On Maximal Abelian Self–adjoint Subalgebras of Factors of Type II1

Abstract

In this note, we show that if \({\fancyscript N}\)is a proper subfactor of a factor ℳ of type II1 with finite Jones index, then there is a maximal abelian self–adjoint subalgebra (masa) \({\fancyscript A}\)of \({\fancyscript N}\)that is not a masa in ℳ. Popa showed that there is a proper subfactor ℛ0 of the hyperfinite type II1 factor ℛ such that each masa in ℛ0 is also a masa in ℛ. We shall give a detailed proof of Popa's result.