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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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