DIAGONALS IN TENSOR PRODUCTS OF OPERATOR ALGEBRAS

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.AMS 2000 Mathematics subject classification: Primary 46L06. Secondary 46L05