A separation property of positive definite functions on locally compact groups and applications to Fourier algebras
One says that a locally compact group \(G\) has the \(H\)-separation property, where \(H\) is a closed subgroup of \(G\), if for every \(x\in G\) which is not in \(H\) there exists a continuous positive definite function \(\phi\) on \(G\) which is 1 on \(H\) and such that \(\phi(x)\not=1\). This occurs when \(H\) is either normal, or compact, or open in \(G\) (see \textit{A. T. Lau} and \textit{V. Losert} [Pac. J. Math. 123, 149-159 (1986; Zbl 0591.43004)] and the references therein). It is shown that \(G\) has the \(H\)-separation property if and only if there exists a projection (of a special kind) from the von Neumann algebra \(VN(G)\) onto \(VN_H(G)=I(H)^\perp\). As a nice application, the authors show, that if \(G\) is further amenable, then \(I(H)\) has an approximate identity with norm \(\leq 2\). Moreover, this bound 2 cannot be improved when \(G/H\) is infinite. That follows previous works of \textit{B. Forrest} [Illinois J. Math. 34, 1-25 (1990; Zbl 0712.43002)] and of \textit{J. Delaporte} and \textit{A. Derighetti} [Proc. Am. Math. Soc. 124, 1159-1169 (1996; Zbl 0848.43002)]. Under the hypothesis that \(G\) is an almost connected locally compact group, the authors show that \(G\) has the \(H\)-separation property for every closed subgroup \(H\) if and only if \(G\) contains an open normal subgroup \(N\) of finite index such that \(N\) is a direct product of a compact group and a vector group, and if and only if \(G\) is a SIN-group (i.e. \(G\) has a basis of invariant neighbourhoods \(V\) of the identity: \(V=xVx^{-1}\) for every \(x\in G\)). To do that, they boil down to the case where \(G\) is a connected Lie group, and then prove their theorem in several steps. The paper ends with applications to Ditkin sets.
scientific article; zbMATH DE number 1507746
scientific article