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Abstract

We consider quantum analogues of locally convex spaces in terms of the non-coordinate approach. We introduce the notions of a quantum Arens–Michael algebra and a quantum polynormed module, and also quantum versions of projectivity and contractibility. We prove that a quantum Arens–Michael algebra is contractible if and only if it is completely isomorphic to a Cartesian product of full matrix $C^{\ast}$-algebras. Similar results in the framework of traditional (non-quantum) approach are established, at the moment, only under some additional assumptions.