Diagonals in Tensor Products of Operator Algebras

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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, H¹(A,X) = 0 for every bounded, Banach A-bimodule X, if and only if A is isomorphic to a finite dimensional C*-algebra.