Rearrangements of Tripotents and Differences of Isometries in Semifinite von Neumann Algebras

Abstract

Let τ be a faithful normal semifinite trace on a von Neumann algebra ℳ, and ℳ^u be a unitary part of ℳ. We prove a new property of rearrangements of some tripotents in ℳ. If V ∈ ℳ is an isometry (or a coisometry) and U − V is τ-compact for some U ∈ ℳ^u then V ∈ ℳ^u. Let ℳ be a factor with a faithful normal trace τ on it. If V ∈ ℳ is an isometry (or a coisometry) and U − V is compact relative to ℳ for some U ∈ ℳ^u then V ∈ ℳ^u. We also obtain some corollaries.