REND1CONTI DEL CIRCOLO MATEMATICO DI PALERMO Scrie 11, Tomo XLVIII (1999), pp. 123-134
A C H A R A C T E R I Z A T I O ...
6 downloads
270 Views
440KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
REND1CONTI DEL CIRCOLO MATEMATICO DI PALERMO Scrie 11, Tomo XLVIII (1999), pp. 123-134
A C H A R A C T E R I Z A T I O N O F A C L A S S O F [Z] G R O U P S VIA KOROVKIN THEORY MANJU RANI AGRAWAL - U.B. TEWARI
We characterize all the central topological groups G for which the centre
Z(LI(G)) of the group algebra admits a finite universal Korovkin set. |t is proved that Z(LI(G)) has a finite universal Korovkin set iff (~ is a finite dimensional, separable metric space. This is equivalent to the fact that G is separable, metrizable and G/K has finite torsion free rank, where K is a compact open normal subgroup of certain direct summand of G.
1. Introduction. Let A be a commutative Banach algebra with continuous involution. An eminent problem in Korovkin approximation Theory is to characterize those A which admit a finite universal Korovkin set. Here, a subset S of A is said to be a universal Korovkin set iff the following analogue of the classical Korovkin T h e o r e m ([9]) is true: For every commutative Banach algebra B with continuous symmetric involution, every * - h o m o m o r p h i s m T : A --+ B and every uniformly b o u n d e d net {T,~} of positive linear operators from A to B, the convergence lim II(T=x - Tx)^l[~ = 0 Yx ~ S implies lim II(T,~y - Ty)^ll~ = 0
'r
E A.
In [1], we had characterized the central topological groups (or [Z] groups) G having a compact open normal subgroup K such that G =
Key words and phrases: Universal Korovkin set, central topological group, continuous irreducible unitary representation, induced representation, centre of group algebra, Segal algebra.
124
MANJU
RANI
AGRAWAL
-
U.B.
TEWARI
K Z , where Z is the centre of G. for which the centre Z(LI(G)) of the group algebra has a finite universal Korovkin set. In this connection it is appropriate to mention (see, [4, Theorem 4.4]) that every [Z] group is of the form G = V • H, where V is an Euclidean group and H has a compact open normal subgroup. Further, the group algebra of any Euclidean group has a finite universal Korovkin set. Therefore to prove that Z(LI(G)) has a finite universal Korovkin set, it suffices to prove that Z ( L I ( H ) ) has a finite universal Korovkin set, see [1]. Thus the basic problem is to investigate it for the [Z] groups which have a compact open normal subgroup. In this paper we characterize such [Z] groups for which Z ( L I ( G ) ) has a finite universal Korovkin set. If G is a [Z] group having a compact open normal subgroup K. then we may assume, without loss of generality, that G / K is abelian [4, Cot. 2, p. 331]. Moreover, in this case there exists a finite chain of open normal subgroups of G such that G : Gn ~_ Gn-I ~_ ' ' . D Gl ~_ Go = K Z and Gi/Gi-1 is a cyclic group of prime order 'v'i : 1 , . . . , n (see, [7, Section 1.4, p. 70]). In [1] we had already settled the problem for the case n : 0. In this paper we settle the problem for any [Z] group which has a compact open normal subgroup and prove the following. THEOREM 1.1. Let G be a [Z] gronp having a compact open normal subgroup K such that G / K is abelian. Then the following statements are equivalent.
i) Z(LI(G)) admits a finite universal Korovkin set. ii) G is separable, metrizable and G / K has finite torsion free rank. iii) G is a finite dimensional, separable metric space.
2. Notations and preliminaries. We shall follow the notations used in [1]. A locally compact group G is said to be a [Z] group if G / Z is compact, where Z is the centre of G. Throughout the paper G will be a [Z] group. It is known [5, Theorem 2.1] that every continuons irreducible unitary representation of G is finite dimensional. G will denote the set of equivalence classes of
A CHARACTI-RIZATION OF
A ('I_ASS OF
IZ[
GROUPS
VIA
KOROVK[N
THI-ORY
125
continuous irreducible unitary representations of G. For o- 9 (~, let X,~ and do be its character and dimension respectively and let r,~ be the multiplicative linear functional on Z ( L t ( G ) ) defined as r~,(f) =
f ( x ) X~(X) dx, f 9 Z ( L J ( G ) ) .
It is shown in [6, Section 6] that every multiplicative linear functional on Z ( L I(G)) arises in this manner and the set Y =
~'cs
e(~
equipped with the topology of uniform convergence on compact subsets of G coincides with the maximal ideal space of Z ( L I ( G ) ) . The topology of ~ can be transported to G in a natural way. Let H be a closed normal subgroup of G and cr 9 /4 then cr (; will denote the representation of G induced by or, see [8]. Let S ( a ) = {s 9 G : CGx.~-t = cr~-u E H} denote the stability group of ~7. For p 9 G, PjH will denote its restriction to H. By the dimension of a topological space we mean the covering dimension (see [10], p. 9).
3. Proof of the main result. To prove Theorem 1.1, let us first collect some auxilliary results. Since each quotient G i / G i _ I has a prime order, there is a nice relation^ ship among Gi's. We quote the following useful Lemma in this connection. LEMMA A [7, Lemma 1.1] Let H be a normal subgroup (of prime index p) in a locally compact group G such that all the continuous irreducible unitary representations o f G are finite dimensional. We define
(G)t = {P ~ G " PlH is irreducible}, ( G ) I I = {P E G 9 p = (yG
f o r some a ~ Igl},
(121)1 = {or 9 121 " S(cr) = G} and (ft)
1 =
9 121 "
=
HI.
Then (~ is the disjoint union of ((~)t and (~;)lt; H is the disjoint union of (/4)I and (/-))t/. Moreover we have
126
MANJU
RANI
AGRAWAL
-
U.B.
TEWARI
(i) If cr 6 /-), then cr 6 ( H ) t iff cr = PtH for some p 6 (G)I- In this case all the extentions of cr are of the form 2 | for X 6 ( G / H ) , where )~ denotes the lift of X to G. (ii) If cr ~ / ~ , then ~r E (,0)1/ iff cr c is irreducible. Further, for p 6 (~, p 6 (0)11 iff Xp(t) = 0 for t r H . We shall also need the following. LEMMA 3.1. Let H and G be as in Lemma A. I f p ~ G and PlH is irreducible then Xp(X)7~0 Vx c G. P r o o f By (i) of L e m m a A, PlH 6 (`0)I and therefore the stability group of PlH is G. Thus we have G = {s ~ G "(plH)sxs-i = (PlH)xVX C H} = {s E G 9 Psxs-l = pxVX E H} = {s G G 9 psPxPs-i = pxVX E H} = {s E G " PsPx = PxPs v x E H}. Since {p~ : x E H} is an irreducible set of operators, it follows that Ps = Cp(S)lp, where Ip is the dp-dimensional identity operator and Cp(S) is a scalar depending on p and s. It is easy to check that Cp(St) -= cp(s)cp(t) for s , t E G and cp(e) = 1. Since Xp(X) = Cp(X)dp, Xp can not vanish on G.
(3.2) Proof o f Theorem 1.1 (i)=:~(iii). It has been shown in [1] that if Z ( L l ( G ) ) has a finite universal Korovkin set then its maximal ideal space ~ is a finite dimensional separable metric space. Consequently d; is also a finite dimensional separable metric space. (iii)=*(ii). Since G / K is abelian, K contains the closure of the commutator group G'. N o w as in the proof of [1, Theorem 4.2], it can be shown that G / K has finite torsion free rank. Further, by Theorem 2.3 of [7], G is separable and metrizable.
A CHARACTERIZATION OF A CLASS
OF
[Z]
GROUPS VIA
KOROVKIN THEORY
127
(ii)=*(i). Since Z ( L I ( G ) ) has a bounded approximate identity, in view of Theorem 2.9 of [1] it suffices to show that there exist finitely many functions in Z ( L I ( G ) ) such that their Gelfand transforms separate the points of ~ . As discussed in Section 1, there exists a finite chain of open normal subgroups of G such that G = Gn D G,_I D . . . D_ GI D__Go = K Z
and Gi/Gi-I is a cyclic group of prime order 'v'i = 1 . . . . , n. Further, since G / K has finite torsion free rank, each G i / K (i = 0 ..... n) has finite torsion free rank. We shall prove the assertion by induction on the length of the normal series. The case n = 0 has already been taken care of [1, Theorem 4.2].
Step 1. We assume that n = 1, that is G = G1 __. Go = K Z and G / G o is a cyclic group of prime order p. Let y be an arbitrary but fixed element in G such that the coset yGo is a generator of G / G o . Since G is separable and metrizable, so is K. Hence /( is countable. Since K is a compact subgroup of G, by Theorem 5.1 of [5], /( _ {crlX :or E d;}. Thus we may choose a sequence {or,}/~7 = 1 in (~ such that /~ = {~r,ix},=lo~ " Let {otn}~ be a sequence of distinct positive numbers such that n=l OXo(Y) = Xp(Y) ==~ X~|
= Xp(Y)
X(Y)Xp(Y)
=
Xp(Y)
~(y) = 1, since by Lemma 3.1 Xp(y)r generator of the cyclic group cr=p.
Case
Therefore x(yGo) = 1. Since yGo is a G/Go, it follows that )~ = 1. Consequently
(iii) a, p 6 ( G ) I / .
The assumption implies that there exist /z, 0 c (~0 such that a = # c and p = r/~. Now for f ~ Z(LI(G)), r~(f) =
f(x)x~(x)dx l fof(X)px.(x)dx pd1,
l fGof(X)X,,(x)dx
d.
and similarly
f(x)xq(x)dx.
r p ( f ) -- ~ 0
Thus the equations r ~ ( f i ) = rp(fi) Vi = 0 ..... r + l yield that tz = q. Since a and p are respectively the representations on G induced by /z and O, we have a = p.
Step 2. Assuming that the assertion is true for Z(LI(Gj_I)) we shall establish the assertion for Z(LI(Gj)). Let Pi be the prime order of the cyclic group Gi/Gi-1. For each i = 1 . . . . . j , we fix Yi E Gi such that yiGi-1 is a generator of the cyclic group Gi/Gi-l. Since K is a compact subgroup of Gj and /s is countable, as in Step 1, we may choose
A CHARACTERIZATION OF A CLASS
OF
[Z]
GROUPS VIA
KOROVKIN THEORY
131
OC 1. Note that the rea sequence {an}~__l in (~j such that /~ {O"n[K}n= striction of each an to each Gi, 0 < i < j -- 1, is irreducible.
Let [~n},~__t be a sequence of distinct positive numbers such that O0
EOlnd2n < (X). n=l
N o w Go/K has finite torsion free rank, say r. Let xl . . . . . Xr, {Si}~= 1 , {ei}ioC__l and {3i}ooi=l be as in Step 1. We define r + j + 2 functions in Z(LI(Gj)) as follows: (3O
fo = Z
Otnda. xa,,~K
n=l
ft'(X) : fO(XXZ1)~Gj_I ,
1< i < r
O0
L+I(X)
: Z
~ifo(XS?I)~Gj-I
i=1 =
fo(xy~ l)~Gj-I
fr+3(X) =
fo(xY2 l)~Gj_l
fr+2(x)
L+j (x) = fo(xyfl ) Gi_i fr+i+, (x) = fo(xyf') in
Note that the restrictions of the functions fo . . . . . fr+j tO Gj-I are Z(LI(Gj_1)) and their Gelfand transforms separate the points of =
"a e Gj-I
points o f ~(aj =
{
9 We claim that f0 . . . . .
d---~- " a e Gj
u = O, 1 . . . . . r + j + 1. Since are three possibilities: (i') a e ( G j ) I and
(ii') a, p e (Gj)I (iii') a, p e (Gj)I1,
p e (Gj)I1
/
fr+j+l
separate the
. Let a, p e Gj satisfy r ~ ( f i ) = r p ( f i )
Gj/Gj_I
has prime order, as before there
132
MANJU
RANI
AGRAWAL
-
U.B.
TEWARI
Case (i') (7 E (Gj) 1
and
p
~
(Gj)II
that is
is irreducible and p = r/C J,
alcj_ ~
for s o m e O E (];j-l. N o w as in Step 1,
ro(f) = ~
f(x)x~(x)dx J
and rp(f) = ~
1 fo J-' f(x)x~(x)dx.
Thus r ~ ( f i ) =-- r p ( f i ) Vi = 0 . . . . . z ~ "6CrIGj_l ( f i ) = r o ( f i )
r 3- j
Vi--=0 . . . . , r 3 - j
atGj_ 1 = 0. This leads to a contradiction as in Step 1.
Case (ii') a, p c ( 0 j ) t that is crlcj_ , and PlGj_~, are irreducible. T h e equatios r o ( f i ) = r p ( f i ) Vi = 0 . . . . , r + j imply that alcj_ J = PlGj_~. T h e r e f o r e , a =)~| where )~ is the lift o f X c (Gj/Gj_~) ^to Gj. Again as in Step 1, we can c o n c l u d e f r o m the equation r~(fr+j+l) = rp(fr+j+l) that )~ =-- 1. Thus a -----p.
Case (iii')
that is a =
tzGJ
and p = r;6J, w h e r e # , r/
e
G j - I - AS
in step 1, r
f(x)x ,(x)dx
(f) =
_
1 pjdI~
f(x)pjXx (x)dx -1
j.
f(x)x, (x)dx. -1
and
~p(f) = ~ 1 fc j_, f(x)x~(x)dx. T h e equations r o ( j q ) = ~p(J}) 'v'i = 0 . . . . , r + j /, = r/ and h e n c e a = p. This completes the proof.
now imply that
A CHARACTERIZATION OF A CLASS OF [Z] GROUPS VIA KOROVKIN THEORY
133
Remarks 3.3. (1) Suppose a separable metrizable group G contains a normal series of length n, G = Gn ~ Gn-I ~ . . . ~ Gl ~ Go = K Z, where K is a compact open normal subgroup of G and each G i l G i _ l ( i = 1, ..., n ) is a cyclic group of prime order. Further, let r be the (finite) torsion free rank of Go/K. Let f 0 , . . - , fr+,+l be the functions constructed as in the proof of Theorem 1.1. It is easy to check that fo > 0. Since fo ..... f r + . + l separate the points of S , by Corollary 2.8 of [1], {fo, fo * f o , ' " , fo * fr+n+l, r+n+l
s0, ]E
is a universal Korovkin set in Z ( L I ( G ) ) . Note that
i=0
this set contains r + n + 4 elements. (2) Let G be as in (1) and f0 . . . . . fr+n+l be the functions constructed as in the proof of Theorem 1.1. Since j~ > 0, the functions /0 ..... .Pr+n+i separate the points of ~ strongly. By [3, Cor. 4.5], r+n+l
f;*f/} is a universal Korovkin set in Z(LI(G)) with
{f0, ..., fr+n+l, Z i=0
respect to positive contraction operators. (3) Let G be a nondiscrete [Z] group. Let S(G) be a Segal algebra on G such that it is closed under the involution inherited from LI(G), (see, [13], [14] for the definition and relevant properties of Segal algebras). Then the centre Z(S(G)) is a commutative Banach algebra with continuous symmetric involution, its maximal ideal space is identified with ~
=
~
"~ ~ G
and Z(S(G)) does not have a bounded
approximate identity. Theorem 1.1 and [2, Cor. 2.5] can be applied to resolve the problem of existence of finite universal Korovkin set w. r. t. positive spectral contraction operators in Z(S(G)), (see, 12] or [121 for the definition of universal Korovkin set w. r. t. positive spectral contraction operators). The arguments similar to those as used in the proof of Corollary 3.1 of [2] yield the following: COROLLARY Let G be a [Z] group having a compact open normal subgroup K such that G / K is abelian. Then for a Segal algebra S(G) which is closed under the involution, the following statements are equivalent. (i) Z(S(G)) has a finite universal Korovkin set w.r.t, positive spectral contraction operators.
134
MANJU R A N I
AGRAWAL
-
(ii) G is separable, metrizable and G / K
U.B. TEWARI
has finite torsion free rank.
(iii) G is a finite dimensional, separable metric space.
REFERENCES [1] Agrawal M. R., Tewari U. B., On existence of finite universal Korovkin sets in the centre of group algebras. Mh. Math. 123 (1997), 1-20. [2] Agrawal M. R, Tewari U. B., On universal Korovkin sets w.r.t, positive spectral contractions. Rend. Circ. Mat. Palermo 46 (1997), 361-370. [3] Altomare F., On the universal convergence sets. Ann. Mat. Pura Appl. (4) 138 (1984), 223-243. [4] Grosser S., Moskowitz M., On central topological groups. Trans. Amer. Math. Soc. 127 (1967), 317-340. [5] Grosser S., Moskowitz M., Representation Theory of central topological groups. Trans. Amer. Math. Soc. 129 (1967), 361-390. [6] Grosser S., Moskowitz M., Harmonic Analysis on central topological groups. Trans. Amer. Math. Soc. 156 (1971), 419-454. [7] Grosser S., Mosak R., Moskowitz M., Duality and Harmonic analysis on central topological groups, Indag. Math. 35, 65-91 (1973). Correction to (duality and Harmonic analysis...) Indag. Math. 35 (1973), p. 375. [8] Kirillov A. A., Elements of the theory of Representations (translated from Russian by E. Hewitt). Springer Verlag, Berlin-Heidelberg-New York, (1976). [9] Korovkin P.P., On convergence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR (N.S.) 90, (1953) 961-964. [10] Nagata Jun-iti., Modem Dimension Theory. Amsterdam: North Holland, (1965). [11] Pannenberg M., A characterization of a class of locally compact abelian groups via Korovkin Theory. Math. Z. 204 (1990), 451-464. [12] Pannenberg M., When does a commutative Banach algebra possess a finite universal Korovkin system? Atti. Sem. Mat. Fis. Univ. Modena 40 (1992), 8999. [13] Reiter H., Classical Harmonic Analysis and locally compact groups. Oxford University Press (1968). [14] Reiter H., Ll-algebras and Segal algebras. Lecture Notes in Math. Vol. 231, Springer Verlag. Berlin-Heidelberg-New York (1971). Pervenuto il 30 giugno 1997.
Mathematical Sciences Division Institute of Advanced Study in Science and Technology Khanapara. Guwahati-781022, India Department of Mathematics Indian Institute of Technology Kanpur- 208016, India