Papers

The usual dual of an infinite-dimensional order-unit space need not carry an order unit, and this is the main reason why duality for operator systems is notoriously hard. Rather than trying to compute a canonical dual object, we introduce a relative notion of duality for operator systems: Two systems are in duality when a pairing of their underlying vector spaces induces conic pairings at every matrix level. We develop basic properties of this notion, showing that beyond finite-dimensional operator systems, also separable $B(H)$ and $C^*$-algebras with a faithful trace admit duals, while some other operator systems do not. We prove that subsystems, quotients and suitable tensor products of systems with duals again admit duals. As consequences, we show that finite-level maximality is dual to finite-level minimality, and we establish weakly continuous and weak$^*$ closed realizations into $\ell^\infty$-products of matrix algebras.

Towards a question of Voiculescu, two notions of $1$-bounded entropy, $h$ and $h^\text{top}$, are defined for $C^*$-algebras. The quantity $h$ involves the tracial completion with respect to all traces while the quantity $h^\text{top}$ depends on operator norm microstates. It is demonstrated that $h^\text{top}(\mathscr{A})$ does not exceed $h(\mathscr{A})$ for all $C^*$-algebras $\mathscr{A}$. Moreover, a variational principle enabling the computation of $h$ is introduced. The quantity $h$ is computed in various examples, and the computation is used to justify structural conclusions such as $C^*$-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.

We show that if M is a full II_infinity factor such that the fundamental group of a finite corner is R and Out(M) is a locally compact second countable group, then M admits either a unique continuous trace scaling flow or a continuum of pairwise distinct ones, up to cocycle conjugacy. Equivalently, there is either a unique III_1 factor or a continuum of pairwise non-isomorphic III_1 factors with continuous core M. We apply this dichotomy result to factors constructed in [Dep12] and [Cha25] where Out(M) is a Lie group. We also give two independent constructions of full factors admitting a continuum of pairwise distinct trace-scaling flows where the outer automorphism group is unknown.

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local “Hadamard equivalence” game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak–Sołtan–Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

Let $M$ be a $\mathrm{II}_1$ factor, $N$ a tracial von Neumann algebra, and $\Phi: M \rightarrow N$ a subtracial completely positive map. For an irreducible $\mathrm{II}_1$ subfactor $P \subseteq M$, we characterize when $\Phi$ exhibits a flattening property under conjugation by unitaries in $P$. To be specific, we show that the failure of a Pimsner-Popa type inequality for $E_P \circ \Phi^* \circ \Phi$ is the precise obstruction, equivalently characterized by left weak mixing of a naturally associated $P$-$N$ bimodule. As an application, we obtain an asymptotic orthogonalization result generalizing a result of Popa.

We show that an inductive AH-system with diagonal connecting maps describes an action of the canonical AF-groupoid on the spectrum of the canonical C$^*$-diagonal, and that the canonical groupoid model is given by the transformation groupoid associated to this action. This divides the groupoid structure into two distinct aspects: the acting AF-groupoid (which is well-studied) and the unit space on which it acts. We describe the unit space by enriching the Bratteli diagram with extra topological information, and apply these descriptions to examples of interest, including Villadsen algebras of the first kind.

We complete the construction of tracial spectral nonassociative $\mathrm{L}^p$-spaces for general $\mathrm{JBW}^*$-algebras. More precisely, if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra equipped with a normal finite faithful trace $\tau$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (\tau[(x^* \circ x)^{\frac p2}])^{\frac1p}$, where $x \in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for $\mathrm{JW}^*$-algebras. The main difficulty is the complexified Albert algebra $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. To treat this case, we establish a Jordan analogue of the joint convexity of the Kiefer map $\mathrm{M}_n \times \mathrm{H}_n^{++} \to \mathrm{H}_n^{+}$, $(a,h) \mapsto a^*h^{-1}a$, where $\mathrm{H}_n$ is the space of Hermitian matrices, and a Jordan version of a variational formula of Carlen and Lieb. This provides a complex Banach space framework for Jordan-algebraic models arising in some generalized probabilistic theories.

Channel–state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the opposite algebra of $\mathcal{N}$, or equivalently with a commutant; returning to $\mathcal{N}$ itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi–Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is $*$-anti-isomorphic to itself. Naturality cannot be dropped: whenever $\mathcal{M}$ is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.

It is shown that there is a simple unital separable AH algebra which is not singly generated.

We prove that every PPT linear map has finite entanglement breaking index, thereby establishing the eventual entanglement breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly contains the class of 2-superpositive maps, has entanglement breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide evidence that the PPT-cubed conjecture may hold in full generality.

We prove dynamical comparison for Deaconu–Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C^*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen–Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable “large subgroupoids” and establish comparison properties for these using the paradoxical towers technique developed by Gardella–Geffen–Kranz–Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.

In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian–Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian–Pask algebras, and also the class of crossed product Kumjian–Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.

We define two notions of Morita equivalence for graded C*-algebras (graded Morita equivalence and homogeneous Morita equivalence) and provide Brown-Green-Rieffel Stabilization Type Theorems for both notions of graded equivalence. We apply our results to finite regular graphs by establishing an explicit connection between graded C*-algebras and coactions. Lastly, we incorporate Cartan subalgebras with totally disconnected spectra and obtain Brown-Green-Rieffel Stabilization Type Theorems for these cases.

Under a growth condition on an increasing function $g$, we prove that integrability of a freely infinitely divisible distribution with respect to $g$ is equivalent to that of the large-jump part of its free Lévy measure. For every increasing freely submultiplicative function $g$, integrability of the distribution implies integrability of the large-jump part of its free Lévy measure. We also obtain two-sided tail comparisons between freely infinitely divisible distributions and their free Lévy measures, uniform integrability along free convolution semigroups, and integrability results and tail estimates for fractional free convolution powers.

We show that $C^*$-algebras with complexity rank one have real rank zero, as an application, the uniform Roe algebra $C_u^*|\mathbb{Z}|$ has real rank zero.

Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.

We give a complete characterization of Cartan inclusions of finite dimensional $C^*$-algebras in terms of their inclusion matrices. More precisely, for a unital inclusion $\mathcal{B}\subseteq\mathcal{A}$ with inclusion matrix $\Lambda=(\Lambda_{ij})$, where Cartan means that $\mathcal{B}$ is a generalised Cartan subalgebra of $\mathcal{A}$ in the sense of Exel, we prove that the inclusion is Cartan if and only if \[ \sum_i \Lambda_{ij}\leq 1 \] for every $j$. We call matrices satisfying this condition multiplicity free. Thus, our characterization provides a purely combinatorial criterion for determining when a finite dimensional inclusion is Cartan. We further prove that every Cartan inclusion admits a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$. Conversely, we show that, for unital inclusions of finite dimensional $C^*$-algebras, the uniqueness of the conditional expectation is sufficient for the inclusion to be Cartan. Consequently, a unital inclusion $\mathcal{B}\subseteq\mathcal{A}$ of finite dimensional $C^*$-algebras is Cartan if and only if there exists a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras $A$ and $B$, are tomographically complete, purifications of a state on $A$ need not be related by unitaries in $B$. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant $B'$ is strictly larger than $A$. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on $A$ by measurements on $B$ is equivalent to the existence of a state-preserving conditional expectation from $B'$ onto $A$. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from $A$ to $B'$. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion $A\hookrightarrow B'$, a left inverse is precisely a conditional expectation, yielding the characterization above.

We introduce and study a notion of decomposition of a C$^*$-algebra over a Coxeter system based on Tits' definition of a $W$-distance. When such a decomposition comes with suitable conditional expectations, we build an associated Fock Hilbert module and reduced C$^*$-algebra $A^r$. This unifies constructions of Voiculescu [Voi85] and Caspers–Fima [CF17] and provides a noncommutative analogue of the situation of a discrete group $G$ acting on a building, in which case $A^r \cong C^*_r(G)$. We construct covariance C$^*$-algebras $\mathscr{C}(i)\supset A^r$ as a noncommutative analogue of the crossed product $C(\Omega)\rtimes_r G \supset C^*_r(G) $ where $\Omega$ is Caprace and Lécureux's minimal combinatorial compactification [CL11] of the locally finite building. Following the approach of Hasegawa [Has17] and Klisse [Kli25], we prove a universal property for the covariance algebras. This structural result yields different approximation properties, some of which are new even for the group case.

We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein–Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick et al(\cite{douglus_plosker_ramsey_povmintegration_1}).

We introduce a new amenability property for étale groupoids, termed fiberwise amenability, along with a stronger variant termed ubiquitous fiberwise amenability. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on étale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for étale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a Følner–paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.

Given a finite group G, we develop a generalization of the fundamental classification of Rokhlin G-actions on C*-algebras to the setting of anomalous G-actions with the Rokhlin property. For Rokhlin G-kernels on C*-algebras covered by the classification program, this implies that the induced G-action on some known invariants determines the conjugacy class. Extending a result of Izumi, we show that G-kernels with the Rokhlin property on Kirchberg algebras are classified by their anomaly and the induced module structure on the K-theory groups. We explore K-theoretic obstructions for the existence of Rokhlin G-kernels on general separable C*-algebras and use these to characterise which G-module structures arise as K-groups of Kirchberg algebras admitting a Rokhlin G-kernel with a given lifting obstruction.

Let $A$ be a unital $C^*$-algebra and $I$ be a symmetrically normed ideal of $A$. I introduce and study lexicographic functional calculus (LFC), a kind of multivariate functional calculus for tuples $(a_1,\ldots,a_m)$ of noncommuting self-adjoint elements of $A$ ”acting in lexicographic order,” i.e., from left to right, with an element $b_i \in I$ ”inserted” between the action of $a_i$ and $a_{i+1}$ for each $i=1,\ldots,m-1$. The reason for its introduction is an application to ”functional calculus calculus,” the differential calculus of maps induced by single-variate (continuous) functional calculus. Specifically, I prove that if $f\colon\mathbb{R}\to\mathbb{C}$ is sufficiently regular and $a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$, then $f_{a,I}(b):=f(a+b)-f(a)\in I$ for all $b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$, the map $f_{a,I}\colon I_{\mathrm{sa}}\to I$ is Fréchet $C^k$, and the $k^{\text{th}}$ Fréchet derivative of $f_{a,I}$ may be written in terms of LFC applied to the $k^{\text{th}}$ divided difference of $f$, a function of $k+1$ variables. This result recovers or vastly generalizes nearly all comparable results in the literature. For example, it simultaneously recovers the following three highly related results on the regularity of the function $f_A\colon A_{\mathrm{sa}}\to A$ defined by $a\mapsto f(a)$: (1) If $A$ is commutative and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; (2) if $A$ is finite dimensional and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; and (3) if $f\colon\mathbb{R}\to\mathbb{C}$ is ”slightly better than $C^k$,” e.g., belongs to the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$, then $f_A$ is Fréchet $C^k$ no matter the choice of $A$. Before LFC, there was no unifying framework for results (1)–(3); in particular, there was no single result of which they were all corollaries.

We investigate the relationship between the tame factorization property, denoted by $\mathfrak{TF}$ and introduced in the companion paper \cite{CDI}, and the DN-$\Omega$ type linear topological invariants of Fréchet spaces. Combining the basic properties of $\mathfrak{TF}$ with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess $\mathfrak{TF}$. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing $\mathfrak{TF}$ none of whose individual pairs are tame. We then investigate triples consisting of an arbitrary Fréchet space $X$, a nuclear Fréchet space $Y$ satisfying the properties $\underline{DN}$ and $\Omega$, and a power series space of finite type $\Lambda_1(\mathcal{E})$ or infinite type $\Lambda_\infty(\mathcal{E})$. We show that requiring such a triple to possess the tame factorization property $\mathfrak{TF}$ characterizes the corresponding linear topological invariants of $X$; in some cases this holds without any restriction on $Y$, while in others it requires the coincidence of the approximate diametral dimension of $Y$ with that of $\Lambda_1(\mathcal{E})$ or $\Lambda_\infty(\mathcal{E})$.

We prove that every countable inner amenable group has vanishing $\ell^2$-Betti numbers in all degrees. This was previously open in every degree $n\geq2$.

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

The quantum-mechanical two-sided Bogoliubov inequality provides upper and lower bounds for the free energy required to separate a system of interacting particles into independent subsystems. The bounds can be calculated straightforwardly from the ensemble average of the interface energy, bypassing the direct evaluation of the free energy. In this work, we generalize the two-sided Bogoliubov inequality to arbitrary von Neumann algebras by employing the Araki-Uhlmann relative entropy and the framework of unbounded perturbation theory of KMS states. Furthermore, we obtain variational expressions for the relative free energy that extend existing bounded-perturbation principles to the unbounded setting. Crucially, these mathematical developments yield a physically well-founded thermodynamic criterion for the quantification of entanglement in infinite-dimensional systems.