The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappié transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappiè's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

We prove a Bishop--Phelps--Bollobás theorem for operators into spaces of continuous scalar-valued functions on extremally disconnected compact Hausdorff spaces over both the real and complex scalar fields. The main result applies whenever the density character of the domain is strictly smaller than the Baire number of the underlying compact space. The proof also yields an explicit quadratic Bishop--Phelps--Bollobás modulus. In particular, every separable Banach space paired with such a function space has the Bishop--Phelps--Bollobás property for operators.

Given $n \in \mathbb{N}$, an Ahlors--David $n$-regular set $S \subset \mathbb{R}^{n+1}$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_{1}$-condition, we present a complete intrinsic description of the trace-space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ of the weighted first-order Sobolev space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$ to $S$. Furthermore, we construct a new family of nonlinear bounded extension operators acting from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$. Finally, we find conditions on $\gamma$ that sufficient for the existence of a bounded linear extension operator from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$.

We show that if $M$ is a metric space of Nagata dimension at most $d$, then there exists an atlas on $M$ modeled on $\mathbb R^d$ such that every Lipschitz map $f:M\to Y$ (with values in an arbitrary Banach space $Y$) can be uniformly approximated by maps that are affine, and thus $\mathcal{C}^1$-smooth, with respect to this atlas. The construction relies on random metric partitions and stochastic retractions inside Lipschitz-free spaces. As an application, we introduce approximate continuous upper gradient $X$-structures (ACUG $X$-structures) on metric spaces and prove that every space of finite Nagata dimension carries an ACUG structure modeled on a superreflexive Banach space. Finally, adapting a proof due to Bourgain, we show that if $M$ has an ACUG superreflexive-structure, then the Lipschitz-free space $\mathcal{F}(M)$ has Pelczyński's property (V*). In particular, at least in the compact case, our result recovers all previously known examples of metric spaces $M$ for which $\mathcal{F}(M)$ has property (V*).

For a uniformly locally finite coarse space $(X,\mathcal{E})$, we prove that for every $p\in\{0\}\cup[1,\infty]$, the lattice of geometric ideals in the $\ell^p$ uniform Roe algebra $B^p_u(X,\mathcal{E})$ is isomorphic to the lattice of ideals of $\mathcal{E}$ (equivalently, to the lattice of ideals in the associated family of controlled partial coverings of $X$). In particular, the lattices of geometric ideals for different values of $p$ coincide. Using limit operators, we establish a canonical isometric isomorphism between $B^p_u(X,\mathcal{E})$ and the reduced $L^p$ operator algebra of the coarse groupoid for $p\in[1,\infty]$, and show that it induces an isomorphism between lattices of ideals that preserves inner support. In particular, geometric (resp. ghostly) ideals correspond precisely to dynamical (resp. restrictive) ideals under this isomorphism. Using equivalent formulations of property A for coarse spaces, we prove that for $p\in(1,\infty)$, property A implies that $B^p_u(X,\mathcal{E})$ admits a multiplier approximate identity with controlled propagation, that all ideals are geometric, and that all ghosts are trivial. For the extreme cases $p\in\{0,1,\infty\}$, these properties hold for every uniformly locally finite coarse space without assuming Property A. Finally, for $p\in[1,\infty)$, a Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra is shown to preserve the lattice of geometric ideals.

We develop a new technique to upgrade relative biexactness in general von Neumann algebras: suppose that $\{N_i\}_{i\in I}\subset M$ are mixing and biexact subalgebras of a separable von Neumann algebra with expectation, and if $M$ is biexact relative to $\{N_i\}_{i\in I}$, then $M$ is biexact. This result yields several new examples of biexact von Neumann algebras, notably including amalgamated free products. By generalizing the relative biexactness results of Hoshino to the von Neumann algebra setting and applying our result above along with certain bimodule computations, we in fact obtain, as an application, a new classification result for biexactness for graph products of finite dimensional von Neumann algebras. This yields significant extensions of prior works of Caspers-Borst, and Blufstein-Goldman-Oyakawa.

In this paper, we obtain order estimates for the Kolmogorov widths of an intersection of a finite family of Besov classes $SB_{p_j,\theta_j}^{\overline{r}_j}(\mathbb{T}^d)$ with dominating mixed smoothness in a Besov space $B_{q,\sigma}^{\overline{l}}(\mathbb{T}^d)$ in the case $2<q, \, \sigma <\infty$ when the parameters satisfy certain conditions of general position.

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{\psi_n}$ be the Laplacian with word length \[ \psi_n(k) = \min(k,n-k). \] We prove the sharp log-Sobolev inequality \[ \text{Ent}_{\pi}(f^2) \le 2\pi(f A_{\psi_n} f), \qquad f:\mathbb Z_n \to [0,\infty), \] for every $n \ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a 3rd moment estimate; the new point is a blockwise 3rd moment estimate adapted to the word-length multiplier. The same 3rd moment argument also recovers the log-Sobolev inequality for Poisson-semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were also obtained recently by Yao~\cite{Yao2026} by a different method.

It is a study note detailing the connection between Boolean inverse monoids and ample groupoids.

The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of [Devetak, Junge, King, Ruskai, CMP 2006] to multi-index Schatten norms. As an application, we strengthen the additivity statement of [Van Himbeeck and Brown, 2025] thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the Lebesgue null ideal $\mathcal{N}$. Consequently, every infinite-dimensional normed barrelled space has dimension $\ge\mbox{cov}(\mathcal{N})$ and so it is consistent with \textsf{ZFC} that no normed barrelled space has dimension equal to the bounding number $\mathfrak{b}$.

We prove that the conjectured capillary Blaschke-Santaló inequality holds for any unconditional, strictly convex capillary hypersurface when $θ\in \left(0, \tfracπ{2}\right)$. Moreover, for $θ\in \left(\tfracπ{2}, π\right)$, we show that the capillary volume product has no finite upper bound.

We present a unified theoretical framework for kernel-based measures of dependence on product spaces. Building on the ideas underlying distance covariance, distance multivariance, and the Hilbert-Schmidt Independence Criterion (HSIC), we define a new family of kernels on an $n$-fold Cartesian product, termed positive definite independent of order $k$ (PDI$_{k}$ kernels). These kernels extend the concepts of positive definite and conditionally negative definite kernels to higher orders and provide the foundation for generalized independence and interaction tests, such as the generalized Lancaster interaction of order $k$ ($\Lambda_{k}^{n}$), and the Streitberg interaction ($\Sigma$). Our analysis focuses on the continuous setting, where we prove a Kernel Mean Embedding Theorem for PDI$_{k}$ kernels and establish the corresponding integrability restrictions. Based on these results, we characterize how the Kronecker products of PDI kernels behave.

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(\Omega)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(\Omega)$ for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.