Journal of Operator Theory

Volume 39, Issue 1, Winter 1998 pp. 113-121.

The central Haagerup tensor product of a $C^*$-algebra

Authors: D.W.B. Somerset
Author institution: Department of Mathematical Sciences, University of Aberdeen, AB24 3UE, U.K.

Summary: Let $A$ be a $C^*$-algebra with an identity and let $\theta_Z$ be the canonical map from $A\otimes_Z A$, the central Haagerup tensor product of $A$, to $CB(A)$, the algebra of completely bounded operators on $A$. It is shown that if every Glimm ideal of $A$ is primal then $\theta_Z$ is an isometry. This covers unital quasi-standard $C^*$-algebras and quotients of AW$^*$-algebras.

Keywords: $C^*$-algebra, Haagerup tensor product, primal ideal

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