Normalizers and approximate units for inclusions of C^*-algebras

Abstract

For an inclusion of C*-algebras $D\subseteq A$ with $D$ abelian, we show that when $n\in A$ normalizes $D$, $n^*n$ and $nn^*$ commute with $D$. As a corollary, when $D$ is a regular MASA in $A$, every approximate unit for $D$ is also an approximate unit for $A$. This permits removal of the non-degeneracy hypothesis from the definition of a Cartan MASA in the non-unital case.