Spectral synthesis for \(L^1\)-algebras and Fourier algebras of locally compact groups (Q2751519)

In these notes the author reports on progress during the last 20 years in spectral synthesis for \(L^1\) and Fourier algebras of locally compact groups \(G\). In particular results on local and spectral synthesis and on the union of spectral and Ditkin sets in \(A(G)\) are explained, as well as projection and injection theorems for these algebras. Concerning \(L^1(G)\), an example of a noncompact group \(G\), which is a semi-direct product of a compact with a normal abelian subgroup, is mentioned, for which every closed subset in \(\widehat G\) is of synthesis. However, if \(G\) contains a compact normal subgroup \(K\), such that \(G/K\) is a finite extension of a nilpotent group, then if spectral synthesis holds for \(L^1(G)\), \(G\) must be compact. It is also recalled that for a group \(G\) of polynomial growth and symmetric \(L^1\)-algebra, there exists for every closed subset \(E\) of \(\text{Prim}_*L^1(G)\) a smallest ideal in \(L^1(G)\) with hull \(E\). The author explains the injection and projection theorems, which relate spectral sets in \(\text{Prim}_*L^1(G)\) and in \(\text{Prim}_*L^1(G/N)\), where \(N\) is a closed normal subgroup of \(G\). Finally some examples of spectral and nonspectral subsets in \(\text{Prim}_*L^1(G)\) are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].