arXiv subjects

Paweł Kasprzak

Publications and source records attributed to Paweł Kasprzak.

At least 19 recordsLinked to original sources

arXiv2026

Laplace operators on quantum graphs

We introduce a class of Laplace operators associated with quantum graphs in the operator-system framework. To this end, we investigate structural properties of quantum graphs related to the Schur product and transposition. Our main innovation is the identification of a specific inner product on the space of matrices with respect to which the projection onto the operator system underlying a quantum graph is orthogonal. This enables us to define a quantum analogue of the classical incidence operator and the associated Laplace operator. We compare this construction with a recently proposed alternative definition and examine the resulting Laplace operators for several families of quantum graphs. These results extend fundamental constructions from spectral graph theory to the operator-algebraic setting, providing a step toward a spectral theory of quantum graphs that generalizes classical graph-theoretic concepts while revealing new connections with quantum information theory and noncommutative geometry.

math.OA↗

arXiv2025

Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings

We investigate a quantum generalization of the Mycielski construction for quantum graphs. In particular, we analyze the quantum symmetries of quantum Mycielskians and their relation to the symmetries of the underlying quantum graphs. We introduce a quantum analogue of graphs with twin vertices and show that, for classical graphs with a small number of vertices, this notion reduces to the classical one. Finally, we propose a quantum extension of the distinguishing number and examine its behavior under the quantum Mycielski construction.

math.OA↗

arXiv2023

Quantum Mycielski Graphs

The classical Mycielski transformation allows for constructing from a given graph the new one, with an arbitrarily large chromatic number but preserving the size of the largest clique contained in it. This particular construction and its specific generalizations were widely discussed in graph theory literature. Here we propose an analog of these transformations for quantum graphs and study how they affect the (quantum) chromatic number as well as clique numbers associated with them.

math.OA↗

arXiv2021

Quantum correlations on quantum spaces

For given quantum (non-commutative) spaces $\mathbb{P}$ and $\mathbb{O}$ we study the quantum space of maps $\mathbb{M}_{\mathbb{P},\mathbb{O}}$ from $\mathbb{P}$ to $\mathbb{O}$. In case of finite quantum spaces these objects turn out to be behind a large class of maps which generalize the classical $\mathrm{qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove a number of important functorial properties of the mapping $(\mathbb{P},\mathbb{O})\mapsto\mathbb{M}_{\mathbb{P},\mathbb{O}}$ and use them to study various operator algebraic properties of the $\mathrm{C}^*$-algebras $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ such as the lifting property and residual finite dimensionality. Inside $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ we construct a universal operator system $\mathbb{S}_{\mathbb{P},\mathbb{O}}$ related to $\mathbb{P}$ and $\mathbb{O}$ and show, among other things, that the embedding $\mathbb{S}_{\mathbb{P},\mathbb{O}}\subset\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ is hyperrigid, $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ is the $\mathrm{C}^*$-envelope of $\mathbb{S}_{\mathbb{P},\mathbb{O}}$ and that a large class of non-signalling correlations on the quantum sets $\mathbb{P}$ and $\mathbb{O}$ arise from states on $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})\otimes_{\rm{max}}\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ as well as states on the commuting tensor product $\mathbb{S}_{\mathbb{P},\mathbb{O}}\otimes_{\rm{c}}\mathbb{S}_{\mathbb{P},\mathbb{O}}$. Finally we introduce and study the notion of a synchronous correlation with quantum input and output sets, prove several characterizations of such correlations and their relation to traces on $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$.

math.OA↗

arXiv2020

Noncommutative Furstenberg boundary

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

math.OA↗

arXiv2020

Resolution Enhancement in Protein NMR Spectra by Deconvolution with Compressed Sensing Reconstruction

Multidimensional NMR spectroscopy is one of the basic tools for determining the structure of biomolecules. Unfortunately, the resolution of the spectra is often limited by inter-nuclear couplings. This limitation cannot be overcome by common ways of increasing resolution, i.e. non-uniform sampling (NUS) followed by compressed sensing (CS) reconstruction. In this paper, we show how to enrich CS processing with virtual decoupling leading to an increase in resolution, sensitivity, and overall quality of NUS reconstruction. A mathematical description of the decoupling by deconvolution approach explains the effects of noise, modulation of the sampling schedule, and reveals relation with the underlying assumption of the CS. The gain in resolution and sensitivity is demonstrated for the basic experiment used for protein backbone assignment 3D HNCA applied to two large protein systems: intrinsically disordered 441-residue Tau and a 509-residue globular bacteriophytochrome fragment.

physics.data-an↗

arXiv2019

Kawada-Itô-Kelley Theorem for Quantum Semigroups

Idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be Haar states on a certain sub-object described by an operator system with comultiplication. We also give a characterization of the situation when this sub-object is actually a compact quantum subgroup. In particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation.

math.OA↗

arXiv2018

The Lattice of Idempotent States on a Locally Compact Quantum Group

We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal ($σ$-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group $\mathbb{G}$ and on its dual $\widehat{\mathbb{G}}$ is established. Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups.

math.OA↗

arXiv2018

Generalized (co)integrals on coideal subalgebras

Given a a Hopf algebra $H$, its left coideal subalgebra $A$ and a non-zero multiplicative functional $μ$ on $A$, we define the space of left $μ$-integrals $L^A_μ\subset A$. We observe that $\dim L^A_μ=1$ if $A$ is a Frobenius algebra and we conclude this equality for finite dimensional left coideal subalgebras of a weakly finite Hopf algebra. In general we prove that if $\dim L^A_μ>0$ then $\dim A <\infty$. Given a group-like element $g\in H$ we define the space $L^g_{ A}\subset A'$ of $g$-cointegrals on $ A$ and linking this concept with the theory of $μ$-integrals we observe that: - every semisimple left coideal subalgebra $A\subset H$ which is preserved by the antipode squared admits a faithful $1$-cointegral; - every unimodular finite dimensional left coideal subalgebra $A\subset H$ admitting a faithful $1$-cointegral is preserved by the antipode square; - every non-degenerate right group-like projection in a cosemisimple Hopf algebra is a two sided group-like projection. Finally we list all $\varepsilon$-integrals for left coideals subalgebras in Taft algebras and we list all $g$-cointegrals on them.

math.QA↗

arXiv2018

Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\hat{\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\mathbb{G}}$. Using our results we characterize coideals in $L^\infty(\hat{\mathbb{G}})$ admitting an atom preserved by the scaling group in terms of idempotent states on ${\mathbb{G}}$. We also establish a one to one correspondence between integrable coideals in $L^\infty({\mathbb{G}})$ and group-like projections in $L^\infty(\hat{\mathbb{G}})$ satisfying an extra mild condition. Exploiting this correspondence we give examples of group like projections which are not preserved by the scaling group.

math.OA↗

arXiv2018

Group-like projections for locally compact quantum groups

Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.

math.OA↗

arXiv2017

Hopf images in locally compact quantum groups

This manuscript is devoted to the study of the concept of a generating subset (a.k.a. Hopf image of a morphism) in the setting of locally compact quantum groups. The aim of this paper is to provide an accurate description of the Hopf image of a given morphism. We extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups. In particular, we provide a characterization of fullness of Hopf image in the language of partial actions as well as in representation-theoretic terms in the spirit of representation $C^*$-categories, extending some known results not only to the broader setting of non-compact quantum groups, but also encompassing a broader setting of generating subsets.

math.OA↗

arXiv2017

Induction for locally compact quantum groups revisited

In this paper we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

math.OA↗

arXiv2016

Quantum actions on discrete quantum spaces and a generalization of Clifford's theory of representations

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e. when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R.Vergnioux.

math.OA↗

arXiv2016

Fundamental isomorphism theorems for quantum groups

The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-Hölder theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa. The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact / discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.

math.QA↗

arXiv2016

The canonical central exact sequence for locally compact quantum groups

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.

math.OA↗

arXiv2016

Open quantum subgroups of locally compact quantum groups

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C*-algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence with normal compact quantum subgroups of the dual quantum group.

math.OA↗

arXiv2016

Quantum families of quantum group homomorphisms

The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.

math.OA↗