The crossed product by a pointwise unitary action on a \(C^*\)-algebra with continuous trace (Q2716991)

A C\(^*\)-algebra \(A\) is said to be with continuous trace if the linear span of the set \(T(A^+)\) is dense in \(A\): here, \(T(A^+)\) consists of all positive elements \(a \in A\) for which the function mapping a representation \(\pi \in \hat{A}\) to \(\text{ tr}(\pi(a))\) is finite and continuous on \(\hat{A}\). NEWLINENEWLINENEWLINENext, if a locally compact group \(G\) acts continuously by automorphisms \((\alpha_g)\) on \(A\), the action \(\alpha\) is called pointwise unitary if any representation of \(A\) can be extended to a representation of the full crossed product. \textit{D. Olesen} and \textit{I. Raeburn} [J. Funct. Anal. 93, No. 2, 278-309 (1990; Zbl 0717.46055)] proved, among other things, that, if \(\alpha\) is a pointwise unitary action of a separable abelian locally compact group on a separable C\(^*\)-algebra with continuous trace, then the full crossed product has continuous trace. NEWLINENEWLINENEWLINEIn the paper under review, the author provides a simpler prove of the same result and removes the separability condition.