Previous issue ·  Next issue ·  Most recent issue · All issues

# Journal of Operator Theory

Volume 65, Issue 1, Winter 2011  pp. 87-113.

Injectivity of the module tensor product of semi-Ruan modules

Authors Gerd Wittstock
Author institution: Fachrichtung 6.1 Mathematik, Universitaet des Saarlandes, 66041 Saarbruecken, Germany

Summary:  We show that the projective module tensor product of a certain class of contractive left respectively right modules over properly infinite \Cstar-algebras is injective, i.e. the module tensor product of isometric morphisms is an isometric linear map. Helemskii introduced \textit{Ruan} \cB-bimodules and left or right \textit{semi-Ruan} \cB-modules, where $\cB = \B(L)$ and $L$ a separable Hilbert space. Then he shows that certain \cB-modules have a flatness property with respect to \textit{(semi-) Ruan} \cB-modules. We generalize this program to properly infinite \Cstar-algebras $\cA$ and show that the projective module tensor product of arbitrary left and right \textit{semi-Ruan} \cA-modules is injective; i.e.\ they are flat in the sense of Helemskii. The proof starts with the special case of \textit{cyclic} semi-Ruan modules and then uses an exhaustion argument. As an application we obtain a generalization of the extension theorem for completely bounded \Cstar-bimodule morphisms and a proof for the injectivity of the module Haagerup tensor product of operator \Cstar-modules. Semi-Ruan modules have a minimal isometric isomorphic representation as a submodule of $\B(K,H)$ for some Hilbert spaces $H, K$.

Contents    Full-Text PDF