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On abelian, triangularizable, total reduction algebras

Suppose that ℌ is a complex Hilbert space and that 𝔅(ℌ) denotes the bounded linear operators on ℌ. In this paper we show that if 𝒜 ⊆ 𝔅(ℌ) is an abelian algebra with elements that admit representations under which they have a spanning set of eigenvectors, and if 𝒜 has 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 an invariant subspace of ϱ(𝒜), then 𝒜 is similar to an abelian C*‐algebra. From this it follows that any amenable Banach algebra of triangular operators on a separable Hilbert space is similar to an abelian C*‐algebra.