Using the recently developed notion of a Herz-Schur multiplier C*-dynamical system we introduce weak amenability C*- and W*-dynamical systems. As special case recover Haagerup's characterisation discrete group. We also consider generalisation Fourier algebra its multipliers to crossed products.

We fix a finitely presented group Q and consider short exact sequences 1 → N → Γ → Q → 1 with Γ finitely generated. The inclusion N ↪→ Γ induces a morphism of profinite completions N̂ → Γ̂. We prove that this is an isomorphism for all N and Γ if and only if Q is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residua...

A multilinear version of Schur’s test is obtained for products of L spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the linear operators on univariate polynomials preserving realrootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal dist...

In the study of Zeilberger’s conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let (t)n denote the rising factorial, and let ΛR denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from ΛR to R defined by φ(hn) = 1/((t)nn!) for some t > 0, then for any Schur function sλ, the value φ(sλ) is positive...

We prove that the duality operator preserves the Frobenius–Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius–Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius–Schur indicators of certain real-valued, i...