2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61829In 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.10 pages, latex fileOperator Algebras47L30text