Abelian, amenable operator algebras are similar to C*algebras
Abstract
Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C*algebra. We do this by showing that if A is an abelian subalgebra of B(H) with the property that given any bounded representation $\varrho: A \to B(H_\varrho)$ of A on a Hilbert space $H_\varrho$, every invariant subspace of $\varrho(A)$ is topologically complemented by another invariant subspace of $\varrho(A)$, then A is similar to an abelian $C^*$algebra.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 arXiv:
 arXiv:1311.2982
 Bibcode:
 2013arXiv1311.2982M
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Functional Analysis
 EPrint:
 Duke Math. J. 165, no. 12 (2016), 23912406