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