Pedersen ideal and group algebras

Abstract:

For a locally compact ${T_2}$ group $G$ which has an open subgroup of polynomial growth (e.g., $G$ a group that has a compact neighbourhood invariant under inner automorphisms or $G$ a compact extension of a locally compact nilpotent group) the intersection of the Pedersen ideal of the group ${C^ * }$-algebra with ${L^1}(G)$ is dense in ${L^1}(G)$ (Theorem 1). For groups with small invariant neighbourhoods this intersection is the smallest dense ideal of ${L^1}(G)$, and it consists exactly of those $f \in {L^1}(G)$ whose "Fourier transform" vanishes outside some (closed) quasicompact subset of $\hat G$ (Theorem 3); the Pedersen ideal of ${C^ * }(G)$ is described as the set of all $a \in {C^ * }(G)$ for which $\left \{ {\pi \in \hat G:\pi (a) \ne 0} \right \}$ is contained in some (closed) quasicompact subset of $\hat G$ (Theorem 2).