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. We give examples of singular MASA inclusions: for some, every approximate unit for $D$ is an approximate unit for $A$, while for others, no approximate unit for $D$ is an approximate unit for $A$. Our results imply that if the unitization of an inclusion $D\subseteq A$ is a C*-diagonal, then $D$ is regular in $A$. In contrast, we give an example of a non-regular inclusion whose unitization is a Cartan inclusion. If $D$ is a MASA in $A$, we ask when $A$ is a subalgebra of $B$ with $D$ a regular MASA in $B$. When $D$ is a MASA in $\mathcal B(\ell^2(\mathbb N))$, no such $B$ exists.