Counting topologically invariant means on L_∞(G) and VN(G) with ultrafilters

2019/06/24 by John Hopfensperger, Hopfensperger, John
Mathematics · #20F24 #43A07 #43A30 #54A20 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:20F24 #msc:43A07 #msc:43A30 #msc:54A20

paper · pdf · doi:10.48550/arxiv.1906.09706

10 pages, completely rewritten from v2

arxiv created 2020/02/29 · arxiv updated 2020/03/03

Abstract

In 1970, Chou showed there are |ℕ^*| = 2^{2^ℕ} topologically invariant means on L_∞(G) for any noncompact, σ-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on L_∞(G) and VN(G) were determined for any locally compact group. Each paper on a new case reached the same conclusion -- "the cardinality is as large as possible" -- but a unified proof never emerged. In this paper, I show L₁(G) and A(G) always contain orthogonal nets converging to invariance. An orthogonal net indexed by Γ has |Γ^*| accumulation points, where |Γ^*| is determined by ultrafilter theory. Among a smattering of other results, I prove Paterson's conjecture that left and right topologically invariant means on L_∞(G) coincide iff G has precompact conjugacy classes.

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