In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten S_p class p 2 [2; 1/, then the n-fold tensor power H_Γ^˝n for n ≥ ^p₂ is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in S_p for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-S_p property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation.

Let G be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup K. Let ΩK be minus the radial Casimir operator. Let 1 4 dim(G/K) < SG < 1 2 dim(G/K), s ∈ (0, SG] and p ∈ (1,∞) be such that(1 p - 1 2 )< s 2SG . Then, there exists a constant CG,s,p > 0 such that for every m ∈ L∞(G) ∩ L2(G) bi-K-invariant with m ∈ Dom(Ωs K) and Ωs K(m) ∈ L2SG/s(G) we have, (0.1) ∥Tm : Lp(G) → Lp( G)∥ ≤ CG,s,p∥Ωs K(m)∥ L2SG/s(G), where Tm is the Fourier multiplier with symbol m acting on the noncommutative Lp-space of the group von Neumann algebra of G. This gives new examples of Lp-Fourier multipliers with decay rates becoming slower when p approximates 2.

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C _b(G) be a p-multiplier on G with 1≤p<∞ and let T _m:L _p(G^)→L _p(G^) be the corresponding Fourier multiplier. Similarly, let Tm| _Γ:L _p(Γ^)→L _p(Γ^) be the Fourier multiplier associated to the restriction m| _Γ of m to Γ. We show that (Formula presented.) for a specific constant 0≤c(U)≤1 that is defined for every U⊆Γ. The function c quantifies the failure of G to admit small almost Γ-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1 when G has small almost Γ-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(B _ρ ^G)≥ρ ^-d/4 where B _ρ ^G is the ball of g∈G with ‖Ad _g‖<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G with c(Γ)=1. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

For a real Hilbert space H_R and −1 < q < 1 Bozejko and Speicher introduced the C^∗-algebra A_q(H_R) and von Neumann algebra M_q(H_R) of qGaussian variables. We prove that if dim(H_R) = ∞ and −1 < q < 1, q ∕= 0 then M_q(H_R) does not have the Akemann-Ostrand property with respect to A_q(H_R). It follows that A_q(H_R) is not isomorphic to A₀(H_R). This gives an answer to the C^∗-algebraic part of Question 1.1 and Question 1.2 in raised by Nelson and Zeng [Int. Math. Res. Not. IMRN 17 (2018), pp. 5486–5535].

Let π_α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space Aα2(Ω) on a bounded symmetric domain Ω , of formal dimension dπα>0. It is shown that if the Bergman kernel kz(α) is a cyclic vector for the restriction π_α| _Γ to a lattice Γ ≤ G (resp. (πα(γ)kz(α))γ∈Γ is a frame for Aα2(Ω)), then vol(G/Γ)dπα≤|Γz|-1. The estimate vol(G/Γ)dπα≥|Γz|-1 holds for kz(α) being a p_z-separating vector (resp. (πα(γ)kz(α))γ∈Γ/Γz being a Riesz sequence in Aα2(Ω)). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G= PSU (1 , 1).

Let G be a locally compact unimodular group, and let φ be some function of n variables on G. To such a φ, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative L _p-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes S _p(L ₂(G). We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called multiplicatively bounded (p1,....,pn)-norm"of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map L _p1(R,S _p1) × L _p2 (R,S _p2), → L ₁(R,S ₁), whenever p1 and p2 are such that {equation presented}. A similar result holds for certain Calderón-Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.

One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call 'approximate linearity with almost commuting intertwiners'. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-S2 condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann-Ostrand property; in particular, the same strong solidity results follow again (now following [27]).