ON THE SET OF TOPOLOGICALLY INVARIANT MEANS ON THE VON NEUMANN ALGEBRA VN(G)

摘要

The study of the cardinality of the set of invariant means on a group was initiated by Day [3] and Granirer [8]. In 1976, Chou [1] showed that for a discrete infinite amenable group G the cardinality of the set ML(G) of all left invariant means on l~∞(G) is 2~(2~(|G|)). Later, Lau and Paterson [20] proved that if G is a noncompact amenable locally compact group, then the set MTL(G) of all topologically left invariant means on L~∞(G) has cardinality 2~(2~(d(G))), where d(G) is the smallest cardinality of a covering of G by compact sets. (Of course, when G is compact, MTL(G) is the singleton containing only the normalized Haar measure of G). For results on the size of the set ML(G)MTL(G), see Granirer [9], Rudin [29], and Rosenblatt [26]. See also Yang [32] and Miao [21] for some recent developments in certain related aspects. We refer the readers to the books of Pier [23] and Paterson [22] for more details on the study of the size and the structure of the set of invariant means on groups and semigroups.