Functorial methods in the theory of group representations. I (Q1890940)

It is a well-known classical result that, for a finite group \(G\) and for any linear representation \(\varphi : G \to GL (V)\), \(V/ \mathbb{C}\) a finite-dimensional vector space, there is a unique algebra homomorphism \(R : \mathbb{C} [G] \to \text{End}_ \mathbb{C} (V)\), \(\mathbb{C} [G]\) the group algebra of \(G\). This result was extended by Gelfand and Naimark in 1943 and by Rieffel in 1967 to locally compact groups with different methods. The authors go one important step further and give a solution for the corresponding problem even for Hausdorff topological groups. In order to introduce a suitable notion of group algebra \(M(G)\) in the case of a Hausdorff topological group \(G\), the authors first define the category of topological Banach balls \({\mathcal T} {\mathcal B}\), i.e., the full subcategory of the category of topological totally convex spaces [cf. \textit{H. Kleisli} and \textit{H.-P. Künzi}, Topological totally convex spaces. I, Appl. Categ. Struct. 2, No. 1, 45-55 (1994; Zbl 0810.18003); \textit{H. Kleisli} and \textit{H.-P. Künzi}, Topological totally convex spaces. II, Cah. Topologie Géom. Différ. Catégoriques 36, No. 1, 11-52 (1995; see the preceding review)] generated by the unit balls of Banach spaces equipped with an additional Hausdorff, locally convex topology. The full subcategory \({\mathcal B}\) of \(T {\mathcal B}\) generated by Banach balls with the norm topology is dual to the full subcategory \({\mathcal C}\) of \(T {\mathcal B}\) generated by the compact Banach balls. This duality gives rise to a pre-\(*\)-autonomous situation [cf. \textit{M. Barr}, \(*\)-autonomous categories, Lect. Notes Math. 752 (1979; Zbl 0415.18008)] and leads to a natural enlargement of the category \({\mathcal C}\) to a bigger full subcategory \({\mathcal E}\) of \(T {\mathcal B}\). \({\mathcal E}\) possesses a canonical tensor product and, for a Hausdorff topological group \(G\), the dual of the unit ball of the Banach space of all bounded, \(k\)-continuous, \(\mathbb{C}\)-valued functions on \(G\), is an object \(M(G)\) of \({\mathcal E}\). The tensor product in \({\mathcal E}\) permits the definition of an algebra structure on \(M(G)\), which makes \(M(G)\) an algebra object over \({\mathcal E}\). That \(M(G)\) solves the problem stated at the beginning is shown by theorem 6.7: every \(k\)-continuous unitary representation of \(G\) in a complex Hilbert space \(H\) can be uniquely extended to a continuous representation \(R\) of \(M(G)\) in \(H\).