Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters

Abstract

In 1970, Chou showed there are |ℕ∗|=22ℕ 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 if and only if G has precompact conjugacy classes. Finally, I discuss some open problems arising from the study of the sizes of sets of invariant means on groups and semigroups.