Journal of Operator Theory

Volume 47, Issue 2, Spring 2002 pp. 389-412.

Relative tensor products and infinite $C^*$-algebras

Authors: Claire Anantharaman-Delaroche (1), and Ciprian Pop (2)
Author institution: (1) Univeersite d'Orleans, Departement de Mathematiques, B.P. 6759, 45067 Orleans Cedex 02, France
(2) Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70000 Bucharest, Romania

Summary: Let $A, B, C$ be $C^*$-algebras. Given $A$-$B$ and $B$-$C$ normed bimodules $V$ and $W$ respectively, whose unit ball is convex with respect to the actions of the $C^*$-algebras, we study the reasonable seminorms on the relative tensor product $V\oB W$, having the same convexity property. This kind of bimodule is often encountered and retains many features of the usual normed space. We show that the classical Grothendieck program extends nicely in this setting. Fixing $B$, we then establish that there exists an unique such seminorm on $V\oB W$ for any $V, W$ if and only if $B$ is infinite in a weaker sense than proper infiniteness and stronger than the non existence of tracial states (the equivalence of these two latter notions still remaining open). Applying this result when $B$ is a stable $C^*$-algebra, we show that the relative Haagerup tensor product of operator bimodules is both injective and projective.

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