By methods of harmonic analysis, we identify large classes of Banach spaces invariant of periodic Fourier multipliers with symbols satisfying the classical Marcinkiewicz type conditions. Such classes include general (vector-valued) Banach function spaces Φ and/or the scales of Besov and Triebel–Lizorkin spaces defined on the basis of Φ.

We apply these results to the study of the well-posedness and maximal regularity property of an abstract second-order integro-differential equation, which models various types of elliptic and parabolic problems arising in different areas of applied mathematics. In particular, under suitable conditions imposed on a convolutor c and the geometry of an underlying Banach space X, we characterize the conditions on the operators A, B, and P on X such that the following periodic problem

\begin{equation*}\partial P \partial u + B \partial u + {A} u + c \ast u = f \qquad \textrm{in } {\mathcal D}'({\mathbb{T}}; X)\end{equation*}

is well-posed with respect to large classes of function spaces. The obtained results extend the known theory on the maximal regularity of such problem.

For a finite group G, let $\operatorname { {AD}}(G)$ denote the Fourier norm of the antidiagonal in $G\times G$. The author showed recently in [‘An explicit minorant for the amenability constant of the Fourier algebra’, Int. Math. Res. Not. IMRN 2023 (2023), 19390–19430] that $\operatorname { {AD}}(G)$ coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants from Johnson [‘Non-amenability of the Fourier algebra of a compact group’, J. Lond. Math. Soc. (2) 50 (1994), 361–374], we determine exactly which numbers in the interval $[1,2]$ arise as values of $\operatorname { {AD}}(G)$. As a by-product, we show that the set of values of $\operatorname { {AD}}(G)$ does not contain all its limit points. Some other calculations or bounds for $\operatorname { {AD}}(G)$ are given for familiar classes of finite groups. We also indicate a connection between $\operatorname { {AD}}(G)$ and the commuting probability of G, and use this to show that every finite group G satisfying $\operatorname { {AD}}(G)< {61}/{15}$ must be solvable; here, the value ${61}/{15}$ is the best possible.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.

Starting from a uniquely ergodic action of a locally compact group G on a compact space $X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product $C(X_0)\rtimes _\alpha {\mathbb Z}$, through a $1$-cocycle of G in ${\mathbb T}$, with $\alpha $ commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with ${\mathcal P}({\mathbb T})$, the Borel probability measures on the one-dimensional torus ${\mathbb T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.

Let G be the Lie group ${\mathbb{R}}^2\rtimes {\mathbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of G, and consider the Laplacian $\Delta=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_i\Delta^{-1/2}$, $i=0,1,2$. We first show that the atomic Hardy space H1 in G introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from H1 to H1.

Let $\varphi$ be a normal semifinite faithful weight on a von Neumann algebra $A$, let $(\sigma ^\varphi _r)_{r\in {\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exists another von Neumann algebra $M$ equipped with a normal semifinite faithful weight $\psi$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $\sigma ^\psi \circ J=J\circ \sigma ^\varphi$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb {E}}_JU^kJ$ for all integer $k\geq 0$, where $ {\mathbb {E}}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here, $VN(G)$ is equipped with its Plancherel weight $\varphi _G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be nonunimodular, and hence, $\varphi _G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1\lt p\lt \infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a noncommutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.

We establish hyperweak boundedness of area functions, square functions, maximal operators, and Calderón–Zygmund operators on products of two stratified Lie groups.

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.

In Caspers et al. (Can. J. Math. 75[6] [2022], 1–18), transference results between multilinear Fourier and Schur multipliers on noncommutative $L_p$-spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier multipliers for non-unimodular groups and show that the aforementioned transference results also hold in this more general setting.

Entropy of measure-preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are those given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein–Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein–Weiss lemma due to Gromov.

If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

Let $\Gamma $ be a finite group, let $\theta $ be an involution of $\Gamma $ and let $\rho $ be an irreducible complex representation of $\Gamma $. We bound ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$-representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$, where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is the maximal normal p-subgroup of $\Gamma $.

This implies, in particular, that if $\mathbf {G}$ is a group scheme over $\mathbb {Z}$ and $\theta $ is an involution of $\mathbf {G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf {G}(\mathbb {Z}_p)/ \mathbf {G} ^{\theta }(\mathbb {Z}_p) \right)$ is bounded, uniformly in p.

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group $\mathcal {F}$. We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup $\mathcal {S}$ and the free semigroup $\mathcal {F}_n$ on n ($\geq 2$) generators. Let $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal {S}$ and $\mathcal {F}_n$, respectively. We prove that all derivations on $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ are automatically continuous, and every derivation on $\mathfrak {B}(\mathcal {S})$ is induced by a bounded linear operator in $\mathcal {L}(\mathcal {S})$, the weak-operator closed Banach algebra consisting of all bounded left convolution operators on $l^2(\mathcal {S})$. Moreover, we prove that the first continuous Hochschild cohomology group of $\mathfrak {B}(\mathcal {S})$ with coefficients in $\mathcal {L}(\mathcal {S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$. We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$. Arens regularity of $L_E^1(G)$, the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$), which is not a $\Lambda (1)$-set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.

The main purpose of this paper is to prove Hörmander’s $L^p$–$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$–$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$–$L^q$ norms of the heat kernel for generalised radial Laplacian.

We show that $\ell ^1(\mathbb {N}_\wedge )$ is $\varphi $-amenable for each multiplicative linear functional $\varphi :\ell ^1(\mathbb {N}_\wedge )\rightarrow \mathbb {C}.$ This is a counterexample to the final corollary of Jaberi and Mahmoodi [‘On $\varphi $-amenability of dual Banach algebras’, Bull. Aust. Math. Soc. 105 (2022), 303–313] and shows that the final theorem in that paper is not valid.

We prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G.

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.