Quasi-Hermitian locally compact groups are amenable

Samei, Ebrahim; Wiersma, Matthew

doi:10.1016/j.aim.2019.106897

“…Actually, we claim that C can be seen as (a copy of) the enveloping C * -algebra of ℓ 1 (C) . In [30,Corol. 4.8] it is shown that a symmetric group (as our rigidly symmetric G) is surely amenable.…”

Section: The Abstract Theory 21 the Main Resultsmentioning

confidence: 99%

Symmetry and Spectral Invariance for Topologically Graded C*-Algebras and Partial Action Systems

Jaure,

Mantoiu

2021

Preprint

0

View full text Add to dashboard Cite

A discrete group G is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ 1 (G) and any C * -algebra A is symmetric. We show that in each topologically graded C * -algebra over a rigidly symmetric group there is a ℓ 1 -type symmetric Banach * -algebra, which is inverse closed in the C * -algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the ℓ 1 -decay are included. Various concrete examples are presented.

show abstract

“…Of course, it was later shown that von Neumann's conjecture is false and there are many torsion nonamenable groups. Nonetheless, the conjecture regarding the amenability of (quasi-)Hermitian groups remained as a reasonable one for many years which was recently solved in the affirmative by the second name author and M. Wiersma [12].…”

Section: Introductionmentioning

confidence: 99%

“…

Motivated by the recent result in [12] that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups.

…”

mentioning

confidence: 99%

Uniformly quasi-Hermitian groups are supramenable

Azam¹,

Samei²

2021

Can. Math. Bull.

Self Cite

0

View full text Add to dashboard Cite

Motivated by the recent result in [12] that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

show abstract

“…Neither arrow 3 nor arrow 4 can be reversed: Any compact connected semisimple real Lie group is rigidly symmetric, but in [7] it is shown that it contains a (dense) free group on two elements. The discretization will not be amenable; by [24] it cannot be symmetric.…”

Section: Introductionmentioning

confidence: 99%