Spectral synthesis and topologies on ideal spaces for Banach \(^{*}\)-algebras (Q1865318)

Let \(A\) be a Banach algebra and \(\text{Id}(A)\) the set of closed two-sided ideals of \(A\). The topology \(\tau _\infty \) on \(\text{Id}(A)\) introduced by \textit{F. Beckhoff} in [Stud. Math. 115, 189-205 (1995; Zbl 0836.46038)] is compact but seldom Hausdorff. It is known that in case of a commutative Banach algebra \(A\) the topology \(\tau _\infty\) is Hausdorff if and only if \(A\) has spectral synthesis. The second topology \(\tau _r\) on \(\text{Id}(A)\) introduced by \textit{D. W. Somerset} in [Proc. Lond. Math. Soc. (3) 78, 369-400 (1999; Zbl 1027.46058)] is also compact and it is Hausdorff whenever \(\tau _\infty \) is. Hence, there are Banach algebras for which \(\tau _r\) is Hausdorff but \(\tau _\infty\) is not. Properties of \(\tau _\infty \) and \(\tau_r\) in the case when \(A\) is a Banach *-algebra are studied in the paper under review. It is shown that spectral synthesis is equivalent to the Hausdorffness of \(\tau _\infty\) for a class of Banach *-algebras which includes the \(L^1\)-algebras of \([FC]^-\)-groups. An example of a non-compact non-abelian group \(G\) for which \(L^1(G)\) has spectal synthesis are given. It is shown that (a) for nilpotent groups, \([FC]^-\)-groups and Moore groups, spectral synthesis for \(L^1(G)\) is equivalent to the compactness of \(G\), (b) if \(G\) is locally compact but not compact, then \(\tau_r\) is not Hausdorff on \(\text{Id}(L^1(G))\), and (c) if \(G\) is a non-discrete locally compact group, then \(\tau_r\) is not Hausdorff on the ideal lattice of the Fourier algebra \(A(G)\).