Arnaud Brothier, Aidan Sims, Dilshan Wijesena
We develop new techniques for the construction and classification of representations of row-finite and locally convex higher-rank graph C*-algebras O. This class includes Cuntz--Krieger algebras associated to row-finite directed graphs. Our approach relies on the representation theory of a certain non-self-adjoint algebra and a lifting process of representations. We introduce a novel dimension vector for representations of O yielding a countable partition of the spectrum. Given a Cuntz--Krieger algebra and a finite dimension vector, we construct a smooth manifold parametrising the corresponding spectral component. Our techniques are both explicit and functorial.
Tomasz Kania
We develop a unified framework for locating natural properties of algebraic and analytic structures within the Borel hierarchy. Objects are presented as quotients of a universal generator and definability is read directly from the quotient data. For separable Banach-type structures (Banach algebras, $C^*$-algebras, Banach lattices, TROs) the kernel space is Polish under the Wijsman topology, and the quotient-norm functional $K\mapsto \|x+K\|$ is continuous, yielding a uniform definability scheme whose Borel ranks are bounded by quantifier alternation depth. For countable algebraic structures (groups, rings, lattices) we work on compact Polish spaces of congruences where atomic predicates are clopen. We obtain explicit Borel upper bounds: in the \emph{unital} $C^*$-algebra coding based on $C^*_{\max}(F_\infty)$, stable finiteness is closed, nuclearity is Borel, simplicity is~$G_δ$, AF-ness lies in~$Π^0_3$, nuclear dimension~$\le n$ lies in~$Π^0_3$, and for fixed exact~$D$, $D$-absorption is analytic. For countable groups, soficity is~$G_δ$; for abelian groups, slenderness is~$Π^0_3$. We give an internal Borel coding of the $K_0$-assignment in the quotient/Wijsman framework; for each fixed coordinate the corresponding section is $F_σ$, and suspension together with Bott periodicity yields Borel codings of all higher $K$-groups. We also show that several bounds are optimal ($Σ^0_2$- and $Π^0_2$-complete). To calibrate the method's reach, we exhibit a $Π^1_1$-complete property (separable dual in the commutative $C^*$-setting), provably outside the Borel hierarchy.
Jintao Deng, Ryo Toyota
Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$.
Raphael Ponge
Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It's only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly open the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this memoire are illustrated by semiclassical Weyl's laws and integration formulas in the following settings: (i) Dirichlet and Neumann problems on Euclidean domains with smooth boundaries; (ii) closed Riemannian manifolds; (iii) open manifolds with conformally cusp metrics of finite volume; (iv) quantum tori; and (v) sub-Riemannian manifolds.
Jacek Krajczok, Piotr. M. Sołtan
We continue the study of the braided compact quantum group $\mathrm{SU}_q(2)$ for complex $q$ satisfying $0<|q|<1$ introduced by Kasprzak, Meyer, Roy and Woronowicz (J. Noncommut. Geom. 10(4):1611-1625, 2016). We address such aspects as existence of the Haar measure, construct the scaling group, the antipode and its polar decomposition and describe the related braided Hopf algebra. We also study when the braided flip extends to a completely bounded map and establish equivalence between the two approaches to bosonization and braided tensor product taken in the literature (Kasprzak, Meyer, Roy, Woronowicz J. Noncommut. Geom. 10(4):1611-1625, 2016 vs. Meyer, Roy Woronowicz Internat. J. Math. 25(2):1450019, 37, 2014, Roy Int. Math. Res. Not. (14):11791--11828, 2023 and De Commer, Krajczok arXiv:2412.17444, to appear in J. Operator Th.).
David Muñoz-Lahoz
Let $X$ be a Dedekind complete Banach lattice, and let $P\colon X\to X$ be a positive projection for which the largest central operator below $P$ is $α\operatorname{id}_X$, for some $α\ge 0$. Wickstead conjectured that $α$ must either be $0$ or $1/n$, for some $n \in \mathbb{N}$, and proved it for finite-dimensional $X$. In this paper, we show that the conjecture holds in general. As a consequence, we settle the representation problem for Banach lattice algebras: we show that there exist Banach lattice algebras of dimension $2$ that do not admit a faithful representation as regular operators on any Dedekind complete Banach lattice.
David Gao, Srivatsav Kunnawalkam Elayavalli
We prove a new "Toeplitz exactness" theorem for strong convergence. This is a machine to upgrade strong convergence in the general setting of $C^\ast$-correspondences, and has several applications.
Jan Kochanowski, Omar Fawzi, Cambyse Rouzé
Apr 15, 2026·quant-ph·PDF We define 2-indexed $(q,p)$-Schatten quasi-norms for any $q,p > 0$ on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that $|\frac{1}{q} - \frac{1}{p}| \leq 1$. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the $q \to p$ completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for $q \geq p>0$, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for $α\geq \frac{1}{2}$ or the Sandwiched Rényi Umlaut information for $α< 1$. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-$α$-entropy for $α\geq\frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-$α$ entropy for $α\geq \frac{1}{2}$.
Shuhei Tsujie, Yuki Ueda
We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we establish $t$-deformed analogues of the law of large numbers and the central limit theorem, revealing structural parallels with classical, free, and finite free probability theories. We show that the case $t=-1$ recovers classical convolution at the level of moment generating functions, thereby connecting the theory directly to classical probability. We further investigate the infinitesimal generators associated with $\boxplus^t$-continuous semigroups, deriving explicit representation formulas that clarify how these generators describe the infinitesimal evolution of the semigroup. In the case $t = d$, our results yield explicit formulas for finite free infinitesimal generators. In the case $t = -1$, we relate these generators to those of one-dimensional Lévy processes by identifying the corresponding terms in their representations. This establishes a direct connection between $\boxplus^t$-convolution semigroups and classical Lévy-Khintchine-type generators.
Lei Li, Siyu Liu, Antonio M. Peralta
We prove that a Le Page-type inequality is also valid for metrically characterizing those JB$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists $γ>0$ satisfying $$\big\|\{a,b,\{x,y,z\}\}\big\|\leq γ\ \! \big\|\{x,y,\{a,b,z\}\}\big\|, \hbox{ for all } a,b,x,y,z\in E.$$
Ralf Meyer, George Nadareishvili
We prove a Universal Coefficient Theorem for objects in the bootstrap class in the equivariant Kasparov category for a finite cyclic group of square-free order.
Tianyi Lou
We study the simplicity of $C^{*}$-algebras built from group actions. For a faithful isometric action of a group $G$ on a countable metric space $X$, we use the associated action representation on $\ell^2(X)$ to define the action-based $C^{*}$-algebra $C^{*}_{X}G$. We define generalized versions of the properties $P_{\text{naive}}$ and $P_{\text{analytic}}$ relative to the action and show that the naive form implies the analytic form. We also prove that the properties $P_{\text{analytic}}$ associated with a continuous action ensure the simplicity of the action-based $C^*$-algebra. As an application, we deduce that big mapping class groups satisfy the property $P_{\text{naive}}^{\mathbb{X}}$ and the associated action-based $C^*$-algebra is simple.
Malik Amir
We present an expanded expository account of the $K$-moment problem for polynomial algebras over \(\R^d\), with special emphasis on compact basic closed semialgebraic sets. The central question is to characterize those linear functionals on \(\R[x_1,\dots,x_d]\) which admit representation by integration against a positive Radon measure supported on a prescribed set \(K\subseteq\R^d\). We begin with the classical background and with Haviland's formulation of the multidimensional moment problem, then explain how real algebraic geometry enters through quadratic modules, preorderings, and Positivstellensätze. The compact case is treated in detail from two complementary perspectives. The geometric route through Schmüdgen's theorem and the operator-theoretic route through a Gelfand--Naimark--Segal construction and the spectral theorem. We also discuss Putinar's refinement, compare the roles of \(T(f)\) and \(Q(f)\), and explain how Archimedeanity provides the algebraic shadow of compactness. In order to place the subject in a broader context, we survey determinacy and uniqueness questions, the truncated \(K\)-moment problem and flat extension phenomena, the relation with sums of squares and Hilbert's seventeenth problem, and the special case of algebraic varieties, where positivity modulo an ideal becomes especially transparent.
David Jekel
We study entropy and optimal transport theory in the free probabilistic setting motivated by the large-$n$ theory of random tuples of matrices. We define a new version of free entropy $χ_{\operatorname{chron}}^{\mathcal{U}}$, which is concave along geodesics in the corresponding Wasserstein space. Moreover, the heat evolution satisfies the evolution variational inequality, which means that the heat evolution is the Wasserstein gradient flow for entropy in the metric sense. It also has further desirable properties such as a chain rule for iterated conditioning, and an expression in terms of stochastic control problems. This entropy is defined using microstate spaces of matrix approximations with respect to an expanded class of test functions called chronological formulas, which are constructed so as to be closed under taking partial suprema and infima and application of a free heat semigroup. These formulas are part of a novel framework for studying noncommutative filtrations and stochastic processes as metric structures in the sense of continuous model theory.
Tomasz Kania, Mariusz Niwiński
We introduce a norm-controlled notion of semiprojectivity for Banach lattices, requiring liftability of contractive lattice homomorphisms through inductive limits of closed ideals with arbitrarily small loss of norm control. Our main result establishes that, for a compact metric space $X$, the Banach lattice $C(X)$ is semiprojective if and only if $X$ is an absolute neighbourhood retract. Notably, this characterisation is strictly more permissive than its $C^*$-algebraic counterpart: by a theorem of S\orensen and Thiel, $C(X)$ is semiprojective in the category of $C^*$-algebras and $*$-homomorphisms if and only if $X$ is an ANR of dimension at most one. The dimensional obstruction disappears in the Banach-lattice setting because lattice homomorphisms between $C(K)$-spaces are automatically weighted composition operators, and therefore no commutation relations need to be lifted. We also show that uncountable $\ell_1$-sums of $1^+$-projective Banach lattices with topological order units are semiprojective but need not be $1^+$-projective, establishing that the two notions are genuinely distinct. On the negative side, we prove that $\ell_p$ and $L_p([0,1])$ for $p \in (1,\infty)$ as well as Orlicz spaces are not semiprojective.
Clément Cren, Jean-Marie Lescure, Omar Mohsen
We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a Heisenberg group structure on the fibers. This generalizes an approach due to Guillemin and Melrose in the symplectic case.
M. H. M. Rashid
This manuscript presents a systematic study of Calkin algebras -- the quotients $\mathcal{L}(X)/\mathcal{K}(X)$ of bounded operators modulo compact operators on a Banach space $X$ -- and establishes a framework for realizing commutative $C^*$-algebras as such quotients while preserving geometric and topological information. Building on Motakis's reflexive version of the Bourgain--Delbaen construction, we prove that for every compact metric space $K$, there exists a reflexive Banach space $\mathfrak{X}_{C(K)}$ whose Calkin algebra is isomorphic to $C(K)$ as a Banach algebra. Our contributions advance this result in several directions: we establish stability under finite products, enabling the realization of finite direct sums of $C(K)$ spaces and matrix algebras $M_m(C(K))$ as Calkin algebras; we prove a localization principle showing compact operators on $\mathfrak{X}_{C(K)}$ can be approximated by finite-rank operators whose support respects the metric structure of $K$; we demonstrate that the diagonal function $\varphi_T\colon K\to\mathbb{C}$ of any bounded operator $T$ is Hölder continuous with optimal exponent $1/2$, revealing a deep analytic constraint; we prove a rigidity theorem showing the Banach algebra structure of $\mathcal{L}(\mathfrak{X}_{C(K)})$ completely determines the topology of $K$, extending the classical Banach--Stone theorem; we classify all closed two-sided ideals and prime ideals in $\mathcal{L}(\mathfrak{X}_{C(K)})$ in terms of open subsets and points of $K$; and we resolve longstanding problems, notably by constructing the first reflexive Banach spaces with infinite-dimensional reflexive Calkin algebras. These results forge a deep connection between Banach space geometry, operator algebras, and topological invariants, revealing how Calkin algebras can be precisely engineered through the geometry of their underlying spaces.
Kirill Kartvelishvili
We study two notions of largeness for closed submodules of Hilbert C*-modules: essentiality and topological essentiality. While the analogous properties are known to be equivalent for closed two-sided ideals of C*-algebras, the one-sided case is more subtle. We prove that these two notions remain equivalent for closed right ideals of an arbitrary C*-algebra. Next, using the correspondence between submodules and right ideals of the algebra of compact operators, we extend this result to closed submodules of Hilbert C*-modules. In the commutative case, where a Hilbert module can be realized as a continuous field of Hilbert spaces, we give a geometric reformulation of essentiality and derive a fiberwise criterion.
Dawn Archey, Julian Buck, Javad Mohammadkarimi, N. Christopher Phillips, Apurva Seth
We establish comparison and divisibility properties for crossed product C*-algebras arising from automorphisms of algebras C (X, D) which lie over minimal homeomorphisms, from actions of compact groups which have finite Rokhlin dimension with commuting towers, and from actions of compact groups which have the restricted tracial Rokhlin property with comparison. We deduce that these crossed products we consider are pure, and conclude they have stable rank one, and in certain cases have real rank zero. We give examples in which these properties do not follow from previous results, in the case of C (X, D) due to the lack of Z-stability of D, the underlying topological spaces not being finite dimensional, or both.
Shreema Subhash Bhatt, Vinay Deshpande, Bipul Saurabh
In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(SO_0(4)/SO_0(2))$. We show that all four orbits of the unit space $\mathcal{G}_{\mathrm{tight}}^{(0)}$ under the natural action of $\mathcal{G}_{\mathrm{tight}}$ are locally closed, and that the associated isotropy groups are isomorphic to $\mathbb{Z}$. Consequently, every irreducible representation of $C^*(\mathcal{G}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^*(\mathbb{Z})$, which are parametrized by $\mathbb{T}$. In this way, we obtain four families of irreducible representations parametrized by $\mathbb{T}$, and we explicitly construct their equivalence with the corresponding Soibelman irreducible representations of $C(SO_q(4)/SO_q(2))$.