14,420 papers in this slice of arXiv.
Sang-Jun Park
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 containing the class of 2-superpositive maps, has entanglement-breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide strong evidence that the PPT-cubed conjecture may hold in full generality.
Shirly Geffen, Shanshan Hua, Julian Kranz
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.
Roozbeh Hazrat, Huanhuan Li, Promit Mukherjee
In this paper, we explore the idea that the graded Grothendieck group K0gr, 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.
John Quigg, Efren Ruiz, Aidan Sims
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.
Yu Kitagawa
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.
Qingnan An, Zhichao Liu
We show that C∗-algebras with complexity rank one have real rank zero, as an application, the uniform Roe algebra Cu∗∣Z∣ has real rank zero.
Mario Klisse
Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C∗-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak∗-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≥2 the canonical spectral triple of the Lamplighter group (Z/2Z)≀Fd, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.
Indrajit Ghosh, Sumit Kumar
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 B⊆A with inclusion matrix Λ=(Λij), where Cartan means that B is a generalised Cartan subalgebra of A in the sense of Exel, we prove that the inclusion is Cartan if and only if i∑Λij≤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 A onto 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 B⊆A of finite dimensional C∗-algebras is Cartan if and only if there exists a unique conditional expectation from A onto B.
Tattwamasi Amrutam
A discrete group Γ has the C∗--ISR property if every unital Γ--invariant C∗--subalgebra A⊆Cr∗(Γ)
Lauritz van Luijk, Amine Marrakchi, Tobias Osborne +2
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↪B′, a left inverse is precisely a conditional expectation, yielding the characterization above.
Arghya Chongdar
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 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(douglus_plosker_ramsey_povmintegration₁).
Xin Ma, Jianchao Wu
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.
Sergio Girón Pacheco, Gábor Szabó
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.
Thibaut Lescure
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 Ar. 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 Ar≅Cr∗(G). We construct covariance C∗-algebras C(i)⊃Ar as a noncommutative analogue of the crossed product C(Ω)⋊rG⊃Cr∗(G) where Ω 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.
Evangelos A. Nikitopoulos
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 (a1,…,am) of noncommuting self-adjoint elements of A ''acting in lexicographic order,'' i.e., from left to right, with an element bi∈I ''inserted'' between the action of ai and ai+1 for each i=1,…,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:R→C is sufficiently regular and a∈Asa:={c∈A:c∗=c}, then fa,I(b):=f(a+b)−f(a)∈I for all b∈Isa:=I∩Asa, the map fa,I:Isa→I is Fréchet Ck, and the kth Fréchet derivative of fa,I may be written in terms of LFC applied to the kth 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 fA:Asa→A defined by a↦f(a): (1) If A is commutative and f∈Ck(R), then fA is Fréchet Ck; (2) if A is finite dimensional and f∈Ck(R), then fA is Fréchet Ck; and (3) if f:R→C is ''slightly better than Ck,'' e.g., belongs to the homogeneous Besov space B˙1k,∞(R), then fA is Fréchet Ck 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.
Buket Can Bahadır, Nazlı Doğan
We investigate the relationship between the tame factorization property, denoted by TF and introduced in the companion paper CDI, and the DN-Ω type linear topological invariants of Fréchet spaces. Combining the basic properties of TF with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess TF. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing 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 DN and Ω, and a power series space of finite type Λ1(E) or infinite type Λ∞(E). We show that requiring such a triple to possess the tame factorization property 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 Λ1(E) or Λ∞(E).
Robin Tucker-Drob
We prove that every countable inner amenable group has vanishing ℓ2-Betti numbers in all degrees. This was previously open in every degree n≥2.
Rupert L. Frank, Paata Ivanisvili
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.
Cyril Houdayer, Corentin Le Bars
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 the corresponding classification for the full flag action of SL3(Z) would imply ordinary ITAP.
Benedikt M. Reible, Albert Much, Rainer Verch +2
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.