arXiv subjects

Martijn Caspers

Publications and source records attributed to Martijn Caspers.

At least 19 recordsLinked to original sources

arXiv2026

Strongly convergent matrix models for $q$-Gaussian algebras

We construct strongly convergent finite-dimensional random matrix models for finite $q$-Gaussian family in the range $\vert q \vert < \sqrt{2}-1$. The construction has two stages. First, we show that normalized sums of graph-product semicirculars over an Erdos-Renyi graph satisfy the $q$-Toeplitz relations up to an operator norm error converging to zero in probability. Using ultraproduct methods, this yields complete strong convergence, uniformly over all matrix coefficient dimensions, for noncommutative polynomials with bounded degree. Second, we use a quantitative tensor-GUE for graph-product semicirculars to convert these operator models into finite-dimensional random matrices. For every fixed polynomial degree, the convergence is uniform over coefficient dimensions that may be larger than the dimension of the random matrices. As applications, in the above range of $q$, the C$^\ast$-algebra generated by a finite $q$-Gaussian family is MF, and the Brown-Douglas-Fillmore extension semigroup of the nontrivial C$^\ast$-algebra generated by a finite $q$-Gaussian family is not a group.

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arXiv2025

On the best constants of Schur multipliers of higher order divided difference functions

Let $f \in C^n(\mathbb{R})$ be such that $\Vert f^{(n)} \Vert_\infty < \infty$. Let $f^{[n]} \in C(\mathbb{R}^{n+1})$ be the $n$th order divided difference. A special case of our main result states that for $1 < p < \infty$ we have \[\Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert \lesssim p^\ast p^n \Vert f^{(n)} \Vert_\infty, \] where $p^\ast = p/(p-1)$ is the H\"older conjugate of $p$ and $T_{f^{[n]}}$ is the multilinear Schur multiplier with symbol $f^{[n]}$. In case of the generalized absolute value map $f(\lambda) = \lambda^{n-1} \vert \lambda \vert, \lambda \in \mathbb{R}$, we show that \[p^\ast p^{n} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] This provides an alternative proof to one of the key theorems in the solution of Koplienko's problem on higher order spectral shift [Invent. Math. 193, No. 3, 501-538 (2013)], which is moreover sharp as $p \searrow 1$ and as $p \to\infty$ for any $n$.

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arXiv2025

Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets

Let $n \in \mathbb{N}_{\geq 1}$. Let $1 \leq p_1, \ldots, p_n < \infty$ and set the H\"older combination $p := (p_1; \ldots ; p_n) := \left( \sum_{j=1}^n p_j^{-1} \right)^{-1}$. Assume further that $0 < p \leq 1$ and that for the H\"older combinations of $p_2$ to $p_n$ and $p_1$ to $p_{n-1}$ we have, \[ 1 \leq (p_2; \ldots ; p_n), (p_1; \ldots ; p_{n-1}) < \infty. \] Then there exists a constant $C> 0$ such that for every $f \in C^n(\mathbb{R}) \cap \dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}$ with $\Vert f^{(n)} \Vert_\infty < \infty$ we have \[ \Vert T_{f^{[n]}}: S_{p_1} \times \ldots \times S_{p_n} \rightarrow S_p \Vert \leq C ( \Vert f^{(n)} \Vert_\infty + \Vert f \Vert_{\dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}}). \] Here $S_q$ is the Schatten von Neumann class, $\dot{B}_{p,q}^s$ the homogeneous Besov space, and $T_{f^{[n]}}$ is the multilinear Schur multiplier of the $n$-th order divided difference function. In particular, our result holds for $p=1$ and any $1 \leq p_1, \ldots, p_n < \infty$ with $p = (p_1; \ldots; p_n)$.

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arXiv2025

Internal graphs of graph products of hyperfinite II$_1$-factors

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph ${\rm Int}(\Gamma)$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_{\Gamma}$. In particular, we can classify $R_{\Gamma}$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite II$_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.

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arXiv2024

Rigid Graph Products

We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. Furthermore, we show that for many graph products of II$_1$-factors, including the hyperfinite II$_1$-factor, we can, up to a constant 2, retrieve the radius of the graph from the graph product. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products.

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arXiv2024

On the best constants of Schur multipliers of second order divided difference functions

We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko's conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the H\"ormander-Mikhlin-Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. More precisely, we prove that for $f \in C^2(\mathbb{R})$ and $1 < p, p_1, p_2 < \infty$ with $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$ we have \[ \Vert M_{f^{[2]}}: S_{p_1} \times S_{p_2} \rightarrow S_p \Vert \lesssim \Vert f'' \Vert_\infty D(p, p_1, p_2), \] where the constant $D(p, p_1, p_2)$ is specified in Theorem 7.1 and $D(p, 2p, 2p) \approx p^4 p^\ast$ with $p^\ast$ the H\"older conjugate of $p$. We further show that for $f(\lambda) = \lambda \vert \lambda \vert$, $\lambda \in \mathbb{R}$, for every $1 < p < \infty$ we have \[ p^2 p^\ast \lesssim \Vert M_{f^{[2]}}: S_{2p} \times S_{2p} \rightarrow S_p \Vert. \] Here $f^{[2]}$ is the second order divided difference function of $f$ with $M_{f^{[2]}}$ the associated Schur multiplier. In particular it follows that our estimate $D(p, 2p, 2p)$ is optimal for $p \searrow 1$.

math.CA↗

arXiv2023

Classification of right-angled Coxeter groups with a strongly solid von Neumann algebra

Let $W$ be a finitely generated right-angled Coxeter group with group von Neumann algebra $\mathcal{L}(W)$. We prove the following dichotomy: either $\mathcal{L}(W)$ is strongly solid or $W$ contains $\mathbb{Z} \times \mathbb{F}_2$ as a subgroup. This proves in particular strong solidity of $\mathcal{L}(W)$ for all non-hyperbolic Coxeter groups that do not contain $\mathbb{Z} \times \mathbb{F}_2$.

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arXiv2023

A Sobolev estimate for radial $L^p$-multipliers on a class of semi-simple Lie groups

Let $G$ be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup $K$. Let $\Omega_K$ be minus the radial Casimir operator. Let $\frac{1}{4} \dim(G/K) < S_G < \frac{1}{2} \dim(G/K) , s \in (0, S_G]$ and $p \in (1,\infty)$ be such that \[ \left| \frac{1}{p} - \frac{1}{2} \right| < \frac{s}{2 S_G}. \] Then, there exists a constant $C_{G,s,p} >0$ such that for every $m \in L^\infty(G) \cap L^2(G)$ bi-$K$-invariant with $m \in {\rm Dom}(\Omega_K^s)$ and $\Omega_K^s(m) \in L^{2S_G/s}(G)$ we have, \[ \Vert T_m: L^p(\widehat{G}) \rightarrow L^p(\widehat{G}) \Vert \leq C_{G, s,p} \Vert \Omega_K^s(m) \Vert_{L^{2S_G/s}(G)}, \] where $T_m$ is the Fourier multiplier with symbol $m$ acting on the non-commutative $L^p$-space of the group von Neumann algebra of $G$. This gives new examples of $L^p$-Fourier multipliers with decay rates becoming slower when $p$ approximates $2$.

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arXiv2022

On the isomorphism class of $q$-Gaussian W$^\ast$-algebras for infinite variables

Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeq M_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraic counterpart of this result was obtained recently in [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.

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arXiv2022

Overcompleteness of coherent frames for unimodular amenable groups

This paper concerns the overcompleteness of coherent frames for unimodular amenable groups. It is shown that for coherent frames associated with a localized vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.

math.FA↗

arXiv2022

Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces

Let $G$ be a locally compact unimodular group, and let $\phi$ be some function of $n$ variables on $G$. To such a $\phi$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Further, we prove that the bilinear Hilbert transform is not bounded as a vector valued map $L_{p_1}(\mathbb{R}, S_{p_1}) \times L_{p_2}(\mathbb{R}, S_{p_2}) \rightarrow L_{1}(\mathbb{R}, S_{1})$, whenever $p_1$ and $p_2$ are such that $\frac{1}{p_1} + \frac{1}{p_2} = 1$. A similar result holds for certain Calder\'on-Zygmund type operators. This is in contrast to the non-vector valued Euclidean case.

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arXiv2022

On the isomorphism class of $q$-Gaussian C$^\ast$-algebras for infinite variables

For a real Hilbert space $H_{\mathbb{R}}$ and $-1 < q < 1$ Bozejko and Speicher introduced the C$^\ast$-algebra $A_q(H_{\mathbb{R}})$ and von Neumann algebra $M_q(H_{\mathbb{R}})$ of $q$-Gaussian variables. We prove that if $\dim(H_{\mathbb{R}}) = \infty$ and $-1 < q < 1, q \not = 0$ then $M_q(H_{\mathbb{R}})$ does not have the Akemann-Ostrand property with respect to $A_q(H_{\mathbb{R}})$. It follows that $A_q(H_{\mathbb{R}})$ is not isomorphic to $A_0(H_{\mathbb{R}})$. This gives an answer to the C$^\ast$-algebraic part of Question 1.1 and Question 1.2 in [NeZe18].

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arXiv2022

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Let $π_α$ be a holomorphic discrete series representation of a connected semi-simple Lie group $G$ with finite center, acting on a weighted Bergman space $A^2_α (Ω)$ on a bounded symmetric domain $Ω$, of formal dimension $d_{π_α} > 0$. It is shown that if the Bergman kernel $k^{(α)}_z$ is a cyclic vector for the restriction $π_α |_Γ$ to a lattice $Γ\leq G$ (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ}$ is a frame for $A^2_α(Ω)$), then $\mathrm{vol}(G/Γ) d_{π_α} \leq |Γ_z|^{-1}$. The estimate $\mathrm{vol}(G/Γ) d_{π_α} \geq |Γ_z|^{-1}$ holds for $k^{(α)}_z$ being a $p_z$-separating vector (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ/ Γ_z}$ being a Riesz sequence in $A^2_α (Ω)$). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $G = \mathrm{PSU}(1, 1)$.

math.FA↗

arXiv2022

Local and multilinear noncommutative de Leeuw theorems

Let $\Gamma < G$ be a discrete subgroup of a locally compact unimodular group $G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p < \infty$ and let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding Fourier multiplier. Similarly, let $T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma})$ be the Fourier multiplier associated to the restriction $m|_{\Gamma}$ of $m$ to $\Gamma$. We show that \[ c( {\rm supp}( m\vert_{\Gamma} ) ) \Vert T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma}) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \] for a specific constant $0 \leq c(U) \leq 1$ that is defined for every $U \subseteq \Gamma$. The function $c$ quantifies the failure of $G$ to admit small almost $\Gamma$-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, $c(\Gamma) =1$ when $G$ has small almost $\Gamma$-invariant neighbourhoods. Our result thus extends the De Leeuw restriction theorem from [CPPR15] as well as De Leeuw's classical theorem [Lee65]. For real reductive Lie groups $G$ we provide an explicit lower bound for $c$ in terms of the maximal dimension $d$ of a nilpotent orbit in the adjoint representation. We show that $c(B_\rho^G) \geq \rho^{-d/4}$ where $B_\rho^G$ is the ball of $g\in G$ with $\Vert {\rm Ad}_g \Vert < \rho$. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs $\Gamma<G$ with $c(\Gamma) = 1$. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

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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.

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arXiv2021

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

In deformation-rigidity theory it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule $H$ over the group algebra $\mathbb{C}[\Gamma]$, with $\Gamma$ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of $H$ is contained in the Schatten $\mathcal{S}_p$ class $p \in [2, \infty)$ then the $n$-fold tensor power $H^{\otimes n}_\Gamma$ for $n \geq p/2$ is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carr\'e du champ of a symmetric quantum Markov semi-group. For Coxeter groups we give a number of characterizations of having coefficients in $\mathcal{S}_p$ for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-$\mathcal{S}_p$ property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups: (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by T. Sinclair for discrete groups admitting a proper cocycle into a $p$-integrable representation.

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arXiv2021

Riesz transforms on compact quantum groups and strong solidity

One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semi-groups (QMS's) and non-commutative Riesz transforms. We introduce a property for QMS's of central multipliers on a compact quantum group which we shall call approximate linearity with almost commuting intertwiners. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a QMS that is approximately linear with almost commuting intertwiners, satisfies the immediately gradient-$\mathcal{S}_2$ condition from [Cas21] and derive strong solidity results (following [Cas21], [OzPo10], [Pet09]). Using the non-commutative Riesz transform we also show that these quantum groups have the Akemann-Ostrand property; in particular the same strong solidity results follow again (now following [Iso15b], [PoVa14]).

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arXiv2020

BMO spaces of $\sigma$-finite von Neumann algebras and Fourier-Schur multipliers on $SU_q(2)$

We consider semi-group BMO spaces associated with an arbitrary $\sigma$-finite von Neumann algebra $(\mathcal{M}, \varphi)$. We prove that the associated row and column BMO spaces always admit a predual, extending results from the finite case. Consequently, we can prove that the semi-group BMO spaces considered are Banach spaces and they interpolate with $L_p$ as in the commutative situation, namely $[\mathrm{BMO}(\mathcal{M}), L_p^\circ(\mathcal{M})]_{1/q} \approx L_{pq}^\circ(\mathcal{M})$. We then study a new class of examples. We introduce the notion of Fourier-Schur multiplier on a compact quantum group and show that such multipliers naturally exist for $SU_q(2)$.

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