*- Regularity of locally compact groups

, - R E G U L A R I T Y OF LOCALLY COMPACT GROUPS

E b e r h a r d Kaniuth F a c h b e r e i c h M a t h e m a t i k / Informatik der U n i v e r s i t ~ t - G e s a m t h o c h s c h u l e

Paderborn

D-4790 P a d e r b o r n

Let A be a Banach ,-algebra and Prim,A the set of all p r i m i t i v e of A, i.e.

of all kernels

of t o p o l o g i c a l l y irreducible

of A. Prim, A is endowed with the h u l l - k e r n e l - t o p o l o g y : E ~ Prim,A is given by E = h(k(E)), where k(E) {P £ Prim,A;

ideals

*-representations the closure of

= n {P;P E E] and h(1)=

I ~ P} for I ~ A. The ideal theory of A is based on this

structure space rather than on the space Prim A of a l g e b r a i c a l l y A-modules.

Every r e p r e s e n t a t i o n

simple

~ of A extends u n i q u e l y to a r e p r e s e n t a -

tion ~ of the e n v e l o p i n g C*- algebra C*(A)

of A. Thus there is a

continuous m a p p i n g ¢ : Prim C (A) ~ Prim,A, from Prim C*(A)

= Prim, C*(A)

¢ is a h o m e o m o r p h i s m .

P ~ P n A

onto Prim •A. A is called

If A is commutative,

algebra of Gelfand transforms

*-regular, _

if

then this means that the

of A is a regular function algebra on

the h e r m i t i a n part of the s p e c t r u m of A. A locally compact group G is called *-regular if its L l - a l g e b r a LI(G) is ~-regular.

For a unitary r e p r e s e n t a t i o n w of G, we also denote by N

the c o r r e s p o n d i n g

, - r e p r e s e n t a t i o n of LI(G)

e x t e n s i o n to C (G) = C*(LI(G)). equivalence

and then by ~ the

The dual space ~ of G is the set of

classes of irreducible unitary r e p r e s e n t a t i o n s

equipped with the inverse image of the h k - t o p o l o g y ^ ~ Prim C* (G),w ~ ker N~. Evidently, the m a p p i n g G then equivalent

ot the following:

~ E, there exists

f C LI(G)

p E E. If G is abelian,

* -regularity^ of G is

given a closed subset E of G and

such that w(f) ~ 0 and p(f)

= 0 for all

A

then G can be i d e n t i f i e d with the dual group

of G and this r e g u l a r i t y c o n d i t i o n is well known to hold, • -regular.

of G,

on Prim C*(G) under

i.e. G is

The i n v e s t i g a t i o n of ,-regularity of locally compact groups

has been started in

[2]. As a first step, the authors verify the follow-

ing Lemma 1. The f o l l o w i n g conditions on G are equivalent: (i)

¢ is a h o m e o m o r p h i s m ;

(ii)

ker w c ker 0 ~ ker w ~ ker ~ for all r e p r e s e n t a t i o n s

(iii) ker w ~ ker 0 ~ representations

ll0(f)ll

w and 0 of G.

The m a i n result of [2] is

w and 0 of G;

~ llw(f)ll for all f £ LI(G) and all

236

T h e o r e m 1. (i) If G is ,-regular, (ii) If G has p o l y n o m i a l growth,

then G has to be amenable; then G is ,-regular.

(i) follows from the above lemma and the fact that G is amenable if the left r e g u l a r r e p r e s e n t a t i o n of LI(G)

extends f a i t h f u l l y to C*(G). Before

i n d i c a t i n g the proof of (ii), we recall the d e f i n i t i o n of a p o l y n o m i a l l y growing group:

G has p o l y n o m i a l ~ r o w t h if for every compact subset K of

G there is a p o l y n o m i a l PK such that the Haar measure of K n is bounded by PK(n)

for all n C ~.

For instance,

compact extensions

locally compact groups are p o l y n o m i a l l y

Suppose that ~ and p are unitary r e p r e s e n t a t i o n s ker [ c k e r

p and

II~(f)II < lip(f)

of nilpotent

growing. of G such that

for some f*: f E

C (G). Now an imC

--

portant

functional calculus due to Dixmier

E C~(~)

s a t i s f y i n g ~(0)

for the support

[6] can be applied.

Take

: 0. Then, using the above growth condition

of f, Dixmier has shown that the integral

~{f}

A

= j~ exp(ilf)~(~dl,

A

where ~ denotes the Fourier t r a n s f o r m of f, converges in LI(G). over,

More-

for every unitary r e p r e s e n t a t i o n w of G, the equation ~(~{f])

: ~(~(f))

holds, where the right hand side is defined by the usual functional culus on the h e r m i t i a n operator w(f) in the Hilbert way, Di~mier's

functional calculus

studying ideal theory of LI(G).

cal-

space of w. By the

turned out to be a very good tool in

Now,

choose ~ such that ~(t)

= 0 for

t ~ Iiw(f)II and ~( II0(f)II ) = 1. Then it follows that IIw(~{f})II = o, but IIp(~{f})II

~ 1, i.e. ~{f} E ker w, but ~{f} { ker p, a contradiction.

In view of T h e o r e m 1, the following problems arose: (i)

Do there exist amenable groups which fail to be *-regular and

(ii)

Find,

*-regular groups which are not p o l y n o m i a l l y growing? at least for special classes of locally compact groups,

and only if conditions (iii)

if

for *-regularity.

Find further classes of ,-regular groups.

Of course,

the candidates to look at are the solvable groups.

wer to (i) is yes.

The ax+b-group

is *-regular,

and the group G

consisting of all matrices

0

a

y

0

0

i

The ans-

, x, y, z E JR, a > O,

237

turned

out

connected

to be n o n - , - r e g u l a r . solvable

c a n be

checked

Boidol

[4],

Lie g r o u p

In fact,

which

by a p p l y i n g a v e r y

for

,-regularity

G is the

is not

deep

s~allest

,-regular.

and

powerful

of c o n n e c t e d

dimensional

These

assertions

criterion,

due

to

groups. A

Suppose

that

G is a c o n n e c t e d

unique

closed

group,

such

U

,i.e.

G is s a i d

nomial

growth

Theorem it has

for all

2 [4].

exponential

Lie

According

fail

to be

(see

[5, T h e o r e m

,-regular.

are

Every

3 [5~ T h e o r e m

W e are First

now we

going

show

metabelian

that

groups

a representation

tation

of N d e f i n e d

center

Lemma

in

is

a sub-

representation

= {x C G; dual

~(x)

if N /K

hand,

2 below)

results

: I}.

has

poly-

if and

only

if

with

groups

of

the

as a c o n s e q u e n c e

one

one

length

in

3 may

of T h e o r e m

2

obtains

group

is * - r e g u l a r .

for m e t a b e l i a n assumption

groups.

The

on the a c t i o n

first of A

[2].

Suppose G is

that

that

G : A~
is a s e m i d i r e c t

of

*-regular.

metabelian

result

"weakly

of

discrete

[9]

monomial"o

implies

: %(x-lnx).

groups that

are

~x d e n o t e s

Moreover,

,-regular.

second

If N is a n o r m a l

of N and x C G, t h e n

by %X(n)

*-regular

coincides

solvable

metabelian

the m a i n are

of G,%

the

other

Then

to s h o w

criterion

restrictive

3].

A a n d N.

compact

example,

On the

one has u n d e r a s o m e w h a t ^ on N a l r e a d y b e e n p r o v e d

groups

commutator

induced

. Let K induced

the

exists

dual.

this

above

positive

abelian

to the

U

locally

induced

connected

Theorem

of G c o n t a i n i n g

there

'~ E G.

2] and L e m m a

two m o r e

~ E G. T h e n

equivalent

~ : ker

groups

to the

N

and

a polynomially

A connected

a polynomially

Corollary.

ker

to h a v e

[3].

There

subgroup

~ is w e a k l y

(~ ~ U

Then

For

normal

that

group

countable

subgroup

the r e p r e s e n -

we w r i t e

Z(G)

for

of G.

2. Let

G be a m e t a b e l i a n

second

countable

locally

compact

group

A

a n d N the N%

closure

= [x C N; %(x) H%

Then,

of the

given

and ~ ~ U× .

~ C G,

commutator

= i},

G%

= (x E G; xN% there

exist

subgroup

= (x E G; %x

of G. F o r = %}

% E N set

and

C Z(G~/N%)}. a % C ~ and

a X C H%

s u c h that

×I N = %

238

Proof.

By

[9, Theorem

presentations

4.3] there are I 6 ~ and a homogeneous

~ of G X satisfying

a representation

of Gz/Nx,

olN ~ X and ~ ~ U °. o is in fact

and GI/N X is nilpotent

the kernel

of a homogeneous

C*-algebra

is a primitive

representation

ideal

re-

[7,

of class 2. Now

of a separable

(3.9.1)

and

(5.7.6)].

Thus

A ~ U~ follows

for some • E Gx/N ~ such that ~IN N X

The assertion

then

from [11, Lemma 2].

Lemma 3. Let G be a second countable mapping Prim C*(G) ~ Prim,Ll(G)

metabelian

group.

Then the

is injective.

A

Proof.

Suppose

that Wl' w2 E G such that ker 71 = ker ~2" and choose

Xj and Xj, J = 1,2, according pf the corresponding

to Lemma 2. For f E LI(N),

Radon measure

denote by

on G. If f 6 ker wlIN,

then

~1 IN g * pf E ker U

E ker ~1 : ker ~2'

and hence ~2(g)~21N(f) shows that ker WllN weakly

equivalent

clude that ~

= v2(g • pf)

= ker w21N.

to the G-orbit : G--~.

Using the notations

~

This

[8, Theorem

5.3], we con-

we obtain G~I : G~2.

of Lemma 2, we have

: ~

#~ x<EG

Nx

:

Z(G~/ x/'~eG Nxx)"

implies

/~ N x 6 G 11 x Therefore,

G(Xj)

G/N being abelian,

GAx : GX, Hxx : H X and HX/ Moreover,

= 0 for all g £ Cc(G).

Since N is abelian and wjlN is

:

/~ N . x E G 12 x

by the usual reduction,

we can assume that

~ x6G

N

A

Xj x

Then HAl = HX2 as GII : GI2 , and wj N U Xj for some Xj E H, where H : HX. , j : 1,2. Finally, the above argument J that ~ : ~ and hence Wl N w2. Theorem Proof. can

4. Every m e t a b e l i a n Notice

assume

discrete

group G is ,-regular.

first that by [5, Theorem

1] or [10, Lemma

that G is finitely generated.

remains

to show that the m a p p i n g

closed.

Suppose that ~ E G and that

applied to H shows

1.1] we

In view of Lemma 3 it

Prim C*(G) ~ Prim~Ll(G)

is

A

(7) I 16 1

is a net in ~ such

={e}.

239

that

wl ~ w. Let N d e n o t e

every w

choose

the c o m m u t a t o r

11 and XI a c c o r d i n g

subgroup

to L e m m a

2:

:

{x

of G, and for

A

X

I I E N, N I : {x E N; It(x)

: l}, G I

6

G;

I

:

I

I

I },

wK and XI 6 H I s a t i s f y i n g

H I = {x 6 G; xN I E Z ( G I / N I ) } , XI wl ~ U

and x I I N

Let S ( G ) d e n o t e a topology

= 11 .

the set of all s u b g r o u p s

on S(G),

U(C,V)

: {H E S(G);

finite

(see[8,

p.

a subbasis

of w h i c h

is g i v e n by the sets

H n C : 9, H n V , ~}

427]).

fore we can a s s u m e

of G. F e l l has i n t r o d u c e d

S(G)

that H

, where

is a c o m p a c t

~ H in S(G).

C, V _~ G and C is

(Hausdorff)

i.e.

space.

I

all

i > 1(x).

Now H

Setting

K :

{~ I E I

we can a s s u m e abelian Now,

generated,

assume

that

~

{~ H I. I E1 i, so that

E G/K for all I

But t h e n is H a b e l i a n

since H/{~

N x is

x EG ker ~ I H

and wlH ~ G(X)

~here

and abelian.

that H :

i.

ker w~ ~ ker w i m p l i e s

wllH~ G(XIIH)

generated

and we can a s s u m e

~ N x we h a v e x E G I'

K = {6}.

for all

for I

D N and G/N is f i n i t e l y

Thus H/N is f i n i t e l y

There-

x E H iff x E H

exists

~ ker^wlH.

Moreover,

for some X E H. H b e i n g a b e l i a n ,

a net

(x) l

we can

in G s u c h that I ~

I

X

(xi]H)

I ~ X. O b v i o u s l y ,

then X

(HI,x I ) -~ (H,x) in F e l l ' s

subgroup

is c o n t i n u o u s

r~presentation

in this

topology

topology

[8, T h e o r e m

[8, § 2]. 4.2],

Since

inducing

it f o l l o w s

that

X

N

U Xll -~ U X .

l

Finally, that

w is

~ is

weakly contained

We c o n c l u d e w i t h Remarks. cally

i n U~IH a n d U~IH ~ Ux .

This

shows

closed. some

a) It is,

compact

of course,

group

b) P o g u n t k e

[12]

exponential

Lie g r o u p

,-algebra.

expected

that

every metabelian

lo-

is , - r e g u l a r .

recently

proved

the r e m a r k a b l e

is , - r e g u l a r

iff L](G)

result

that an

is a s y m m e t r i c

Banach

240

c) It has been shown in [10] that, with relatively

compact

function on G with is ,-regular d) Barnes

classes 1

compact group

and ~ a symmetric

then the Beurling

weight

algebra LI(G)

iff ~ is non-quasianalytic.

[1] has defined

regular Banach Banach

conjugacy

rate of growth

if G is a locally

,-algebra

has polynomial

the interesting

,-algebra and shown that is ,-regular,

growth

concept

of a locally

(i) a locally regular

and (ii) LI(G) is locally regular if G

(compare Theorem

1).

References 1.

Barnes•

B.A.:

Ideal and representation

of a group with polynomial Colloq.

Math. 45, 301-315

2. Boidol,

J., Leptin•

theory of the Ll-algebra

growth.

(1981)

H., Sch~rmann,

J., Vahle•

D.: R~ume primiti-

ver Ideale von Gruppenalgebren. Math. Ann. 3. Boidol,

236, 1-13

(1978)

J.: ,-regularity

Invent.

Math.

4. Boidol,

of exponential

56, 231-238

J.: Connected

J. Reine Angew.

Lie groups.

(1980)

groups with polynomially

Math.

331, 32-46

(1982)

5. Boidol, J.: *-regularity of some classes Math. Ann. 261, 477-481 (1982) 6. Dixmier, taires.

J.: Op@rateurs

P~bl. Math.

induced dual.

of solvable

groups.

de rang fini dans les repr@sentations

Inst. Hautes Etudes Sci.

6, 305-317

uni-

(1960)

7. D~xmier, J.: Les C*-alg@bres et leurs representations. Paris: Gauthier-Villars 1964. 8. Fell, J.M.G.: groups II. Trans.

Weak containment

Amer. Math.

9. Gootman,

Soc.

E., Rosenberg•

and induced representations

110, 424-447

(1964)

J.: The structure

C*-algebras: a proof of the generalized Invent. Math. 52, 283-298 (1979) 10. Hauenschild, W., Kaniuth, E., Kumar, ling algebras on [FC]- groups. J. Functional 11. Kaniuth, Monatsh.

Analysis

E.: On primary Math.

51, 213-228

Effros-Hahn

conjecture.

A.: Ideal structure

(1983)

(1982)

12. Poguntke, D.: Algebraically irreducble Ll-algebras of exponential Lie-groups. preprint

of crossed product

ideals in group algebras.

93, 293-302

of

representations

of

of Beur-