## Duke Mathematical Journal

### Abelian, amenable operator algebras are similar to $C^{*}$-algebras

#### Abstract

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.

#### Article information

Source
Duke Math. J., Volume 165, Number 12 (2016), 2391-2406.

Dates
Revised: 7 October 2015
First available in Project Euclid: 6 September 2016

https://projecteuclid.org/euclid.dmj/1473186403

Digital Object Identifier
doi:10.1215/00127094-3619791

Mathematical Reviews number (MathSciNet)
MR3544284

Zentralblatt MATH identifier
1362.46048

#### Citation

Marcoux, Laurent W.; Popov, Alexey I. Abelian, amenable operator algebras are similar to $C^{*}$ -algebras. Duke Math. J. 165 (2016), no. 12, 2391--2406. doi:10.1215/00127094-3619791. https://projecteuclid.org/euclid.dmj/1473186403

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