REND1CONTI DEL CIRCOLO MATEMATICO DI PALERMO Scrie 11, Tomo XLVIII (1999), pp. 123-134
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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.
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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
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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
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GROUPS VIA
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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
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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
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MANJU
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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.
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MANJU R A N I
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(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