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Abelian, amenable operator algebras are similar to \(C^{\ast}\)-algebras. (English) Zbl 1362.46048

Summary: Suppose that \(H\) is a complex Hilbert space and that \(\mathcal{B}(H)\) 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 \(\mathcal{A}\subseteq\mathcal{B}(H)\) is an abelian algebra with the property that given any bounded representation \(\varrho:\mathcal{A}\to\mathcal{B}(H_{\varrho})\) of \(\mathcal{A}\) on a Hilbert space \(H_{\varrho}\), every invariant subspace of \(\varrho(\mathcal{A})\) is topologically complemented by another invariant subspace of \(\varrho(\mathcal{A})\), then \(\mathcal{A}\) is similar to an abelian \(C^{*}\)-algebra.

MSC:

46J05 General theory of commutative topological algebras
47L10 Algebras of operators on Banach spaces and other topological linear spaces
47L30 Abstract operator algebras on Hilbert spaces
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