On the type of a discrete crossed product (Q1073927)

It is shown that a discrete \(W^*\)-crossed product \({\mathcal M} \times_{\alpha} {\mathbb{Z}}\) where \({\mathcal M}\) is semifinite, is semifinite (respectively of type III) if and only if the action \(\alpha\) leaves a faithful, normal, semifinite trace on \({\mathcal M}\) invariant (respectively admits no normal, semifinite invariant trace). The main part of the proof is a reduction to the classical method for abelian \({\mathcal M}\), here applied to a certain maximal abelian *-subalgebra of \({\mathcal M}\).