Convex combinations of unitary operators in von Neumann algebras (Q1079159)

The unitary rank of an element T in a von Neumann algebra \({\mathfrak A}\) is defined as the smallest number u(T) for which there is a convex combination of unitaries from \({\mathfrak A}\) of length u(T) and equalling T. We determine the unitary rank for every element T in the closed unit ball of \({\mathfrak A}\) in terms of the index i(T) of T and the distance \(\alpha\) (T) from T to the group of invertible elements in \({\mathfrak A}\). If \(i(T)=0\), then u(T)\(\leq 2\); if i(T)\(\neq 0\), then \(u(T)=n\) when \(n-1<2(1- \alpha (T))^{-1}\leq n\), and \(u(T)=\infty\) when \(\alpha (T)=1\). We also determine precisely which asymmetric convex decompositions of T can be realized. We show that only an approximate version of these results holds in a general \(C^*\)-algebra.