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 $C^{\ast }$-algebra.

References in corpus (1)

• A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra

Cited by in corpus (1)

• An algebra whose subalgebras are characterized by density