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Journal of Operator Theory

Volume 89, Issue 2, Spring 2023  pp. 521-570.

Crossed products of dual operator spaces and a characterization of groups with the approximation property

Authors:  Dimitrios Andreou
Author institution: Department of Mathematics, National Kapodistrian University of Athens, Athens, 157 84, Greece

Summary:  Let $ G $ be a locally compact group. We prove that every $ L^{\infty}(G) $-comodule is nondegenerate and saturated, whereas every $ L(G) $-comodule is nondegenerate if and only if every $ L(G) $-comodule is saturated if and only if $ G $ has the approximation property of Haagerup and Kraus. This allows us to extend known results from the duality theory of crossed products of von Neumann algebras to the recent theory of crossed products of dual operator spaces. Also, we obtain a characterization of groups with the approximation property involving crossed products improving a recent result of Crann and Neufang and we generalize a theorem of Anoussis, Katavolos and Todorov providing a less technical proof of it.

Keywords:  crossed products, dual operator spaces, locally compact groups, the approximation property

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