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arXiv · 1311.2982

Abelian, amenable operator algebras are similar to C*-algebras

Laurent W. Marcoux·Alexey I. Popov
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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.

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Laurent W. Marcoux, Alexey I. Popov. 2013-11-12. Abelian, amenable operator algebras are similar to C*-algebras. https://doi.org/10.1215/00127094-3619791

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