Papers

Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$ belonging to the noncommutative $L^p$ space $ L^p(\hat \Gamma)$, we obtain convergence in the $L^p$ norm. The proof combines uniformly bounded finite-support Fourier cutoffs with an elementary Bernoulli even-moment estimate. We give a self-contained free-group argument, then establish a general criterion and verify it for the two classes above. The operator-norm conclusion establishes the Révész property for these groups. In particular, it confirms the polynomial-growth conjecture of Hamm, Hayes, and Petrosyan and disproves their conjecture that the free group $\F$ does not have this property.

Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $\psi:S_V\to S_{\ell_2}$ such that the restrictions $\psi|_{S_{V_j}}$, $j=1,2,\ldots$, are isomorphisms from $S_{V_j}$ onto $S_{H_j}$. The Kasparov-Yu problem asks whether the Banach space $c_0$ of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that $c_0$ admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum $\mu_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\ell_\infty^n}$ to $S_{\ell_2^n}$ satisfies $\lim_{n\to\infty} \mu_n/\log n=1/2$. (2) Let $\lambda_n$ and $C_n$ denote the infima of the Lipschitz constants of homeomorphisms from $S_{\ell_\infty^n}$ onto $S_{\ell_2^n}$ and from $B_{\ell_\infty^n}$ onto $B_{\ell_2^n}$, respectively. Then $\lim_{n\to\infty} \lambda_n/\log n=\lim_{n\to\infty} C_n/\log n=1/2$. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.

In this paper, we resolve the convolution inequality problem posed by Naor and Schechtman in \cite[Question 6.1]{N-S2016} (see also \cite{Na2016}). More precisely, we establish the following strong convolution inequality on products of finite cyclic groups: For each $1\leq p<\infty$ there exists $c_{p}>0$ such that for every $n$, $m\in \mathbb{N}$ the following holds \[ \begin{split} &\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|E_{\{1,\cdots,n\}}f(x+\varepsilon)-E_{\{1,\cdots,n\}}f(x-\varepsilon)\right|^{p}\\ \leq& c^{p}_{p}\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|\varepsilon_{j}\left[E_{\{1,\cdots,n\}\setminus\{j\}}f(x+e_{j})-E_{\{1,\cdots,n\}\setminus\{j\}}f(x-e_{j})\right]\right|^{p}, \end{split} \] $E_{A}f(x)=\frac{1}{2^{n}}\sum_{\delta\in \{-1,1\}^{n}}f\left(x+\sum_{j\in A}\delta_{j}e_{j}\right)$ for every $A\subseteq \{1,\cdots,n\}$. The growth order we obtain here $c_{p}=O(p)$ as $p\to \infty$. For the endpoint $p=\infty$, we show by a counterexample that the strong convolution inequality fails for the $L_{\infty}$, and we prove that the appropriate endpoint for $p=\infty$ is the martingale $BMO$ space.

For every reproducing kernel Hilbert space $\mathcal{H}_K$ with bounded kernel $K$, we prove that the linear sampling widths $g_m^{\text{lin}}$ and Gelfand widths $c_n$ in the uniform norm satisfy $$ g_m^{\text{lin}}(B_{\mathcal H_K})_\infty \leq \frac{m+1}{m-n+1}\, c_n(B_{\mathcal H_K})_\infty\quad , \quad m\ge n. $$ In a certain sense, the leading factor $(m+1)/(m-n+1)$ on the right-hand side is sharp. We transfer the problem into the selection of a maximal volume subset of kernel translates (kernel $D$-optimal design). Apart from basic linear algebra, our proof employs low rank approximation techniques and a version of the Eckart-Young-Mirsky theorem. The same argument gives corresponding bounds for the selection of a reduced basis in Hilbert spaces.

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$ \text{$t_{\mathrm{opt}}(2,q)=\frac12\log(q-1)$ if and only if $q\ge q_0$},$$ whereas the strict inequality $t_{\mathrm{opt}}(2,q)>\frac12\log(q-1)$ holds for $2<q<q_0$.

In this paper, we present an application of matrix-valued truncated Toeplitz operators to the matrix version of the Nevanlinna-Pick interpolation problem. We then investigate the unitary part of a matrix-valued truncated Toeplitz operator induced by a contractive matrix-valued symbol. We further prove that, for a pure matrix-valued inner function, the corresponding model space is invariant under multiplication operator if and only if the symbol is constant. In addition, we give an algebraic proof of a characterization of the invertibility of matrix-valued asymmetric truncated Toeplitz operators using Toeplitz operators with 2 by 2 block matrix symbol. We also discuss the solvability of matrix-valued asymmetric truncated Toeplitz operators.

We study absolutely $r$-summing Toeplitz operators $T_f:F_\varphi^p\to F_\varphi^q$ with complex symbols. The weight has a uniformly positive and bounded real Hessian. The main difficulty is that local distance to holomorphic functions does not detect an entire part of the symbol, while the Toeplitz operator does. Using the complete classification of absolutely summing diagonal maps, we characterize absolutely summing Fock–Carleson embeddings. An IDA decomposition, a Rademacher argument, and local scalar recovery then detect the missing analytic part of a complex symbol. We obtain equivalent criteria in terms of the local $L^q$ mean, the complex ball average, and the Berezin transform for every $1\le p,q,r<\infty$. At $q=1$ the criterion is $L^{s_p}$, where $s_p=2p/(3p-2)$ for $p\le2$ and $s_p=1$ for $p>2$. We also obtain a sectorial-symbol criterion without an IDA assumption and prove sharpness by separated models.

Let $\omega: \mathbb{N} \to [1, \infty)$ be a weight on the multiplicative semigroup of natural numbers. We investigate the Bochner-Schoenberg-Eberlein (BSE) property for the weighted Banach algebra of Dirichlet series $A_\omega = \ell^1(\mathbb{N}, \omega)$ under Dirichlet convolution. By employing Bohr's correspondence, we establish a weak-$\ast$ approximation framework for $\omega$-bounded semicharacters using scaled truncations that lie within the predual space $c_0(\mathbb{N}, 1/\omega)$. Utilizing the canonical $L$-projection decomposition of the bidual space $A_\omega^{\ast\ast} = A_\omega \oplus c_0(\mathbb{N}, 1/\omega)^\perp$ alongside the weak-$\ast$ compactness of the unit ball via the Banach-Alaoglu theorem, we prove that $A_\omega$ is a BSE-algebra for any weight. Furthermore, we show that the intrinsic BSE-norm on the Gel'fand transforms is isometric to the natural weighted algebra norm $\|\cdot\|_\omega$.

For \(\omega\in\calD\), we give necessary and sufficient symbol conditions for the boundedness and compactness of the Riemann–Liouville family\(V^\varphi_{\alpha,\beta}:A^p_\omega\to H^q\) for every \(0<p,q<\infty\) and admissible \(\alpha,\beta>0\): without any additional weight assumption when \(\alpha\ge\beta\), and, when \(\alpha<\beta\), under the condition \(p(\beta-\alpha)<d_-(\omega)\), where \(d_-(\omega)\) is the critical reverse-doubling exponent. We prove that this inequality is exactly equivalent to the naturally shifted source weight retaining the two-sided doubling and tail geometry required by the reduction; an integrable logarithmic example shows that the endpoint fails. The extension from power weights is not formal, because the reduction replaces \(\omega\) by \(\omega(z)(1-|z|^2)^{p(\alpha-\beta)}\), and a negative shift may destroy even integrability. Combining moment-induced Littlewood–Paley theory for doubling Bergman spaces with an exact coefficient-multiplier comparison, we establish the required Riemann–Liouville transfer, including the finite-dimensional exceptional modes at positive integral shift orders. A finite-codimensional range decomposition and a fractional \(g\)-function reduction then reduce both questions to a single weighted area-map problem. Within the admissible parameter set, the effects of \(\alpha\) are absorbed by the shifted source weight and bounded model corrections, so the final symbol conditions depend on \(\beta\). The thresholds \(p=2\) and \(p=q\) yield pointwise, Carleson, tent-integral, and non-tangential maximal criteria. In the tent-integral range, boundedness already implies compactness.

Let $B(\mathcal H)$ be the $*$-algebra of all bounded linear operators on a Hilbert space $\mathcal H$. A classical result due to Stamplfi shows that, for any $a\in B(\mathcal H)$, the norm of the inner derivation $\delta_a=[a,\cdot]$ on $B(\mathcal H)$ is given by $\|\delta_a\|_{B(\mathcal H)\to B(\mathcal H)}=2\inf\{ \|a-c{\bf 1}\|_{B(\mathcal H)}:c\in \mathbb{C}\}$. In the present paper, we provide a formula for the norm of a generalized derivation implemented by self-adjoint operators from an ideal in $B(\mathcal H)$, which partially answers a question posed by Fialkow and Loebl in the 1980s. We also consider this question in a more general setting of a symmetrically normed (not necessarily complete or separable) space affiliated with a properly infinite semifinite von Neumann algebra.

Inspired by Grothendieck's foundational work, we establish that the sequence of functors $(Ext^n_{pBan}(E,-))_{n \ge 0}$ on the category of $p$-Banach spaces ($0 < p < 1$) forms a universal $\delta$-functor. To this end, for any pair of $p$-Banach spaces $E$ and $X$, we construct a $p$-Banach space $\mathcal N_E^X$ that renders each functor $Ext^n_{pBan}(E,-)$ effaceable. Furthermore, we show that this space serves as a relative injective object for $E$ and $X$, enabling the construction of relative injective presentations and proving that the functors $Ext^n_{pBan}(E,-)$ are the right derived functors of $\mathcal{L}(E,-)$. We will also examine the situation in the category of quasi-Banach spaces.

The estimates of intermediate derivative play important role in the theory of partial differential equations. The modern approach to this problem is via pointwise estimates, since this allows to obtain results for various function norms. The paper aims to extend the pointwise estimate for intermediate derivative in terms of the averaging operator for higher orders derivatives.

The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We also give a randomized algorithm that finds a signing of discrepancy below $12\sqrt n$ with failure probability at most $p$. The algorithm uses $n^{3+o(1)}\operatorname{polylog}(1/p)$ arithmetic operations in the real-arithmetic model. This matches the size $n^3$ of the dense input up to subpolynomial factors. In the other direction, we prove that for every $n$ there are collections of symmetric matrices such that every signing has discrepancy at least $(2-o(1))\sqrt{n}$. We present three different proofs of the matrix Spencer conjecture. The key to every proof is a hereditary small-ball estimate. This is a lower bound on the Gaussian measure of the spectral body $\{x\in \mathbb{R}^n:\|\sum_ix_iA_i\|\le R\}$ that holds for every subfamily of the matrices. The other ingredient turns that Gaussian measure into a partial signing. We give three approaches to obtain such a signing. The first one covers the cube by partially signed faces through Gaussian concentration with a constant $7\cdot10^9$. The second proof replaces the covering by a projection lemma with explicit parameters for a constant $156000$. The third proof turns Gaussian measure into signs by a lossless coding, with no union bound. It proves the estimate at the right radius with smooth spectral barriers and certified coefficients. It gives a constant below $7.88$. For algorithms, the main idea is to project Gaussian points onto a smoothed spectral body. The $n^{3+o(1)}$ time algorithm tracks the Gibbs matrix of that body across coordinate-descent steps with sketched increments and random refreshes.

We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.

The polynomial Hardy–Littlewood inequalities bound coefficient norms of homogeneous polynomials on $\ell_p$ balls by their supremum norms, with constants independent of the dimension. Their $p=\infty$ endpoint is the Bohnenblust–Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving $D_{m,p_m}\le(1+o(1))D_m$ whenever $p_m/m^2\to\infty$, where $D_{m,p}$ and $D_m$ are the optimal complex polynomial Hardy–Littlewood and Bohnenblust–Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, $D_{m,m}=m$. We then prove the full first-order asymptotic $D_{m,p_m}\sim m$ whenever $p_m\ge m$ and $p_m-m\to0$, and the exact polynomial growth $D_{m,p_m}=m^{1+o(1)}$ on the wider scale $p_m-m=o((m/\log m)^{1/3})$.

We determine the optimal exponential scale in Erdős's robust polynomial interpolation problem and identify the leading exponential coefficient for Chebyshev–Lobatto grids. Given $C>0$ and $0\le\rho<1$, put $H=C/(1-\rho)$. We prove that every configuration of $n\ge n_0$ nodes in $[-1,1]$ admits sign data for which every polynomial of degree at most $(1+\epsilon)n$ and uniform norm at most $C$ has error greater than $\rho$ at more than $\epsilon n$ nodes. One may choose $\epsilon\ge e^{-A(1+H)}$ and $n_0\le e^{A(1+H)}$, where $A$ is an absolute constant, and a matching upper bound shows that the exponential order is optimal. For the same obstruction restricted to Chebyshev–Lobatto grids, the optimal parameter $\epsilon_{\mathrm{Ch}}(C,\rho)$ satisfies $\log(1/\epsilon_{\mathrm{Ch}}(C,\rho))=\frac{\pi}{2}H+O(\log(H+1))$ as $H\to\infty$. The proofs combine finite interpolation estimates in Bernstein spaces with periodic sign constructions and averaging. We also establish obstructions for random signs and stability under angular perturbations of the nodes.

In this article, we show that every norm-increasing $3$-concave weighted shift on a directed tree admits Wold-type decomposition. We then classify the $4$-isometries within a class of weighted shifts on a rootless directed tree introduced by S. Chavan and S. Trivedi, and use this classification to construct analytic norm-increasing strict $4$-isometries without the wandering subspace property. This answers a question of S. Shimorin in the negative for every $m\in \mathbb{Z}_{\geqslant 4}$.

We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical \(L_p\)-spaces. Our main result is a sphere version of the classical Banach–Kadec–Paley theorem: for \(1\le p,q<\infty\), the unit sphere \(S_{L_q}\) admits an almost Lipschitz embedding into \(L_p\) if and only if \(L_q\) admits a linear embedding into \(L_p\). Consequently, the almost Lipschitz geometry of \(L_p\)-spheres completely determines the underlying linear structure of \(L_p\)-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton–Randrianarivony's concentration inequalities, Mendel–Naor's metric cotype inequality, and Naor's sharp metric \(X_p\) inequality. They also yield several sharp results on the Hölder geometry of \(L_p\)-spheres.

By constructing an appropriate quadratic form, we prove that if an analytic Toeplitz operator commutes with a co-analytic Toeplitz operator on the harmonic Bergman space, then at least one of the symbols is constant. This resolves an open question posed by Choe and Lee in 1999.

Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $\tau(T)=0$ for all $\tau\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal{A}$ has unique trace; $(2)$ $\mathcal{A}$ is separable and has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

For two-dimensional norms that generate rotationally-symmetric norms in all higher dimensions, necessary and sufficient conditions are established for them to be subspaces of $L_1$. \

For $a,b,c\in\mathbb R$, we consider the Forelli–Rudin operators $$ T_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{(1-z\overline w)^c}f(w)\,dA(w) $$ and their positive counterparts $$ S_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{|1-z\overline w|^c}f(w)\,dA(w). $$ We obtain a complete and sharp classification of their weak- and restricted weak-type mapping properties in the hypersingular regime $$ \Omega_{\mathcal H}:=\{(p,q):1\leq p,q\leq\infty,\ p>q\}, $$ thereby substantially extending the recent work of the first and fourth authors on hypersingular Bergman projections. One of the main discoveries of this work is an intrinsic cancellation phenomenon associated with the complex Forelli–Rudin kernel: at certain critical endpoints, cancellation creates a sharp separation between the two operators, with $T_{a,b,c}$ remaining bounded while its positive counterpart $S_{a,b,c}$ fails to be bounded. Perhaps surprisingly, this cancellation is invisible in the strong $L^p$–$L^q$ theory established by Zhao and Zhou in 2022, where the two operators have the same boundedness range, and emerges only at the weak- and restricted weak-type levels. Allowing the parameters $a,b,c$ to vary, we show that the collection of all weak-type Forelli–Rudin pairs in $\Omega_{\mathcal{H}}$ is precisely $$ \mathcal{FR}_w=\{(p,q)\in\Omega_{\mathcal H}:1<p\leq2\}, $$ whereas the collection of all restricted weak-type Forelli–Rudin pairs is $$ \mathcal{FR}_{rw}=\{(p,q)\in\Omega_{\mathcal H}:p\neq\infty\}. $$ These ranges, together with all corresponding endpoint estimates and failures, are sharp. Our approach combines dyadic decompositions, probabilistic constructions, and weak-type Hardy estimates.

We present an explicit construction of examples showing that the estimate $ \|S(f)\|_{L^2(w)\to L^{2,\infty}(w)}\lesssim \big([w]_{A_2} \log(1 + [w]_{A_2})\big)^{1/2}$ for the dyadic square function is sharp in terms of the characteristic $[w]_{A_2}$.

V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_\tau(X)$ is $C$-embedded in $P_\sigma(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(\beta X)$ if and only if $X$ is pseudocompact, where $P_\sigma$, $P_\tau$, and $P_R$ are the functors of probability $\sigma$-additive on the Baire $\sigma$-algebra, $\tau$-additive, and Radon measures on the space $X$? The answers to these questions are negative. However, if instead of probability measures we consider the corresponding alternating measures $M_\sigma$, $M_\tau$, and $M_R$, the situation changes. It is proved that (i) $M_\tau(X)$ is $C$-embedded in $M_\sigma(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(\beta X)$ if and only if $X$ is pseudocompact. The question of $C$-embedding of measure spaces is an extension of the question of coincidence of measure spaces, which is a development of the classical concepts of universally measurable and universal measure zero sets. A general theorem is obtained, which leads to the mentioned results.

We establish a constructive and quasi-Banach framework for approximation spaces $A_\mu^\rho$ of compact $H$-operators between Banach and quasi-Banach spaces. By leveraging delayed Riesz means $V_{2^{bn}}$ generated by self-adjoint differential operators $P(D)$, we replace abstract best approximants with explicit linear operator decompositions. Furthermore, we extend the theory of $H$-operator approximation spaces to quasi-Banach settings ($0 < p < 1$), bypassing the collapse of Peetre's $K$-functional by employing localized moduli of smoothness $\omega_\varphi^r(f,t)_p$. Integrating seminal spectral bounds due to Markus \cite{M1966} and factorization techniques for operator ideals, we prove that operators belonging to $A_\mu^\rho$ possess complete systems of root vectors whose expansions are Abel summable.

We develop applications of a Jensen-type inequality for pairs of positive measure spaces. The framework yields $L^p$ and operator inequalities for positive integral operators, norm and entropy estimates for Fejér means, and probabilistic moment and variance bounds. We also obtain robustness estimates under random and deterministic erasures. These results provide a unified connection between Jensen-type inequalities, harmonic analysis, operator theory, probability, and robust reconstruction.

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},\tau)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $\tau$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.

In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-1}\|_\mathrm{r} \leq 9+6\sqrt3$. The method also gives positive versions of classical nonmetrizable results for Miljutin and Dugundji compacta. In particular, if $G$ and $H$ are infinite compact groups of the same weight, then $C(G)$ and $C(H)$ are positively and regularly isomorphic with the same uniform quantitative estimate.

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $\kappa$-barrelled and $\omega$-barrelled groups. Our first main result establishes that every $\omega$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $\sigma$-compact, $\omega$-barrelled group, then its dual group $\widehat{G}_\kappa$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $\omega$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.