The Extensions of an Invariant Mean and the Set LIM ∽ TLIM

Canadian journal of mathematics, Tome 46 (1994) no. 4, pp. 808-817

Résumé

Let with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L ∞(G) which are mutually singular to every element of TLIM(L ∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L ∞(G). It follows that the cardinalities of LIM(L ∞(G)) ̴ TLIM(L ∞(G)) and LIM(L ∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G 1 ⨯ G 2, where G 1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G 2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.

Détail

DOI : 10.4153/CJM-1994-046-x

Mots-clés : 43A07, locally compact groups, amenable groups, invariant means, the set of left invariant means

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