Operator Space Complexification Transfigured

Abstract

Abstract Given a finite group $G$, a central subgroup $H$ of $G$, and an operator space $X$ equipped with an action of $H$ by complete isometries, we construct an operator space $X_{G}$ equipped with an action of $G$ that is unique under a “reasonable” condition. This generalizes the operator space complexification $X_{c}$ of $X$. As a linear space $X_{G}$ is the space obtained from inducing the representation of $H$ to $G$ (in the sense of Frobenius). Indeed a main achievement of our paper is the induced representation construction in the category of operator spaces. This has been hitherto elusive even for Banach spaces since it is not clear how to norm the induced space. We show that for a large class of group actions the induced space has a unique operator space norm.