Groups with finite dimensional irreducible representations

Abstract

Abstract. It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that //"modulo its center is compact.1. Let G be a locally compact group; a (unitary) representation of G is by definition a strongly continuous homomorphism n of G into the group of unitary operators on some Hubert space H(tt) [11]. One says that 77 is irreducible if the only closed subspaces of 7/(77) invariant under all the operators 77(g), ge G, are (0) and H (it). We shall denote by G the set of equivalence classes under unitary equivalence of irreducible unitary representations [11], and we shall use 77 to denote both a representation and its equivalence class. Associated with any representation 7T we have a cardinal number d(n), the degree of 77, which is by definition the cardinality of an orthonormal basis of the Hubert space H(w). It is not assumed here that the topology of G satisfies the second axiom of countability nor that d(tr) is restricted to be ^ X0.Although, in general, examples show that representations with d(n) finite are rather rare, the purpose of this paper is to investigate two closely related hypotheses involving finiteness conditions on d (-rr). To be precise, we say that G satisfies (1) if there is an integer M<oo such that d(n)-¿M for every -rr eG. We say that G satisfies (2) if d(n) < 00 for every neG. Our results, which are stated below, supply simple necessary and sufficient conditions in terms of the structure of G that a group satisfy either (1) or (2). These questions have been investigated previously and many results obtained. In particular, Freudenthal [3] in 1936 showed that a separable (i.e. second countable) connected group G satisfies

Continuous Dense Embeddings of Strong Moore Algebras

“…Our terminology is chosen in honour of Calvin C. Moore who characterized the locally compact groups all of whose irreducible *-representations are finite dimensional [13].…”

Extension and separation properties of positive definite functions on locally compact groups

“…These groups are named after C.C. Moore, who completely clarified their structure [30]. In particular, Moore groups are SIN-groups.…”

“…We first show that G is a SIN-group. Since G/N is compact and N is compactly generated and every Moore group is a projective limit of Lie groups [30], by [18, Theorem 4] G is a projective limit of Lie groups G/C α . Since a projective limit of SIN-groups is a SIN-group, it then suffices to show that each G/C α is a SIN-group.…”

Weak Frobenius reciprocity and compactness conditions in topological groups

“…By [7,Thm. 2.11;16,Lemma 4.3] any SIN-group G is a protective limit of Lie groups G/K jf j e J, K § compact normal. In particular, every G/K 3 is first countable.…”

“…In particular, every G/K 3 is first countable. By Proposition 2.3 in [16], there exists j e J such that π(K ά ) = {/}. Since KjH/Kj is contained in (G/Kj) F , by Proposition 2.8 we may assume G to be first countable.…”