Papers

Let $M$ be a nonzero type III von Neumann algebra with separable predual, and let $N$ be a semifinite von Neumann algebra. We prove that $M_*$ is not isomorphic to a subspace of $N_*$. For $1<p<\infty$, $p\ne2$, we prove that $L_p(M)$ is not isomorphic to a complemented subspace of $L_p(N)$. In particular, the Banach isomorphism class of $L_p(M)$ distinguishes type III algebras from semifinite algebras for $1\le p<\infty$, $p\ne2$.

We obtain the necessary and sufficient conditions on input functions for a family of positive exponent power weight $L^1$ Stein-Weiss inequalities. Earlier work by De Nápoli and Picon showed that a certain cocanceling condition was necessary and sufficient in $L^1$, but it was unknown if this condition remained necessary in the positive exponent power weight setting. This present article identifies weaker necessary conditions for positive exponent power weights, some of which were shown to be sufficient for positive, non-integer exponent power weights in the author's earlier work.

Given a matrix $A\in M_{k\times N}(\Bbb{F}),$ we examine finite frames $\{f_j\}_{j=1}^N$ with vanishing $A$-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing $A-$moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix $A$. In particular, we show that every frame with redundancy $k$ admits a dual frame with vanishing $A$-moments for some totally nonsingular matrix $A$. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy $k$ (erasure robustness is not needed) allows perfect signal reconstructions for any number of $\ell$-erasures ($\ell \leq k$) which only involves inverting a matrix of small size ($\ell$) instead of inverting the subframe operator which is a matrix of large size ($N-\ell$). A special case of this type of frames are $k$-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with $k-th$ vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.

When two quantum systems are prepared independently and each is driven to an idealized limit, the expectation of a bipartite observable may depend on which system is idealized first. We call an observable stable when it does not. We show that stability is equivalent to two operational properties. First, every idealized preparation of one system can be simulated to arbitrary accuracy, uniformly over all probes, by finite mixtures of actual preparations, provided that the probing system is idealized first. Second, any two idealized preparations have a unique, symmetric independent product. Under a uniform form of stability, we prove, the number of preparations that $n$ adaptive probes can tell apart at a fixed margin is bounded by a polynomial in $n$, and responses can be learned with a bounded number of mistakes; otherwise, at some margin, protocols with $n$ probes tell apart $2^n$ preparations. The swap test, Hong–Ou–Mandel coincidences and discrete equality tests turn out to be uniformly stable, with bounds independent of the dimension, while order comparisons of discrete spectra unbounded above are unstable. The proofs rest on the classical fact that a single double limit condition appears, under different names, in topology (Grothendieck), functional analysis (Krivine–Maurey), algebra (Arens), probability (Simons) and combinatorics (Pták), and in model theory as Shelah's stability, where the independent product corresponds to Harrington's lemma. Model theory thus serves as a Rosetta stone, pairing each physical question with its mathematical formulation.

We prove that a smooth Dunkl harmonic function in the upper half-space whose Dunkl gradient satisfies a Carleson measure condition is the Poisson integral of a function in $\mathrm{BMO}(dw)$, defined using Euclidean balls. The result holds for arbitrary finite root systems and nonnegative multiplicities. The proof uses an estimate for increments from the origin, obtained from a finite-dimensional system indexed by the reflection group, together with the atomic characterization of the Dunkl Hardy space.

We introduce the notion of a projection scheme on a Banach space and develop an abstract framework for studying the fixed point property (FPP) for nonexpansive mappings. Within this framework, we obtain a general criterion ensuring the FPP for a broad class of nonreflexive Banach spaces. In particular, every uniformly convex Banach space admits an isometric embedding into a nonreflexive Banach space having the FPP. This criterion unifies and extends several previously known fixed point results for nonreflexive Banach spaces with Schauder bases. As an application of our general result, we characterize the FPP for variable Lebesgue spaces. More precisely, we prove that if $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite measure space and $p:\Omega \to [1,\infty]$ is a measurable function, then $L^{p(\cdot)}(\Omega)$ has the FPP if and only if it does not contain an isometric copy of $\ell_1$. Consequently, as in the classical Lebesgue space framework, the presence of an isometric copy of $\ell_1$ completely characterizes the failure of the FPP in variable Lebesgue spaces. This result yields a class of nonreflexive Banach spaces with the fixed point property that contains isometric copies of infinitely many $L^p$-spaces, $1<p<\infty$. Although nonreflexive sequence spaces with the FPP are known, to the best of our knowledge, these variable Lebesgue spaces provide the first examples of nonreflexive Banach function lattices with the FPP.

We prove that the analytic Besov space $B_{p,p}^{1/p}(\C_+^n)$ embeds continuously into Chang–Fefferman product BMO for every finite $n$ and $1\le p<\infty$. Consequently, every small Hankel operator in a finite Schatten class has a product BMO symbol. The proof uses smooth product wavelets and an estimate for dyadic rectangles. We also give a condition on Fejér means that yields a BMO symbol for a bounded Hankel operator on the polydisc.

In this paper, we characterize surjective nonlinear composition operators acting between the space of sequences of bounded variation and the space of bounded sequences. We show that their generators exhibit highly pathological behavior. To complement this, we construct explicit examples, confirming that such exotic objects do in fact exist.

We establish a geometric rigidity theorem for a class of monomial varieties in finite-dimensional unit balls. Every ball automorphism mapping one such variety onto another is unitary, and whenever such an equivalence exists, it can also be realized by a permutation of the ambient coordinates. As a consequence, we obtain a complete isometric classification of the multiplier algebras of integrally generated complete Nevanlinna–Pick Dirichlet series kernels in terms of their defining weight and frequency data. We also obtain an explicit isometric classification for a class of complete Nevanlinna–Pick kernels on bounded Reinhardt domains.

We characterize Fourier quasicrystal measures with positive integer masses as real sections of effective Lee-Yang cycles, with weighted local intersection degrees. The section matrix has positive maximal minors, and the mass at each root is its weighted local intersection degree. For Delone support the reduced support of the cycle can be chosen strict. The support of every integer-mass Fourier quasicrystal measure is the common complex zero set of finitely many real-valued trigonometric polynomials. For unit masses the representing cycle can be chosen reduced, with local degree one at every physical root. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches and their integer coefficients from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee-Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property.

We determine the hundredths digit of the real Grothendieck constant by proving $1.773 \leq K_G \leq 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G \approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bounds are obtained by a heuristic primal/dual search for these objects, followed by rigorous certification of their values. The search and certification are performed using AI tools, and in particular the rigorous certificates are very large (analytical reductions to thousands of numerical inequalities).

In this article, we study three properties of a bounded $\mathbb{C}$-convex domain $\Omega \subset \mathbb{C}^n$, $n \geq 2$, with smooth boundary, namely Gromov hyperbolicity of its Kobayashi distance, finiteness of the D'Angelo type of $\partial \Omega$, and the existence of a subelliptic estimate for the $\bar{\partial}$-Neumann problem on $(0,1)$-forms. We prove that these three properties are equivalent. The subelliptic estimates are obtained without Catlin's construction of plurisubharmonic weights. Our weights are explicit bounded transforms of the logarithm of the Bergman kernel. A single quantity controls all three properties, namely the radius of the largest disc through a point of $\Omega$ in a given complex direction. Gromov hyperbolicity forces a power bound for these radii. Finite type gives the power $1/M$, where $M$ is the maximal type, and this power cannot be improved. A subelliptic estimate gives a power bound through the canonical solution operator applied to normalized Bergman kernels. The analytic estimates need no boundary regularity. On a bounded pseudoconvex domain, a bounded weight whose induced Hessian on $(0,q)$-forms is bounded below by a negative power of the boundary distance gives a Sobolev estimate for the extension by zero of $(0,q)$-forms, and also eigenvalue bounds for the $\bar{\partial}$-Neumann operator. On smooth $\mathbb{C}$-convex domains of finite type, we show that the directional expansion exponent, the normal expansion exponent, and the supremum of the subelliptic gains are all equal to $1/M$. Next, we prove that on a Lipschitz $\mathbb{C}$-convex domain whose Kobayashi distance is Gromov hyperbolic, the Euclidean boundary and the Gromov boundary are bi-Hölder equivalent. Finally, we give a Hardy-type criterion for Gromov hyperbolicity of collar metrics.

For a normed space $E$, we denote by $E_w$ and $E'_{w^*}$ the space $E$ and its dual $E'$ endowed with the weak topology $w$ and the weak$^*$ topology $w^*$, respectively. It is well known that for a Banach space $E$, both spaces $E_w$ and $E'_{w^*}$ are Fréchet–Urysohn topological spaces if and only if $E$ is finite-dimensional. This statement remains true for all linear subspaces of $E_w$ and $E'_{w^*}$. Moreover, for any normed space $E$, every Fréchet–Urysohn subgroup of $E_w$ is locally finite-dimensional; that is, it has an open subgroup contained in a finite-dimensional linear subspace. Similarly, for any Banach space $E$, every Fréchet–Urysohn subgroup of $E'_{w^*}$ is locally finite-dimensional. Analogous results are obtained for the free locally convex space $L(X)$, its weak version $L_p(X)$, and the free topological vector space $V(X)$ over a Tychonoff space~$X$. The same conclusions hold for Baire subgroups and Baire linear subspaces of $E_w$, $E'_{w^*}$, $L(X)$, $L_p(X)$, and $V(X)$. We emphasize that even for metrizable subgroups, these results are new and nontrivial. In fact, all results are established for the much wider family of $\kappa$-Fréchet–Urysohn subgroups of topological vector spaces. The class of $\kappa$-Fréchet–Urysohn spaces is the natural framework in which the proofs of our results work.

We construct an explicit binary linear system game that separates $C_{qa}$, the closure of the set of finite dimensional quantum correlations, from $C_{qc}$, the set of commuting operator correlations. The game admits a perfect commuting operator strategy, while every correlation in $C_{qa}$ has a success probability strictly less than 1. This provides a concrete counterexample to Tsirelson's problem in its approximation form. The defining system has 1417152 equations in 1889684 variables, with exactly three nonzero coefficients per equation and a single nonzero entry on the right hand side. We also compute the classical value exactly. The complete system is specified in Lean 4, and the separation and exact classical value are formalized using Mathlib.

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(\mathbf 1_Ef)$ is compact on $L^2(\mu)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein–Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

For a convex body $K$ its parallel section function in the direction $\theta\in S^{n-1}$ is defined by $$A_{K,\theta}(t) = \mathrm{vol}_{n-1} \left( K \cap \{ x \in \mathbb{R}^n ~ | ~ \langle x, \theta \rangle = t \} \right) .$$ We study to what extent the body $K$ is determined by partial information about these functions. As applications, we characterize convex bodies with locally separable section functions, establish partial results on the homothety conjecture for bodies of flotation, and answer a question of Barker and Larman concerning the determination of convex bodies from section functions at infinitely many distances from the origin.

For every $1<p<\infty$, the Hilbert transform on $L^p(\mathbb{R};\mathcal{S}_p)$ and the ordered Hilbert Schur multiplier on $\mathcal{S}_p$ have norm $C_p=\cot(\pi/(2p^*))$, where $p^*=\max\{p,p/(p-1)\}$. The scalar Hilbert transform and the Schur multiplier have the same canonical completely bounded norm. This proves the Gohberg-Krupnik sharp Volterra conjecture. A simultaneous finite-dimensional realization also resolves the Riesz-Titchmarsh conjecture and Laeng's Conjecture 5.7 for all shifts. The norm identities extend to finite families of real Hilbert-transform pencils with $\ell^p$ direct-sum ranges. The proof uses a spectral-defect identity with two nonnegative remainders, based on Heinävaara's tracial joint spectral measure. It classifies trace inequalities for finite signed real ridge sums and identifies the least homogeneous superharmonic majorant as the pointwise least additive eigenvalue correction. The associated angular cone is characterized by nonnegative potentials and realized by weak-* limits of weighted angular measures of finite nilpotent matrices.

We prove that the peripheral spectrum of every positive operator on a complex Banach lattice is cyclic, thereby resolving the cyclicity conjecture without additional growth assumptions. Our proof is based on a spectral domination theorem derived from Vesentini's theorem on the subharmonicity of the spectral radius: when two positive operators have the same spectral radius and one dominates the other, the peripheral spectrum of the smaller operator is contained in that of the larger one. This extends an earlier result of Räbiger and Wolff. After passing to a suitable lattice extension, an averaging procedure based on torsion operators produces a decreasing sequence of positive minorants which retain the spectral radius and become asymptotically rotationally self-similar. An ultrapower argument turns this asymptotic relation into exact rotational self-similarity, allowing us to deduce cyclicity from spectral domination. The same approach also yields an alternative proof of Lotz's cyclicity theorem for Abel solvable operators.

Mathematical programming formulations are routinely modified through algebraic reformulations, perturbations of constraint functions, and the addition or removal of constraints. Some of these modifications preserve the feasible set, whereas others reflect newly imposed modeling requirements and therefore change the optimization problem itself. This paper develops a metric framework for quantifying the robustness of formulation-dependent properties. Formulations are modeled as typed collections of constraint functions. Within each fixed typed-cardinality class, their distance is defined by optimally matching constraints of the same type and measuring the resulting discrepancies in a norm on the underlying function space. As a concrete demonstration of the framework, we study the Linear Independence Constraint Qualification (LICQ). We introduce the pointwise LICQ radius, defined as the distance from a feasible formulation to the set of formulations that remain feasible at a prescribed point but fail LICQ there. Under suitable assumptions, we derive an explicit formula for this radius and characterize the effect of adding a single equality or inequality constraint.

Let $E$ and $F$ be separable symmetric Banach sequence spaces, and let $C_E$ and $C_F$ be the corresponding ideals of compact operators on a separable Hilbert space. We prove that if $C_F$ is isomorphic to a complemented subspace of $S_E$, then $F=\ell_2$. As a consequence, if $C_E$ is isomorphic to $C_F$, then $E=F$ with equivalent norms. This answers Pełczyński's 1979 question concerning the uniqueness of symmetric structure. We also strengthen Arazy's result by showing that if $E$ has finite cotype and $C_F$ embeds isomorphically into $S_E$, then $F=\ell_2$. For reflexive $F\ne\ell_2$ with non-trivial Boyd indices, the embedding theorems of Lindenstrauss and Szankowski yield a separable reflexive symmetric sequence space $E$ with trivial cotype such that $C_F$ embeds isomorphically into $S_E$. This gives negative answers to several questions raised by Arazy in 1981.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

In this paper we are going to discuss compactness in combinatorial Banach spaces. After discussing the definition and structure of these spaces, we proceed to propose compactness criteria for them and for their extensions. Finally, we demonstrate their connection with theory of hypergraphs.

The main result of the paper states that if $0\neq v\in\mathbb C^m\otimes\mathbb C^n$, $d=\min\{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf U)$ denotes the operator Schmidt rank of $\mathbf U$. Furthermore, if $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ is unitary and $OSR(\mathbf U)\le k$, then, for every unit vector $x\in\mathbb C^m\otimes\mathbb C^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.

We provide a description of the real variable Hardy spaces $H^p$, $p\in (0,1]$ associated to a regular polyhedral open convex cone, devised as real variable analogues of the holomorphic Hardy spaces on the corresponding tube domain.

We study commutativity and zero products of slant Toeplitz operators on the Bergman space of the unit disk. For bounded polyanalytic symbols and for bounded harmonic symbols with boundary values in $W^{1,2}(\mathbb T)$, we prove that two such operators commute if and only if their symbols are linearly dependent over $\mathbb C$, with both symbols belonging to the same class. In the polyanalytic case, no boundedness assumption is imposed on the individual analytic components. We also show that every finite-rank commutator in this class is zero and that every finite product associated with nonzero polyanalytic symbols has infinite rank. Finally, for $s\ge2$, we prove that a product of $s$ slant Toeplitz operators with bounded harmonic symbols whose boundary values belong to $W^{s-1,2}(\mathbb T)$ is zero if and only if at least one symbol vanishes identically.

Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.

We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p>1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the $L^p$ norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For $L^p$ changes of Gaussian measure, the factor in the bounded-target estimate is $d^{(p-1)/(4p-2)}$. For each $p$, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.

Using the theory of symmetric polynomials, we prove that the spectrum of every Toeplitz operator with an analytic polynomial symbol on the harmonic Bergman space can be represented as the union of a closed curve and a finite set of at most $(n-1)2^{n-1}$ isolated eigenvalues, where $n$ is the degree of the symbol. This implies that Weyl's theorem holds for such operators.

We determine the sharp $L_p$ Brunn–Minkowski range for planar dual quermassintegrals on origin-symmetric convex bodies: $0<q\leq 2$ for $0\leq p<1$, and $0<q\leq4$ for $p=1$. The main new results are the Minkowski inequality for $2<q<4$, proved using a weighted spectral inequality on the projective circle, and parallelogram counterexamples for $0<p<1$ and $q>2$. Together with the known logarithmic and endpoint results, we obtain the complete equality classification: only positive dilates occur, except at $(p,q)=(0,2)$, where parallelograms with corresponding sides parallel also give equality.