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Diagonals in Tensor Products of Operator Algebras

Vern Paulsen·Roger Smith
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Abstract

In this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra A possesses a diagonal in the Haagerup tensor product of A with itself, then A must be isomorphic to a finite dimensional $C^*$-algebra. Consequently, for operator algebras, the first Hochschild cohomology group, $H^1(A,X) = 0$ for every bounded, Banach A-bimodule X, if and only if A is isomorphic to a finite dimensional $C^*$-algebra.

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Vern Paulsen, Roger Smith. 2001-07-10. Diagonals in Tensor Products of Operator Algebras. https://arxiv.org/abs/math/0107077

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