Marcoux, Laurent W.Popov, Alexey I.2020-04-012020-04-012016-12https://doi.org/10.1215/00127094-3619791http://hdl.handle.net/10012/15731Originally published by Duke University PressSuppose 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.enabelian operatorBanach algebraC∗-algebratotal reduction propertyArticle