Convex combinations of unitary operators in von Neumann algebras

Abstract

The unitary rank of an element T in a von Neumann algebra U is defined as the smallest number u ( T ) for which there is a convex combination of unitaries from U of length u ( T ) and equalling T . We determine the unitary rank for every element T in the closed unit ball of U in terms of the index i ( T ) of T and the distance α ( T ) from T to the group of invertible elements in U . If i ( T ) = 0, then u ( T ) ⩽ 2; if i ( T ) ≠ 0, then u ( T ) = n when n − 1 < 2(1 − α ( T )) −1 ⩽ n , and u ( T ) = ∞ when α ( 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.