ACTA issues

A note on a question by T. Ando

E. Andruchow

Acta Sci. Math. (Szeged) 80:3-4(2014), 651-658

Abstract. In 1979, T. Ando posed the following question: suppose $E$ and $F$ are two projection-valued measures defined on an algebra $\Sigma $ of subsets of $\Omega $, which satisfy $$ \|E(\Delta )-F(\Delta )\|\le1-\delta, \Delta\in \Sigma, $$ for some $\delta >0$. Does there exist a unitary operator $u$ such that $u^*E(\Delta )u=F(\Delta )$ for all $\Delta\in \Sigma $? He knew that the answer was affirmative if both measures were strongly $\sigma $-additive and maximal (i.e. $E$ and $F$ have cyclic vectors). In this note, we show that the answer is also affirmative if both measures take values in a common finite von Neumann algebra.

DOI: 10.14232/actasm-012-080-1

AMS Subject Classification (1991): 47B15, 47C15

Keyword(s): spectral measure, unitary equivalence, finite algebra

Received October 10, 2012. (Registered under 80/2012.)