A characterization of unitary duality

Author:
David W. Roeder

Journal:
Trans. Amer. Math. Soc. **148** (1970), 129-135

MSC:
Primary 22.60

MathSciNet review:
0255737

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Abstract | References | Similar Articles | Additional Information

Abstract: The concept of unitary duality for topological groups was introduced by H. Chu. All mapping spaces are given the compact-open topology. Let *G* and *H* be locally compact groups. is the space of continuous finite-dimensional unitary representations of *G*. Let denote the space of all continuous maps from to which preserve degree, direct sum, tensor product and equivalence. We prove that if *H* satisfies unitary duality, then and are naturally homeomorphic. Conversely, if and are homeomorphic by the natural map, where *Z* denotes the integers, then *H* satisfies unitary duality. In different contexts, results similar to the first half of this theorem have been obtained by Suzuki and by Ernest. The proof relies heavily on another result in this paper which gives an explicit characterization of the topology on . In addition, we give another necessary condition for locally compact groups to satisfy unitary duality and use this condition to present an example of a maximally almost periodic discrete group which does not satisfy unitary duality.

**[1]**Richard F. Arens,*A topology for spaces of transformations*, Ann. of Math. (2)**47**(1946), 480–495. MR**0017525****[2]**Hsin Chu,*Compactification and duality of topological groups*, Trans. Amer. Math. Soc.**123**(1966), 310–324. MR**0195988**, 10.1090/S0002-9947-1966-0195988-5**[3]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****[4]**John Ernest,*Notes on the duality theorem of non-commutative non-compact topological groups*, Tôhoku Math. J. (2)**16**(1964), 291–296. MR**0169949****[5]**Kazuo Suzuki,*Notes on the duality theorem of non-commutative topological groups.*, Tôhoku Math. J. (2)**15**(1963), 182–186. MR**0148788****[6]**Shuichi Takahashi,*A duality theorem for representable locally compact groups with compact commutator subgroup*, Tôhoku Math. J. (2)**4**(1952), 115–121. MR**0053947****[7]**T. Tannaka,*Über den dualitotsatz der nicht kommutativen topologischen gruppen*, Tôhoku Math. J.**45**(1938), 1-12.**[8]**Nobuhiko Tatsuuma,*A duality theorem for locally compact groups. I*, Proc. Japan Acad.**41**(1965), 878–882. MR**0206161****[9]**A. Weil,*L'intégration dans les groupes topologiques et ses applications*, Hermann, Paris, 1953.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0255737-X

Keywords:
Unitary duality,
locally compact groups,
compact-open topology,
finite-dimensional unitary representations,
category,
functor,
maximally almost periodic group

Article copyright:
© Copyright 1970
American Mathematical Society