Representations of Compact Right Topological Groups

Milnes, Paul

Canadian mathematical bulletin, Tome 36 (1993) no. 3, pp. 314-323

Résumé

Compact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centre or a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that: (i) for the simplest (perhaps) G generated by Z, (l, H) decomposes into one copy of each irreducible representation of Z and c copies of the regular representation. (ii) for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0. (iii) when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

Détail

DOI : 10.4153/CMB-1993-044-1

Mots-clés : 22D10, compact right topological group, Haar measure, regular representation