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 the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on 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 $\mathcal B(H)$ 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. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules.

中文翻译：

---

C⁎-运营商系统和交叉产品

本文的目的是通过操作员系统和操作员空间上的组的动作和协作来介绍通用和约简交叉乘积的一致概念。我们将特别强调揭示万能交叉积的万能性质的全部力量。事实证明，为了使事情保持一致，在一些更大的类别上执行我们的构造似乎很有用，这些类别允许研究通用属性的正确框架，并且即使对于非离散群，在交叉积的构造下也是稳定的。在算子系统的情况下，这个更大的范畴就是我们所说的 $C^*$-operator 系统，即某个 $\mathcal B(H)$ 的自伴随子空间 $X$，它包含一个 $C^*$ -代数 $A$ 使得 $AX=X=XA$。在操作符空间的情况下，更大的类别由我们所说的 $C^*$-operator bimodules 给出。在我们介绍了各自的交叉积之后，我们证明了经典的 Imai-Takai 和 Katayama 对偶定理通过 $C^*$-代数上的群（共）作用的交叉积将一对一扩展到我们的交叉积概念： $C^*$-operator 系统和 $C^*$-operator bimodules。