Unitary representability of free abelian topological groups

Vladimir V. Uspenskij


For every Tikhonov space X the free abelian topological group A(X) and the free locally convex vector space L(X) admit a topologically faithful unitary representation. For compact spaces X this is due to Jorge Galindo.


Unitary representation; Free topological group; Positive-definite function; Michael selection theorem

Full Text:



W. Banaszczyk, On the existence of exotic Banach – Lie groups, Math. Ann. 264 (1983), 485–493.


B. Bekka, P. de la Harpe, and A. Valette, Kazhdan property (T), available at http://poncelet.sciences.univ-metz.fr/˜bekka/.

Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol. 1, AMS Colloquium Publications, vol. 48 (AMS, Providence, RI, 2000).

J. Galindo, On unitary representability of topological groups, arXiv:math.GN/0607193.

S. Gao and V. Pestov, On a universality property of some abelian Polish groups, Fund. Math. 179 (2003), 1–15; arXiv:math.GN/0205291.

M. Megrelishvili, Reflexively but not unitarily representable topological groups, Topology Proceedings 25 (2000), 615–625.

E. Michael, Continuous selections I, Ann. Math. 63 (1956), 361–382.


V. Pestov, Dynamics of infinite-dimensional groups and Ramsey-type phenomena, (Publica¸c˜oes dos Colóquios de Matemática, IMPA, Rio de Janeiro, 2005).

V. Pestov, Forty-plus annotated questions about large topological groups, arXiv:math.GN/0512564.

I. J. Shoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. Math. 38 (1937), 787–793.


I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522–536.


M. G. Tkachenko, On completeness of free abelian topological groups, Soviet Math. Dokl. 27 (1983), 341–345.

D. Repovˇs and P. V. Semenov, Continuous selections of multivalued mappings, Kluwer Academic Publishers, Dordrecht–Boston–London, 1998.

V. V. Uspenskii, Why compact groups are dyadic, in book: General Topology and its relations to modern analysis and algebra VI: Proc. of the 6th Prague topological Symposium 1986, Frolik Z. (ed.) (Berlin: Heldermann Verlag, 1988), 601–610.

V. V. Uspenski˘ı, Free topological groups of metrizable spaces, Izvestiya Akad. Nauk SSSR, Ser. Matem. 54 (1990), 1295–1319; English transl.: Math. USSR-Izvestiya 37 (1991), 657–680.


V. V. Uspenskij, On unitary representations of groups of isometries, in book: Contribuciones Matem´aticas. Homenaje al profesor Enrique Outerelo Domínguez, E. Martin-Peinador (ed.), Universidad Complutense de Madrid, 2004, 385–389; arXiv:math.RT/0406253.

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt