Diagonals in Tensor Products of Operator Algebras

2001/07/10 by Vern I. Paulsen, Vern Paulsen, Roger Smith +3
Mathematics · #47L30 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA #msc:47L30

paper · pdf · doi:10.48550/arxiv.math/0107077

10 pages, latex file

arxiv created 2001/07/10 · openalex publication_date 2001/07/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^*-algebra. Consequently, for operator algebras, the first Hochschild cohomology group, H¹(A,X) = 0 for every bounded, Banach A-bimodule X, if and only if A is isomorphic to a finite dimensional C^*-algebra.

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