A note on the structure of Moore groups

Robertson, Lewis C.

doi:10.1090/s0002-9904-1969-12252-4

Cited by 47 publications

(14 citation statements)

References 9 publications

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“…Weil [24] extended this result to arbitrary locally compact connected groups. Finally recent work of L. Robertson [20] expands and amplifies our results concerning condition (2). Kaplansky [10] has considered groups G which satisfy condition (1) and proved many interesting results.…”

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confidence: 74%

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Groups with finite dimensional irreducible representations

Moore¹

1972

Trans. Amer. Math. Soc.

112

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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

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confidence: 74%

“…In [20] L. Robertson has improved considerably the criterion characterizing non-Lie groups G for which cf(7r) < oo for all ireô. The following example shows that Theorem 2 is false in general even for separable groups.…”

mentioning

confidence: 99%

Groups with finite dimensional irreducible representations

Moore¹

1972

Trans. Amer. Math. Soc.

112

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“…As in [ [17], it is locally compact and is clearly an FC-group in the sense of [4]. By [15] (for a proof, see [8]), B(G) is an /N-group. It then follows from [10] that G(~) is compact.…”

Section: Some Consequences Of Theoremmentioning

confidence: 92%

Automorphisms, orbits, and homogeneous spaces of non-connected lie groups

1974

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“…It should be noted that this is not a simple consequence of the fact that the co-representations of G give rise to (ordinary) representations of the central extension G", since not all representations of G a are obtained this way. However, looking at the central extension is a successful approach and we use it in conjunction with Robertson's characterization of Moore groups [7].…”

Section: I) V(g) Is Type I (Ii) (G Co) Is Type I (Iii) G Has An Abmentioning

confidence: 99%

“…LEMMA Discrete groups (7] it: g -» T(e n p(g~'))* of G, where A* denotes the adjoint of A as an operator on the dual space of the Hubert space on which A acts. After taking determinants, u"(x,y) det ir(xy) = det -TT(X) det ir(y), we see that w" is trivial, so by Theorem 3.…”

Section: The Type / Part Of V(g)mentioning

confidence: 99%