On the normalizing algebra of a MASA in a II$_1$ Factor

Abstract

Let $A$ be a maximal abelian subalgebra (MASA) in a \II1 factor $M$. Sorin Popa introduced an analytic condition that can be used to identify the normalizing algebra of $A$ in $M$ and which we call \emph{the relative weak asymptotic homomorphism property}. In this paper we show this property is always satisfied by the normalizing algebra of $A$ in $M$ and as a consequence we obtain that $\bar{\bigotimes}_{i\in I}(\mathcal{N}_{M_{i}}(A_{i})^{\prime\prime})= (\mathcal{N}_{\bar{\bigotimes}_{i\in I}M_{i}}(\bar{\otimes}%_{i\in I} A_{i}))^{\prime\prime}$ .