Amenability notions of hypergroups and some applications to locally compact groups
Mahmood Alaghmandan
备注：
Significant revisions to the paper. Abstract revised, some typos corrected, some references added. The exposition has been improved while the paper has been shortened significantly. An error in the proof of the Leptin theorem is corrected by restricting the case to discrete hypergroups
Different notions of amenability on hypergroups and their relations are studied. Developing Leptin's theorem for discrete hypergroups, we characterize the existence of a bounded approximate identity for hypergroup Fourier algebras. We study the Leptin condition for discrete hypergroups derived from the representation theory of some classes of compact groups. Studying amenability of the hypergroup algebras for discrete commutative hypergroups, we obtain some results on amenability properties of some central Banach algebras on compact and discrete groups.
In this paper, for a locally compact commutative hypergroup $K$ and for a pair $(Φ_1, Φ_2)$ of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of $K,$ for the convolution $f\ast g$ to exist a.e., where $f$ and $g$ are arbitrary elements of Orlicz spaces $L^{Φ_1}(K)$ and $L^{Φ_2}(K)$, respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from $L^1_w(K)$ into $L^Φ_w(K)$ for a…
We consider amenability constants of the central Fourier algebra $ZA(G)$ of a finite group $G$. This is a dual object to $ZL^1(G)$ in the sense of hypergroup algebras, and as such shares similar amenability theory. We will provide several classes of groups where $AM(ZA(G)) = AM(ZL^1(G))$, and discuss $AM(ZA(G))$ when $G$ has two conjugacy class sizes. We also produce a new counterexample that shows that unlike $AM(ZL^1(G))$, $AM(ZA(G))$ does not respect quotient groups, however the class of groups that does has $\frac{7}{4}$ as the sharp amenability constant bound.
Compact and weakly compact multipliers on Fourier algebras of ultraspherical hypergroups
局部紧群$ G $是离散的当且仅当傅立叶代数$ A(G) $有非零（弱）紧乘子。我们将这个结果部分地推广到超球超群的情形。设$H$是超球超群，$A(H)$表示相应的Fourier代数.我们将根据$A(H)$的代数性质给出$ H $离散性的几个刻画。我们还研究了$ A(H)$的闭理想的Arens正则性.
Let $ω$ be a weight function on a locally compact group G mand let $ M_* (G, ω) $ be the subspace of $ M(G , ω)^* $ consisting of all functionals that vanish at infinity. In this paper, we first investigate the Arens regularity of $ M_* (G, ω)^* $ and show that $ M_* (G, ω)^* $ is Arnes regular if and only if G is finite or $ ω$ is zero cluster. This result is an answer to the question posed and it improves some well-known results. We also give necessary and sufficient criteria for the weight function spaces $ Wap(G , 1/ ω) $ and $ Wap(G , 1/ ω) $ to be equal to $ C_b (G , 1/ ω) $. We prove th…
The notion of Fréchet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fréchet locally $C^*$-algebra is ideally amenable.
We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). By using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.
In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G, ω)$ is weakly amenable if and only if every non-inner quasi-additive function in $L^\infty(G, 1/ω)$ is unbounded. This provides an answer to the question concerning weak amenability of $L^1(G, ω)$ and improve some known results in connection with it.
Finie dimensional invariant subspace property and amenability for a class of Banach algebras
受Ky Fan在1965年的一个结果的启发，我们利用有限维不变子空间的性质，建立了左顺从F-代数（包括局部紧群的群代数和Fourier代数，量子群代数，或者更一般地说，Hopf von Neumann代数的前对偶代数）的一个刻画.这是通过首先揭示一个不动点性质的半群的范数为1的正线性泛函的代数。我们的结果回答了第一作者于1993年在东京提出的一个公开问题（见[25，问题5]）。我们还证明了F-代数中理想的左顺从性可以决定该代数的左顺从性。
设G是局部紧群，且1 < p < \infty .本文研究了左L^1（G）-模L^p（G）的内射性.我们定义了一个顺从性类型的条件族，称为（p，q）-顺从性，对于任何1 <= p <= q。对一般的局部紧群G，证明了如果L^p（G）是内射的，则G必是（p，p）-顺从的.对于离散群G，我们证明了l^p（G）是单射的当且仅当G是（p，p）-顺从的.
We establish several new characterizations of amenable $W^*$- and $C^*$-dynamical systems over arbitrary locally compact groups. In the $W^*$-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of $(M,G,α)$ converging point weak* to the identity of $G\bar{\ltimes}M$. In the $C^*$-setting, we prove that amenability of $(A,G,α)$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$, as well as a particular case of the pos…
