Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T02:53:26.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Haar measure and compact right topological groups

Published online by Cambridge University Press:  17 April 2009

Paul Milnes
Paul Milnes
Affiliation:
Department of MathematicsUniversity of Western OntarioLondon Ontario N6A 5B7, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 3 , June 1992 , pp. 399 - 413
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Anzai, H. and Kakutani, S., ‘Bohr compactifications of a locally compact abelian group I, II’, Proc. Imp. Acad. Tokyo 19 (1943), 476–480, 533–539.Google Scholar
[2]Auslander, L. and Hahn, F., ‘Real functions coming from flows on compact spaces and concepts of almost periodicity’, Trans. Amer. Math. Soc. 106 (1963), 415–426.CrossRefGoogle Scholar
[3]Berglund, J.F., Junghenn, H.D. and Milnes, P., Analysis on semigroups: function spaces, compactifactions, representations (Wiley, New York, 1989).Google Scholar
[4]Ellis, R., ‘Distal transformation groups’, Pacific J. Math. 9 (1958), 401–405.CrossRefGoogle Scholar
[5]Ellis, R., Lectures on topological dynamics (Benjamin, New York, 1969).Google Scholar
[6]Hewitt, E. and Ross, K.A., Abstract harmonic analysis I (Springer-Verlag, Berlin, Heidelberg, New York, 1963).Google Scholar
[7]Knapp, A.W., ‘Distal functions on groups’, Trans. Amer. Math. Soc. 128 (1967), 1–40.CrossRefGoogle Scholar
[8]Milnes, P., ‘Ellis groups and group extensions’, Houston J. Math. 12 (1986), 87–108.Google Scholar
[9]Milnes, P., ‘Enveloping semigroups, distal flows and groups of Heisenberg type’, Houston J. Math. 15 (1989), 563–572.Google Scholar
[10]Milnes, P., ‘Extension of flows, Ellis groups and groups of Heisenberg type’, Math. Nach. 150 (1991), 97–108.CrossRefGoogle Scholar
[11]Milnes, P. and Pym, J., ‘Haar measure for compact right topological groups’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[12]Milnes, P. and Pym, J., ‘Homomorphisms of minimal and distal flows’, (submitted).Google Scholar
[13]Namioka, I., ‘Right topological groups, distal flows and a fixed point theorem’, Math. Systems Theory 6 (1972), 193–209.CrossRefGoogle Scholar
[14]Namioka, I., ‘Ellis groups and compact right topological groups’, in Contemporary Mathematics 26: Conference in modern analysis and probability, pp. 295–300 (Amer. Math. Soc., 1984).CrossRefGoogle Scholar
[15]Nielsen, O.A., Unitary representations and coadjoint orbits of low-dimensional nilpotent lie groups (Queen's Papers in Pure and Applied Mathematics, no. 63, Kingston, Ont., 1983).Google Scholar
[16]Ruppert, W.A.F.,‘Über kompakte rechtstopologische Gruppen mit gleichgradig stetigen Linkstranslationen’, Sitzungsber. Österreichischen Akad. Wiss. Math.-Natur. Kl. 184 (1975), 159–169.Google Scholar
[17]Zappa, G.,‘Sulla construzione dei gruppi di due dati sottogruppi permutabili tra loro’, Atti Secundo Congresso Unione Matematica Italiana, Bologna (1940), 119–125.Google Scholar