Related papers: Reiter's properties (P_1) and (P_2) for locally co…

The notion of an action of a locally compact quantum group on a von Neumann algebra is studied from the amenability point of view. Various Reiter's conditions for such an action are discussed. Several applications to some specific actions…

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of…

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine…

We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological…

Given a locally compact quantum group $\mathbb{G}$ and a closed quantum subgroup $\mathbb{H}$, we show that $\mathbb{G}$ is amenable if and only if $\mathbb{H}$ is amenable and $\mathbb{G}$ acts amenably on the quantum homogenous space…

In this short note, further to Ng's study, we extend Bekka amenability and weak Bekka amenability to general locally compact quantum groups. We generalize some Ng's results to the general case. In particular, we show that, a locally compact…

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of…

We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space…

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a…

As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the…

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…

We prove that a closed subgroup $H$ of a second countable locally compact group $G$ is amenable if and only if its left regular representation on an Orlicz space $L^\Phi(G)$ for some $\Delta_2$-regular $N$-function $\Phi$ almost has…

We study relative amenability and amenability of a right coideal $\widetilde{N}_P\subseteq \ell^\infty(\mathbb{G})$ of a discrete quantum group in terms of its group-like projection $P$. We establish a notion of a $P$-left invariant state…

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied…

Let $K$ denote a locally compact commutative hypergroup, $L^1(K)$ the hypergroup algebra, and $\alpha$ a real-valued hermitian character of $K$. We show that $K$ is $\alpha$-amenable if and only if $L^1(K)$ is $\alpha$-left amenable. We…

We introduce a new combinatorial condition that characterises the amenability for locally compact groups. Our condition is weaker than the well-known F{\o}lner's conditions, and so is potentially useful as a criteria to show the amenability…

Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove…

In this short note we introduce a notion called "quantum injectivity" of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of…

We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result…

It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable.…