A double commutant theorem for purely large $C^*$-subalgebras of real rank zero corona algebras
Volume 190 / 2009
Studia Mathematica 190 (2009), 135-145
MSC: Primary 46L35.
DOI: 10.4064/sm190-2-3
Abstract
Let ${\cal A}$ be a unital separable simple nuclear $C^*$-algebra such that ${{\cal M}({\cal A} \otimes {\cal K})}$ has real rank zero. Suppose that $\mathbb C$ is a separable simple liftable and purely large unital $C^*$-subalgebra of ${\cal M}({\cal A} \otimes {\cal K})/ ({\cal A} \otimes {\cal K})$. Then the relative double commutant of $\mathbb C$ in ${{\cal M}({\cal A}\otimes {\cal K})/({\cal A} \otimes {\cal K})}$ is equal to $\mathbb C$.
