ""
In fact, for
all
B,
~ Li u i «l/#G)Lg,G 0g(ui*f))
"i uiE(ui*f)
" • 'g «#G)v- 1 f x(~) . )1/2(1/#G)o «#G)v- 1 f . )1/2" (f) g x(~) g -1/2 -1/2 112 'i Lg v 0g(v )(fX(i)"g(fx(i)) ng(f) -1 -1 -1 ~i v fX(i)£::= v (L i fX(i»f ::= v vf = f.
Hence E is of index-finite type.
Moreover 11-1
2
Index E = [1 Uill i * : : [i u i ::= [i (G)v 1 1 (#G)v- (L. f (.)) ~ #Gv- v ~ #G. x
~
~
Next we shall show that c(E) :::: l/IlG. E(b) ~ (l/IIG)L the enequality c(E)
fX(i)
g
ng(b) < (l/IIG)b
~ l/#G is clear.
Since for bE B+,
By definition
E(f x ) ~ (l/#G)!g "g(fx) < c(E)f x ' evaluating at x 6 X, we have l/IIG ~ c(E). Thus c(E)
:=
l/II G.
*-homomorphism Ij;: B,.:lG-"7'«'£A) such that 1fiO: b A) ,). ~ g g g ~~ ~,,-;-fE-g--';b-gu"g-'-Smee ElSOfl.riaex-fInHe type, -tTE: ) ~ ~ctAl ~ C* - . - - - - - - - A A Hence 1fi : B >;;!G -7C* is a surjective * homomorphism. We shall show A o. Let that \jI is injective. Consider
EB : C*
a
A>--) B be
the dual conditional
In fact, let
~ °g,l'
l~
gt:G.
Then
expectation.
for
Since
f<-C(X),
have that
0g(f)E(U g )
~
EB("g(f)u g )
~
EB(uif)
~
EB(ug)f.
Since B "" C{X) is abelian, (a (f) - f)E(u ) : : : 0 g g For any x €: X. we have that
for all f
~
EB(U g )
C(X).
we
70
YASUO WA'l'ATANI
EB(Ug)(X) "
-«"g(f»(X)
This shows that ES(u ) '" 0 if g ~ 1.
Therefore,
g
bk"EB«Igbgug)Uk")
Put ~-,-:v -1
•"- u;omorp . h'lsm.
o.
- fx(x»cB(ug)(X)
EB(O)
0
for kEG,
O.
This shows that
..
'rh en 0 ( U)
g
is
~
a
onto
Therefore
g
~(xeAY) = ¢(X(l/dG)(~g Ug)Y)
(l!#G)X~g AgY "" (l(dG)E g Remark.
a
By
X0
Q.e.D.
g (Y)'g
e"" -manifold
If a finice group G acts on a compact
similar
proof,
can
we
show
that
M freely.
E
is
of
index-finite type and Index E "" li G.
Next we shall describe when E is not of i.ndex-fini.t-e type.
2.8.2.
Proposition Hausdorff
space X.
Proposition
Assume that a finite group G acts on a
Put B "" C(X).
Suppose
2.8.1.
(~
Let
that
and E be the same as
the action of G is
not
{l} } is dense in X.
Xo "'" {x €: X ; the isotropy group Gx
compact those
free
Then E is
in
but not.
of index-finite type.
Suppose that E were of index-finite type. i
~
1, ... ,n } be a quasi-basis for E.
Let
t(ui,u i *) ;
Then we have
for f"'B" C(X), that is,
G ui*(g-lX)f(g-lX» = f(x) for x eX. g --------F'Dy--x-E"-X~lTeT"e___J.-~-(-X-)-su_c-h____rh_ar__t__( -1.) 0' x + x ~~ (*)
that
(1/'C)[i Ui(X)(I
(l/"GH i ui(x}ui(x)* "" and
(constant).
Xo is dense in X, we have
=
for g
0
that EB(u g ) "" 6 g,l'
that
Since Index
that
E
Bya similar argument as that in Proposition 2.8.1.., we have
chat EB(ug)(x)
2.8.1.
for XE:X O'
further.
~
1 and xEXO'
Since
Xo
is dense in X, we have
Therefore we can follow the argument in
Thus there is a *-isomorphism
i{,.
Proposition such
71
INDEX FOR C"-SUBALGEBRAS
Since
Ci~
has an identity 1 "" L1
l' Il,-Ie ~ ~i
1
(ljffG)\"~gcG
ul,oAul'>'()
This implies that ' i Ui"g(U i *) "
Since the action of G is not free,
~ g~ G such that g-lt "" t.
1
(ii G )c g ,1' there exists t if X\ XO'
Then
'G = L1 ui(t)ui(t)* = [1 ui(t)ug(ui*)(t) This is a contradiction,
Example
2.8.3.
by
(b + a (b»/2. g
=
2
"" {l,g}
above.
a
= f2" Consider a
gox "" -x.
Let
u :2
2
free
action
-7 Aut(Sl)
C(S1)-; Ba
E , B
Let X = SI and G "" 1. ' 2
on X by g·x '"
the
be
by
of
E(b)
x
Consider a non-free action of
(the complex conjugate).
Let
and E
0.:
be
as
Then E is not of index-finite type by Proposition 2.8.2.
Remart< be
51 and G
Then E is of index-finite type by Proposition 2.8.1.
Example 2.8.4. 2
=
Define
action.
c.orresponding
O.
=
Therefore E is not of index-finite type. Q.E,D.
Let X
on
Hence there is
Let E : B-7 A be a conditional expectation.
faithful
representation. of
Ef":1T(B)"--::>1T(A)"
covering
E.
Suppose
that
there
index-finite
type but
E~
iT
is
an e:xtension
The above two examples suggest
space may be captured by the condition such
B~ B(H)
Let
:
"brancheoi"
that
that E is
is of index-finite type in the
not
of
non-commutative
situation.
Next we shall consider the case when B is non-commutative...
We must
replace the freeness of the action of G on X by an appropriate one.
The
notion of freeness have been considered by many authors deeply [12J. [43J. Here we shall discuss a condition given by
[50), (56), (64),
spectrum sp'-
(el)
the
strong
following Kishimoto (43J and the notion of saturation
by
Rieffel (cf.[56) ,)
Definition 2.8.5. (Rieffel) Let B be a C*-algebra and an
action.
For
xG:: B
put x
L
g
u (x»). f g
g
BAG.
G-)Aut B be
0:
Then
a
is
called
72
YASUO WATATANI
saturated B>4G
if
the elements x-*y-. for )(,y 6,B, span a dense
(see Phillips
J
x
~
and
Let
~, A = )' "g AgXA g g g Ug(X)A g x~*y- = (1I G)2 x *py.
algebraic *-ideal of
" is
saturated
Proposition action
of
E : B-7A
=
(1
(IlliG))'
'g G Ag'
A
g
g
a linear span of { x-*Y-G, B -A G J
be
L
a =
of
[56; 7.14, 7.15J.)
Define the projection p '" B ~ G by P ,
subspace
€=<
~ L'O) 1
finite
(1IG)px.
x,YGo B J •
Then
L
is
an
be
an
B1 G.
L71~ L
Let B be a C*-algebra and
G.
group
Ci B by E(b)
=
Thus,
B~G.
2.8.6.
)x
Then
(lIIIG)'i
g
Define
a
If
Cig(b).
0.
(..t
:
G-)Aut\3
conditional
expectation
is saturated, then E is of
index-finite type.
Proof.
Consider
a
~1(l.:gG:G bgAg) "" Lg bgU g .
w(x'*y')
*-homomorphism Since
= w«#G)2 x *py) Since
C*
also has an identity.
L
Clearly we have that
"g(b) ~ (l/IIG)b
fact
saturated,
is
0.
for
bE B+.
~
1.
Therefore
c(E) ;:; lin.
Therefore
E is
In of
Q.E.D.
index-finite type by Proposition 2.1.6.
Now we shall assume that G is a finite abelian group.
For
/'
y E.
G, let
for all g '" G J.
By = ( bt;B ; "g(b) = b
------Definition 2.8.7. (Kishimoto (43»
The strong spectrum
sp-(a) of
a
is given by B *BB Y Y
Proposition action
of
expectation type.
2.8.8.
B ).
Let B be a C*-algebra and a : G --7 Aut B
a finite abelian group.
Consider the
canonical
be
an
conditional
/'
E : B-1 A "" Bo..
If sp-(o.)
= G, then E is
of
index-finite
INDEX FOR C*-SUBALGEBRAS Proof. Theorem]
sp~(a)
Assume that
=
73
~
G.
Then a is saturated by l56;
7.2.7.
Hence the conclusion follows immediately from Proposition 2.8.6.
Q.E.D.
Finally
we
consider the case of covering spaces more
generally
by
giving different conditional expectations. Let X and Y be compact Hausdorff spaces and
projection.
Put B " C(Y) and A
'IT
:
",-"(CU.)) ,
Y ~ X be
Assume
that
X "9 Xl-jd
x
GIN lJ{oo}
is bounded.
Let
covering
n*, C(X) ->C(Y)
where
is the canonical inclusion induced by the onto map n.
a
Let d
=
x
#n-1(x).
w: Y-.?]O,l}
be
a
continuous function such that L (w(z)
for all x. '" X.
Define a conditional expectation Ew Ew(f)(y) "
Proposition
2.B.9.
type and Index Ew
Vx
1/w
=
Since nei5hborhood
Z (w(z)f(z)
of
1r
of
x
f y IV y "1 •
finite wf
.
----y-(-L) .
A by
Ec , - lI1l,,-))
, z
for £'-B.
In the above situation, Ew is ~
of
index-finite
B.
is a covering map, for any x E:: X there exist a x and open nei~hborhoods Vy of
n\vy :Vy-7Vx is a homeomorphism. Ux
:B~
and fy € C(Y)+
Moreover there exist open
for y G n
where Dy"" n-1(Dx)CVy'
-1 y E 1r (x)
-1
(x)
such
that
such
supp(f ) C Vy y
" =~!Uy(i)'
~
choose
and a
Put
. 1/2(:- C(Y) ¥ (-J._) •
is positive and {(u i ' u i ) ; i "'" 1, ... , n} is a quasi-basis for Ew'
Then
U.1 ____________ _
In fact
for any h G C(Y) and y," Y, we have that
ZEn -.(y) 1I
Therefore (Index E)(y)
that
nei~hborhoods
Since Y is compact we can
subset {Y(l)''''~Y(n)} CY such that Y Since v is invertible, we put u; :::: v- 1/2 f
open
74
YASUO WATATANI
Q.E.D.
I/w(y) .
2.9. Some other examples The following example is due to Jones.
Example
2.9.1.
Let
e ,e 2 ,e ,.·· be Jones' projections 1 3
hyperfinite factor M of type Ill'
and
M
!i-j\,
for
"" ej€i
n"'" 3,!4, ... } U [4,00(.
2
\3",t
-I
B --) A such that
for x,y(CB.
1Xy
E(xe1y) 1
in
Let B "" C:"i'1,e1,e Z ''''} and A "" O'\1,e 2 ,8 •... ). 3
Then there exists a conditional expectation E
If
4cos 2 n/n, then E is of index-finite type and Index E= tl.o $'"%. 4, then E is !l2.!. of index finite type, if 1 -1 In fact, let
,
,
Bn = C*{l ,e l ,e Z ""
,en} and A 11
C*-algebras
En::;; An'
wich
C';'{l,e Z ,e ,··· ,en} be finite tlHYlenSi.onal 3 are Thus B "" lim Bn and A "" lim A
?)
n"
be
Let
Let
expectation,
En be the restriction of
the E~
canonical
to B ,
conditional
Since
n
En(xely)
En -7 An the
usual
tr : Bn -7 C,
conditional
II En lJ S
-II ~
conditional
to"
n
~A=VA n
E(x);;
IX
of
E-.
an
identity by Proposition 2.1.6,
n
,In
fact,
the
trace
Thus E(xe y) "'" 'lxy 1
for
for x t-B-t, because E is the
8
0
,e 1 ,e 2 ""
Let eO
also satisfy
~
e A,
Put
is
the
x,Yt:-B,
By
E : B -) A
Therefore E is of index-fi.nite type if and only if
then
by
7' An
-----...1'1'
II
expectation.
f'imsner and Popa [58]
, .. }.
determined
l, there exists an onto positive Ii-near map
E,B=UB desired
expectation
coincides
Since the following diagram conunute: Bn
and
n
Consider the canonical trace tr on M and its restriction to
AF-algebras.
with
a
So we have that
l e If e 2 ,·,,· }," where T~l is a fixed constant
=
in
"" Te i
e i ei:.!:.l e i
eie j
{36]
C
=
following relations:
restriction C~'
€A>
has
C*{1,e O,el,e2'
INDEX FOR C*-SUBALGEBRAS
and Since
2.
Ceo C),
"" (the closure of BeaB)"" (the closure of
C* is A Now we can use a beautiful analysis due
the ideal generated by eO in C. to
!i-j\ '
for
eie j "'" ej€i
75
Jones [36J.
Suppose that
t
-1
~
By Jones
4.
C*{1,e ,e ,e , ... } O 1 2
the following Bratteli diagram and the ideal C* of it is A
has
determined
by the shadow:
1'
"-2
/"
2
3
I
"1
V
"-
~
'L
;"-
S' 9
VV
5
Thus C* is a proper ideal. And C* does not have an identity. A In fact suppose that C* had an identity p. Then p is a non-trivial central projection in C and C>'c "" pC. A
This contradicts that
Center
C is the scalars.
Thus C* does not have an identity. so that E not of index-finite type. Next, Bratteli
suppose diagram
that ,-I
because
will not grow any wider from
some
level.
shall
Hence E is of index-finite type. show that Index E = 4cos 2 n/n in this case. Let
i"" 1, ... ,n} restriction of for
E~.
be
a
E~
quasi-basis
for E.
M-"7N, {(ui,u *)
i
;
Since
M:: B"
and
the
Therefore
C has an identity.
we
is
Finally
E
is
the
i"" 1, ... ,n} is also a quasi-basis
Hence Index E
Index
E~
Jones also suggested that the construction of subfactors due to Wenzl also give a construction of simple C*-subalgebras of index-finite type.
76
We
YASUO WATATANI
also
follow the nice description by Goodman, de 1a Harpe
and
Jones
[29) .
Example
2.9.2.
Let B1 C B2 C . ..
dimensional C*-algebras and tr -;;->
of
Ale A2 C
Similarly consider an
u
subalgebras
An
of
En
with
ascending
the
same
sequence
unit.
Put
Furthermore we assume the following conditions (C) and (P):
A '" lim A -;;-'
finite
a faithful tracial state on its inductive
C*-algebra B '" lim B .
limit
be an ascending sequence of
u
(C)
Aj+! For each j,
C
Bj+!
Aj
(P) There is a jo
Bj
C
0 and a p > 1 and a suitable
~
:\
is a commuting square.
lJ
V
orderin~
of the factors
in Aj 's and Bj 's such that for all j , jo' (i)
The
Bj +p
inclusion
C
Bj +p +1 '
matriX
for
same
the
Bj C. Bj+l is
as
that
for
similarly for Aj C Aj + 1 and Aj +p C Aj +p +1 '
(ii) The inclusion matrices 1> j
for Bj C B + and 1¥ j j p
for Aj C A
j
+p
are
primitive. (iii) The inclusion matrix Aj for Aj C Bj is the same as that for Aj
+p C
Then
Bj
+p
C*-algebras
A and
conditional expectation E : tr(aE(b)) In
~
B
B~A
tr(ab)
are
simple
by
(P)(ii).
for a
By the condition (C) , E
be extended to E
,
B~
A.
Let
have
a
such that ~
A and b G B.
fact let En : Bn-7 An be the conditional expectations
trlBu'
We
; (j)
~
~
by
is well-defined and
can
(resp. t(j») be the trace vector
for
UB h"'\
n
~
VA ., ... \ n
determined
Bj (resp. Aj) . Proposition and
for all j
G:
jO'
2.9.3. In the above situation, E is of index-finite type
-~
INDEX FOR C*-SUBALGEBRAS
Proof.
77
Let n be the GNS representation of tr.
We regard B (resp. A)
as a subalgebra of a factor M "" n(B)tI (resp. N"" n(A)") Let E- : M"'"""':)N be the
canonical
[M , NJ '
and C
conditional expectation Let
w
Popa
1 ""
(M : N]-l.
Then E-IB
Then E(x) ~
tX
=:
By Wenzl
E.
for x ~ B+
By Proposition 2.1.6. it is sufficient
[58J.
[76],
Pimsner
by
to
show
that
C* has an identity. Let
=
von
Neumann
L "" <M,e N> be the basic construction as Consider finite dimensional C*-subalgebras
algebras. Then C
l
C Cz C C
3
C .... and C •
lim C • .....
We
n
also
consider
finite dimensional C*-subalgebras Ln "" TI(Bn)eNn(B ) of L. By n Proposition 2.2.11. there are onto *-isomorphisms fn : Cn -) Ln such that lim Ln be norm
closure of
show
that
L tI.
Since C is isomorphic to Loo' it is sufficient Let Zj be the central support of
Then
n
by
V L . ...... \ n
has an identity.
L~
the
proves
i~
by (2Q,4.3.1.J and Zj •
proof of [29;Theorem 4.3.3.J. that
E
the
-,;->
Hence L has
is of index-finite type.
an
2.5.1. and Proposition 2.5.4., Index E "" (M: NJ.
in
for
, j 0
j
identity.
Furthermore
by
to
This
Proposition
By [29;Theorem
4.3.3.J
we have that Index E
[M
Q.E.D.
NJ
2.10. Radon Nikodym derivatives In
this section we consider some relations between
expectations.
two
conditional
Let E and F : B -) A be two conditional expectations.
Even
if E is of index-finite type, F may fail to be of index-finite type.
Example fGC[O,lJI.
2.10.!. Define
Let
a
• «f+g)/2,(f+g)/2). Proposition
2.8.1.
conditional
E(hO for
aGA faithful.
F(a) '" a
expectation
and E:
A' [( f , f) "B , by
B~A
E(f,g)
Then E is of index-finite type and Index E Let hI (x) "" 2x and h2 ex) "" 2-2x for
Define F(x)
B' C[O,l]\llC[O,lJ
another conditional
xc::- [0, 1 j
expectation
Since h ~ CB(A). F is a A-A bimodule because Suppose that Feb) "'" 0 for all
map.
by For
F
b
by
Put
.
F: B -) A
f"B.
In fact let bf.B.
2
is
78
YASUO WATATANI
Therefore b i "" b 2 "" O.
h1b 1 "" 0 and h 2 b "" O. 2
condi.tional expectation.
Therefore F is a faithful
But F is not of index-finite type by Propositon
1.4.1, because h is nOt invertible.
if
Nevertheless
property
of
tole
assume
that dim C CA) is
s
finit'e,
being index-finite type does not depend on
conditional expectations, \(~. ?ropos~""'~V'\
Proposition index-finite
2.10.2.
then
the
the
choice
of
expectation
of
2,,'"\. S.)
Let E : B-7 A be a conditional
Take another conditional expectation F : B-) A.
type.
If
dim CSCA) is finite, then F is also of index-finite type.
Proof. ~
F(x) were
Proposition
By
E(hx) for xof:.B. not
1..4.1., there
exists
h(:,CSCA)
We now show that h is invertible.
invertible.
Since CS(A) is a finite
such
Suppose that h
dimensional
C*-algebra,
there would exist a non-zero positive element x of E with hx "" O.
F(x) " E(hx) " O. is invertible.
that
This contradicts to that F is faithful.
Hence
Therefore
h
Thus F is of index-finite type by Proposition 1.4.1.(2). Q. E.D.
The following proposition is a variant of the uniqueness condition of the
normal
conditional expectation in
a von Neumann algebra,
cf.
for
example [66; 10.17)
B that CE(A) C A.
If there exists a conditional expectation E : B-7 A,
is uniquely determined. uAu* "A
Proof. part,
let
"" u"i(E(uxu*)u
"-=)
be
E(uxu*)
~
uE(x)u*
a unitary in B.
for x
~
B.
it
Moreover, for a unitary u in B, we have that for all x 6.8.
The former part is proved in Corollary 1.4.3. u
such
Suppose
that
uAU"k
For the lat ter A.
Put
Then E1 : B-:l A is a positive linear
that
i1 (a) "" u*E(uau*)u "" u""uau""u
~
also
a
Therefore El
conditional expectation.
a
for aE A. ~
Hence E,
that
El (x)
map
El : B-"7 A is,
such is
E(uxu*)
INDEX FOR C*-SUBALGEBRAS
uE(x)u*
for KGB.
Conversely~
in
suppose that E(uxu*) '" uE(x)u*
particula.r.
we
for x GB.
Then E(uau*) "" uE{a)u* "" uau"<.
Put
shall
ae
x '"
G- A.
Hence uau*
A.
shows that uAu"':
Now
79
A
This
Q.E.D.
investigate interrelations
between
two
conditional
expectations.
Lerruna
2.10.4.
Let E and F : B -7 A be
index-finite type.
By
Proof. cb
;S
condi.tional
expectations
of
Then there exists a constant r > 0 such that
Lemma 2.1.5., there exists a constant
E(b) for bE- B+,
c > 0
such
that
Hence
eFCb) , FCeb) , FCECb)) • ECb). By
symmetry. there also exists a constant d > 0 such
for bEB+,
We
that
dE(b) 5 F(b)
Q.E.D.
Put r "'" min {c,d} > O.
need
some
conditional
preparation
expectation.
completion
of £:0
>:>
B.
to
further.
Le t
E : B -? A
Recall that the Hilbert A-module
Let Endk
consider -[A (£J "'" {T G Endk(c)
:
L-, L
is
; T has an adjoint}, ~(8
)
{TE[ACU ;
go
T is of "rank one"},
--." andK.c8·;J CV
a
""
£:
be
a
is
the
linear}.
We
(linear
span
of
Then
3- C8 Now
C /(C'0C1-.Ac8 C EndACCA) C EndkC\:). assume that E is of index-finite type. Then
we
£:.=
f
Band
algebra homomorphisms
and for
A
B-. EndkCL)
such that ,Cb) n Cx)
oCbx)
p
B-J End k CL)
such that pCb)nCx)
n Cxb)
b,x "E.
For
* -homomorphism. Lemma 2.10.5.
bEB
>(b)
is in
f. AC8
and
,
;
B -itA (1)
is
a
But pCb) need not be in -LA < ~) in general.
Assume that E is of index-finite type.
the following are equivalent:
Then for
h~B
80
YASUO WATATANI
p(h)Ei.A(i:)
(1)
(2) p(h) con~utes with the right action of A. (3) h6C (B) A
(4) F
Eh is an A-A-bimodule map
(5) F
hE is an A-A-bimodule map.
Furthermore
if
satisfies one of the above
hE- B
involuted p(h)i' of pCh) iniAC
8
where (ui,u i
Proof.
;
sufficient
to
*»
and (2)~ (3) are trivial sincep is injective. that
show
h is in CE(A).
Put
m"" I:
that pCh) has pem) as its
uiE(h""u *). It is i i adjoint. In fae t for
we have that '"
I:
i
E(E(x~~yui)h*ui *)
E(h*I i E(x*yui)u *) i
E(h*xy) =
We
F(xa) "" E(hxa)
(3),
have
"" E(x*ym)
L:i E(x~~yui)E(h*ui*)
E(x*yt i uiE(h*u *» i
(4h~
the
i "" 1 •... tn} is a quasi-basis for E.
(l)~(2)
Assume
(3)=:7(1):
x,y~Bt
,)
then
is calculated by
p(h)* - p(E i uiE(h*u i i
conditions,
=
F(ax) "" E(hax) - ECahx) - aE(hx) aF(x)
E(hx)a
F(x)a
for a6A.
Since E(hax)
F(ax)
aF(x)
is non-degenerate, ha "" ah
=
aE(hx)
and
E{ahx) for x
E
for aG: A.
(3)~(5)
is similar to (3)~(4).
p(CB(A»
is exactly the centralizer C;(2)(A(B)) of A(B)
Q.E.D.
A
p(h) = ' i ACuih)eAA(u *) i
Proof.
Let h be in CB(A).
for hE:CB(A).
Lemma 2.10.5. says that pCh) is ini-A(t.).
Since p(h)A(b)n(x) = n(bxh) = A(b)p(h)n(x)
pCh)
is in the centralizer of ;\.(B) in;lA([)'
have that
for x,b e B, By Proposition 1.3.3.,
we
a
INDEX FOR C*-SUBALGEBRAS
Let T be in CiA(£)(l(B)). since [ ~
Co "" B.
81
Then there exists h cB such that
We claim that T
~
p(h).
In fact for bE: B,
Tn(b) - Tl(b)n(1) - l(b)Tn(1) - l(b)n(h) - n(bh).
By Lemma 2.10.5 .• h is in CA(B).
Lemma 2,10.7.
Let
a positive element in
Proof.
Q.E.D.
a Hilbert A-module and T ~ fA «().
E be iA
Then T is
if and only if
Suppose that T is positive, i.e. T "" S*S for some Sf-fA (€).
Then it is clear that
that ;;: 0 for all £;
such that T+,T_
~
f
<S*Ss,£;> = <S£;,Ss> G O. Conversely suppose
E
Then T '"" T*.
Decompose T as
T "" T
t
0 are positive elements in C*(T) and T+T_
O.
-
T_
Since.
fort,6c[,
o
;£
"" «1'+ -
we have that
T_)T_<,T_~> - _ > 0,
"" O.
H€~ce T 3 ~ 0, so that T
o.
Thus
is positive.
Q.E.D.
Lemma 2.10.8. Suppose that E is of index-finite type. be an A-A-bimodule with F "" hE for h ~CB (A).
Let
Then the following hold:
(!)
F
hE is *-preserving if and only if p(h) is self-adjoint.
(2)
F
hE is positive if and only if p(h) is positive.
(3)
Assume
that
index-finite
type
E(h)
=
1.
Then a conditional
expectation
if and only if p{h) is invertible in
Proof. (1) F - hE is *-preserving
E (x"h)
~ E(h*x)
E(xh)*
for xE'B
E(xh)
for xEcB
~ E(h*x*y) - E(x*yh) ~)
p(h)
(2)
-
-
for x,y <:-B
p(h)i~.
F - hE is positive ~
E(x*xh)
~
<
for x
0
<
0
for xc: B
for x,y
F
LA ('Z-.),
also if and only if h is invertible in B.
4=1
F: B~A
~
B
is
of and
YASUO WATATANI
82
<'<:=) (3)
(by Lemma 2.10.7)
p (h) is positive
Q.E.D.
See Proposition 1.4.2.
The following is a certain analogue of Sakai's Radon-Nikodym
theorem
[65 J • Proposition index-finite
2.10.9.
type
Let E : B.-:y A be a conditional
with
a
quasi-basis
{(ui,u i *)
expectation
;i "" 1, ... ,n}.
of Take
non-degenerate) conditional expectation necessarily (not another F : B~A. Then there exists q~CA(B) such that Fex) "" E(q*xq) for x~B.
If
F
is
of
index-finite
type, then the .
invertible.
Furthermore in thLS case {(uiq
a quasi-basis for F and Index F
By Lemma 1. 4.2.
Proof. is
p(h)
positive,
p(CB(A»C l..A(~·J·
p(q) ~ O.
Assume Lemma
=
in
a
)*)
C~~-algebra
such
-1
be
F
Since
C;C([)(B) that
p(h)1/2
=
=
E(q*xq).
Then h is
Hence p(q) "" p(h)1/2 is invertible in
p , CB(A)-;,p(CB(A»
to
A
exists q~CA(B)
furthermore F is of index-finite type. 2.10.8.
chosen
i""l, ... ,n}is
we have that
E(xh) =
is
C (B) with F = hE. A
there exists h
there
Hence
,(uiq
q
L;" u.(q*q)-lu.*. L L
positive
is
Then for xE'B
~
above -1-1
p(CB(A».
is an algebra isomorphism, q is also
Then {(v.
invertible
by
Since
invertible.
*) ; i"" 1, ... ,n} is a quasi-basis for F.
In fact, for xlfB, we have that -1 -1 uiq E(q*q* ui*xq)
( by q&CA(B»
Q.E.D.
Proposition Assume
that
2.10.10.
Let F : B~A be a conditional
there exists another conditional expectation
index-finite type.
Then c(F) >
a
expectation. E: B -yA
of
implies that F is of index-finite type.
INDEX FOR C*-SUBALGEBRAS
Proof. and
Put c "" c(F).
p(h)
Then there exists h fCB(A) such
is positive by Lerruna 2.10.8.
cE(x*x)
~
F(x~'<"x)
Since
taking E, we have that F(x*x) G cE(x*x) E(x*xh)
83
that
cx*x
Ii:
F
for
hE
=
xE. B,
or
for xE-B.
Hence
:::::
-
c
E(x*xh) - cE(x*x) ? O.
By Lerruna 10.2.2., we have that p(h) - c :;; O.
II
Hence F ""
Hence p(h) is invertible.
Q.E.D.
is of index-finite type by Lemma 2.10.8.
Finally
we shall show that basic constructions do not depend on
the
choice of conditional expectations.
PrOEosition. 2.10.11. Let E and F of
index-finite
for
constructions
e ,
C*
type. E
Let and
,
B --7 A be conditional expectations
C* A F. Then
-? C* such that 6(b)
~
and
there
C* exists
for x
If;
B.
a
*-isomorphisrn
invertible
We may assume that
C"k
fA>
Define U : ~ -?
t
by
UnF(x) "" nE(xq).
basic
the
b for bGB.
By proposition 2.10.9. there exists an such that F(x) "" E(q*xq)
be
Since q is invertible, U is a
and moreover U is a Hilbert A-module isomorphism.
a
map
bijection
In fact, we have that
------~IL(.I)F·(.bj-"-a)--"'-.lI.(.llF·(.ba)J--"'--"'E-<.haqj'-----------------
"E(bqa) ~ nE(bq).a ~ (UnF(b))'a and
=
x,y f.
UTU- 1
Then e is a ir-isomorphism and
8(b)"E(x) UnF(bxq
for bE-B.
F(x*y} "" .
Put
B.
TE:-C*. A
-1
~
UbU
-1
nE(x)
) ~ "E(bxq
-1
~
Ub"F(xq
-1
)
q) ~ "E(bx) ~ bnE(x)
Thus O(b) "'" b for bE-B.
Q.E.D.
for
YASUO WATATANI
84
Remark.
The
isomorphism
above
e
explicitly given by e (xfAy) =- xqe q'ky A this~
To see
z
for x,yf: B.
it is enough to show that S(f )
A
=
qeAq*.
We have that,
for
~B.
-1
-1
8(f A)"E(z) = UfAU nE(z) = UfA"F(zq ) -1 -1 -1 U(hF(F(zq )) = nE(F(zq )q) = "E(E(q*zq q)q)
Q.E.D.
= "E(qE(q*z)) = qeAq*"E(z)
"E(E(q*Z)q)
2.11. Reduction
In
this
algebra
Z
section
we shall consider a
= Center A A Center B.
expectation. C*-algebras
For B
t
~
each
point
E ; B
Let t
reduction
in the dual
---~
over
A
Z' ,
be
we
At ~ 1 and a conditional expectation E
the a
conditional
shall Bt
:
t
abelian
associate
~
At'
Let
Kt be the maximal ideal of Z and Xt its character on Z corresponding to
in ZA,
generated
by Kt'
the ideal of B
Then It is the norm closure of AK t and J t is the
norm
Then we have that
closure of BK . t
E(J ) C I t
(II)
In
Jt
Let It be the ideal of A generated by K and t
t
t
fact.
since KtC A. we have
that
E(BK t )
= E(B)K t
CAK .
t
Hence
E(J t ) C It· Let 'ITt : A -jA/l Let
and P
fact,
only
t
be the canonical quotient map?
non-trivial
Hence
a
Then there is
thing is to show that it is injective.
injective
clear.
The
Suppose
that
shows
that
Then
Pti(a) "" i t
is in J t and
a
= itn t
since It C J t , the existence of such a map it is
pt(a)
t
: B-7 B/J
A/lt-? B/J t with pti
it(nt(a)) "" 0 for a€::.A.
ll
t
i : A -1 B be the canonical embedding.
it In
t
ll
t (a) "" O.
a = E(a)
E: E(Jt) C It by Uf).
This
(a) :::: O. so that it is injective. Next we shall show that there exists a positive linear map
.. INDEX FOR C*-SUBALGEBRAS In
fact, the existence of such a linear map F
85
is clear by
t
(d),
We
have
to
show that F is positive. Let p be a positive element of t BOt· there exis ts a positive element b in B with p " P (b) . t Ft(p) " Ftpt(b) " ntE(b) is positive.
Definition
2.11.1.
For
in
t
subalgebra At and a positive map E Bt ::::: B/J t ,
Lemma 2.11.2.
At "" it(A/l t )
Z" :
t
a
C*-algebra
Hence
Bt'
its
Bt~At by
and
The map E
define
Then
E
::::: itF ,
t
t
Bt --7 At is a conditional expectation with
t
Etp t ::::: itTIcE.
Proof. with
x :::::
We shall show that Et(x) "" x for x in At'
it(Y)'
Take a in A with y
=
Et(x) : : : icFtitTItCa) itFcpc(a) ::::: it-tE(a)
= ITt(a).
There is y in A/It
Then
itFtpti(a)
=
itTIt(a)
x.
Since Et is a norm one projection, Et is a conditional expectation by Tomiyama [73].
J.
Moreover we have that
Q.E.D.
Proposition 2.11.3. index-finite in Z"
type, then E t
n Center
A.
If E ; B-:>A is of
: Bt ~ At is of index-finite type for all t
and Index E
Proof. b
Let Z "" Center B
Let {(ui,u i ''')
Li uiE(ui*b) pt(b)
for b ~
=
t
; i "" 1.2 •...• n} be a quasi-basis for E.
B.
Then
Hence we have that
ti pt(ui)pt(E(ui*b»
= ri
Pt(ui)itTItE(ui*b)
; i = 1.2, ... n} is a quasi-basis for E . t is of index-finite type and
This shows that {(Pt(ui),Pt(uil") Therefore Et Index E
t
Q.E.D.
YASUO WATATANI
86
The
general.
converse
of
Proposition 2.11.6.
Consider Example 2.8.4.
does
not
hold
in
Then E in not of index-finite type but
= 1 or
is of index-f in ite type for each t E': Z" . Moreover Index Ee c and Z" -7 t I-J Index Ec is not continuous at the branched points. E
The example 2.8.4. al so suggests that we can define a weaker
2
notion
of index even if E is not of index-finite type.
Definition
2.11.4.
Let
conditional expectation.
B?A 9 1
be
C*-algebras
and
E ;B---?A
a
Define a numerical invariant norm-index
n-index E of E by n-index E
c
Sup {l\Index Ec ll , ,
if Ee : Bc---' At is of index - finite type for all
Z....
t (;..
index finite type for some t62", then we put n -index E
When Ee is not
=
~
For instance, consider Example 2.8.4. , then E is not of type but n-index
Proposition
~
2.11,5.
Index Et
En.
~ l\Index
Center B.
Hence
~
$ (lnd ex E)
Let: E : B -) A be a conditional
expectation
of
Then .n-index E "" \) Index En.
Since
n - index E
index-finite
= 2.
index-finite type.
Proof .
of
=
pt(lndex E)
by
Recall that Index E is a
there exists a character
$
:
Pr op osition positive
Center B-) It
2.11.3. ,
element such
of that
\\ Index E 11- , -T"ke~-Z"'-Wi-th=~X1:'c"':="----'$"--I,--,Z,-",--~c~o~n:'.S::id~e::r,---~i:,:d'.:'e'.:'a:.:l.:S_ _
t '" Ker xtC Z,L "" Ker ~c Center B, J t '" BK t C Band N ~ JtnCenter B. We shall show that N C L, Let 4l~ : B --7 C be an extended state with K
,y "'ICenter B
""~.
Sin ce NeJ , it is enough to show that q, -(J ) "" O. t t this follows from the inequality: j~(bv)1
Let q since
~
~(bb~'()Q(v'f(v)
::::
0
for
b~B
I
v"K
t
But
·
Center B-)(Center B)fN and P t : 8-7 B/J be the quotient maps. t NC.L, there exists a * - homomorphism f ; (Center) / N --} (Center B) / L
such tha t
~
= f ' q,
Therefore
Ulndex Ell = 1I$(Index E)\\ • ~ f "q(Index E)IIS l\q(lndex E)l\ ~ lilndex Ell
INDEX FOR C*-SUBALGEBRAS
Since
N "" Jc()Center B,
g : (Center E)/N"-7 B/J
e
there such that P
exists =
t
a
g'q.
8)
injective
E)I\
iiIndex Etl!" I!ptondex E)II "i\goq(Index E)\\" \\q(Index Hence n-index E ""
Theorem
I(
Index E 1\
2.11.6.
conditional
Let
*-homomorphism
Then
lIIndex Ell
.
Q.E.D.
B ':JA:Y 1 be C*-algebras
expectation of index-finite type.
and
E: B-7 A
be
If Index E is in A,
a
then
the spectrum sp(Index E) of Index E6 B is included in 2 {4cos n/n j n = 3,4,5, .. ,} U [4,00).
Proof.
Let A be in sp(Index E).
in fact in Z "" Center All Center B.
with A Let j
Since Index E is in A, Index E
Therefore there exists a point
t <=
= xt(Index E), where Xt is the character on Z corresponding to
: 2-7 B be the canonical embedding. Then there is a
is
Z"
t.
*-homomorphism
Z/Kt -) Bt such tho..t j tXt ptj· because j (K t ) CJ , Then we have that t Index E " pt(Index E) pt(j(Index E» t jtXt(lndex E) " "By Theorem 2.3.6. , A " Index E 6: {4cos 2 Tl/n n " 3,4, ... }U[4,oo). Q.';.D. t j t:
If Index E not
included
2.6.5.
in
is not in A, then the spectrum
the above set in general.
In fact
sp(Index E) consider
and the case of covering space, say Proposition 2.8.9.
A ~ [l. J, there is a conditional expectation E such that oo
any
sp(Index E),
More over in this caSe
weight
w
2.12.
Minimizing index
;
a
pair
of
Example Then
for
A is
in
Index Ew is in A if and only if the
Y -7 J 0 ,1 [ are equal on each fiber
Consider
is
factors
M':) N,
11
-1 (x) for x
Hiai [35]
G::
X.
This shows
characterize
a
conditional expectation EO whose index is the minimum of (Index E ; E is a normal conditional expectation of M onto N). In
this
section
C*-algebras.
we
shall
consider
a
similar
problem
for
simple
Our proof is essentially a modification of Hiai[35]
Let B be a C'''-algebra and A a C*-subalgebra with the same unit, sO(B,A)
be
the
set
of
all
conditional
expectations
E: B4 A
Let of
88
YASUO WATATANI
index-finite type. Throughout this section we shall assume that
Center A
and Center B consist of scalars. Simple C*-algebras will do. Therefore if E
is in £O(B,A), then E!CB(A) : CB(A)-) CA(A ) "" Center A is
a
fai t hful
state.
Lenuna 2.12.1. If ~O(B,A)~ $, then there exist s EE; €O(B, A) suc h
that
EjCB(A) is a trace.
Proof .
Since
dimensional norma lized
positive
Center A
Proposition
by
trace
x
~CB CA)
=
F(h- l / 2xh- l / 2 ) .
elemen t a
Define
U,
the
2.7.3.
on CB(A) .
1
invertible .
=:;
C*·algebra
Hence
there
CB(A) exists
Take any F in t o(B.A).
h in CBCA) s uch
that
expectation
conditional
=
Eh .
F(x} E
,
t
=:;
B-.?A
is
for
(h-x.)
by
a
E(x)
Q. E.D.
By Then
Proposition
E«h-l)x)
=
faithf u l and h- l EO C (A ) , h-l
B
The
fo l lowing
c onditional
Theorem
= FiCB(A),
that
1.4.1., there
Eh (x) ~
O.
-
E(x)
that
hECB(A)
0 for x
Hence F
shows
exists
=
we
E.
HE
= Index E
tha t
such
Since
E
is
Q.E.D.
can
minimize
exp e ctati ons of s imple C*- a lgebras onto simple
Pr opos i tion 1.2.9 and HE(l) shows
faithful
= F.
Proof.
F
a
finite
Then there
Lemma 2.12.2. Let E and F be in 'O(B,A). If EiCB(A) then E
is
indices
C* - algebras ,
Remark after Proposition
is exactly coincides with &- 1 on CB(A) if B
of
and
2.5.3.
A
are
Suppose that Center B and Center A ar e scalar s
and
factors.
Theorem 'O(B.A) ~
(1)
$.
2.12 . 3.
Then the following hold,
There exists a unique conditional expectation EOf. ': O(B,A) such
Index EO
= min{Index
E ;
E ~ €O(B.A)}.
that
INDEX FOR C*-SUBALGEBRAS
89
(2) If E6s 0 (B,A), then the fOllowing are equivalent: EO'
(i)
E =
(ii)
EICB(A) and HE are traces on CBCA) and HE = (Index E)EICB(A).
= cEjCS(A)
(iii) HE
for some constant c.
(3) I f CB(A)\C, then (Index E ; E<;sO(B,A)} = [Index EO'oo).
First we shall show that there exists EO~ EOeStA)
Proof.
(iii) of (2). is =
a
By Lemma 2.12.1 there exists E<& 'O(B,A) such that
trace.
E)CBCA)
Then
HE is also a trace.
In
fact,
since
for any unitary u in CB(A), uEu* :::: E by Lemma
HECK) "" HE(u"'~xu) for XC-CBCA) ,by PropoSition 1.4.1. central
satisfying
projections
£1'£2""
,fm
positive invertible element h ~ Li aif
i
uEu* I e (A) B
2.12.4.
Hence
Now choose a minimal
.ti fi :::: 1.
in CB(A) with
EICB(A)
Define
- = "'1.
(E. E(f.)1/2H (f.)1/2)-l H (f.)1/2 E (f.)-1/2 (1"< ) J J E J E 1. 1. dl~m 1 2 1 2 h / Eh / . Then EO ESO(B,A) and HE (x) = HE(h-l/2xh-l/2)
=
Let EO
for XbCB(A). cEOCf i )
=
Put c = cE(hf i )
(L.
J
a
in Center CBCA) by •
E(f.)1/2HE(f.)1/~)2 J
= CaiE(f i
)
=
Then J -1 a HE(f ):::: HEo(f )· i
i
i
Since HEo and EO I CB (A) are traces, HEo "" cEO IC (A).
Then
B
c = cEO (l) = HE o( 1) = Index EO' Hence HE Now
o we
(Index EO)EOICB(A). shall show that the above EO satisfies
(1).
E €-'O(B,A).
Then there is a positive invertible element
that
EO(hx)
E(x)
for
x<=CB(A).
--~E-(-hlL2xh.lf-2.) for xc;- B. 0
{(Ui,U i *) ; i
=
1, ... ,n}
Since FICB(A) be
Define
Take
another
h('CB(A)
FE- 'O(B,A)
by
such F(x)
EICB(A) , F - E by 2.12.2.
a quasi-basis for EO' Then
by
Let
PropoSition
1.4.1. and 1.4.2., ;;;Index EO' by
nequality. some
Hence
cons tant A.
the equality hold if and only if This means that A "" h ::: EO (h) "" 1.
hold if and only if E
schwarz. =
Ahl/2
Hence the
for
equality
EO'
We shall show the equivalence in (2). (iii)-=7 (i):
h- 1/2
Let E~ sO(B,A) satisfy (iii).
(i)'9 (ii) ~ (iii) are clear. Then c
=
HE(l) ""
Index E.
90
YASUQ WATATANJ
Define =
a
positive
invertible
element
h
c
EO(hl / 2Xhl/2') for x6 B as in the proof of tI).
GS(A)such
t hat
E(x)
Then
HE(x) = HE . (h - l/ZXh-l / Z) for xCCB(A) . And
Index EO
=
'Z
= (Index E)EO(h ) Hence 1
E
a
EO(h Z ).
= HE (h)
HE (1) ~
(Index E)E(h)
Z
(Index EO)EO(h ) .
Therefore E ((h-l)Z) = O
EO(hZ) - 1 , 0 .
Sinc e
h = 1,
= EO' (3):
Assume that GS(A)
~!C.
PI and P2 in CS(A). with PI +
Pz .:
Then there exists non - zero
1.
and 0:: 2 such that h '" alPl + (l.2 P 2 and 1 E(x) = ( h 1 / Zxh l / Z) for x" B. Then
0.
Choose a certain positive E(h) '" 1.
scalars
Define E ~ cO(B ,A)
Hence Index E can take any real number in [ Ind ex EO' w) .
Definition 2 . 12.'"
projections
by
Q.E . D.
In the above setting we define the index
[B'A\,= min[lndex E; E e oO(B,A)) .
Example Z. l Z.S.
(1 )
[Mn(C),C).= n Z .
(2)Let B and A be simple C*a l gebras generated by Jones project i ons as 2 Example 2.9.1. Since c O(B , A) is o ne pOint , [ B:A)o= 4cos 1f/n .
in
CHAPTER
m.
TRANSFER
3,1. Induced and co-induced modules Let
Consider a
A C B be k-algebras with a common identity.
module M.
left
A
Define induction and co-induction from A to B as usual by Ind! M
~ B@A
M
and COind! M ""'HoffiA (AB'AM).
Then
Ind~ M and COind1 M are left B-modules b ,(x@m)
bx®m
(b.f) (x)
f(xb)
for b,x E: B,
f 6- HOffi (B,A), A
E, put B "'" f[C] and A
He G are groups and k
If
by
~
l[H].
Then we
find
ourselves standing in the setting of the (co)homology theory of groups. The
following
Proposition is essential to define transfer maps
(co)homology theory of groups.
Proposition finite,
then
3.1.1.«(7J
in
the
See for example [7;p.8l and p.lO).
5.8.)
Let M be a H-module.
If
IG,H]
is
Ind~ M ~ Coind~ M, that is,
EiG]0zIHJM'2 HomlIH](.£[G],M)
as left ZIG]-modules.
We shall generalize the above proposition to meet our situation.
Proposition 3.1.2. Let M be a A-module and E : B7A be a conditional expectation
of index-finite type.
Then we have Ind! M ~ COind! M,
that
is, there is a left B-module isomorphism ~ : B @A M"'""7HomA.~(:cB,-,M=),-._ _ _ __
Proof. ¢
,
Let {(ui,v i )
Defin8
B0A M->HomA(B,H) by ¢(b0m)(x) " E(xb)m
Then
1, ... ,n} be a quasi-basis for E.
i
~
is a left B-module homomorphism.
~(d(b@m))(x)
Define
for x';B, b0m ~B0A M.
¢
,
~ .(db@m)(x)
In fact for d t B
~ E(xdb)m" (d •• (bQ)m))(:<).
HomA(B,M)-clB ®A M by ¢(O ~ Li ui<Sf(v i ) for fEHomA(B,M) 91
1
I~
i
92
YASUQ WATATANI
We shall verify that tP "" $-1,
For b0m,:;;.BGDA M and fE HOffiA(B,M),
liH ji(b(6lm) "" Li u @1J(bCom)(v )
i
i
(L
[i ui@E(vib)m
uiE(v b) 10m· b@m. i
i
(¢~(f))(x)· ¢(~i
And
=
ui@f(vi))(x) • ' i E(xui)f(v i )
(since f is a left A-module homomorphiSm)
[i f(E(xui)v ) i
f(x)
for xf:.B.
Q.E.D.
Remark.(l) ------
.
The definition !/J(f) "'" t. u. ®f(v.;) of the above ~I does not ~ ~ -I
depend on the choice of the quasi-basis, since (2 )
by
B-module
HOffi/"I-(BA,M ) A == mE(bx)
and b,xGB.
for
isomorphism factors
x E:B
~
Let M be a
We also have a right B-module version:
Then
~
right
(f·b) (x) •
B-module.
f(bx) for
f '"
Define ¢
m
and
and $-1 is given by $-1(£)
Then
':i
¢
is
a
f(ui)'$v i ,
right
B-module
If H and
of type III and M "" B. then ¢ is exactly the map
A
are
biG b 2 \-oj b i eb 2
defined by Jones in [37].
Corollary
Let
3.1.3.
index-finite type.
E: B-)A be a
expectation
for b EoB and" "'B.
Then ¢ is an isomorphism as
Put M
B-h.-bimodules.
Q.E.D.
A.
In [29], the authors called E is~ faithful if the
Remark.
of:
Define $
¢(b) (x) • E(xb)
Proof.
conditional
right
B-module version of Corollary 3.1.3. holds.
Corollary
3.1.4.
Let E : B-'7 A be a conditional
the following (1) and (2)(=
(2~)
+ (2r»
expectation.
Then
are equivalent.
(1) E is of index-finite type. (2)
(2t)
B
is
a
finitely
the left B-module map ¢k,
generated 1JlQ,.
:
projective
left
A-module
B-?HomA(AB'AA) defined by
(b) (x) "" t:(xb) is an isomorphism.
and
.,
INDEX FOR C*-SUBALGEBRAS
93
(2r) B is a finicely generated projective right A-module and
the right B-module map ~r : B-7 HOffiA(BA,AA) defined by iJlr(b) (x) "" E(bx) is an isomorphism.
Proof.
(1)~(2):
This is exactly Corollary 3.1.3.
and
its
right
B-module version. (2)~(1): Since B is a finitely generated projective left A-module, there exist vi E:-B and f1 !CHomA (AB'AA) for i
x Eo B.
for
= [(XU )
i
Since
for i
0/)1. is surjective,
there exist
£1 f1 (X)v
such
lli
that
Hence
1, ... ,no
=
"" 1, ... ,n such that
]
]
Therefore E is of
fiCx)
index-finite
for x,,; B.
]
Thus B<1JA B has a left identity and a right identity, so that
identity.
"" x
Similarly,
there exist s]' ,t].':;B for j = 1, ... ,m such that L. S.E(Lx) "" x
an
i
type
b,
B
B
has
c..oyollQ..\'j 1,2.."/. Q,IO,O.
Remark,
If we consider C.,Lalgebras, then either one of (2£.) or
(2r)
is enough to show that (2)"'9 (l) .
3.2.
Transfer maps in Tor and Ext. Let
H be a subgroup of a group G and MaG-module.
the canonical inclusion. Res G H and
Cor
If
G
H
[G,B]
,
Let i
Hn(G,M) --> Hn(H,M)
called restriction map
Hn(H,M) -) Hn(G,M)
called corestriction map.
00
,
H-> G
be
Then there are natural maps
then there exist maps called transfer maps,
going
in
____~t~l~'e~r~e~v~e~r~s~e~d~i~r~e~c~t~i~o~n~'~--_:~--------------_____________________ _______ __
and In or'iginai
~ Hn(G,H)
Corg
Hn(H,M)
Res~
Hn(G,M) -:7 Hn(H,M).
particular
V "'"
Res~
group-theoritic
: Hi (G,Z)-;:G/[G] -)H
1
(H,l)~H/[HJ
transfer map:
is
the
(disjoint
union). For x f:-G, we have xg i Then V
G/iGj -,) Hf(H] is given
Reea 11 that lIn(G,M) " section
for hi E:-H, 1 ; ;: i ;;; n, 1 ~ Tr(i) ::>n.
;:, gl!(i)h i
sxt~(G' ~
J
by V(x[G]) " (IT (l,M)
i
hi) [H].
and H (G,M) " Tor ZIG ] (l,M). n
n
In this
we shall extend transfer maps in the homology and cohomology
of
YASUO WATATANI
94
groups to transfer maps in Tor and Ext of modules. case
of Ext.
the
First we consider the
We need the following Ext version of "Shapiro's lemma"
cohomology
of groups, whi.ch might be known.
In
the
in
following
we
assume that A is a k-subalgebra of a k-algebra B.
Lemma 3.2.1. Let N be a left A-module and M a left B-module. that
B is projective as a left A-module.
EXt:~(AM'AN)-7 Ext~(BM,COind~
S11
Recall
N
the
Then we have
a
Assume
k-isomorphism
N).
Hom-tensor relation: for
L
AN, we have that HOffi (L A
Put L
B.
C6\
H, N) ~ Hom (M, HOID (L, N» A B
.
Then we have an isomorphism s°
Horn (AM'AN) A
~
HOr.1 (M, Hom (AB,N». B A
Let
1,
(P)
be
a
projective
Coind! N
=
1"
,
... -'> P z -7 P1 -? Po -" BM-) 0
resolution
HOIDA(AB'AM)
as
a
of
BM
as
a
left B-module and
Consider
B-module ..
left
form
the
sequence
of
k-modules
f,~
1,*
0-) Horn CPo' HOffi (AB,N» --:)Hom (P ,HomA (AB, N»-7 A B l B
Then
we have that
r:xt~(BM,COind~
N)
¢n+l~'
Ker
""
hand, since B is projective as a left A-module, projective resolutiOYl
(P)
and Ext~(AM'AN) "" Ker Qn+l~/Im ¢n~'
g~HomA(p,N),
n
(0)
-1
a
can be regarded as
0'-7 Hom (PO' HomA (AB, N» B
leO J..
- - 7 ROffiA(PO,N)
1,'
~ Hom (PI' HomA (AB, N» B
.p,~
l
)
...
(g) (p) (b)
PI2P and b
following commutative diagram:
o
other
Consider isomorphisms
and
that
ff::.HoffiB(Pn,HoffiA(B,N»,
the
of AM as a left A-module, so we have that
f>t'" cp;"" o -7 ROffiA (P 0' N) -'7 Rom (PI' N)---; ROffi A (P 2' N) -> A
such
On
el
HomA(Pl,N)
= g(bp)
Then we
have
4>; ---j
~~
~ ~
Hence we have isomorphisms
Q.E.D.
for the
INDEX FOR C"-SUBALGEBRAS
From
now
on,
we
always
that
expectation
of
Consider
left B-module homomorphism
a by
defined
index-finite
a~sume
xHlbHbxJ
Def:h.0j.:....cion 3.2,2.
for
type.
E
95
B -)A
Let M and
N
is be
a
conditional
left
B-modules.
f
x (;:- Nand b G: B.
£
Then
induces
f'>\:. n
We defj,rle the natural map
by the composice
Res~~ "" (srl)-lof n
"
wher,~ sn
In
j.
s an it:>omorphism defined in Lcmtrna 3.2. 1. • ("Shap i'l:o' s lemma.)
particular, ,-,hen n "" 0,
T(::stl-iction.
More generally,
Res~
Hom g 01,N)-7> HOffiA(H,N) is
He cl.aim t.~",..t R8Sf
j"s just the "reser'ieeion" in the eochain level.
JUSt
the
E:xt~(M,N)-) Ext~(M,N) In fact,
let
(P)
be
a proj0ctiv0 resolution of M as a left B-rnodule.
Since (l) can
also
be
r0garded as a projective resolution of M as a le~t A-module, we
have
the following COlmnut8.ti.ve d.ial_;l~amt:
~
0-----,>, HomB(PO,N)
1
fO
is f
N-)HomA(B,N),
no n (l f
nhiot-lbnj,
Therefore
the
the
eochain
HomB(Pn,N)~HomA(PI1,N) induces a k-homomorphism Hence
HomB(Pn,N),~ HomA(Pn,i~)
we
need
then
("nof l1 Cr )(p)
(0 11 0 (f~r»(p)
« f
lb....,br(p») (1)
1') (p) ) (l)
This s1'1o\-"$ che clai.m.
to
show
induced
by
homomorphism
Res! '" 8110£*-n that
}\ £n
is just the "restriction of scalars" from b to A.
Let r<sHom]3(Pn,N) and P~Pn'
0
only
map
r (p) ,
96
YASUO WATATANI
Nextly we shall define the transfer map B
going
n
n
CorA' Ext A (M,N) --? ExtB(M,N) the reverse direction. We need to define a
in
homomorphism h h ' g •• -
left
B-module
COind! N "" HomA(B,N)~ N by the composite 1 t-, ~ ,HOIDA(B,N)-'-jB
GlA
N-'7N
where CP;B@AN-,}HoffiA(B,N) is defined by lI(b@y)(x) "" E(xb)y for x6.B
b~YG:::BG9A N
as
in PropOSition 3.1.2. and g : B®A N......,N is
g(b@x) "" bx for b~B and x&N.
q;-lCr) "" [i
N•
i
by
defined
Since
Q9r(v ) i
HOffi (B,N)-7 N A
h(r) •
By
U
and
for r GHOID (B,N), A
is given by
g •• -l(r)
Proposition 3.1.2., h does not depend on the choice of quasi-basis.
Ext~(M,COind~.
Then h induces h;\n :
Definition 3.2.3.
Ext~(M,N).
We define the transfer n
B
N)-'7
~
n
CorA' ExtA (M,N) -7 ExtB(M,N) by the composition n n CorAB "" h ~"s :
In particular when n
cor! :
0, the transfer map
HOIDA(M.N)
is given by rl-:>[rnHl:
Remark.
=
i
~
HOffiB(M,N)
~ir(vim)l
for
r~HomA(M.~)
and mf:.M.
The well-definedness of the transfer map corl for n
be also checked directly as follows:
o
can
For r E:Hom (M,N) A
cor!cr)(bm) "'" l:i uir(vibm) "'" Ei uir(l:j E«vibuj)vjm) Lj
l:i UiE(vibuj)r(vjm)
~
l:j bujr(v j
More ~enerally, we shall express the transfer map corr in the cochain level
as
follows:
commutative diagram:
As
in
the case 'of
Res B A
we
have
the
following
INDEX FOR C*-SUBALGEBRAS
h: Hom CB,N)-7N A
such
that
her)
=
Li u i
r(v i )
97
for
rI;';HoIDA(B,N).
the transfer map CorI is induced by the cachain
Therefore
homomorphism
hnO(en)-1 for rG-HomA(Pn,N) and P n n n -1 )(r)(p) n 1 (h 0(9) ~i u «(9 )- (r))(p)(v )) i i
The
following
proposition shows that the composition
is
induced from the multiplication map by Index E.
Theorem F : C -7B
3, 2 . 4 . be
Let
ACBee
be
k-algebras.
Let
E : B_,A
Then
conditional expectations of index-finite type.
and
we
have the following:
(1) Let M and N be left C-modules. B
C
ResA"Res B C
and
n ExtC(M,N) --> ExtAiM,N)
A
C
B
Cor B oCor A
Ext~(M.N) -)Ext~(M.N).
CorA
(2) Let M e.nd N be left B-modules.
01
by the mUltiplication map
Define a left B-module map
q : N -7 N
=
(Index E)n
q(n)
Index E eCenter B. i.e.,
n Let q*n : Ext n CM.N)-"7 ExtBCM,N) be the map induced by q. B
for nf:-N. the
Then
n
C Res
composition Cor! .. Res! : Ext~(M,N) -? Ext~CM,N) is exactly
Then
q~n.
In
particular if Index E is in k, then n
for xlG ExtBCM,N).
Proof.
(Ult is enough to show the equalities in the cochain level.
Let (p)
,
a projective resolution of M as a left C-module.
be
just
Since Res! is
"restriction from B to A" : HomBCPn.N)-) HomA(Pn,N) in the cochain level. it is clear that
B
C
ResA~ResB
~
C Res A.
Let {Cui,v ) ; i'" l, ... n} be a quasi-basis for E and {(Sj,t ) ; j i j 1 •...• m} a quasi-basis for 1<"'. Then {(SjU i , Vi tj) ; i "" 1, ...• n, j
"
I, ... ,m)
is
a quasi-basis for EoF by
B u : HOffi (Pn,N) -7 HOffiS(Pn,N) by cu!cr))(p) A A C Similarly define u B r E:HOffiA(Pn.N).
Proposition
1.7.1.
Define
for p 6- P n
and by
YASUO WATA'fANI
98
(U~(W)(P) We
also define u
level.
= >:i 'j
corZ are
cochain level.
W fHoffiB(Pn,N).
: HOffiC(Pn,N) -)HOffiB(Pn,N) by
A
(u~(r»(p) Cor~ and
Cor~,
Then
C
for p'P n and
'" ):j SjW(tjP)
sjuir(vitjP)·
u~
induced by the maps u!,
Hence It is enough to show that
u~ou!
u~
and
'" u~
in
the
in the cochain
For reHOffiA(Pn,N),
(u~ou!(r»(p)
=
Zj Sj(u!(r»(tjP) = 'j Sj('i uir(vi(tjP»)
Zi L j (sjui)r«vitj)p) C B This shows that COTSoCorA
=
C
uA(r)(p). C
CorA'
Let
(2:)
(P)
iX : HOffiB(Pn,N)
be a projective resolution of M as a left B-module. Let -jHOffiA(Pn,N)
be
just the restriction
HOffiB(Pn,N) be a map given as in (1).
Res!
and
cor! respectively.
HomB(pn,N)---, HomB(Pn,N) the
other
hand
map
B uA
and
Then cochain map
Hence the composed
induces
B
iX
HomA(Pn,N)-j
and u!
map
cochain n
B
induces
u~oi!
n
On
CorA"Res A : Ext B CM,N)--7 ExtB(M,N).
the multiplication map q : N -7 N gives
a
cochain
map
(Index E)r(p) for&r HOffiB(Pn,N) and plS-P , n
It is enough to show that
In
fact, for rG.HoffiB(Pn,N). we have that
(u!oi~(r»
(p)
(Index E)r(p)
(u~(r» (p)
Zi u i Y(viP)
q-(p).
Q.E,D.
This proves the Theorem.
-- - Ne*t: - --we - --s-ha-1-l----0on-s-i-de·); --'t-he---c-ase---·o-f----To-r--.-- --1'-he---_fol_low_i_ng__ is ___ the __ Tor_
version of "Shapiro's lemma"
Lemma 3.2.5.
Let A be a k-subalgebra of a k-algebra B.
B is projective as a right A-module. ri6ht B-module.
sn
Proof.
Let
Assume
that
N be a left A-module and
M a
Then we have a k- isomorphism
TOr~(MA'AN)-? Tor~(MB,Ind!
N).
For n "" 0, we have a trivial isomorphism So
' HI;
Gil,
AN -7 MB®(B@A N).
INDEX FOR C*-SUBALGEBRAS
99
Let (P)
,
be a projective resolution of M as a right B-module. projective
resolution
of
Then CP) is also
M as a right A module
a
Consider
B ®A N as a left a-mOdU;:, and form ;~e sequence of k-modules - ; P2 AN B
Ti2
...
- ; P2
GiJ A
N
1\~
1i,
PI
0A
Ti,
t,-" Po 0 A
N
Hence we have an isomorphism sn
-,;0
N - 7 O.
TOr~(MA'AN)-7 Tor~(MB,Ind!
N).
Q.E.D.
We
still
index-finite
Consider
type,
a
canonical
g(bQ;l;;) "" bx
TOr~(M>N)
assume
that E : B ---YA is
a
conditional
expectation
Let N be a left A-module and M a right left
B-module
foy bf..B and x<:'-N.
map
B-module.
g: B Q>A N ----J N
Then g induces
of
defined
by
g'''n: TOr~(M,B<2)A N)~
.
Definition. 3.2.6.
COT~
:
We define the natural map
T01-~(M,N)-:1 TOr~(M,N)
by the composition
Cor B "" g'''', ~ s
Ann
TorA(M, N) -7 TorE (M, Ind B N) -) TarnB (M, N) , n
n
f'\.
where sn is an isomorphism defined in Lemma 3.2.7. In
particular, when n ::::: 0, CorX :
m@A n t--?ffi
G'h
n
for
mf:M
and
("Shapiro's lemma).
M~N~M ~N is just the n~N.
Similarly
we
natural can
map
write
._~_Q_:f!__ _TQ£~1tLJU _____ 2__ 1'9_~!_Lr:-LJ'lL __l._n.___ t_h_e____cbain ___ l.e_v_e_L ___LeL_ (P), be
... -;p 2 t,p[ t,P 1"'M~' 0 o
a projective resolution of M as a right B-module.
Then we
is the map given by
have
the
100
YASUO WATATANI
for p 6- P, b G Band n Then
the chain map gn in : P n 0
0A
N -) Pn B N induces
TOr~(M,N)4TOr~(M,N).
cor! "" g*npsn:
~
the
We see that
N. k- isomorphism
gnoin : Pn@A N-'7
PnQ)B N is the natural map given by
"%
p ~A n '-> p
1 ~ An.., P @B n.
Next we shall define the transfer map
TOr~(M,N) -7 Tor~(M,N)
Res!
going
in
the
reverse direction.
We need to
define
a
left
B-module
homomorphism w : N -) B ®A N by the composition -1 f 1>-' where
o f , N-c>HomA(B,N)-,-, BOA N,
$
W·
B~A
N-7Hoffi (B,N) is the map given in Proposition 3.1.2. and A f : N-;)HoffiA.(B,N) is given by X 1-) [bl---7hx] for xE-N and bE-B. Since ¢
cp-l(r) '" I.:
i
wen)
By
ui@r(v ) for rG:-HoIDA(B,N). w is given by i
q,-l(f(n»
==
=:
3.1.2 .•
Proposition
Li u
i
@f(n)(v ) i
not
does
w
Definition 3.2.7.
depend
on
the
choice
of
the
Tor~(M,N)-7 TOr~(M,B<©A N).
Thus w induces w* n
quasi-basis.
Ei ui@viu.
=
We define the transfer
~
Res! : TOr~(N,N) -)TOr~(M)N) by the composition
sn~w*n: TOr~(M,N)-:fTOr~(M.B®A N)-7TOr~(M,N).
Res! "" In
particular
when n "" 0, the transfer
map
Res!: M Q), N ...., M@A N
~A v~t')).
given by m@B n.....,1: i (mu i
B
Remark. The well-definedness of the above map Res A can be also checked directly as follows: £:i mbui@A vin "'" 1: i m(£:j ujE(vjbu i » Ej
0 A vin
(muj'@A I: i E(vjbui)vin) "" I: j mUj@A vjbn.
More generally we shall express the transfer map Res! : in the cochain level. N •.. -') P2 "" wB
TOr~(M,N) -)TOr~(M,N) We have the following commutative diagram; 41./\:-.. J
1
i2
1
~
P,")
N
Po (8JB N -'I 0
" 1w,
tW2 -,)P2@B (BGJA N)
- " P20A N
P
I',~ P 1 q(BIi\)A
.p,~)
V,
P 1 0" N
1"-', N)!l; Po
N) -; 0
Fa
'f~ -.!.."
GJ~(BQA
Po
QJ Ie N
--l 0
is
a
INDEX FOR C*-SUBALGEBRAS Therefore
the
chain homomorphism
k-homomorphism Res! :
lnOWn:
TOr~(M,N) ~ TOr~(M,N).
inwn(pQn) "" ino.:i P@(ui®vio»
The
following
Theorem
P
0B
n
101 N -? P2 ®A N
induces
a
Here we have that "" ti pui@vin.
shows that Cor!~Res!
is
induced
from
the
E : B~A
and
multiplication map by Index E.
Theorem
3.2.8.
Let
be
ACBee
k-algebras.
Le t
F : C""""B be conditional expectations of index-finite type.
the
followin~:
(1)
Let M be a right C-module and N a left C-module. ResB"Res C
Res
Cor~oCor!
COT e
A
and (2)
Let
B
C A
Tor~(M.N) -J TOr~(M,N)
A
Tor~(M,N)
--p
TOr~(M,N).
B-module map q : N'-7 N by the multiplication q(n)
=
a
Index E c::::Center B,
i.e., map is
(1)
=
B
(Index E)x
~y X~ Torn (H,N).
It is enough to establish the equations
in
the
chain
Let
(P) be
left
In particular if Index E is in k, then
cor!'Res!x
level.
a
Then the composition Cor~"Res! : Tor~(M,N)-:;l TOr~(M,N)
induced by q.
Proof.
of
Define
Let q*n ;TOr~(M,N)---1TOr~(M,N) be the
(Index E)(n) for nf:N.
exactly q'l
Then
be a right B-module and N a left B-module.
N
Then we have
... --7P -lP ->P ->M-.,0 2 l O projective
resolution of M as a right C-module.
by: the chain ,ffi?P gn°;ln .:. }!n@A N :-7,PnG)B N. such
induced
P~B n, it is clear that Cor~"'Cor!
=
Cor~.
Since that
is ,p
Gb n ~ A
; i "" l .... ,n} and {(Sj,t ) ; j "" l .... ,m} be quasi-bases j E and F. Then { (sjui,vit ) ; i=1, .. ,n, j=l, .. m) is a quasi-basis j C EoF. Define vB Pn Gl N-7 P
Let {(ui'V i )
for for
e vA
,
e
,
,
[i ps/:il tjn Mel VA(p6!>e n) = . v for plOP and n ~N. Then ~,j pSjlii0 i tj B C e and e in the the Res , Res and Res~ are induced by the maps B A B vA' vB vA e = e chain level. Since it is clear that VABoVB VA' we have that
B
e
ResA'Res B
Res C, A
102
YASUO I'ATATANI
(2) Let
be a projective resolution of M as a right B-module, P n @A N map
bYP!(P<SB
be the natur8,l map defined
given
respectively.
Hence
the
composed
chain
. B B B B ~nduce)CorA·ResA : TOTn(M,N)--jTorn(M,N).
Pn@B N7
n) "" P®A n and v! be
cor!
Then chain maps p! and v! induce
as in (1).
p!:
Let
the
and
Res!
B B Rl 9A ovA , Pn "'1lN ---7 Pn,%N
map
hand
On the other
q , N-) N
gives a chain map q~ : MGi1 N-1 M ~ N suchtkt q~(mren) "" m&.Index E)n.
It is enough to show that p!~v!
=
q-
This is in fact
p!v!(m($B n) = o!U i mU i @A Yin) = ' i mU i uivi)n '" m~B
m0s U i
0B
Yin
(Index E)n = q-(m'@B n).
This proves the Theorem.
We shall give
Proposition index-finite
M
Q.E.D.
o.pplication of the transfer maps.
3.2.9.
type.
Let E : B -J A be a conditional
Assume that Index E
expectation
is invertible in B.
of
Then
we
have the fOllowing. (1)
o
If Ext~(M,N)
Let M and N be a left B-module.
for some
n,
then
n ExtB(M,N) = O.
(2)
Let M be a right B-module and N a left B-module.
for some n, then
Tor~(M,N)
=
an
isomorphism.
isomorphism.
Hence
o
O.
Proof. (1 )Since Index E is invertible, q : N"7 n
is
TOr~(M,N)
If
n
'1--)
(Index E)n f N
Ext~(M,N) -) Ext~(M,N)
q....
is
also
an
By Theorem '3.2.4., we have that'
q / "" Cor!"Res! :
Ext~(M.N) ~Ext~(M,N)""
Therefore Ext~(M,N) "" 0.
0 -)
Ext~(MIN).
(2) is similar by using Theorem 3.2.8.
Q.E.D.
If normal F =
we
consider factors B
conditional
(Index E)-lE- 1
~
expectation B"~
A"
is
A on a Hilbert space H E
,
also
B-jA
with
a conditional
and
a
Index E
<
faithful 00
expectation
proj ection of norm one. Note that A' = HOIDA(H,H) and B' = HOIDB(H,H) .
Then
or
a
Now
we
INDEX FOR C*-SUBALGEBRAS
[03
shall generalize the existence of such a projection F to the case
of
higher dimensional functors Ext
D
instead of Hom.
It also gives an analog
of a result in G-functors by Yoshid.a([791 ;Lemma 3.1.).
Proposition index-fintte
3,2.10.
type.
Let E :
B~A
be a conditional
expectation
of
Assume that Index E is an invertible element in k.
Then we have the following: (l)Let
M
direct
summand
Then Ext~(M,N) is isomorphic
and N be a left B-module. of
Ext~(M,N)
and
the
projection
is
to
a
by
given
-[ B B (Index E) ResA,Cor A, (2)Let M be a right B-module and N a left B-module. Then TorBCM,N) is ison
morphic to a direct summand of TOr~(M,N) and the projection is given
(Index E)
Proo~.
By
(1):
Ext~(M,N).
-[
B
by
B
ResA-Cor A ,
B Theorem 3.2.4., CorAB ResA(x)
(Index E)x
for
in
x
Since Index E is invertible in k, Resr is injective and n For ye-ExtA(M,N) ,
is surjective.
Corr(y - (Index E)-[Res~'Cor!)(y))
-
Cor!(y) cor!(y)
(index E)-lCor!oRes!'Cor!Cy) -[ B (Index E) (Index E)CorACy)
o.
Since Y
~
(Index E)
-[
B
B
Res A CorACi) + (y - (Index E)
-[
B
B
Res A CorA(y»,
we have that Ext~(M,N) ""'(Image ReS!)+(Ker corr) . y be in (Image Res~)
Let
that y ~ Res!(Z).
o=
cor!Cy)
(I
(Ker Cor!), then there is z6Ext~(M,N)
such
Then Cor!-Res!(z) = (Index E)z.
=
Since Index E is invertible in k, z lImage Res~)l\lKer Cor!).
=
O.
Thus
o.
Hence
E:;(t~(M,N) ",,(Image Resrl~~er Cor!} and the projection is given
(Index)
-[
B
II
ResAoCorA.
The
i)1"OO£
of (2) is similar to
by
Q.E.D.
(1).
3.3. Transfer maps in the K-theory of C*-algebras Let identity
B be 1.
a C""-algebra and A a C*-subalgebra of
B with
Let i : A -? B be the canonical injection.
Then
p...
i
(OlO\'v>\o'f't
induces
YASUO WATATANI
104
l;l(.: K (A)-7 KO(B). O
The aim of this section is to construct a map
in the other direction, called a transfer
~.
Throughout this
going
section
we shall assume that there exists a conditional expect8.tion E : B ----;> A
index-finite type. and {(ui,u *) ; i i
Recall
in
that W :
fact
A~
of
= 1, ... ,n} is a quasi-basis for E.
C* is an injective *-homomorphism A
let J be an ideal of C* containing
Ae , A
defined
then
by
J:> BeAB
"" C"«B,e >· Therefore 1).! induces an isomorphism <{J"+: KO(A)-;> KO(C*
K (A)-7 KO(C*
cOnstructiOn
directly
reduced
basic
C~;r'
Definition 3.3.1.
The transfer
-1 composition TE :;;; (t)it-) <>
the
by the definition of
j~,
~
where j
TE : KO(B)-7KO(A) is defined : B-7 C* is the A
by
canonical
embedding preserving identity.
~
3.3.2.
The transfer map TE does not depend on the choice of
a
conditional expectation E : B -? A of index-finite type.
Let
Proof. index-finite Then
type
F : B -) A
another
be
conditional
and C* the corresponding basic A
F(x) "" E(q*xq) for some q GCh(B) by PropOSition
expectation
of
construction.
2.10.9.
Consider
the following diagram;
C* A
1
B
9
C* A
where
e
C* -) C* is the isomorphism defined in A A
2.10.11., so that S(f ) A clear
=
qeACJ*.
The diagram is not commutative.
that jE -= ejF by Proposition
-7 KO(C*
2.10.11, so that
On the other hand for a~A,
"" qaeAq* and WE(a) "'" ae A .
Hence
Proposition
t/J
E ~ 81/!F'
jE~
It
is
"" 9o)(:jFt-! KO(S)
8I/JF(a) "" e(a~Ji;.) "" aqeAq*
But they induce the same map
tVE "" 8-tI/!F-+ : KOCA) ---? KO(C*
INDEX FOR C*-SUBALGEBRAS
105
have that
and (qpeAl*(qpeAl ~ eApqi'qpeA ~ peAq*qeAP ~
peAP
Hence
peAP
~
e~WF~([pJ)
peA
=
pE(q*qleAP
~ ~E(pl.
WE~lP])
for a projection p in A, and
equation holds for a projection in A®M n , commutative diagram:
Therefore we have the following
Hence TE
Q.E.D.
Remark. not
the
similarly
We may write T
depend
= TE for the transfer map, because
on the choice of conditional
expectations
of
does
it
index-finite
type.
Example
3.3.3.
E : B "" c'keS ) -7 A 3
we
have
the
Let =
Sn be the symmetric group of
C*(S2)
following
degree
n.
Let
be the usual conditional expectation.
Bratteli
diagram
for
C*(S2)
Then
~ C*(S3)
i.,
C*:
1
/ 1 \
1 /
...... /" 2
3
i""
KO(Al-j KO(Bl is given
by (:
1
3
~ ~t ~£.
Then KO (Al
"
.,-
~
Therefore the natural map
land the transfer
I I 0)
by (0 I something like a "transpose"
observRtiaV'l
I
Hence the
of the natural map
is essentially done in Jones [37].
i~.
The
transfer
T
is
above kind
of
YASUO WATATANI
106
We some
We
shall investigate the transfer map T more precisely. known
facts
Blackadar
which
are
essentially
stated
in
review [62 J ,
Rieffel
(4 j . E(Ui*U ). j
Define p
Then p is a projection,
In fact
E(Ui*U ) "" Pij =(P.,'r)ij' j
p(AD)
Therefore A-module
is
Consider
a Hilbert A-submodule of AD.
2 • (:0
Then o.(B) "" p(An), because
bE;-B has a form b ""
a
Hilbert
for a. f: A and
Lj uja j
J
(E(ui*b»i"" (E(Ui*(Lj ujaj»)i "" (E i E(u i *u j )8. j »i'
1
Therefore
by
given
0. 0.
is an isomorphism as Hilbert A-modules and
-1 «ai)i) ""
neLi uia i )· In fact "" Li E(ui*x)*E(ui*y)
E{x*t:
Hence
0.
i
induces
"" E(~(*y) ""
uiE(ui*y»
an *_isomorphism
'If
equation xeAy* "" 8 n (x) ,nCy) "" a
C*-) p(A<SMn)p
:
-1
the
0.-
e(in(}~) ,0.1I(Y)Ci.
such
that
In particular,
for
b<;-B neb) "" r.(l.:
k
bukeAu *) "" O::k E(ui-/
(~(ui *I.: bukE(uk *u j » ij "'" E(U i *bu j ) ij . k We shall summarize the above discussion, which also includes a result
by
Pimsner and Popa[58].
~
3.3.4.
index-finite
is
E , B.-I A
If
type and {(ui,u *) i
; i
~
a
conditional
for
E.
By
adding
W
B
--:>
A@Mnsuchthat
1, ... m}
Take another quasi-basis 0,
may
assume
that
n "" m.
Define
another
by 1j.J'(b} '" (E(Si*bSj»ij' Then there e~~ists n a partial isometry v€: A«lMn such that web) "" vw' (b)v* and w! (b) "" v*cp(b)v
*_homomorphism
for b G B,
w' : B~
we
~l
1, ... ,n} is a quasi-basis for E.
Then there exists an injective *_homomorphism web) "" (E(ui*buj»ij'
expectation
Al9M
l07
INDEX FOR C"-SUBALGEBRAS
Proof. and
a
Consider two projections p "" (Pij)i.j and q "'" (qij)ij IS AQVI ....
partial isometry E(8 ,,*8),) and v .. 1.J
v "" (v
ij
) ij~ A0Mn
J
1.
by Pij "" Ee u *u ), i i
Then v*v "" q and vv* "" p.
E(u.*s.).
=
defined
In fact
(v*v)ij - Zk E(sl*U k )E(u k *Sj) =
E(si 7Q:k ukE(Uk"~Sj»:::'E(Si*Sj)
= Pij'
and similarly (vv*)ij that VIjJ(b)v* = W' (b).
qij
=
Thus v is a partial isometry.
And we
have
In fact we have that
(V\jJ' (b)v*)ij "" l.k Eh E(u *sk)E(Sk 1(bs
1
h
)E(Sh*u ) j
[k E(ui*sk)E(sk*bE h ShE(Sh*u j » Lk E(ui'':"sk)E(Sk*bu j ) = E(ui*T. k skECsk*bu j )
E(ui*bujl - (.(bll ij · Similarly we have that
Let
E
Q.E.D.
~'(b)
E.
be a Hilbert A-module and E" II E..
Suppose that
Then the "rank one" operators 0s,E, and are projections by Rieffel [S8;PYoposition 2.8.J.
Lemma 3.3.5.
on, f1
In the above situation, if
,
then 8~,~ and
are von Neumann equivalent in K(~).
Proof.
Put v
Oc
=
{" n
c K! f).
v*v ~ (8 c ,n)*0 c ,n = e
'"
"
Similarly we have that vv*
.Lemma be
<~,~>
the
3.3. .
Let
Then we have that ~
e
n, t; S, T1
= 0 n,n
and
~
Q.E.D.
be as above.
Let h
canonical inclusion map and 0 : A --7 A(i;lM
o(al • aQe general:
U
.
n
be the map
such
We have the following diagram which does not commute
C>"«B,e > A
'" p(A@Mn)p ~
h Aq:)M ~
n
~A~ But it induces a commutative diagram in the K-theory:
KO(C;'
~
't'k
n
KO (A)
~
K (M1)M I O
("'~
C"
_______
n
that in
108
YASUO WATATANI
Moreover, maps are all isomorphisms in the K-theory.
Proof. in A)
h)('If,;1J;~
It is enough to show that
'}"-p.,
then
-=.
t-e.p! 'tl
On the other hand o(a)
a®e 11 ;
e c«n(q»,c«n(q»
Lemma 3.3.5.
Let q be a projection
(k'l'\J.
hn~,(q) = hn(qe A ) = hn(8 n (q),n(q»
Since
o~.
= 8 a (n(q),n(q»'
n
8~.~~ K(A ) , where ~
E(q*q) '" ECq)
=
q
projections
and Be,t" =A0M n , , are von Neumann equivalent in I/eAn) N
Hence
h~ 1f-{ 1JI-;(, [q]
by
"The general cas€,(q,A@Mk),is
'" o';lfJq].
similar to the above, because Hilbert A-modules a(n(A» are
(q,O, .. 0) G AU.
two
<1;,1;>,
=
=
isomorphic by the partial isometry v
and
= e x,y • where x
A.@OG,i ... 6)O =
(1,0,0,." ,)
Q.E.D.
Proposition index-finite
restriction M
=
MS'
Let E
type.
Let
Res[MBl
= [MAi
B -:; A be a
conditional
KO(B)-7KO(A) be the map
Res
expectation
given
for a finitely generated projective
by
of
the
B-module
Then the transfer T "" TE coincides with Res.
Proof. generated
3.3.7.
Recall that a projection q~ B0Mk corresponds to the finitely k k -1-1 projective module qB C B Since TE = (W*) .j~ ~ (o~) h~~~j* for a projection qc;;,B@M • qB k k are isomorphic as right A-modules. The Hilbert
by Lemma 3.3.6, It is enough to show that and
«~ I k ) (q) (Ank )
A-module
isomorphism
Ci
:
E. ""
B -7 P (An) defined before also induces
an
Bk __~ (p (An» k = (p <8 1 ) ((An) k) . Then we have k nk that o(k)(qB ) = ((n@Ik)(q)(A ). In fact, let q = (qst)st" B0Mk, then (k)----k-- - (k) k k o (qB) = a . (T(f~q~~b~r;E'jj· ; (01' :.:;15;iEiB ) T
isomorphism a (k)
,,,k
k
{(("t E(ui*qstbt»i)s "'(An)k ; (b l ,· .. ,b k )
€;
Bk).
On the other hand, we have that ((n<$ " k ) (q) (A
nk
)
(E(Ui*qstUj)ij)st(An)k {(tt(O::j E(Uii'qstujat(j »i)s {O::t «E(ui*qstbt)i)s ; b t <2:B
o(k)(qBk )
at(j)GA, t
=
t""l, .. ,k, j=l, .. n}
l,. .. ,kJ
Q.E.D.
INDEX FOR C*-SUBALGEBRAS
Corollary
Let
3.3.8.
E : B-jA
expectations of index-finite type.
From
correspondence
Lemma
between
TEoF'
by
Rierf~l
such
traces
on
C*-algebras
which
trace
are
is
Morita
Then there
= w.
(non-normalized) trace wI on C* such that
'=B.
wl (X€AY)
is
a
w(xy)
=
Moreover we have the following:
w(Index E)
=
Wo
that
(62] which says that there
Let w be a trace on B with wE
3.3.9.
(1) wI (1) (2) Let
conditional
We assume that E is of index-finite type (;..~J. w(\)-:::. I
equivalent.
for x,y
~
be
The following Lemma shows that the existence of a Markov
is deduced from an observation
unique
Then TE·T F
F : C --7 B
now on we shall assume that there is a trace w on B
wE "" w.
a
and
109
be the restriction of w
to A.
Then the following diagram
is
commutative:
If
(3)
Index E
is a scalar, then wi (b) "'" (Index E)w(b)
for
b
~
Band
(Index E)-i w1 is a Markov trace of w in the sense of (29J. that C* =IZ(t). Then the existence of wI is a of Rieffe 1 [58;Proposition 2.2.], because consequence
Proof. direct
w(xy)
= w(yx) = E(yx)
fact that
(2)
Note
wo«n(y*),n(x»), (or one can directly check
the
is a trace since C* is isomorphic to B Q9A B.) (I.)10:i uieAu i *) "" w(t i uiu i *) w(Index E).
This is just a statement in Rieffel [50;Proposition 2.5.)
following direct proof is short:
Let q "'" (q .. )" E A®M 1.J J.J
Then WI'", ([q]) = (wl@Tr)«qijeA)ij) Ei w(qii)
=
wO"([q])·
(3) Suppose that Index E is a scalar.
Then
wI (b) =- wl O:i buieAu i *) "" ti w(buiu i *)
w(b(Index E»
= (Index E)w(b).
n
But
the
be a projection.
YASUQ WATATANI
110
Combining with (1), (Index E)-lw
1
is a Markov trace of
lJ.
Q,E.D.
we shall show that if Index E is a scalar, then the
Now
of
T€,i.(: K (A)-7 KO(A) O
the
composition
natural map i:;ll.: KOCA) -----} KO(B)
and
the
transfer map T£: KO(B)--; KO(A) is just a multiplication map by Index E up
to trace evaluations.
Theorem
3.3.10.
index-finite
Index E
Let Let
type.
is
a
E :B-ry A
KO(A)
Define
scalar.
1Wo
V
IR
Proof.
conditional
a
wE =
multiplication
expectation
w.
map
Assume
of
that
V: [R -7 \R
by
Then both the following diagrams commute:
"c~) KO(A)
wo'l
a
w be a trace on B with
Vet) '" (Index E)t for tE:::. IR.
(1)
be
(2)
0
IR
(1)
For a projection q "" (qij )ijE A~Mn' we have that
wO,~*-lj"i,,([ql)
"O'TEi,([ql) wl-j"i~([ql)
(by Lemma 3.3.9.(2))
(wl@Tr)«joi )(qij)ij ~ (w l =
Li w1(qii)
=
~i
<9 TY)«Qij)ij) (by
(Index E)wO(qii)
Lemma
3.3.9.(3»
(Index E)(wO~Tr)«qij)ij) ~ (Index E)wO'([q]) VWO'([qi) .
(rij)ij E. B@M , we have that n w"-i;f.W*,-lj..,I;,( [rJ)
(2) For a projection r w"i.".T ( [r) E
~O'~ilJ.([:]) (w
l i()Tr)«j(rij»)ij)
[i WI (r
ii
)
~
[rl)
Lemma 3.3.9.("2" ..)".)"." ............ "
~ (w 1 <6)Tr)«r ij )ij)
Li (Index E)w(r ii )
(Index E)(wQ)Tr)«rij)ij) ~ (Index E)w'([rl)
Q.E.D.
Vw"([rJ).
If multiplication
we
do not evaluate by a trace, then
T · i f. E
is
map by Inde:{ E in general, even if Index E is an
and the multipllcqtion map by Index E on KO(A) make Cunz-algebra
sense. (cf.
For
[4]).
not
a
integer example,
Define
an
INDEX FOR C*-SUBALGEBRAS
(\ tC Aut B
automorphism
algebra A "" 02 oJ. by
E(x) -
+
(x
°
;:;
4
,
by o:(si)
-si (i "" 1,2). Then the
=
fixed
Let E : B - j A be a conditional expectation
for
.(x))/2
(1,1) ,(sl*,sl)j
III
is
a
x
quasi-
Then
~B,
basis for
Index E We
E.
have
=
2.
point defined
In
the
fact
following
sequence: KO(A) Hence TE ·
i.~ =
<-:II To; Z/3l--->K O(B)"O 0 -'KO(A)-:;: t!3Z.
S:
0 is not a multiplication by Index E.
Corollary 3.3.11. Assume as above. (1)
(2)
Index
is in wO-(KO(A))nw-(KO(B)).
E
If KO(A)~.zn (resp.
Assume that w is faithful.
Index E
Then we have the following:
is
an
T ,.1* (resp. E
eigenvalue
i';(.1'E)
of
the
integral
KO(B)-;-- Zm),
matriK
then
corresponding
to
and Index E is an algebraic number of degree at most
n (resp. m).
Proof.
(l)Inctex E
= (Index E)l = (Index
E)wO~([lJ)
is in wO"(KO(A», and similarly Index E is in w"(Ka(B».
(2)
Assume
integral
rD" IZr. -::::--!l n . KO(A) "" 1.:J,J. ~"I ~ -
that
t
(X a)i
so
that
Tlten(TE~i%)(ri)
matrix corresponding to TEoi-f'
Put {)i "" wO"(ri)G IR and =:
0;
(Cti)iE1Rn.
""
be
the
"" I~j
n Xn
Xjir j .
Since w is faithful, {)~O.
And
Xjia j ""' L j XjiwO'"'(r j ) '" wO'"'(Zj Xjir j )
Lj
Xta "" (Index E)a.
algebraic number
Let X "" (X ij )ij
Thus Index E is an eigenvalue of
of degree at most n.
X
and
an
The case for Ka(B) is similar.
Q.E.D.
Example 3.3.12.
Let w
,
w
,
Let B '" C'k(S3} and A '" C*(S2) as in Example 3.3.3.
be
C"(S3) -)C
a
trace
such
K (C"(S3)) S: Z6)E El>Z ->IR is given by O w"(x,y,z) (x + 2y + z)/6 ~
And
Wo
,
K (C*(S2)) O wO'(x,y)
~
-
['\)
Z@l-) IR
(x +y)/2
Therefore wO-TEi ii- ~ ~ w O
for
wO:g ag'g) "" all.
Then
(x,y,z), £i3>l$Z.
is given by
for (x,y)€ £Q L
CIl ~ )l~ :)G) 0
that
(3x + 3y)/2.
112
YASUO WATATANI
And VW " \~) O
0::
3(x +y)/2.
wAi~TE \~) and Vw
A
~~) =
on
Suppose
(~ (~
A
w
0
3(x
Example 3.3.13. trace
Similarly we have that
\)
+
\
n\1)
(x + 2y +z)/2
2y+ z)/6.
Let M::>N be factors of type III and w the
Let E : M--r N be the canonical conditional
M.
Index E "" [M:N]
that
<
Consequently
is the identity. KO(N),£lR
is
(w ATE~) (x)
the
3.3.14.
golden number.
xf. lR.
= 4cos 2 (n/S) ~ (3 + (5)/2
Let ,-I
ei€irl€i "" te i and €i€j "" ej€i for Then e , ... ) and A = C*{1,e 2 ,e 3 ,·· .).
3 Fibonacci in Effros-Shen [26] ,(27].
=
$2, where $
Z2
T,,- (,
~ KO(A)
iiAj\'
Band
"
= .2 =
(3
is such
Let
2. A
are
C*-algebras
Let w be a unique trace of B.
Index E
and
+ /5)/2.
of
Then
Moreover
Z2
wo 1
wd
A
V
r + ¢Z
"
Z + ¢E
commutes, TEL" is represented by an integral matrix S
and the spectral radius r(S) of S coincides with
Definition. 3.3.15. tr.
Therefore
Let e 1 ,e 2 '€3' .... be Jones projections in [36]
that
KO (A) "
for
T,,'
and
KO(M)'2I R -7
=
O
Example
TE(X) ""
by
given
expectation.
KO(M)-;:;: KO(N)~IR,
Then
~
canonico.l
$2
~
';(~'~7 Index E.
Let B be a simple C*-algebra with a unique trace
Define B-jA a conditional expectation with 1·E
Corollary. an irrational 8.
3.3.16. (1) Let Ae be the irration~l rotation algebra for Suppose that 8 is EQ! an algebraic number of degree 2.
INDEX FOR C*-SUBALGEBRAS tl,2,3.,.,
,oo}
Let Ae "" C~~(u,v) with uv "'" e
e+
113
2-nie
vu.
By Rieffel [62], Ko(A )=
From Corollary 3.3.11
Ol CIR.
e considering
the C~':-subalgebras Dn generated by un and v, we know that n is int-fCAe).
Q.E.D.
Corollary.
3.3.17 .
Let F2 be the free group of two
generators.
Then JIC*rIFZ)) - 11,Z,3, ... ,00).
Proof.
Pimsner
Tberefore J
[F2:Gni
=
JI C* r I FZ) ) .
n.
2
»
»
and Voiculescu showed thll..t K (C*r(F ":?- &':(cf. (4),[801), O 2 C {l,2,3, ... ,oo}. Let Gn be a subgroup of F2 with
By considering C*-suhalgebras C*rCGn)' we know that n is in Q.
E. D.
REFERENCES
( 1J H. Araki,
2J 3] 4J SJ 6j
Some
properties
of
the
modular
conjugation
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