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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The $ L\sp{1}$- and $ C\sp{\ast} $-algebras of $ [FIA]\sp{-}\sb{B}$ groups, and their representations

Author: Richard D. Mosak
Journal: Trans. Amer. Math. Soc. 163 (1972), 277-310
MSC: Primary 22D12; Secondary 46L05
MathSciNet review: 0293016
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Abstract: Let $ G$ be a locally compact group, and $ B$ a subgroup of the (topologized) group $ \operatorname{Aut} (G)$ of topological automorphisms of $ G$; $ G$ is an $ [FIA]_B^ - $ group if $ B$ has compact closure in $ \operatorname{Aut} (G)$. Abelian and compact groups are $ [FIA]_B^ - $ groups, with $ B = I(G)$; the purpose of this paper is to generalize certain theorems about the group algebras and representations of these familiar groups to the case of general $ [FIA]_B^ - $ groups. One defines the set $ {\mathfrak{X}_B}$ of $ B$-characters to consist of the nonzero extreme points of the set of continuous positive-definite $ B$-invariant functions $ \phi $ on $ G$ with $ \phi (1) \leqq 1$. $ {\mathfrak{X}_B}$ is naturally identified with the set of pure states on the subalgebra of $ B$-invariant elements of $ {C^\ast}(G)$. When this subalgebra is commutative, this identification yields generalizations of known duality results connecting the topology of $ G$ with that of $ \hat G$. When $ B = I(G),{\mathfrak{X}_B}$ can be identified with the structure spaces of $ {C^\ast}(G)$ and $ {L^1}(G)$, and one obtains thereby information about representations of $ G$ and ideals in $ {L^1}(G)$. When $ G$ is an $ [FIA]_B^ - $ group, one has under favorable conditions a simple integral formula and a functional equation for the $ B$-characters. $ {L^1}(G)$ and $ {C^\ast}(G)$ are ``semisimple'' in a certain sense (in the two cases $ B = (1)$ and $ B = I(G)$ this ``semisimplicity'' reduces to weak and strong semisimplicity, respectively). Finally, the $ B$-characters have certain separation properties, on the level of the group and the group algebras, which extend to $ {[SIN]_B}$ groups (groups which contain a fundamental system of compact $ B$-invariant neighborhoods of the identity). When $ B = I(G)$ these properties generalize known results about separation of conjugacy classes by characters in compact groups; for example, when $ B = (1)$ they reduce to a form of the Gelfand-Raikov theorem about ``sufficiently many'' irreducible unitary representations.

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Keywords: Group and $ {C^\ast}$-algebras, structure space, $ \char93 $-operator, unitary representation, $ B$-character, spherical function, $ [FIA]_B^ - $ group, $ {[SIN]_B}$ group, central group, type I group
Article copyright: © Copyright 1972 American Mathematical Society