Relatively weak\(^{\ast }\) closed ideals of \(A(G)\), sets of synthesis and sets of uniqueness (Q2879363)

Let \(G\) be an amenable locally compact group and let \(A(G)\) and \(B(G)\) denote the Fourier and the Fourier--Stieltjes algebras of \(G\), respectively. For a closed subset \(E\) of \(G\), let \(J(E)\) denote the smallest closed ideal in \(A(G)\) with hull \(E\) and \(k(E)\) the largest. A closed set \(E\) is a set of synthesis if \(J(E) = k(E)\), and \(E\) is a set of uniqueness if the weak*-closure of \(J(E)\) in \(B(G)\) is all of \(B(G)\).NEWLINENEWLINERelated to these notions, the author introduces the following definitions. A closed subset \(E\) of \(G\) is a \(J\)-set if NEWLINE\[NEWLINE E = \overline{\bigcup\{ \sigma(\phi) \mid \phi \in J(E)^\perp\cap C^*(G) \} }, NEWLINE\]NEWLINE where \(\sigma(\phi)\) is the spectrum of \(\phi\), \(J(E)^\perp\) is the annihilator of \(J(E)\) in the group von Neumann algebra \({ VN}(G)\cong A(G)^*\) and \(C^*(G)\) is the group \(C^*\)-algebra of \(G\) (considered as a \(C^*\)-subalgebra of \({ VN}(G)\) now that \(G\) is amenable). The notion of \(k\)-set is defined similarly, replacing \(J(E)\) by \(k(E)\). The author shows that a closed set \(E\) is a \(J\)-set if and only if the \(\sigma(A(G), C^*(G))\)-closure of \(J(E)\) is contained in \(k(E)\) and it is a \(k\)-set if and only if the \(\sigma(A(G), C^*(G))\)-closure of \(k(E)\) is equal to \(k(E)\).NEWLINENEWLINEOne of the main results concerning sets of synthesis is the following characterisation. Let \(A_1\) be the unit ball of \(A(G)\) and write \(J(E)_1= J(E)\cap A_1\) and \(k(E)_1 = k(E)\cap A_1\). The author shows that a closed subset \(E\) of \(G\) is a set of synthesis if and only if the norm-closures of \(J(E)_1\) and \(k(E)_1\) in the space \(C_0(G)\) of continuous functions vanishing at infinity coincide if and only if the weak*-closures of \(J(E)_1\) and \(k(E)_1\) in \(B(G)\) coincide.NEWLINENEWLINELet us then consider an example result concerning sets of uniqueness. A subset \(U\) of \(G\) is said to be a set of interior uniqueness if every closed subset of \(U\) is a set of uniqueness. The author gives a characterisation of sets of interior uniqueness using \(J\)-sets and shows that every closed subset of \(G\) decomposes uniquely to a disjoint union of a \(J\)-set and a set of interior uniqueness.NEWLINENEWLINEThe paper contains also many related results and ends with a list of open questions.