Related papers: On the alpha-Amenability of Hypergroups

For each ordinal $\alpha<\omega_1$, we prove the existence of a space with a basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of spaces with Szlenk index not exceeding $\omega^\alpha$. Our proof involves…

We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the…

In this paper, we introduce a weak form of amenability on topological semigroups that we call $\varphi$-amenability, where $\varphi$ is a character on a topological semigroup. Some basic properties of this new notion are obtained and by…

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ with the property that $\varphi(a,b)=0$ whenever $a$ and $b$ commute is of the form…

It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a…

For two Banach algebras $A$ and $B$, the $T$-Lau product $A\times_T B$, was recently introduced and studied for some bounded homomorphism $T:B\to A$ with $\|T\|\leq 1$. Here, we give general nessesary and sufficent conditions for $A\times_T…

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…

Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact…

We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A*P+PA-PBB*P+C*C=0 when (A,B) is exponentially stabilizable and (C,A) is exponentially…

It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a…

Let $K$ be a locally compact hypergroup with a left invariant Haar measure. We show that the Liouville property and amenability are equivalent for $K$ when it is second countable. Suppose that $\sigma$ is a non-degenerate probability…

Associated with two Banach algebras $\mathcal A$ and $\mathcal B$ and a norm decreasing homomorphism $T:{\mathcal B}\rightarrow{\mathcal A}$, there is a certain Banach algebra product ${\mathcal A}\times_T {\mathcal B}$, which is a…

We use $\mathrm{C}^{\ast}$-algebra ultrapowers to give a new construction of the Stone-Cech compactification of a separable, locally compact space. We use this construction to give a new proof of the fact that groups that act isometrically,…

Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $\sigma$-finite $\mathbb{G}$-dynamical von Neumann algebras. We prove that the $\mathbb{G}$-inclusion $A\subseteq B$ is strongly equivariantly amenable if and…

Motivated by the question of Mikael de la Salle, we investigate the problem of the existence of equivalent strictly convex norms on Banach spaces that are invariant with respect to an action of a group by linear isometries. We develop…

We examine the condition that a complex Banach algebra $A$ have dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in…

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these…

We prove some rather precise renorming theorems for Banach spaces with Szlenk index $\omega_0$. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ \gamma $-element and that $ \gamma = 1 $. It follows that free orthogonal quantum groups are $…

In this paper, we introduce a new notion of biprojectivity, called Connes-biprojective, for dual Banach algebras. We study the relation between this new notion to Connes-amenability and we show that, for a given dual Banach algebra $…