Suppose that is a complex Hilbert space and that denotes the bounded linear operators on . We show that every abelian, amenable operator algebra is similar to a -algebra. We do this by showing that if is an abelian algebra with the property that given any bounded representation of on a Hilbert space , every invariant subspace of is topologically complemented by another invariant subspace of , then is similar to an abelian -algebra.
"Abelian, amenable operator algebras are similar to -algebras." Duke Math. J. 165 (12) 2391 - 2406, 1 September 2016. https://doi.org/10.1215/00127094-3619791