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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

Citations:

Zbl 0358.46040
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Full Text: Euclid