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 ${C^*}$-algebras is injective, i.e. the module tensor product of isometric morphisms is an isometric linear map. Helemskii introduced $\textit{Ruan}\ \mathcal{B}$-bimodules and left or right $\textit{semi-Ruan}\ \mathcal{B}$-modules, where $\mathcal{B} = \mathcal{B}(L)$ and $L$ a separable Hilbert space. Then he shows that certain ${\mathcal{B}}$-modules have a flatness property with respect to $\textit{(semi-) Ruan\ } \mathcal{B}$-modules. We generalize this program to properly infinite ${C^*}$-algebras $\mathcal{A}$ and show that the projective module tensor product of arbitrary left and right $\textit{semi-Ruan}\ $ ${\mathcal{B}}$-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 ${C^*}$-bimodule morphisms and a proof for the injectivity of the module Haagerup tensor product of operator ${C^*}$--modules. Semi-Ruan modules have a minimal isometric isomorphic representation as a submodule of $\mathcal{B}(\mathcal{K},\mathcal{H})$ for some Hilbert spaces $\mathcal{H}$, $\mathcal{K}$.

Keywords: Projective module tensor product, properly infinite \Cstar-algebra, operator module, module Haagerup tensor product, extension theorem, completely bounded operator

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