arXiv subjects

K. Bardadyn

Publications and source records attributed to K. Bardadyn.

4 recordsLinked to original sources

arXiv2026

Banach algebra crossed products by inverse semigroup actions

We give a self-contained and simplified presentation of the theory of covariant representations for inverse semigroup actions on Banach algebras, which was recently introduced in the authors and A. Mckee in the twisted case. The main result of this note is a general universal description of the associated Banach algebra crossed product, that allows disintegration of all representations of the crossed product. Such a disintegration was studied so far only for actions on spaces.

math.FA↗

arXiv2023

On $C^*$-algebras associated to transfer operators for countable-to-one maps

Our initial data is a transfer operator $L$ for a continuous, countable-to-one map $φ:Δ\to X$ defined on an open subset of a locally compact Hausdorff space $X$. Then $L$ may be identified with a `potential', i.e. a map $\varrho:Δ\to X$ that need not be continuous unless $φ$ is a local homeomorphism. We define the crossed product $C_0(X)\rtimes L$ as a universal $C^*$-algebra with explicit generators and relations, and give an explicit faithful representation of $C_0(X)\rtimes L$ under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver $C^*$-algebras of Muhly and Tomforde, $C^*$-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid $C^*$-algebras associated to Deaconu-Renault groupoids. We describe spectra of core subalgebras of $C_0(X)\rtimes L$ and use it to characterise simplicity of $C_0(X)\rtimes L$ and prove the uniqueness theorem for $C_0(X)\rtimes L$. We give efficient criteria for $C_0(X)\rtimes L$ to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of $C_0(X)\rtimes L$ and conformal measures for $φ$.

math.OA↗

arXiv2023

Topologically free actions and ideals in twisted Banach algebra crossed products

We generalize the influential $C^*$-algebraic result of Kawamura-Tomiyama and Archbold-Spielberg for crossed products of discrete transformation groups to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalised intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\infty]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

math.FA↗

arXiv2020

Spectrum of weighted isometries: C*-algebras, transfer operators and topological pressure

We study the spectrum of operators $aT\in B(H)$ on a Hilbert space $H$ where $T$ is an isometry and $a$ belongs to a commutative $C^*$-subalgebra $C(X)\cong A\subseteq B(H)$ such that the formula $L(a)=T^*aT$ defines a faithful transfer operator on $A$. Based on the analysis of the $C^*$-algebra $C^*(A,T)$ generated by the operators $aT$, $a\in A$, we give dynamical conditions implying that the spectrum $σ(aT)$ is invariant under rotation around zero, $σ(aT)$ coincides with essential spectrum $σ_{ess}(aT)$ or that $σ(aT)=\{z\in \mathbb{C}: |z|\leq r(aT)\}$ is a disk. We extend classical Ruelle's result and prove that for a general expanding map $φ:X\to X$ and continuous $c:X\to [0,\infty)$ the spectral logarythm of a Ruelle-Perron-Frobenious operator $\mathcal{L}_cf(y)=\sum_{x\in φ^{-1}(y)} c(x)f(x)$ is equal to the topological pressure $P(\ln c,φ)$. As a consequence we get the variational principle for the spectral radius: $$ r(aT)=\max_{μ\in \text{Erg}(X,φ)} \exp\left(\int_{X}\ln(\vert a\vert\sqrt{\varrho })\,dμ+\frac{h_φ(μ)}{2} \right), $$ where $\text{Erg}(X,φ)$ stands for ergodic Borel probability measures, $h_φ(μ)$ is the Kolmogorov-Sinai entropy, and $\varrho:X\to [0,1]$ is the cocycle associated to $L$. In particular, we clarify the relationship between the Kolmogorov-Sinai entropy and $t$-entropy introduced by Antonevich, Bakhtin and Lebedev.

math.FA↗