C⁎-operator systems and crossed products

Abstract

The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the universal crossed products. It turns out to make things consistent, one has to perform the constructions in some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a C⁎-operator system, i.e., a selfadjoint subspace X of some B(H) which contains a C⁎-algebra A such that AX=X=XA. In the case of operator spaces, the larger category is given by what we call C⁎-operator bimodules. We introduce the notion of crossed products in these categories and show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on C⁎-algebras extend to our notion of crossed products by C⁎-operator systems and C⁎-operator bimodules.