Papers

In this paper we obtain all doubly-stochastic solutions to the Yang-Baxter-like matrix equation $AXA = XAX$ when $A$ is a permutation matrix. We characterize the permutation solutions, proper doubly stochastic solutions, and we accompany our results with suitable examples.

Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of \(ab\) and \(ba\) differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain–Delbaen space \(X\) obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to \(C(S^4)\), we construct \(S,T\in\B(X\oplus X)\) such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, \(\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2\), which is the least possible stable rank for such an example. More generally, if \(K\) is compact metrisable and \(X\) is any space arising from Motakis' construction for \(C(K)\), then, for every \(n\geqslant1\), \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.

We develop a finite-corner framework for convex cones of normal maps between \(\mathcal B(\mathcal H)\) and \(\mathcal B(\mathcal K)\). The basic structural assumption is stability under finite cut–pad operations. We prove that every point-ultraweakly closed cut–pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut–pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.

In this note, we give an explicit description of countable products in $KK$- and $E$-theory and provide several applications.

In this article, we characterize when a Toeplitz operator and a Hankel operator on the vector-valued Hardy space are skew commutators of each other, and determine necessary and sufficient conditions under which their product is self-adjoint. These characterizations extend the results in \cite{LZD}. In addition, we completely classify the skew commutators of the unilateral shift $S$, its adjoint $S^*$ and $S\oplus S^*.$ We also characterize the class of bounded linear operators that have $S$, $S^*$ and $S\oplus S^*$ as skew-commutators.

We consider an alternative notion of minimality for C*-dynamical systems, which we refer to as strong minimality. This notion coincides with the usual definition of minimality in the commutative setting, but diverges sharply from it in the noncommutative setting in a way which we argue addresses several fundamental defects in the usual definition. We demonstrate that strong minimality is deeply connected with Rørdam's notion of a C*-irreducible inclusion, giving a new perspective on the study of such inclusions. We provide various characterizations of this property, including for reduced crossed products, tensor products, and subhomogeneous C*-algebras.

Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

We study the category of dual operator spaces equipped with an action of a locally compact quantum group $\mathbb{G}$. The Fubini crossed product functor $-\rtimes^\mathcal{F} \mathbb{G}$ and the weak$^*$-crossed product functor $-\bar{\rtimes}\mathbb{G}$ are shown to be equal if and only if $\mathbb{G}$ has the approximation property of Haagerup and Kraus. Using the natural isomorphism $-\rtimes^\mathcal{F}\mathbb{G}\cong {}_{L^1(\mathbb{G})}\mathcal{CB}(B(L^2(\mathbb{G}))_*, -)$, this leads to a characterization of the approximation property of $\mathbb{G}$ via an $L^1(\mathbb{G})$-module approximation property for $B(L^2(\mathbb{G}))_*$. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of $\mathbb{G}$.

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.

Let \(\Omega\) be a compact Hausdorff space, and let \(\cA\) be a unital \(C^*\)-algebra. In this study, we continue our examination of the comparison between \(C^*\)-extreme points and linear extremal structures of the unit ball, in the vector-valued \(C^*\)-algebra \(C(\Omega, \cA)\), building upon the work initiated in \cite{HR}. We first enlarge the class of $C^\ast$-algebras in which a $ C^\ast$-extreme point is an extreme point. We demonstrate that if \(\cA\) has a faithful tracial state, then any \(C^*\)-extreme point of the unit ball \(C(\Omega, \cA)_1\) is a unitary. Additionally, we identify a classes of \(C^*\)-algebras where the concepts of \(C^*\)-extreme and pointwise \(C^*\)-extreme points in \(C(\Omega, \cA)_1\) coincide. We show this holds if a von Neumann algebra has a separable predual with the Radon-Nikodým property.

We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of finitely generated projective graded modules, rather than from spaces of Fredholm operators or Kasparov cycles. We define a connective spectrum $k^{\mathrm{ABS}}_A$ refining the Atiyah–Bott–Shapiro construction as a cofiber, together with its periodification $K^{\mathrm{gr}}_A$, and show that both are lax symmetric monoidal and functorial in bimodules, not merely in homomorphisms. The failure of graded $A$-modules to present all cocycles in $K^{\mathrm{gr}}_0(A)$ is shown to be purely a $\pi_0$-phenomenon on $k^{\mathrm{ABS}}_A$. We obtain a natural equivalence $K^{\mathrm{gr}}_{A\widehat{\otimes} \mathrm{Cl}_{p,q}} \simeq \Sigma^{p-q}K^{\mathrm{gr}}_A$, which links topological Bott periodicity with the Morita equivalence between $\mathrm{Cl}_8$ and $\mathbb{R}$. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor $\operatorname{Pic}(\operatorname{Bim}(\mathrm{sBan}_{\mathbb{R}})^{\mathrm{fd}}) \to \operatorname{Pic}(\mathrm{Mod}(KO))$ which splits off the bottom three Postnikov layers of $\operatorname{Pic}(\mathrm{Mod}(KO))$, giving a direct link between super division algebras and invertible $KO$-modules. We also give spectral refinements of Karoubi's and van Daele's graded $K$-groups, with explicit comparison equivalences, therefore connecting to $KK$-theory. We also provide an extensive general treatment for $K$-theory of ungraded topological rings that might be of independent interest. In particular, we characterize connective topological $K$-theory of ungraded Banach algebras by a universal property.

This paper is about disjoint F-transitive operators on Hilbert C*-modules, where F is a finitely invariant Furstenberg family of subsets of N. We characterize disjoint F-transitivity for generalized bilateral weighted shift operators on the standard Hilbert module over C*-algebra of compact operators on the separable Hilbert space. Then we give a sufficient condition for these operators to be disjoint chaotic and give concrete examples. We also characterize disjoint F-transitivity for generalized translation operators on the non-unitary C*-algebra and provide some applications.

The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional $C^*$-algebras throughout, modeling finite quantum systems that may have superselection sectors.

In this paper, we present a model for classical and quantum reference frames for $C^*$-algebras by utilizing Hilbert modules and equivariant correspondences of $C^*$-dynamical systems. We show the classical model provides a commutative baseline for the quantum model, and we discuss how the quantum model encodes the Hilbert space operational formulation of quantum reference frames. We also include various explicit examples for both classical and quantum reference frames.

An electrical network can be summarized at its boundary by a response matrix: prescribed boundary voltages determine boundary currents. A superport network adds a constraint by grouping boundary terminals into superports, requiring the total current in each group to be zero and making voltage differences inside the groups the natural coordinates. Earlier work determined forest formulas for a single response entry and for the determinant of the whole response matrix. The missing case was an arbitrary subdeterminant, or minor: one needs to know not only which spanning forests contribute, but also the sign carried by each forest. This paper supplies that sign rule. After choosing one reference vertex in each superport, the response is \[ L=\left(D^{T}K^{-1}D\right)^{-1}, \] with \(K\) the grounded weighted Laplacian and \(D\) recording the selected voltage differences. Contracting the components of a physical spanning forest \(F\) produces a much smaller quotient port graph \(H_F\). Its reduced incidence matrix \[ B_F=Q_FD \] has columns only of the forms \[ 0,\qquad \pm e_a,\qquad e_a-e_b. \] Hence every square incidence minor is exactly \(0\) or \(\pm1\). For a \(k\)-set of response coordinates \(I\), append to \(B_F\) the selector rows \(E_I^{T}\) and define the completed quotient-incidence determinant \[ \widehat{\chi}_F(I) = \det \begin{pmatrix} B_F\\ E_I^{T} \end{pmatrix}. \] For coordinate sets \(I,J\) of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply \[ \widehat{\chi}_F(I)\widehat{\chi}_F(J). \] Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives \[ \widehat{\chi}_F(I)\in\{0,\pm1\}, \] with nonvanishing occurring exactly when the complementary quotient edges \(N\setminus I\) form a spanning tree of \(H_F\).

We show that a superselction sector describing a single nonabelian anyon violates Haag duality, which is equivalent to the fundamental quantum-information theoretic principle of uniqueness of purifications. We provide a simple physics argument as well as a rigorous proof using sector theory, and instantiate our result concretely in Levin-Wen models. We also show that the associated ground state does not allow for quantum steering despite being a pure state. Finally, we show that this ground state violates approximate Haag duality, disproving the conjecture that all gapped ground states satisfy this condition. Our result implies that there are gapped phases of matter where Haag duality fails at every point in the phase.

We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.

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 the $K$-theory classes of higher Kazhdan projections of a discrete group under a spectral-gap assumption and recall how they are obtained, via the Baum–Connes assembly map, from equivariant Euler classes. After reviewing the case of virtually free groups, where the universal space for proper actions has a one-dimensional model, we prove the main new result: an explicit description of the $K$-theory class of the first higher Kazhdan projection for finitely generated infinite one-relator groups whose first Laplacian has a spectral gap. If moreover the one-relator group is hyperbolic, this also gives corresponding formulas for delocalised $\ell^2$-Betti numbers.

Which vertex-transitive graphs of prime order have quantum symmetry? The question of Banica, Bichon and Chenevier is open in the dense regime of Paley graphs, where coherent-algebra methods give no information. To each such graph we attach a two-basepoint Terwilliger algebra of its cyclotomic scheme and study the module it generates from the basepoints: fullness forces the quantum permutation algebra to be commutative, and the module admits no intermediate state, containing either exactly two point masses or all $p$ of them. One point mass, captured at any depth, therefore suffices, and Chassaniol's orbital criterion is the depth-one case. Three consequences follow. A sharp counting argument replaces the Banica–Bichon–Chenevier threshold $p>6^{\varphi(k)}$ by the quadratic bound $p>(k-1)(k-2)+2$, where $k$ is the type. Four certificates, each a short list of additions modulo $p$, settle $C_{31}(2,4,8,15)$ and $C_{41}(4,10,16,18)$, the two graphs left open by Chassaniol, and complete the classification for type at most $10$ without machine assistance. An exact computation extends the dichotomy “quantum symmetry if and only if complete or empty” to all prime orders $p\le250$, settling the Paley graphs $P_{p}$ with $p\le241$, the first beyond $P_{17}$. What remains is the capture of a single explicit vector: the midpoint $2^{-1}$ of the two basepoints.

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.

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 show that the Hao-Ng isomorphism for full crossed products fails to hold. More precisely, for a non-degenerate $C^*$-correspondence $X$ over a $C^*$-algebra $\mathcal{A}$ and a generalized gauge action $G \curvearrowright X$ by a locally compact group $G$, we isolate three structural obstructions for the canonical commutation $\mathcal{O}_X\rtimes G \cong \mathcal{O}_{X\rtimes G}$ between full crossed product and Cuntz-Pimsner $C^*$-algebra. The first concerns hyperrigidity, the second concerns WEP of the coefficient $C^*$-algebra $\mathcal{A}$ for trivial actions, and the third concerns the coincidence of full and reduced crossed products by the action of $G$ on $\mathcal{A}$.

In this paper, we consider the Rosenthal – Halmos problem of almost commuting matrices with respect to the normalized Hilbert – Schmidt norm $\|\cdot\|_{2,d}=d^{-1/2}\|\cdot\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\|X\|\le 1$, $\|Y\|\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\|X-X'\|_{2,d}+\|Y-Y'\|_{2,d}\le 5\|[X,Y]\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.

Problem 8.4 of Pino, Goodearl, Perera and Molina in [arxiv.org/abs/0806.4156] asks whether a dense subalgebra $A_{0}$ of a $C^*$-algebra $A$ that is purely infinite as a ring forces $A$ to be purely infinite as a $C^*$-algebra. We call this the $\textit{pure infiniteness transfer problem}$, the transfer being from the ring to its $C^*$-completion. The problem is open, even when $A_{0}$ is unital and simple. We settle it under two hypotheses: $A$ has real rank zero, and $A_{0}$ is closed under holomorphic functional calculus. Under these hypotheses $A_{0}$ is purely infinite simple as a ring if and only if $A$ is purely infinite simple as a $C^*$-algebra. We also prove an obstruction: a unital $C^*$-algebra with a nonzero finite projection has no dense hfc-closed purely infinite simple unital subring. Finally, the hypotheses hold for proper subalgebras, for the gauge action of $\mathbb{T}$ on a Cuntz algebra $\mathcal{O}_n$ , the smooth subalgebra $\mathcal{O}_n^\infty$ is a proper dense hfc-closed subalgebra that is purely infinite simple as a ring.

In this paper we continue the investigation of generalizations of the strong law of large numbers to operator means which has been initiated in the earlier paper "Strong law of large numbers for generalized operator means", Adv. Math. 457 (2024), 109933 of the authors. In particular, we clarify an inaccuracy in the statement and proof of the strong law for means associated with uniformly exponentially contracting flows in Thompson metric spaces. We then significantly improve the variant of this strong law given in the mentioned paper for operator means by relaxing the restrictive required moment condition to $L\log^+(L)$ by proving almost sure convergence of implicit stochastic approximations for monotone operators under this condition.

Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce \(\mathcal R\)-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra \(\mathcal R\). Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak–Kasprzak–Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.

Under mild assumptions, the main results of this paper characterize a (real or complex) symmetric operator space affiliated with a semi-finite atomless von Neumann algebra $(\mathcal M, \tau)$ possessing the Daugavet property as $L_1(\mathcal M,\tau)$ or $\mathcal M$ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of \cite{AKM12, KMMW13, AKM15} concerning the Daugavet property for real symmetric function spaces.