Diagonals in Tensor Products of Operator Algebras

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 $C^{\ast }$-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 $C^{\ast }$-algebra.

10 pages, latex file