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DIAGONALS IN TENSOR PRODUCTS OF OPERATOR ALGEBRAS

Published online by Cambridge University Press:  14 October 2002

Vern I. Paulsen  and
Roger R. Smith
Vern I. Paulsen
Affiliation:
Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3476, USA (vern@math.uh.edu)
Roger R. Smith
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (rsmith@math.tamu.edu)
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Abstract

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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 $H^1(A,X)=0$ 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

Keywords

diagonaltensor productHaagerupcohomology
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 3 , October 2002 , pp. 647 - 652
Copyright
Copyright © Edinburgh Mathematical Society 2002