## On nuclearity of the algebra of adjointable operators.(English)Zbl 1334.46042

S. Wassermann [J. Funct. Anal. 23, 239–254 (1976; Zbl 0358.46040)] characterized nuclear $$W^*$$-algebras by showing that a $$W^*$$-algebra $$A$$ is nuclear if and only if it is a direct sum of finitely many type I $$W^*$$-algebras of the form $$Z\otimes M_n(\mathbb{C})$$, with $$n < \infty$$ and $$Z$$ an abelian $$W^*$$-algebra. When $$A$$ is a von Neumann algebra and $$E$$ is a self-dual and full Hilbert $$C^*$$-module over $$A$$, the authors prove that the $$C^*$$-algebra $$B(E)$$ of all adjointable operators on $$E$$ is nuclear if and only if $$A$$ is nuclear and $$E$$ is finitely generated. They also show that if $$A$$ is a factor, then the nuclearity of $$B(E)$$ implies that $$E, A$$ and $$B(E)$$ are finite dimensional.

### MSC:

 46L08 $$C^*$$-modules 46L10 General theory of von Neumann algebras

### Keywords:

Hilbert $$C^*$$-modules; nuclearity; Morita equivalence

Zbl 0358.46040
Full Text: