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Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters
- Source :
- Rocky Mountain J. Math. 50, no. 6 (2020), 2103-2115
- Publication Year :
- 2020
- Publisher :
- Rocky Mountain Mathematics Consortium, 2020.
-
Abstract
- In 1970, Chou showed there are $|\mathbb{N}^*| = 2^{2^\mathbb{N}}$ topologically invariant means on $L_\infty(G)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_\infty(G)$ 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 $L_1(G)$ and $A(G)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\Gamma$ has $|\Gamma^*|$ accumulation points, where $|\Gamma^*|$ is determined by ultrafilter theory. Among a smattering of other results, I prove Paterson's conjecture that left and right topologically invariant means on $L_\infty(G)$ coincide iff $G$ has precompact conjugacy classes.<br />Comment: 10 pages, completely rewritten from v2
- Subjects :
- General Mathematics
Ultrafilter
01 natural sciences
Combinatorics
Conjugacy class
Cardinality
FOS: Mathematics
0101 mathematics
Mathematics
Conjecture
43A07
20F24
010102 general mathematics
Amenable group
Locally compact group
Invariant (physics)
43A07, 43A30, 20F24, 54A20
Net (mathematics)
Functional Analysis (math.FA)
Mathematics - Functional Analysis
010101 applied mathematics
cardinality
FC groups
amenable groups
43A30
invariant means
ultrafilters
Fourier algebras
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Rocky Mountain J. Math. 50, no. 6 (2020), 2103-2115
- Accession number :
- edsair.doi.dedup.....c904c3c7fddca367502bc4e973cdca4e