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 Tr. Mat. Inst. Steklova, 2010, Volume 271, Pages 224–240 (Mi tm3233)

Duality between compactness and discreteness beyond Pontryagin duality

A. I. Shternab

a Moscow State University, Moscow, Russia
b Research Institute for System Studies, Russian Academy of Sciences, Moscow, Russia

Abstract: One of the most striking results of Pontryagin's duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height $2$ whose dual spaces are quasicompact and non-Hausdorff (they are $T_1$ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a $T_1$ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 271, 212–227

Bibliographic databases:

UDC: 517.986.6+517.986.66

Citation: A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Differential equations and topology. II, Collected papers. In commemoration of the centenary of the birth of Academician Lev Semenovich Pontryagin, Tr. Mat. Inst. Steklova, 271, MAIK Nauka/Interperiodica, Moscow, 2010, 224–240; Proc. Steklov Inst. Math., 271 (2010), 212–227

Citation in format AMSBIB
\Bibitem{Sht10} \by A.~I.~Shtern \paper Duality between compactness and discreteness beyond Pontryagin duality \inbook Differential equations and topology.~II \bookinfo Collected papers. In commemoration of the centenary of the birth of Academician Lev Semenovich Pontryagin \serial Tr. Mat. Inst. Steklova \yr 2010 \vol 271 \pages 224--240 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3233} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2847767} \elib{https://elibrary.ru/item.asp?id=15524643} \transl \jour Proc. Steklov Inst. Math. \yr 2010 \vol 271 \pages 212--227 \crossref{https://doi.org/10.1134/S0081543810040164} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000287921200016} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79952239562} 

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