A note on a question by T. Ando

Abstract

In 1979 T. Ando posed the following question: suppose E and F are two projectionvalued measures defined on an algebra Σ of subsets of Ω, which verifykE(∆) − F(∆)k ≤ 1 − δ, ∆ ∈ Σ,for some δ > 0. Does there exist a unitary operator u such that u∗E(∆)u = F(∆) for all∆ ∈ Σ? He knew that the answer was affirmative if both measures were strongly σ-additiveand maximal (i.e. E and F have ciclic vectors). In this note, we show that the answer isalso affirmative if both measures take values in a common finite von Neumann algebra.