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Paper IPM / M / 16210  


Abstract:  
he 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 nondiscrete groups. In the case of operator systems, this larger category is what we call a Câoperator system, i.e., a selfadjoint subspace X of some B(H) which contains a Câalgebra A such that AX=X=XA. In the case of operator spaces, the larger category is given by what we call Câoperator bimodules. We introduce the notion of crossed products in these categories and show that the classical ImaiTakai and Katayama duality theorems for crossed products by group (co)actions on Câalgebras extend to our notion of crossed products by Câoperator systems and Câoperator bimodules.
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