On nuclearity of the algebra of adjointable operators

Abstract

We study nuclearity of the $C^*$-algebra $\mathbb B(\mathcal E)$ of adjointable operators on a full Hilbert $C^*$-module $\mathcal E$ over a $C^*$-algebra $\mathcal A$. When $\mathcal A$ is a von Neumann algebra and $\mathcal E$ is full and self dual, we show that $\mathbb B(\mathcal E)$ is nuclear if and only if $\mathcal A$ is nuclear and $\mathcal E$ is finitely generated. In particular, when $\mathcal A$ is a factor, then nuclearity of $\mathbb B(\mathcal E)$ implies that $\mathcal E$, $\mathcal A$ and $\mathbb B(\mathcal E)$ are finite dimensional.