Search arXiv⌕ Search

arXiv subjects

Adam Skalski

Publications and source records attributed to Adam Skalski.

At least 19 recordsLinked to original sources
arXiv2026

Crossed product functors associated to $\ell^p$-pseudofunctions

We show that the $\ell^p$-pseudofunctions, which were recently shown to lead to exotic completions of group $C^*$-algebras by Wiersma and the second named author, can be used to construct well-behaved crossed product functors in the sense of Buss, Echterhoff and Willett. The construction proceeds via introducing certain Banach algebras, related to operators acting on Hilbert valued $\ell^p$-spaces, which a priori depend on the choice of a Hilbert space representation of the underlying C*-algebra. We prove that, in fact, the resulting algebras are isomorphic (with the isomorphism constant depending only on $p$), and hence their C*-envelopes are isometrically isomorphic. This, in particular, means that the construction genuinely generalises the one studied earlier in the group case. The tools we develop allow us to show that for certain non-amenable actions, the resulting crossed product completions must indeed be exotic.

math.OA↗
arXiv2026

Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras

We compute the quantum isometry groups of Cuntz-Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz-Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic action of a compact matrix quantum group on the Cantor space.

math.OA↗
arXiv2025

Delocalised traces on Hecke algebras of right-angled, hyperbolic type

For every Hecke C*-algebra of right-angled, hyperbolic type, we construct a smooth subalgebra to which traces associated with arbitrary conjugacy classes in the associated Coxeter group extend. We calculate the pairing with K-theory of the delocalised traces arising this way, showing that it is faithful on K-theory.

math.OA↗
arXiv2025

Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

We study the ISR (von Neumann invariant subalgebra rigidity) property for certain discrete groups arising as semidirect products from algebraic actions on certain 2-torsion groups, mostly arising as direct products of $\mathbb{Z}_2$. We present, in particular, the first example of an amenable group with the ISR property that admits a non-trivial abelian normal subgroup. Several other examples are discussed, notably including an infinite amenable group whose von Neumann algebra admits precisely one invariant von Neumann subalgebra which does not come from a normal subgroup. We also investigate the form of invariant subalgebras of the group von Neumann algebra of the standard lamplighter group.

math.OA↗
arXiv2025

A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs

We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual $\widehat \Gamma$ of any non-abelian finite group $\Gamma$ is ``quantum rigid''. That is, $\widehat \Gamma$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\widehat \Gamma$.

math.OA↗
arXiv2025

Asymptotic invariants for fusion algebras associated with compact quantum groups

We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous concepts studied for classical discrete groups. Specifically we introduce uniform F\o lner constants and the uniform Kazhdan constant for a regular representation of a fusion algebra, and establish a relationship between these, amenability, and the exponential growth rate considered earlier by Banica and Vergnioux. Further we compute the invariants for fusion algebras associated with % discrete duals of quantum $SU_q(2)$ and $SO_q(3)$ and determine the uniform exponential growth rate for the fusion algebras of all $q$-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups.

math.OA↗
arXiv2024

Gaussian Generating functionals on easy quantum groups

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.

math.QA↗
arXiv2024

Tracial central states on compact quantum groups

Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on q-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups.

math.OA↗
arXiv2024

Fourier--Stieltjes category for twisted groupoid actions

We extend the theory of Fourier--Stieltjes algebras to the category of twisted actions by \'etale groupoids on arbitrary C*-bundles, generalizing theories constructed previously by B\'{e}dos and Conti for twisted group actions on unital C*-algebras, and by Renault and others for groupoid C*-algebras, in each case motivated by the classical theory of Fourier--Stieltjes algebras of discrete groups. To this end we develop a toolbox including, among other things, a theory of multiplier C*-correspondences, multiplier C*-correspondence bundles, Busby--Smith twisted groupoid actions, and the associated crossed products, equivariant representations and Fell's absorption theorems. For a fixed \'etale groupoid $G$ a Fourier--Stieltjes multiplier is a family of maps acting on fibers, arising from an equivariant representation. It corresponds to a certain fiber-preserving strict completely bounded map between twisted full (or reduced) crossed products. We establish a KSGNS-type dilation result which shows that the correspondence above restricts to a bijection between positive-definite multipliers and a particular class of completely positive maps. Further we introduce a subclass of Fourier multipliers, that enjoys a natural absorption property with respect to Fourier--Stieltjes multipliers and gives rise to `reduced to full' multiplier maps on crossed products. Finally we provide several applications of the theory developed, for example to the approximation properties, such as weak containment or nuclearity, of the crossed products and actions in question, and discuss outstanding open problems.

math.OA↗
arXiv2023

Convolution semigroups on Rieffel deformations of locally compact quantum groups

Consider a locally compact quantum group $\mathbb{G}$ with a closed classical abelian subgroup $\Gamma$ equipped with a $2$-cocycle $\Psi:\hat{\Gamma}\times\hat{\Gamma}\to\mathbb{C}$. We study in detail the associated Rieffel deformation $\mathbb{G}^{\Psi}$ and establish a canonical correspondence between $\Gamma$-invariant convolution semigroups of states on $\mathbb{G}$ and on $\mathbb{G}^{\Psi}$.

math.OA↗
arXiv2023

Separation properties for positive-definite functions on locally compact quantum groups and for associated von Neumann algebras

Using Godement mean on the Fourier-Stieltjes algebra of a locally compact quantum group we obtain strong separation results for quantum positive-definite functions associated to a subclass of representations, strengthening for example the known relationship between amenability of a discrete quantum group and existence of a net of finitely supported quantum positive-definite functions converging pointwise to $I$. We apply these results to show that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-$w^*$-CPAP, which we call the matrix $\epsilon$-separation property.

math.OA↗
arXiv2023

RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we completely characterize the RFD property for crossed products by amenable actions of discrete groups on C*-algebras in terms of the action. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and the latter can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. %The result leads among other consequences to a characterization of when a semidirect product by an amenable group has RFD full C*-algebra. As byproduct of our methods we characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. These descriptions allow us to obtain the properties MAP, RF, RFD and FD for various new examples and generalize some results of Lubotzky and Shalom. Secondly, as another generalization of Bekka's theorem, we provide a sufficient condition and a necessary condition for the C*-algebra of an amenable \'etale groupoid to be RFD.

math.OA↗
arXiv2023

Full solution of the factoriality question for $q$-Araki-Woods von Neumann algebras via conjugate variables

We establish factoriality of $q$-Araki-Woods von Neumann algebras (with the number of generators at least two) in full generality, exploiting the approach via conjugate variables developed recently in the tracial case by Akihiro Miyagawa and Roland Speicher, and abstract results of Brent Nelson. We also establish non-injectivity and determine the type of the factors in question. The factors are solid and full when the number of generators is finite.

math.OA↗
arXiv2022

Coupling capacity in C*-algebras

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras our parameters are related to quantitative versions of Arveson's Null Set Theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalise quantum versions of Strassen's Theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

math.OA↗
arXiv2022

Connectedness and Gaussian Parts for Compact Quantum Groups

We introduce the Gaussian part of a compact quantum group $\mathbb{G}$, namely the largest quantum subgroup of $\mathbb{G}$ supporting all the Gaussian functionals of $\mathbb{G}$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.

math.QA↗
arXiv2022

Hecke algebras and the Schlichting completion for discrete quantum groups

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions.

math.QA↗
arXiv2021

Relative Haagerup property for arbitrary von Neumann algebras

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

math.OA↗