For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

We construct inner amenable groups$G$with infinite conjugacy classes and such that the associated II_{1}factor has no non-trivial asymptotically central elements, i.e. does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.

If$X$is a closed subset of the real line, denote by$G$_{$X$}the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on$X$. In this paper we attempt to find$G$_{$X$}in terms of$X$.