The \(C^ \ast\)-algebra generated by operators with compact support on a locally compact group
Let \(VN(G)\) be the von Neumann algebra of a locally compact group \(G\). The \(C^*\)-subalgebra of \(VN(G)\) generated by operators with compact support is denoted by \(UCB(\widehat G)\). Alternatively, if \(A(G)\) is the Fourier algebra of \(G\), and thus the predual of \(VN(G)\), \(UCB(\widehat G)\) can be identified with the norm-closed span of \(A(G).VN(G)\), the subspace of \(VN(G)\) obtained from the natural action of the algebra \(A(G)\) on it. \(B_ \rho(G)\) is the closure in the compact open topology of the linear span of the positive definite functions of compact support on \(G\). When \(G\) is commutative, \(UCB(\widehat G)\) can be identified with the space of bounded uniformly continuous functions on \(\widehat G\), and \(B_ \rho(G)\) with \(M(\widehat G)\). The space \(B_ \rho(G)\) contains \(A(G)\) and, since positive definite functions have a natural realization as linear functionals on \(VN(G)\), it can be regarded as a subspace of \(UCB(\widehat G)^*\). The second dual \(A(G)^{**}= VN(G)^*\) has an algebra structure induced by \(A(G)\), and this in turn provides an algebra multiplication on \(UCB(\widehat G)^*\). \(B_ \rho(G)\) turns out to be naturally embedded in the algebraic centre of \(UCB(\widehat G)^*\). The main aim of this paper is to discuss the question of whether \(B_ \rho(G)\) is equal to the centre of \(UCB(\widehat G)^*\); this is known to be the case if \(G\) is discrete or commutative. The principal result asserts that if \(G\) is second countable and the closure of its commutator subgroup is not open, then \(B_ \rho(G)\) is the centre. As a consequence, the authors find that if \(G\) is in addition amenable then the centre of \(VN(G)^*\) is \(A(G)\), and they show that this conclusion also holds for amenable discrete, and for commutative, groups. Many other interesting results are presented in the paper. For example, if \(VN(G)\) has a unique topologically invariant mean then \(G\) must be discrete, and \(UCB(\widehat G)= A(G).VN(G)\) if and only if \(G\) is amenable.
positive definite functions of compact support
scientific article; zbMATH DE number 152033