arXiv subjects

Malte Gerhold

Publications and source records attributed to Malte Gerhold.

At least 19 recordsLinked to original sources

arXiv2025

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than $2$. Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.

math.FA↗

arXiv2024

Dilation distance and the stability of ergodic commutation relations

We revisit and generalize the notion of dilation distance ${\rm d_{D}}(u,v)$ between unitary tuples and study its relation to the natural Haagerup-R{\o}rdam distance ${\rm d_{HR}}(u,v) = \inf\{\|\pi(u) - \rho(v)\|\}$, where the infimum is taken over all pairs of faithful representations $\pi \colon C^*(u) \to B(\mathcal{H})$, $\rho \colon C^*(v) \to B(\mathcal{H})$. We show that ${\rm d_{HR}}(u,v)\leq 10\operatorname{d_{rD}}(u,v)^{1/2}$, where ${\rm d_{rD}}(u,v)$ is a relaxed dilation distance, improving and extending earlier results. For an antisymmetric matrix $\Theta$, we show via a concrete dilation construction that a tuple of unitaries $u$ that almost commutes according to $\Theta$ (i.e., $\|u_\ell u_k - e^{i \theta_{k,\ell}} u_k u_\ell\|$ is small) can be nearly dilated to a tuple of unitaries $v$ that commutes according to $\Theta$ (i.e., $v_\ell v_k - e^{i \theta_{k,\ell}} v_k v_\ell = 0$). We show that the dilation can be "reversed" by a second application of the dilation construction, which leads to a rotated version of the original tuple. Thus, a gauge invariant almost $\Theta$-commuting unitary tuple can be approximated (in some faithful representation) by a $\Theta$-commuting unitary tuple. Moreover, when $\Theta$ is ergodic, a $\Theta$-commuting tuple is shown to be {\em almost} gauge invariant, and it follows from the results above that these can be approximated in norm by $\Theta$-commuting tuples. In particular, we obtain the following counterpart of Lin's theorem on almost commuting unitaries: if $q \in \mathbb{T}$ is {\em not} a root of unity, then for every $\varepsilon >0$ there exists $\delta > 0$ such that for every pair of unitaries $u_1,u_2 \in B(\mathcal{H})$ for which $\|u_1 u_2 - qu_2 u_1\| < \delta$, there exists two $q$-commuting unitaries $v_1, v_2 \in B(\mathcal{H} \otimes \ell^2)$ such that $\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$).

math.OA↗

arXiv2024

Linear Deformations of Heisenberg Modules and Gabor Frames

Heisenberg modules over noncommutative tori may also be viewed as Gabor frames. Building on this fact, we relate to deformations of noncommutative tori a bundle of Banach spaces induced by Heisenberg modules. The construction of this bundle of Banach spaces rests on deformation results of Gabor frames with windows in Feichtinger's algebra due to Feichtinger and Kaiblinger. We extend some of these results to Heisenberg modules, \eg we establish an analog of the results by Feichtinger-Kaiblinger and a Balian-Low theorem. Finally, we extend our results to several generators on the bundle of Heisenberg modules and show that they provide a generalized Fell's condition on the bundle of noncommutative tori.

math.OA↗

arXiv2023

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with 1-dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter-Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, H\"artel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then we make use of the fact that the free unitary quantum groups and some of its non-Kac versions can be realized as a glued free product of a (non-Kac) free orthogonal quantum group with $\mathbb Z_2$, the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a byproduct of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi-Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter-Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

math.QA↗

arXiv2023

Additive Deformations of Hopf Algebras

Additive deformations of bialgebras in the sense of Wirth are deformations of the multiplication map of the bialgebra fulfilling a compatibility condition with the coalgebra structure and a continuity condition. Two problems concerning additive deformations are considered. With a deformation theory a cohomology theory should be developed. Here a variant of the Hochschild cohomology is used. The main result in the first partad of this paper is the characterization of the trivial deformations, i.e. deformations generated by a coboundary. Starting with a Hopf algebra, one would expect the deformed multiplications to have some analogue of the antipode, which we call deformed antipodes. We prove, that deformed antipodes always exist, explore their properties, give a formula to calculate them given the deformation and the antipode of the original Hopf algebra and show in the cocommutative case, that each deformation splits into a trivial part and into a part with constant antipodes.

math.QA↗

arXiv2023

Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach

We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).

math.FA↗

arXiv2023

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is, computing distributions of operators in the algebra generated by the left- and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and type I bimonotone independences. In this paper, we extend the preLie calculus pertaining to free, Boolean, and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above-mentioned two-faced equivalents.

math.OA↗

arXiv2023

Towards a classification of multi-faced independences: a combinatorial approach

We determine a set of necessary conditions on a partition-indexed family of complex numbers to be the "highest coefficients" of a positive and symmetric multi-faced universal product; i.e. the product associated with a multi-faced version of noncommutative stochastic independence, such as bifreeness. The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence. We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients. Furthermore, we give a quite explicit description of such families of coefficients, thereby producing a list of candidates that must contain all positive symmetric universal products. We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant relations that give rise to symmetric universal products; to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem.

math.FA↗

arXiv2022

Categorial Independence and Lévy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

math.CT↗

arXiv2022

Bounded perturbations of the Heisenberg commutation relation via dilation theory

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows.

math.FA↗

arXiv2021

Dilations of unitary tuples

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric \[ d_D(u,v)=\log\max\{c(u,v),c(v,u)\} \] on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_Θ$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{iθ_{k,\ell}} u_k u_\ell$, where $Θ=(θ_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_Θ, u_{Θ'})\leq e^{\frac{1}{4}\|Θ-Θ'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $Θ\mapsto U(Θ)\in B(H)^d$ such that $U(Θ)\sim u_Θ$ and \[ \|U(Θ)-U({Θ'})\|\leq K\|Θ-Θ'\|^{1/2}. \] Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] From this, we recover Passer's upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.

math.OA↗

arXiv2021

Schoenberg correspondence for multifaced independence

We extend the Schoenberg correspondence for universal independences by Sch\"urmann \& Vo{\ss} to the multivariate setting of Manzel \& Sch\"urmann, covering, e.g., Voiculescu's bifreeness as well as Bo{\.z}ejko \& Speicher's c-free independence. At the same time, we free the proof in the univariate situation from its dependence on Muraki's ``5 Independences Theorem''.

math.QA↗

arXiv2020

Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas

We point out that a sequence of natural numbers is the dimension sequence of a subproduct system if and only if it is the cardinality sequence of a word system (or factorial language). Determining such sequences is, therefore, reduced to a purely combinatorial problem in the combinatorics of words. A corresponding (and equivalent) result for graded algebras has been known in abstract algebra, but this connection with pure combinatorics has not yet been noticed by the product systems community. We also introduce Cartesian systems, which can be seen either as a set theoretic version of subproduct systems or an abstract version of word systems. Applying this, we provide several new results on the cardinality sequences of word systems and the dimension sequences of subproduct systems.

math.FA↗

arXiv2020

Interacting Fock Spaces and Subproduct Systems

We prove many new results about interacting Fock spaces. We pose many open problems; for most of them we prove that their solutions have no choice but being nontrivial. We ask the kind reader to consult the extended abstract in the paper.

math.OA↗

arXiv2020

On the matrix range of random matrices

This note treats a simple minded question: what does a typical random matrix range look like? We study the relationship between various modes of convergence for tuples of operators, on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric, on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C*-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.

math.OA↗

arXiv2020

Dilations of $q$-commuting unitaries

Let $q = e^{i θ} \in \mathbb{T}$ (where $θ\in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_θ$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that \[ c_θ= \frac{4}{\|u_θ+u_θ^*+v_θ+v_θ^*\|}, \] where $u_θ, v_θ$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q'$-commuting unitaries. The techniques that we develop allow us to give new and simple "dilation theoretic" proofs of well known results regarding the continuity of the field of rotations algebras. In particular, for the so-called "Almost Mathieu Operator" $h_θ= u_θ+u_θ^*+v_θ+v_θ^*$, we recover the fact that the norm $\|h_θ\|$ is a Lipshitz continuous function of $θ$, as well as the result that the spectrum $σ(h_θ)$ is a $\frac{1}{2}$-Hölder continuous function in $θ$ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every selfadjoint $*$-polynomial $p(u_θ,v_θ)$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

math.OA↗

arXiv2017

Homological properties of quantum permutation algebras

We show that $A_s(n)$, the coordinate algebra of Wang's quantum permutation group, is Calabi-Yau of dimension $3$ when $n\geq 4$, and compute its Hochschild cohomology with trivial coefficients. We also show that, for a larger class of quantum permutation algebras, including those representing quantum symmetry groups of finite graphs, the second Hochschild cohomology group with trivial coefficients vanishes, and hence these algebras have the AC property considered in quantum probability: all cocycles can be completed to a Schürmann triple.

math.QA↗

arXiv2017

Bimonotone Brownian Motion

We define bi-monotone independence, prove a bi-monotone central limit theorem and use it to study the distribution of bi-monotone Brownian motion, which is defined as the two-dimensional operator process with monotone and antimonotone Brownian motion as components.

math.OA↗