arXiv subjects

Jacek Krajczok

Publications and source records attributed to Jacek Krajczok.

At least 19 recordsLinked to original sources

arXiv2026

Actions of quantum groups on dual operator spaces and their crossed products

We study the category of dual operator spaces equipped with an action of a locally compact quantum group $\mathbb{G}$. The Fubini crossed product functor $-\rtimes^\mathcal{F} \mathbb{G}$ and the weak$^*$-crossed product functor $-\bar{\rtimes}\mathbb{G}$ are shown to be equal if and only if $\mathbb{G}$ has the approximation property of Haagerup and Kraus. Using the natural isomorphism $-\rtimes^\mathcal{F}\mathbb{G}\cong {}_{L^1(\mathbb{G})}\mathcal{CB}(B(L^2(\mathbb{G}))_*, -)$, this leads to a characterization of the approximation property of $\mathbb{G}$ via an $L^1(\mathbb{G})$-module approximation property for $B(L^2(\mathbb{G}))_*$. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of $\mathbb{G}$.

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arXiv2026

Scaling automorphisms of compact quantum groups

In this short article we verify the conjecture proposed by P. M. Sołtan and the author, by proving that a second countable compact quantum group whose scaling automorphisms are inner, must be of Kac type.

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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.

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arXiv2026

Braided quantum $\mathrm{SU}(2)$ group - a case study

We continue the study of the braided compact quantum group $\mathrm{SU}_q(2)$ for complex $q$ satisfying $0<|q|<1$ introduced by Kasprzak, Meyer, Roy and Woronowicz (J. Noncommut. Geom. 10(4):1611-1625, 2016). We address such aspects as existence of the Haar measure, construct the scaling group, the antipode and its polar decomposition and describe the related braided Hopf algebra. We also study when the braided flip extends to a completely bounded map and establish equivalence between the two approaches to bosonization and braided tensor product taken in the literature (Kasprzak, Meyer, Roy, Woronowicz J. Noncommut. Geom. 10(4):1611-1625, 2016 vs. Meyer, Roy Woronowicz Internat. J. Math. 25(2):1450019, 37, 2014, Roy Int. Math. Res. Not. (14):11791--11828, 2023 and De Commer, Krajczok arXiv:2412.17444, to appear in J. Operator Th.).

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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 Følner 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.

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arXiv2025

Cowling-Haagerup constant of the product of discrete quantum groups

We show that (central) Cowling-Haagerup constant of discrete quantum groups is multiplicative, which extends the result of Freslon to general (not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras $\mathrm{C}(\mathbb{G}), \operatorname{L}^{\infty}(\mathbb{G})$ as operator modules over $\operatorname{L}^1(\mathbb{G})$.

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arXiv2025

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 $ε$-separation property.

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arXiv2024

Braided tensor product of von Neumann algebras

We introduce a definition of braided tensor product $\operatorname{M}\overline{\boxtimes}\operatorname{N}$ of von Neumann algebras equipped with an action of a quasi-triangular quantum group $\mathbb{G}$ (this includes the case when $\mathbb{G}$ is a Drinfeld double). It is a new von Neumann algebra which comes together with embeddings of $\operatorname{M},\operatorname{N}$ and the unique action of $\mathbb{G}$ for which embeddings are equivariant. More generally, we construct braided tensor product of von Neumann algebras equipped with actions of locally compact quantum groups linked by a bicharacter. We study several examples, in particular we show that crossed products can be realised as braided tensor products. We also show that one can take the braided tensor product $\vartheta_1\boxtimes\vartheta_2$ of normal, completely bounded maps which are equivariant, but this fails without the equivariance condition.

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arXiv2024

The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups

The Stone-von Neumann Theorem is a fundamental result which unified the competing quantum mechanical models of matrix mechanics and wave mechanics. It's mechanism of proof ultimately involved the study of unitary group representations on a Hilbert space. In this article, we continue the broad generalization set out in arxiv:1903.09351 and arxiv:2109.08997, analyzing representations of locally compact quantum dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models which appear in the literature, and for certain coactions (G, A, α) recover the multiplicity results of arxiv:2109.08997. As a corollary, we develop a new criterion for identifying strongly regular locally compact quantum groups, related to the study of their dynamics on elementary C*-algebras.

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arXiv2024

Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor

For each $λ\in\left]0,1\right]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra $\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type $\mathrm{III}_λ$ with separable predual. We also show that uncountably many injective factors of type $\mathrm{III}_0$ arise as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for some compact quantum group $\mathbb{G}$. To distinguish between our examples we introduce invariants related to the scaling group modeled on the Connes invariant $T$ for von Neumann algebras and investigate the connection between so obtained invariants of $\mathbb{G}$ and the Connes invariants $T(\mathsf{L}^{\!\infty}(\mathbb{G}))$, $S(\mathsf{L}^{\!\infty}(\mathbb{G}))$. In the final section we show that factors of type $\mathrm{I}$ cannot be obtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compact quantum group $\mathbb{G}$.

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arXiv2024

On certain invariants of compact quantum groups

We introduce and study a number of invariants of locally compact quantum groups defined by their scaling and modular groups and the spectrum of their modular elements. Focusing mainly on compact quantum groups we consider the question whether triviality of one of the invariants is equivalent to the quantum group being of Kac type and show that it has a positive answer in many cases including duals of second countable type $\mathrm{I}$ discrete quantum groups. We perform a complete calculation of the invariants for all $q$-deformations of compact, simply connected, semisimple Lie groups as well as for some non-compact quantum groups and the compact quantum groups $\operatorname{U}_F^+$. Finally we introduce a family of conditions for discrete quantum groups which for classical discrete groups are all equivalent to the fact that the group is i.c.c. We show that the above mentioned question about characterization of Kac type quantum groups by one of our invariants has a positive answer for duals of discrete quantum groups satisfying such an i.c.c.-type condition and illustrate this with the example of $\operatorname{U}_F^+$.

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arXiv2024

Averaging multipliers on locally compact quantum groups

We study an averaging procedure for completely bounded multipliers on a locally compact quantum group with respect to a compact quantum subgroup. As a consequence we show that central approximation properties of discrete quantum groups are equivalent to the corresponding approximation properties of their Drinfeld doubles. This is complemented by a discussion of the averaging of Fourier algebra elements. We compare the biinvariant Fourier algebra of the Drinfeld double of a discrete quantum group with the central Fourier algebra. In the unimodular case these are naturally identified, but we show by exhibiting a family of counter-examples that they differ in general.

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arXiv2023

The approximation property for locally compact quantum groups

We study the Haagerup--Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus--Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the $\mathrm{C}^*$-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak$^*$ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid $\mathrm{C}^*$-tensor categories, building on work of Arano--De Laat--Wahl.

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arXiv2022

On the von Neumann algebra of class functions on a compact quantum group

We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups; in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we construct certain bicrossed products from the quantum group $SU_q(2)$ for which the algebra of class functions is a MASA.

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arXiv2020

Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the $\mathrm{C}^*$-algebra in question is an extension of a non-zero finite direct sum of elementary $\mathrm{C}^*$-algebras by a commutative unital $\mathrm{C}^*$-algebra then it must be finite-dimensional.

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arXiv2020

Modular properties of type I locally compact quantum groups

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on $\mathbb{G}$ and $\widehat{\mathbb{G}}$ act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group $\widehat{\mathrm{SU}_q(2)}$ and the quantum $az+b$ group.

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arXiv2020

The quantum disk is not a quantum group

We show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz C*-algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of type I locally compact quantum groups, but also from the theory of operators on Hilbert spaces.

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arXiv2020

Coamenability of type I locally compact quantum groups

We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space $\operatorname{L}^2(\operatorname{Irr}(\mathbb{G}))$ of functions which are square integrable with respect to the Plancherel measure. The second condition involves spectra of character-like operators associated with direct integrals of irreducible representations. As examples we study special classes of quantum groups: classical, dual to classical, compact or given by a certain bicrossed product construction.

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