Abstract
We introduce a candidate for the group algebra of a Hausdorff group which plays the same role as the group algebra of a finite group. It allows to define a natural bijection betweenk-continuous representations of the group in a Hilbert space and continuous representations of the group algebra. Such bijections are known, but to our knowledge only for locally compact groups. We can establish such a bijection for more general groups, namely Hausdorff groups, because we replace integration techniques by functorial methods, i.e., by using a duality functor which lives in certain categories of topological Banach balls (resp., unit balls of Saks spaces).
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References
Barr, M.: *-Autonomous Categories, Lecture Notes in Mathematics 752, Springer-Verlag, 1979.
Brown, R.: Function spaces and produt topologies,Quat. J. Math. Oxford 15(2) (1964), 238–250.
Cigler, J., Losert, V., and Michor, P.:Banach Modules and Functors on Categories of Banach Spaces, Lecture Notes 46, Marcel Dekker, New York, 1979.
Cooper, J. B.:Saks Spaces and Applications to Functional Analysis, 2nd edn, North-Holland, Amsterdam,Mathematics Studies 139, 1987.
Kelley, J. L.:General Topology, Graduate Texts in Mathematics27, Springer-Verlag, 1975.
Kleisli, H. and Künzi, H.-P.: Topological totally convex spaces I,Applied Categorical Structures 2 (1994), 45–55.
Kleisli, H. and Künzi, H.-P.: Topological totally convex spaces II, to appear inCahiers de Top. et Géom. Diff. Catégorique.
Pumplün, D. and Röhrl, H.: Banach spaces and totally convex spaces I,Comm. Alg. 12 (1984), 953–1019.
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This paper was written while the authors were supported by the Swiss National Science Foundation under grant 21-33644.92.
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Dorofeev, S., Kleisli, H. Functorial methods in the theory of group representations I. Appl Categor Struct 3, 151–172 (1995). https://doi.org/10.1007/BF00877634
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DOI: https://doi.org/10.1007/BF00877634