2021/05/06 by John Hopfensperger, Hopfensperger, John
Mathematics · #43A07 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Rings, Modules, and Algebras #math.FA #math.GR #msc:43A07
paper · pdf · doi:10.48550/arxiv.2105.02768
The author's PhD thesis. Three chapters are derived from previously published papers. v2: Corrected typos
openalex publication_date 2021/05/06 · arxiv created 2021/05/16 · arxiv updated 2021/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose G is an amenable locally compact group. If \Fγ\ = \Fγ\γ∈Γ is a Følner net for G, associate it with the net \χFγ / |Fγ|\ ⊂ L1(G) ⊂ L_∞^*(G). Thus, every accumulation point of \Fγ\ is a topological left-invariant mean on G. The following are examples of results proved in the present thesis: (1) There exists a Følner net which has as its accumulation points a set of 22κ distinct topological left-invariant means on G, where κ is the smallest cardinality of a covering of G by compact subsets. (2) If G is unimodular and μ is a topological left-invariant mean on G, there exists a Følner net which has μ as its unique accumulation point. (3) Suppose L ⊂ G is a lattice subgroup. There is a natural bijection of the left-invariant means on L with the topological left-invariant means on G if and only if G/L is compact. (4) Every topological left-invariant mean on G is also topological right-invariant if and only if G has precompact conjugacy classes. These results lie at the intersection of functional analysis with general topology. Problems in this area can often be solved with standard tools when G is σ-compact or metrizable, but require more interesting arguments in the general case.