REPRESENTATION THEORY OF CENTRAL TOPOLOGICAL GROUPS BY SIEGFRIED GROSSER« AND MARTIN MOSKOWITZ(2) By a central topological group we mean a group G such that G/Z is compact, where Z (or Z(G)) denotes the center of G. A locally compact group satisfying this condition will be called a [Z]-group, and [Z] denotes the class of these groups. In [7] we developed the structure theory of [Z]-groups; in particular, we showed that every [Z]-group G obeys the following Structure Theorem: G=VxH (direct product), where V is a vector group and H contains a compact open normal subgroup K(3). Both the structure theory and, as will be shown in this paper, the representation theory of fZ]-groups generalize and unify in a natural manner the corresponding theories for compact groups on the one hand and for locally compact abelian groups on the other. In fact, many of the features common to these two classes appear in their natural setting only when viewed as being characteristic of [Z]-groups. In addition, there are strong indications that [Z] marks the utmost degree of generality in which all these features are still present; not the least of these is the fact that the representation theory of [Z]-groups is essentially finite-dimensional. By contrast, that of the slightly larger class of [FIA]~-groups is not; an [FIA]~- group being a locally compact group G such that 8(G), the group of inner auto- morphisms, has compact closure in 31(G), the group of all topological group automorphisms of G, in the natural topology. The class [FIA]~ was introduced by R. Godement [4]. (For a full discussion of the relation between [Z] and [FIA]~, see [7].) The finite-dimensionality of representations referred to above in combination with the compactness condition which defines [Z] allows us to obtain less general but considerably sharper results than those laid down by Godement in [6]. The present paper gives complete proofs of the results announced in Bull. Amer. Math. Soc. 72 (1966), 836-841, under the same title. The paper is organized as follows. After establishing the basic definitions and a number of technical results in §1 we proceed to the proof of the fundamental fact that continuous irreducible unitary Hubert space representations of [Z]-groups are finite-dimensional (Theorem 2.1) and we obtain an orthogonality relation for the Received by the editors December 21, 1965. 0) Research partially supported by the National Science Foundation grant GP-3685. (2) Research partially supported by the National Science Foundation, and Office of Army Research, Durham. (3) This is Pontrjagin's classical structure theorem for locally compact abelian groups. 361 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 362 SIEGFRIED GROSSER AND MARTIN MOSKOWITZ [December coefficient functions of such representations. An application is made to a problem in the theory of discrete groups. In §3 we utilize the Gelfand-Raikov Theorem to prove that [Z]-groups are maximally almost periodic, i.e., belong to [MAP]; from this and results of [7] it follows that they satisfy the hypothesis of Takahashi's Duality Theorem [18]. Further results are that continuous functions and continuous central functions on a [Z]-group G can be approximated uniformly on compact subsets of G by representative functions and linear combinations of characters, respectively. In addition, the characters separate the conjugacy classes of G. §4 contains a formula which characterizes, up to normalization, characters of irreducible representations of [Z]-groups. Then follow two irreducibility criteria for continuous finite-dimensional unitary representations (Theorem 4.3) and an application. We conclude this section with a result on nilpotent groups. In §5 we study the possibility of "extending" group characters (continuous homomorphisms into the circle group) from certain central subgroups of a group G to continuous irreducible unitary representations of the whole group; G is assumed to be either an [M^P]-group or a [Z]-group (Theorem 5.1, Corollary 1, and Theorem 5.5, respectively). In the latter case, essential use is made of the structure theorem for [Z]-groups as quoted above. This extension theorem utilizes and generalizes a result of Pontrjagin concerning the extension of group characters in abelian groups. Theorems 5.2 and 5.3 concern representations of bounded degree; they generalize results of I. Kaplansky [10]. Theorem 5.2 as well as the extension theorems depend on a result (Theorem 5.1) which analyses the irreducible repre- sentations of a compact subgroup of an [M^F]-group G in terms of those of G. Next we characterize countable discrete [Z]-groups making use of a recent result of E. Thoma [19]. It should be remarked that the extension theorems derived here shed light on those found in §3 of [7] and yield an alternative proof of Theorem 3.1 independent of infinite-dimensional representation theory. §5 closes with a theorem on locally faithful representations. In §6 an orthogonality relation is derived which augments the one in §2. The final result is a criterion for equivalence of irreducible representations of a [Z]-group. Finally the authors would like to thank J. Alperin for a number of useful discussions on representation theory of discrete groups. 1. Preliminaries. We begin with a section containing basic definitions and some results of a technical nature which we require for our investigation. Definition. (1) Let G be a topological group. Consider continuous finite- dimensional irreducible unitary representations p of G on the complex vector space V0; we denote the degree of p by d„ or deg p and the identity map on V„ by Idp. Form equivalence classes of these representations, with respect to unitary equiv- alence, and choose one representation from each class. We denote by 01 (or .^(G)) the totality of all such representations. (2) If p e 0t we denote by pi} the coordinate functions associated with p relative to some orthonormal basis of Vp, by \p the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1967] REPRESENTATION OF CENTRAL TOPOLOGICAL GROUPS 363 character of p, and by H (or 3E(G))the family of all such characters. (3) We denote by J^ (or J^(G)), J^ (or K{G)), and J^0 (or J^0(G)), respectively, the algebras of complex-valued functions on G which are continuous, (left) uniformly con- tinuous^), and continuous with compact support; by ^(or ^(G)) the subalgebra of 3FUconsisting of representative functions associated with representations in ^, and by ^ (or ^(G)) the subalgebra of ^ consisting of the central functions—a central function being one which is constant on the conjugacy classes of G. (4) As usual, a subset S of G is called invariant if it is stable under the inner auto- morphisms of G. In general, we denote by G ■ S the orbit of S under the inner automorphisms of G. If /e^ and x e G then fx denotes the left translate of / by x, ix.,fx{y)=f{xy); and xa/denotes the conjugate of/by x, i.e., (xAf)(y) =f(xyx~1). The restriction of/ to a subset S of G is denoted by/s. If S is a subset of G on which/is bounded, ||/||s stands for lub {\f(x)\/x e S}. The support of/ is denoted by Supp/ Finally, j"G/zdx denotes the normalized Haar integral on G/Z, and |G dx and Jz dz are left invariant Haar integrals on G and Z respectively; normalized so that JG=JZJG/Z; the associated Haar measures are denoted by H-aiz,Mg, and /¿z. All notation not explicitly defined will be standard. At times elements of ^.(G/Z) will be regarded as elements of ^(G). Theorem 1.1. Assume that G e [Z]. There exists a linear operator #: with the following properties: (1) Iffe&ur\&!thenf#=f. (2) ///e^0 rÄen Supp/#£GBSupp/, andf#s&Co. (3) i/".F «j a compact invariant subset of G then \f#\F<¡ \f\P. (4) Iffearthenf#e&r. Proof. For/in 3FUand j in G, define /#(>>)= f (xAf)(y)dx(?). Jaiz Evidently, for each y, (xAf)(y) is a continuous function on G/Z. One checks easily that # is linear. Now we have f#(tyr1)=¡ (xAf)(tyt-i)dx= f ((xí)-A/)(7)^= f (x/a/Xj)^, Jg/z Jaiz Jaiz (*) Since, as was shown in [7], G has small invariant neighborhoods of 1, i.e., G e [SIN], as is well known, the left and right uniform structures on G coincide. (6) It follows from Theorem 1.1 that in the case of compact groups the # operator as defined above, coincides with an operator used implicitly by Pontrjagin in [16]. (In this context, see the Remark following Theorem 1.2.) Segal [17] explicitly defines an operation like # when considering direct sums of compact and abelian groups while in work of Godement [4], [6] the # operator is defined in a slightly more general setting. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 364 SIEGFRIED GROSSER AND MARTIN MOSKOWITZ [December and by the translation invariance of jGIZdx the last expression equals f#(y). Hence /# 6 ^£(G). Let e > 0 be given and choose a symmetric neighborhood Ue of 1 in G with the property that if y21yi e Ue then |/(ji)—f(y2)\ <«. Since G has small invariant neighborhoods of 1 (by Theorem 4.2 of [7]), we may assume that Us is invariant. Then \f#(yO-f*(yù\* f KxAfXyO-ixAfXy^dx. Jgiz But if y2 1y1 £ Us then one sees easily from the above that KxAfXyi)-(xAfXy2)\ < e, for all x in G/Z.