Papers

We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.

In homage to Hofstadter, we call a compact, citable specification of definitions, theorem, proof mechanism, and examples a mathematical koan when a full paper can be reconstructed and verified from it, including in reader-specific forms generated by AI. Every AI-solvable problem is itself a koan, although reconstruction need not be cheap; the present article is an expansion of one. We introduce Cartan convexity for self-adjoint free functions: locally, such a function agrees with a locally bounded matrix-convex free function. If the variable set has cardinality $\tau$ and $\lambda=\max\{\tau,\aleph_0\}$, every bounded real free set has a universal direct sum on a Hilbert space of dimension at most $2^\lambda$; every point of the set is a reducing summand of an amplification of this sum. Applying the noncommutative Kraus–butterfly theorem at that universal point yields an extension to an open matrix-convex neighborhood of the noncommutative convex hull. For normal affine pencils, the extension domain also contains the bounded strong closure of the hull. We prove an analogous theorem for a graph embedding $\Gamma$. A local convex lift through $\Gamma$ extends to a neighborhood of $\Gamma^{-1}(\operatorname{co}_{\mathrm{nc}}\Gamma(K))$ and admits a butterfly realization in the $\Gamma$-coordinates. We distinguish this lift condition from intrinsic $\Gamma$-convexity. For $\Gamma(x,y)=(x,y,y^2)$, the intrinsically $\Gamma$-affine polynomial $xy+yx$ has no convex lift germ at the origin, whereas a single quadratic coordinate suffices to lift every uniformly real analytic germ.

We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.

We improve the currently best known bound for the norm of the embedding of $PW^2$ into $PW^q$ for all $q>2$. We show that our bound is asymptotically optimal as $q\to \infty$. Our approach combines the recently established theorem of optimal Paley-Wiener concentration with a Hardy-Littlewood-Pólya majorization argument.

We study the $\mathcal F$-transitivity and topological $\mathcal F$-recurrence of left translation semigroups on weighted $L^p$-spaces over directed metric trees. Motivated by the recent work of Mangino and Vargas-Moreno on hypercyclicity and weak mixing for these semigroups, we investigate the different problem of $\mathcal F$-transitivity, where the entire return-time set is required to belong to a prescribed finitely invariant Furstenberg family. Assuming that the weight is $p$-admissible, we obtain necessary and sufficient integral conditions for both rooted and rootless trees, and establish the equivalence between $\mathcal F$-transitivity and topological $\mathcal F$-recurrence. In the rooted case, the criteria depend only on the weights along descendant edges. In the rootless case, an additional ancestor term and a measurable cancellation construction reflect the backward geometry of the tree. Our approach is based on measurable weighted minimization and quantitative estimates over subsets of the edge parameter interval. Examples on homogeneous rooted trees and rootless trees with a free left end show that branching may determine the dynamics, but forward branching alone does not guarantee $\mathcal F$-transitivity in the rootless setting.

We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations $\widehat f$ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators $T_{f,e}$ which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every $\widehat f$ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for $T_{f,e}$ under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.

The Riemann Rearrangement Theorem (RRT) tells us that commutativity of infinite series can differ from finite series. We wish to extend the notion of infinite series rearrangements to weaker forms of convergence, such as partial series convergence on every 2nd or 3rd index, called $2n$- or $3n$-convergence. In the case of $2n$-convergence, it has been shown that the sum range, along with the standard cases in the RRT, can produce a shifted additive subgroup of reals. In this paper, we show cases where the $3n$-sum range is still a subgroup and a case where the $3n$-sum range cannot be a subgroup, proving the existence of more depth for $3n$-convergence.

In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^\mathbb{N})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^\mathbb{N})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H)) = {\omega} + 1$ when taken with respect to the Haar basis for $C(2^\mathbb{N})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.

We provide an example of a projective tensor product whose unit ball contains an exposed tensor which is not elementary. We then transfer the construction to the symmetric projective tensor product and obtain an analogous result in that setting. Finally, we extend both constructions to projective tensor products with an arbitrary number of factors and to symmetric projective tensor products of every degree.

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

Krauel, Shafiq and Wood (\KSW{}) realized torus \(1\)-point functions for the simple affine \voa{}s \(L(k,0)\) built from \(\alg{sl}(2)\) as vector-valued modular forms attached to a cyclic \(R\)-module structure and to the modular tensor category \(\rep{L(k,0)}\). We extend this to torus \(n\)-point functions, defined as traces of chains of \(n\) intertwining operators winding once around the \(\tau\)-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \vvmf{}, lying, whenever a mild lowest weight hypothesis holds, in the very \(R\)-modules of \cite{KSW}. Giving a torus-primarity hypothesis verified below for \(\alg{sl}(2)\), the operator product expansion then reduces leading short-distance behaviour to the classified \(1\)-point theory. For \(\alg{sl}(2)\) we classify the fusion-chain conformal block spaces, identify torus \(n\)-point primary vectors, and treat \(n=2\) in detail, obtaining a \((k+1)\)-dimensional family of vector-valued Jacobi-type forms generalizing the level-\(k\) forms \(\eta^{3k/2}\) of \cite{KSW}. We also extend \KSW{} categorical picture, representations of \(\Bthree=\PMod(\surf{1}{1})\) built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the \(n\)-punctured torus generated by \(S\), \(T\), and adjacent braidings, with a Verlinde-type dimension formula and categorical \(S\)-, \(T\)-operators. The \(\alg{sl}(2)\) matrix form of \(S\) beyond this abstract construction remains open.

We introduce a framework for establishing Blaschke-Santaló-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santaló-type. We call these equivalent inequalities cost-Santaló inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santaló inequalities, including on the space of rectangular matrices and a functional sine Santaló inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santaló inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santaló inequality, which we explicitly instantiate for matrix spaces.

We prove a uniform spectral-gap estimate at every even exponent for isometric representations on arbitrary closed subspaces of real $L_p$-spaces. As an application, this result, combined with the work of Bader, Furman, Gelander and Monod, gives an affirmative answer to their Question 4.1: Kazhdan's property $(T)$ implies property $(T_X)$ whenever $X$ is a closed subspace or a quotient of a real $L_p$-space, for every $1<p<\infty$.

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $\eta>0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\leq n^{-\frac{2}{3}-\eta}$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $\delta>0$ and $\rho\in(0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min\{a,b\}\leq n^{-\frac{1}{2}-\delta}$ and $ab\leq \frac{1}{2}-\rho$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.

Let $\mathcal H_g f(z)=\int_0^1 f(t)g'(tz),dt$, and let $2<p<\infty$. Set $t=2p/(p-2)$ and $X_j=2^{-j/p'}\Delta_j g'$, where $\Delta_j$ is the hard dyadic Taylor projection. We prove that $\mathcal H_g:H^p\to H^p$ is bounded if and only if $h\mapsto(X_jh)_{j\geq0}$ is bounded from $H^t$ to $\ell^t(H^2)$. The square of this embedding norm equals the norm of the positive column operator $b\mapsto\sum_j b_j|X_j|^2$ from $\ell^{p/2}$ to $L^{p/2}$. Coordinate tails yield essential-norm estimates and an exact compactness criterion, while Hardy duality gives an equivalent paraproduct formulation. The criterion is quantitatively invariant under admissible analytic dyadic resolutions and defines a resolution-independent Calderón symbol space equal to the Hilbert-range multiplier space. We construct a bounded noncompact dense-frequency symbol outside the known blockwise sufficient class. We also prove an exact Poisson-atomic testing theorem: finite positive Poisson mixtures recover the full norm, but no fixed atom count suffices. Finally, aggregate probability densities give an intrinsic atomic-complexity formula and a finite-bandwidth testing bound.

In his 2018 paper “On the unique predual problem for Lipschitz spaces”, N. Weaver published proofs that Banach spaces $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$ have strongly unique preduals whenever $M$ has finite diameter or is geodesic. A gap was recently noticed in the proof of a crucial lemma that claimed that the property of having a strongly unique predual passes to 1-codimensional weak$^*$-closed subspaces. In this note, we confirm that the lemma is actually false by providing an explicit counterexample. We also expand on some of Weaver's original arguments to provide a new, valid proof of the following particular case: $\mathrm{Lip}_0(M)$ has a strongly unique predual whenever $M$ is a convex subset of a finite-dimensional normed space.

A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With $\mathrm{NA}_1(X,Y)$ and $\mathrm{NA}_2(X,Y)$ denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that \[ \mathrm{NA}_2(c_0,c_0)=\mathcal{L}(c_0,c_0) \qquad\text{and}\qquad \mathrm{NA}_2(\ell_1,\ell_1)=\mathcal{L}(\ell_1,\ell_1), \] providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for $c_0$, we undertake a systematic study of this equality within a natural family of $\ell_1$-preduals given by hyperplanes of $c$, obtaining a complete characterization in this setting. In particular, we prove that \[ \mathrm{NA}_2(c,c)\neq \mathcal{L}(c,c), \qquad\text{whereas}\qquad \mathrm{NA}_3(c,c)=\mathcal{L}(c,c). \] We also establish Holub–Mujica-type theorems for the classes $\mathrm{NA}_1$ and $\mathrm{NA}_2$. More precisely, under suitable separability and approximation property assumptions, the identity $\mathcal L(X,Y)=\mathrm{NA}_1(X,Y)$ forces every operator from $X$ into $Y$ to be compact, whereas $\mathcal L(X,Y)=\mathrm{NA}_2(X,Y)$ forces every weakly compact operator from $X$ into $Y$ to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.

In this paper, we develop a theory of generalized fractional integration for set-valued mappings for the Katugampola fractional integral, which unifies the classical Riemann-Liouville and Hadamard fractional integrals. The Katugampola fractional integral of a set valued mapping is studied using integrable selections. We study its properties with respect to the Hausdorff metric on the space of nonempty compact subsets of R and several fundamental analytical characteristics including convexity, boundedness and continuity are preserved under Katugampola fractional integration. Furthermore, we establish that both bounded variation and Lipschitz regularity of a set-valued mapping are preserved under its Katugampola fractional integral. We investigate the existence of regular selections associated with the Katugampola fractional integral and show that whenever the original set valued mapping admits a selection with a specified regularity property, the corresponding Katugampola fractional integral does as well.

Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0<p\leq 1$. Specifically, \[ B^s_{p,q}(\Omega)\hookrightarrow \mathcal L_1(\mathbb{D}) \] when $s\geq k+d/p$ for $0<q\leq 1$, and when $s>k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(\Omega), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.

We introduce an operator-valued maximal f-divergence for completely positive (CP) maps between von Neumann algebras, associated with an operator convex function f on (0,+infinity). The construction takes values in the extended lower-semibounded self-adjoint part of the codomain von Neumann algebra. We prove independence of the common CP upper bound used in its computation, joint subadditivity, monotonicity under unital precomposition and normal postcomposition, a martingale convergence theorem for normal CP maps, and joint lower semicontinuity in the point-sigma-weak topology. For normal positive functionals, our construction recovers Hiai's maximal f-divergence. As a consequence, we establish its joint weak lower semicontinuity for arbitrary von Neumann algebras, answering a question left open by Hiai. For eta(t)=t log t, we obtain an operator-valued Belavkin–Staszewski (BS) relative entropy. Moreover, for normal channels with sigma-finite codomain, we prove that the BS channel divergence of Hollands and Ranallo coincides with the extended norm of our operator-valued divergence. Finally, we give finite-dimensional examples and obtain an explicit formula for a finite-index conditional expectation. In the BS case, this formula reduces to the logarithm of the Jones–Kosaki index.

We study operators acting on functions defined on the half-space, with methods inspired by sparse domination. Using the specific link between dyadic grids on the boundary and Whitney regions on the half-space, we obtain a very precise form domination with model operators of Hardy type. We introduce mixed-norm off-diagonal estimates that measures both tangential and transversal decay without requiring pointwise control in the transversal variable, substantially weakening the assumptions used in earlier tent space theories. This allows us to prove optimal weighted tent space extrapolation from local boundedness plus off-diagonal decay. The resulting framework yields new bounds for operators arising from non-autonomous elliptic and parabolic PDEs with non-smooth coefficients.

Let $\mathfrak{c}=2^{\aleph_0}$. We prove that $\ell_\infty$ contains $2^{\mathfrak{c}}$ pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of $c_0$ and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative $\ell_\infty$-Grothendieck property depends on the specified copy of $c_0$ and is not an isomorphic invariant of the subspace. More generally, for every closed $E\subseteq\ell_\infty$ there is an $\ell_\infty$-Grothendieck subspace isomorphic to $\ell_\infty\oplus_\infty E$; taking $E=c_0$ answers a problem by González et al.

Let $(A,\rho)$ be a tracial ${\rm C}^*$-probability space with the rapid decay property relative to a filtration $L$. We show that the associated Sobolev algebra $H_L^{\infty}(A,\rho)$ is closed under the smooth functional calculus of $A$. Some consequences include norm estimates, the ideal separation property, and the fact that the inclusion $H_L^{\infty}(A,\rho)\subset A$ induces an isomorphism in $K$-theory.

We construct a real self-adjoint semigroup on an $\ell^2$-space which is individually eventually strongly positive with respect to a strictly positive vector, but whose operators fail to be positive at arbitrarily large times. Moreover, the generator of this semigroup has compact resolvent. Consequently, individual and uniform eventual positivity are not equivalent for self-adjoint semigroups on $L^2$-spaces, even under the additional assumption that the generator has compact resolvent. The construction consists of a positive self-adjoint rank-one perturbation of a diagonal operator and a sequence of small spectral shifts on antisymmetric two-point modes.

For pairs $(G,H)$, where $G$ is a locally compact group and $H$ is a closed subgroup of $G$, we introduce analogous versions of the full group $\mathrm{C}^*$-algebra and the Fourier–Stieltjes algebra for uniformly bounded representations. We use these objects to characterise relative property (T) in this more general setting in terms of Kazhdan projections and invariant means. We also show that, for semidirect products of the form $G=H\ltimes N$, where $N$ is a nilpotent group, relative property (T) for the pair $(G,N)$ is equivalent to its uniformly bounded version. We first prove this result when $N$ is abelian, and then proceed by studying the behaviour of relative property (T) under quotients by central subgroups, thereby extending a theorem of Serre to the uniformly bounded setting.

We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on $C^*$-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded "Radon-Nikodym" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by $*$-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^\delta) = -\frac{\theta_\delta^2 \pm \theta_\delta}{2} \log N + O(1) \] where $\theta_\delta := \delta$ if $\delta < -1/2$, and $\theta_\delta := \frac 1 \pi \arcsin(\sin(\delta\pi))$ if $\delta \ge -1/2$ and $\arcsin \colon [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the principal branch of $\arcsin$. For $\delta \geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $\delta < - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta < - \frac 12$. Most notably, we obtain for $\delta < -\frac12$ \[ \log \det\bigl(I_N - (H_N^\delta)^2\bigr) = -\delta^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.

We study the algebraic structure of $(6,3)$ biangular Parseval frames. The two-distance property yields adjacency matrices $A_1,A_2$ whose span forms a three-dimensional commutative algebra, and adjoining the commutator $[A_1,A_2]$ produces a three-dimensional Lie algebra $\g$. The Gram matrix $G = I + c_1 A_1 + c_2 A_2$ induces a derivation $\alpha(X) = [G,X]$ on $\g$, equipping $(\g, [\cdot,\cdot], \alpha)$ with a Hom–Lie algebra structure. We compute all structure constants explicitly in terms of the strongly regular graph parameters $(k_1,\lambda_1,\mu_1,k_2,\lambda_2,\mu_2)$ and the frame angles $(c_1,c_2)$, and use this framework to derive explicit bounds for the partial frame operators arising in Marcus–Spielman–Srivastava (MSS) partitions. These bounds depend directly on the structure constants and refine the universal MSS estimate in the regime of highly unbalanced partitions.

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.

In this paper, we shed light on basic questions such as: What is the "right" underlying set on which a function space lives, and is this set uniquely determined? What are the appropriate morphisms to classify function spaces? When does a weighted composition operator induce an isomorphism, and when does it induce an isometric isomorphism? We first formulate suitable notions for domains of reflexive functional Banach spaces and study the corresponding notions of isomorphism between Hilbert function spaces. We then apply this framework to classify certain unitarily invariant spaces of holomorphic functions in several variables. Our main results extend Hartz's results from the complete Pick setting to the general setting of weighted Hardy spaces and establish a simple relation between the kernels of isomorphic spaces, improving also on earlier work of Ofek and Sofer in several directions. Additionally, we apply our results to investigate and estimate the Banach-Mazur distance of Hilbert function spaces.