Spectral synthesis and topologies on ideal spaces for Banach *-algebras

Abstract

This paper continues the study of spectral synthesis and the topologies $τ_{\infty}$ and $τ_r$ on the ideal space of a Banach algebra, concentrating on the class of Banach $^*$-algebras, and in particular on $L^1$-group algebras. It is shown that if a group G is a finite extension of an abelian group then $τ_r$ is Hausdorff on the ideal space of $L^1(G)$ if and only if $L^1(G)$ has spectral synthesis, which in turn is equivalent to $G$ being compact. The result is applied to nilpotent groups, [FD]$^-$-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which $L^1(G)$ has spectral synthesis. It is also shown that if G is a non-discrete group then $τ_r$ is not Hausdorff on the ideal lattice of the Fourier algebra A(G).