  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.
