Duke Mathematical Journal

Abelian, amenable operator algebras are similar to C-algebras

Laurent W. Marcoux and Alexey I. Popov

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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 AB(H) is an abelian algebra with the property that given any bounded representation ϱ:AB(Hϱ) of A on a Hilbert space Hϱ, every invariant subspace of ϱ(A) is topologically complemented by another invariant subspace of ϱ(A), then A is similar to an abelian C-algebra.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

abelian operator Banach algebra $C^{*}$-algebra total reduction property


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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