Abelian, amenable operator algebras are similar to C∗ -algebras
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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 𝒜⊆ℬ(H) is an abelian algebra with the property that given any bounded representation ϱ:𝒜→ℬ(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.
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Laurent W. Marcoux, Alexey I. Popov (2016). Abelian, amenable operator algebras are similar to C∗ -algebras. UWSpace. http://hdl.handle.net/10012/15731