# 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 \( H, K \).

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

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