Voiculescu's Theorem in Properly Infinite Factors

Abstract

In this paper, we investigate Voiculescu's theorem on approximate unitary equivalence in separable properly infinite factors. As applications, we establish the norm-denseness of the set of all reducible operators, prove a generalized Voiculescu's bicommutant theorem and a version of asymptotic bicommutant theorem, and obtain an interesting cohomological result. Additionally, we extend these results to multiplier algebras within separable type $\mathrm{III}$ factors. At last, a concept of the nuclear length is introduced.