For a normal measurable operator a affiliated with a von Neumann factor M we show that if M is infinite, then there is λ₀ ∈ ℂ so that for ε?> 0 there are (Formula presented.) with (Formula presented.). If M is finite, then there is λ₀ ∈ ℂ and u, v ∈ U(M) so that (Formula presented.). These bounds are optimal for infinite factors, II₁-factors and some I_n-factors. Furthermore, for finite factors applying ||.||₁-norms to the inequality provides estimates on the norm of the inner derivation (Formula presented.) associated to a. While by [3, Theorem 1.1] it is known for finite factors and self-adjoint (Formula presented.) that (Formula presented.), we present concrete examples of finite factors and normal operators a ∈ M for which this fails.

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.

For a simple graph Γ and for unital C^*-algebras with GNS-faithful states (A_v,φ_v) for v∈VΓ, we consider the reduced graph product (A,φ)=⁎_v,Γ(A_v,φ_v), and show that if every C^*-algebra A_v has the completely contractive approximation property (CCAP) and satisfies some additional condition, then the graph product has the CCAP as well. The additional condition imposed is satisfied in natural cases, for example for the reduced group C^*-algebra of a discrete group G that possesses the CCAP. Our result is an extension of the result of Ricard and Xu in [28, Proposition 4.11] where they prove this result under the same conditions for free products. Moreover, our result also extends the result of Reckwerdt in [27, Theorem 5.5], where he proved for groups that weak amenability with Cowling-Haagerup constant 1 is preserved under graph products. Our result further covers many new cases coming from Hecke-algebras and discrete quantum 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].

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L_∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L_∞([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0, to be such that ∥g∥_∞ ⩽ (S_V + ε)∥f∥_∞, where S_V is the Steinitz constant corresponding to V.

In 1984, Kwapien announced that every mean zero function f 2 L1[0; 1] can be written as a coboundary f = g o T -g for some g 2 L1[0; 1] and some measure preserving transformation T of [0; 1]. Whereas Kwapien's original proof holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapien's result in full generality. Our method also allows us to improve the original result by showing that for any given ϵ > 0 the function g can be chosen to satisfy ∥g∥1 ≤ (1 + ϵ)∥f∥1.