A Characterization of a Class of [Z] Groups Via Korovkin Theory

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

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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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(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.

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(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 O<3

n=l

Now G o / K has finite torsion free rank, say r. Then there exist Xl . . . . . Xr ~ Go such that {XlK ..... x r K } is a maximal independent set of elements of infinite order in G o / K . Since Go = K Z , xl . . . . . Xr may be chosen to be central elements of G. Let H be the subgroup generated by K and Xl . . . . . Xr. Then H is an open normal subgroup of Go and G o / H is discrete. Since G is separable and metrizable, G o / H is second countable. Therefore G o / H is countable. Further, since Go = H Z , we may choose elements {Si}i= 1 belonging to Z such that G o / H = { s i H } ~ l . Since G o / H is a torsion group, { y ( s i H ) " y ~ (Go~H) ^} is a finite subgroup of the circle group for each i. Therefore for each i, there exists ei, 0 < 6i < 1 and such that ] y ( s i H ) - 11 >_ 'f'iVY E (Go~H) ^ for which

V(siH)~=l.

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L e t {~i}iC~=l be a summable sequence of positive numbers satisfying

en~n - 2 Z

~i > 0 Vn.

i>n+l

1 i-I

(one such sequence is obtained by defining 3i =

4i

I-Isk'

as in [11,

k=l

Section 3, p. 453]). We define r + 3

functions in

Z(LI(G))

as follows:

O(3

fo = Zo~.d~,,,X~..~K n=l

f i ( x ) = fo(xx71)~co,

1< i < r

oo

fr+l (x) ~- Z 3ifo(xsZI)~G~ i=1

fr+2(x) = fo(xy-l), where for a subset E of G, ~E denotes the characteristic function of E. We shall prove that the Gelfand transforms of these r + 3 functions separate the points of ~ . Note that the Gelfand transforms of the restrictions of the functions f0 . . . . . fr+l to Go already separate the points of ~ " a o =

~'a

~Go

(cf. T h e o r e m 4 . 2 o f [ l ] ) . Let a , p ~ d

satisfy

r ~ ( f i ) = rp(fi)gi = 0, 1 . . . . . r + 2. We must show that a = p. In view of Lemma A, there are three possibilities:

(i) a 6 ((~)I, P E (G)II(ii) a, p 6 (G)t, (iii) a, p ~ (G)11.

Case

(i) a E (G)1, P 6 (G)11.

We shall show that this possibility can not occur. The assumption implies that ale 0 is irreducible and p = r/~ (induced representation) for some r/6 (~o- Now

r.(f) = ~

f(x)x~(x)dx

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and

f(x)Xp(X)dx

rp(f) =

pd,j 1 fa f(x)PX~

= @~fof(X)Xo( x)dx"

0

Thus r~(f,-)=rp(fi)

u

.... ,r+l

=> r~lGo(fi) = r~(f,-) O'IG o =

Vi = 0 . . . . , r + 1

r/.

But this is a contradiction because

ala o e (Go)I and r/ e (Go)II = (see, L e m m a A).

=

{or 9

do" S(a)

=

G}

{a e do " S(a) = Go} and the two sets are disjoint

Case (ii) ~, p e (G)1. The assumption implies that O-iGo and PlGo are irreducible. Thus the equations r,,(fi) = rp(f,-)u = 0 . . . . . r + 1 imply that O-tGo = PlGo" It follows from L e m m a A (i) that ~ = 2 |

where X 9 ^ and ,~ is the lift of X to G. We claim that 2 = 1 , Now O'la0 and P16o are irreducible. Since Go = KZ, by L e m m a 4.1 of [1], it follows that crtX and PlK are irreducible. If CrlK = ~mlK = PlK then ro (fr+2) =

~--~olndo,x,,,,(xy-~)~K(xy-1)-Xo(x)dx n=l

--

,.._ ~ , do.. X o , , ( x ) x ~ ( x y ) d x d~l fK ~--" n=l

as can be easily proved. Similarly Olrn Tp(fr+2)

--

-

Xp(J) -

---

d o , , UXo(Y) m~

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Thus Ta(fr+2)

=

Tp(fr+2)

:=>Xo(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

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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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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

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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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(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