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Search: MSC category 43A07 ( Means on groups, semigroups, etc.; amenable groups )

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1. CMB Online first

Ghanei, Mohammad Reza; Nasr-Isfahani, Rasoul; Nemati, Mehdi
A homological property and Arens regularity of locally compact quantum groups
We characterize two important notions of amenability and compactness of a locally compact quantum group ${\mathbb G}$ in terms of certain homological properties. For this, we show that ${\mathbb G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra $L^1({\mathbb G})$; in particular, we improve an interesting result by Hu, Neufang and Ruan.

Keywords:amenability, Arens regularity, co-amenability, locally compact quantum group, homological property
Categories:46L89, 43A07, 46H20, 46M10, 58B32

2. CMB 2015 (vol 58 pp. 415)

Willson, Benjamin
A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability
In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a left Haar measure.

Keywords:invariant measure, Haar measure, hypergroup, amenability, function translations
Categories:43A62, 43A05, 43A07

3. CMB 2012 (vol 57 pp. 37)

Dashti, Mahshid; Nasr-Isfahani, Rasoul; Renani, Sima Soltani
Character Amenability of Lipschitz Algebras
Let ${\mathcal X}$ be a locally compact metric space and let ${\mathcal A}$ be any of the Lipschitz algebras ${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or ${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a consequence of rather more general results on Banach algebras, that ${\mathcal A}$ is $C$-character amenable if and only if ${\mathcal X}$ is uniformly discrete.

Keywords:character amenable, character contractible, Lipschitz algebras, spectrum
Categories:43A07, 46H05, 46J10

4. CMB 2008 (vol 51 pp. 60)

Janzen, David
F{\o}lner Nets for Semidirect Products of Amenable Groups
For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided.

Categories:22D05, 43A07, 22D15, 43A20

5. CMB 2001 (vol 44 pp. 231)

Rosenblatt, Joseph M.; Willis, George A.
Weak Convergence Is Not Strong Convergence For Amenable Groups
Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $(f_\alpha)$ of positive, normalized functions in $L_1(G)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.


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