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Yulia Kuznetsova

Publications and source records attributed to Yulia Kuznetsova.

16 recordsLinked to original sources
arXiv2025

On the Jacobs-de Leeuw-Glicksberg decomposition for representations of semigroups

We develop a systematic study of the Jacobs-de Leeuw-Glicksberg decomposition for JdLG-admissible semigroup representations on Banach spaces, with emphasis on its structural, ergodic, and combinatorial aspects. For a JdLG-admissible representation $\pi$ of a semigroup $S$ on a Banach space $E$, we show that the reversible part is weakly equivalent to a unitary representation on a Hilbert space that decomposes as a direct sum of finite-dimensional representations. We also characterize the almost weakly stable part in terms of the unique invariant mean on the space of weakly almost periodic functions. When $S$ is a bi-amenable measured semigroup, we obtain further characterizations of the almost weakly stable part using invariant means and averages along F\o lner sequences. In addition, we describe, in terms of ultrafilters, the unique projection onto the reversible part whose kernel is the almost weakly stable part, and derive several combinatorial consequences. A range of examples are given to demonstrate the sharpness of our results. Many of our theorems extend known features of the compact-weak mixing decomposition for unitary operators; although some auxiliary results were previously known or part of the folklore, their extension to this general setting, particularly for nonseparable Banach spaces, requires new ideas.

math.FA↗
arXiv2025

Shifted wave equation on noncompact symmetric spaces

Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We carry out a detailed analysis of oscillating integrals involving the Harish-Chandra $c$-function, in the case of real rank $l\ge 2$. This allows to obtain two main applications. Consider the Laplace-Beltrami operator $\Delta$ on the homogeneous space $G/K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp( it\sqrt{|x|}) \psi(\sqrt{|x|}) $ applied to the shifted Laplacian $\Delta+|\rho|^2$. We obtain a polynomial decay in time of the kernel, and of the $L^p-L^q$ norms of the operator, for $1\le p<2<q\le \infty$. For the related distinguished Laplacian, we obtain bounds for the $L^p-L^p$ norms, $1\le p\le\infty$, with a slower growth in time than predicted by earlier results.

math.AP↗
arXiv2024

Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces

Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated with a semisimple Lie group $G$. Assume $F$ is a Borel function on $\mathbb{R}^+$. We give a condition on $F$ such that the kernels of the functions $F(L)$ are uniformly bounded. This condition involves the decay of $F$ only and not its derivatives. By a known correspondence, this implies pointwise estimates for a wide range of functions of the Laplace-Beltrami operator on symmetric spaces. In particular, when $G$ is of real rank one and $F(x)={\rm e}^{it\sqrt x}ψ(\sqrt x)$, our bounds are sharp.

math.AP↗
arXiv2022

Norms of certain functions of a distinguished Laplacian on the $ax+b$ groups

The aim of this paper is to find new estimates for the norms of functions of the (minus) Laplace operator $\cal L$ on the `$ax+b$' groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type $ψ(\sqrt{\cal L})\exp(it \sqrt{\cal L})$, with $ψ\in C_0(\mathbb{R})$. We show that for $t\to+\infty$, the convolution kernel $k_t$ of this operator satisfies $$ \|k_t\|_1\asymp t, \qquad \|k_t\|_\infty\asymp 1, $$ so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of $\cal L$ and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator $\tildeΔ$, closely related to $\cal L$. The functions include in particular $\exp(-t\tildeΔ^γ)$, $t>0,γ>0$, and $(\tildeΔ-z)^s$, with complex $z,s$.

math.FA↗
arXiv2022

Invariant means on subspaces of quantum weakly almost periodic functionals

Let $M$ be a Hopf--von Neuman algebra with the predual $M_*$ and $WAP(M)$ the subspace in $M$ composed of weakly almost periodic functionals on $M_*$. The main example of such an algebra is $M=L^\infty(\mathbb G)$ for a locally compact quantum group $\mathbb G$. We define a pair of left/right spaces $WAP_{iso,l}(M)$ and $WAP_{iso,r}(M)$ inside $WAP(M)$ and prove that they carry invariant means. These spaces are currently the widest known to admit invariant means in the quantum setting. In the case when $M=L^\infty(G)$ and $G$ is a locally compact group, these spaces are equal to $WAP(G)$.

math.OA↗
arXiv2020

$Γ$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups

Let $ω$ be a weight function defined on a locally compact group $G$, $1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, the left translation operator $T_s$ is continuous from the weighted $L^p$-space $L^p(G,ω)$ into itself. For a given set $Γ\subset\mathbb{C}$, a vector $f\in L^p(G,ω)$ is said to be $(Γ,S)$-dense if the set $\{ λT_sf:\, λ\in Γ, \,s\in S\}$ is dense in $L^p(G,ω)$. In this paper, we characterize the existence of $(Γ,S)$-dense vectors in $L^p(G,ω)$ in terms of the weight and the set $Γ$.

math.FA↗
arXiv2018

Quantum semigroups: A survey

Notes of a 8h course given at the University of Göteborg during an Erasmus exchange visit, June 11-15, 2018. It is intended for PhD and graduate students familiar with $C^*$-algebras but not specializing in quantum groups. The proofs, if included, are presented in detail, but most facts are not proved. A special emphasis is put onto motivation of the theory and the intuition behind.

math.OA↗
arXiv2018

On isomorphisms between weighted $L^p$-algebras

Let $G$ be a locally compact group and $ω$ be a continuous weight on $G$. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$-algebras. As a result we extend previous results of Edwards, Strichartz and Parrot on algebra isomorphisms between $L^p$-algebras to the setting of weighted $L^p$-algebras. We then study the automorphisms of certain weighted $L^p$-algebras on integers, applying known results on composition operators on classical function spaces.

math.FA↗
arXiv2017

Homomorphisms with small bound between Fourier algebras

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on second conjugate of Fourier algebras.

math.FA↗
arXiv2017

Density of translates in weighted $L^p$ spaces on locally compact groups

Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,ω)=\{f:\int|fω|^p<\infty\}$, where $ω:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $ω$, $G$ acts on $L^p(G,ω)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,ω)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. Our results are obtained in a more general setting when the translations only by a subset $S\subset G$ are considered.

math.FA↗
arXiv2015

On continuity of measurable group representations and homomorphisms

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, on the following theorem: it is consistent with ZFC that for every null set S in a locally compact group there is a set A such that AS is non-measurable.

math.FA↗
arXiv2014

A duality of locally compact groups which does not involve the Haar measure

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

math.OA↗
arXiv2011

A duality for Moore groups

We suggest a new generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all Moore groups, i.e. groups whose irreducible representations are finite-dimensional. Objects in this category are pro-C*-algebras with a structure of Hopf algebras (in the strict algebraic sense) with respect to a certain topological tensor product.

math.FA↗
arXiv2008

Constructions of regular algebras $L_p^w(G)$

Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a locally compact abelian group $G$ is known by works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$ are constructed on any sigma-compact abelian group $G$. It was proved earlier by the author that sigma-compactness is necessary (in the abelian case) for the existence of weighted algebras $L_p^w(G)$ with $p>1$.

math.FA↗