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

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.