A new characterization of the Haagerup property by actions on infinite measure spaces

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.

P-Tensor Product for Group C*-Algebras

In this paper, we introduce new tensor products ⊗ p ( 1 ≤ p ≤ + ∞ ) on C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ ) and ⊗ c 0 on C c 0 * ( Γ ) ⊗ C c 0 * ( Γ ) for any discrete group Γ . We obtain that for 1 ≤ p < + ∞ C ℓ p * ( Γ ) ⊗ m a x C ℓ p * ( Γ ) = C ℓ p * ( Γ ) ⊗ p C ℓ p * ( Γ ) if and only if Γ is amenable; C c 0 * ( Γ ) ⊗ m a x C c 0 * ( Γ ) = C c 0 * ( Γ ) ⊗ c 0 C c 0 * ( Γ ) if and only if Γ has Haagerup property. In particular, for the free group with two generators F 2 we show that C ℓ p * ( F 2 ) ⊗ p C ℓ p * ( F 2 ) ≇ C ℓ q * ( F 2 ) ⊗ q C ℓ q * ( F 2 ) for 2 ≤ q < p ≤ + ∞ .

On the Haagerup and Kazhdan properties of R. Thompson’s groups

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F, T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero {[F,F]} -invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.

On graph products of multipliers and the Haagerup property for -dynamical systems

We consider the notion of the graph product of actions of discrete groups $\{G_{v}\}$ on a $C^{\ast }$-algebra ${\mathcal{A}}$ and show that under suitable commutativity conditions the graph product action $\star _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FC}_{v}:\star _{\unicode[STIX]{x1D6E4}}G_{v}\curvearrowright {\mathcal{A}}$ has the Haagerup property if each action $\unicode[STIX]{x1D6FC}_{v}:G_{v}\curvearrowright {\mathcal{A}}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.

L p {L_{p}} -representations of discrete quantum groups

Given a locally compact quantum group {\mathbb{G}} , we define and study representations and {\mathrm{C}^{\ast}} -completions of the convolution algebra {L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra {C_{b}(\mathbb{G})} . For discrete quantum groups {\mathbb{G}} , we investigate the left regular representation, amenability and the Haagerup property in this framework. When {\mathbb{G}} is unimodular and discrete, we study in detail the {\mathrm{C}^{\ast}} -completions of {L_{1}(\mathbb{G})} associated with the non-commutative {L_{p}} -spaces {L_{p}(\mathbb{G})} . As an application of this theory, we characterize (for each {p\in[1,\infty)} ) the positive definite functions on unimodular orthogonal and unitary free quantum groups {\mathbb{G}} that extend to states on the {L_{p}} - {\mathrm{C}^{\ast}} -algebra of {\mathbb{G}} . Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.