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math.OA 2026-05-15 2 theorems

Separable type II1 factors gain orthonormal bases of self-adjoint unitaries

by Yixin He, Quanyu Tang +1 more

The separable case of Kadison's problem on orthonormal bases of unitaries for type II₁ factors

This settles Kadison's 1967 question for all separable diffuse finite von Neumann algebras.

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In 1967, Kadison asked ``does every type $\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?'' Using a noncommutative Lyapunov theorem of Akemann and Weaver, we prove that if $M$ is a separable diffuse finite von Neumann algebra with a normal faithful trace $\tau$, then $L^2(M,\tau)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem.

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hep-th 2026-05-15 2 theorems

Rényi QNEC holds for all integer orders n≥2

by Tanay Kibe, Pratik Roy

A general proof of integer R\'enyi QNEC

Second null variation of sandwiched Rényi divergence is non-negative under half-sided inclusions when the divergence is finite.

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The R\'enyi quantum null energy condition conjectures that the second null shape variation of the sandwiched R\'enyi divergence (SRD) of an excited state relative to the vacuum is non-negative in local Poincar\'e-invariant quantum field theory, giving a one-parameter generalization of the quantum null energy condition (QNEC). We prove R\'enyi QNEC for all integer R\'enyi parameters $n\geq 2$ for von Neumann algebras carrying a half-sided modular inclusion structure. The only assumption on the excited state is finiteness of its SRD relative to the vacuum. Concretely, for any $\sigma$-finite von Neumann algebra with such an inclusion, we prove log-convexity, under the associated null-translation semigroup, of the Kosaki $L^n$ norm of any normal positive functional with finite $L^n$ norm.

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math.OA 2026-07-03

All invariant subalgebras in lattice Poisson boundaries are crossed products

by Shuoxing Zhou

On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

They arise exactly from larger parabolic quotients and normal subgroups of the lattice.

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Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two. Let $\Gamma<G$ be an irreducible lattice, let $P<G$ be a minimal parabolic subgroup, and consider the crossed product $L^\infty(G/P,\nu_P)\rtimes \Gamma$. We prove that every $\Gamma$-invariant von Neumann subalgebra of $L^\infty(G/P,\nu_P)\rtimes \Gamma$ is of the form $L^\infty(G/Q,\nu_Q)\rtimes \Lambda$, where $P\leq Q\leq G$ and $\Lambda\lhd\Gamma$. This confirms a conjecture of Amrutam--Hartman.

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math.DS 2026-07-03

Free group minimal actions on Cantor sets can lack dynamical comparison

by Paolo Boldrini, Akshara Prasad

Topologically free minimal actions without dynamical comparison

Such actions exist with or without invariant measures, and crossed product comparison does not imply the dynamical version.

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We show the existence of a topologically free minimal action of $\mathbb F_\infty$ on the Cantor space that does not have dynamical comparison. Moreover, we show that this phenomenon can happen both in the presence and in the absence of invariant measures. We also show that strict comparison of the reduced crossed product C*-algebra does not imply dynamical comparison for minimal actions. Our technique involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid and then realising it as the type semigroup associated to a dynamical system.

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math-ph 2026-07-03

r-deformed Rényi entropy gives tighter Tsallis bound on density operators

by Srikrishna Maity, Shigeru Furuichi +1 more

r-deformed α-z-R\'enyi relative entropy

The three-parameter family lies below an existing upper bound whenever both are applied to quantum states.

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In this article, we consider the $r$-logarithm for defining three-parameter family of R\'{e}nyi relative entropies that are generalization of the $\alpha$-$z$-R\'{e}nyi relative entropies. All the members of $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropies satisfy the necessary axioms to be a divergence. We expose the range of parameters $\alpha$, $z$ and $r$ for which the data processing inequality holds. We also establish that $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy is an upper bound of the Tsallis relative entropy. Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy and the other one, which is discussed in literature. We investigate the order relationship between these two upper bounds of the Tsallis relative entropy. We observe that our new upper bound is more tighter when applicable to the density operators.

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math.OA 2026-07-03

Fuzzy tori converge to Dirac triple on quantum torus

by Frederic Latremoliere

How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale

Extended spectral propinquity for twisted triples with Riesz transform twists enables the approximation of flat spectral triples.

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We prove that the classical and the quantum flat torus can be rigorously approximated at a differential level by finite-dimensional fuzzy tori within the framework of the spectral propinquity. Standard attempts to establish this convergence are traditionally obstructed by the intrinsic non-locality of discrete calculus and the subsequent failure of the Leibniz rule. While contemporary alternatives such as spectral truncations circumvent this issue by abandoning $C^*$-algebras in favor of operator systems, we instead preserve the $C^*$-algebraic category by generalizing the commutator formula. To this end, we introduce a relaxed notion of a twisted spectral triple where the twist is a linear map acting as a discretized Riesz transform that encapsulates the non-locality of the discrete world. By extending the spectral propinquity to this generalized setting of twisted spectral triples with possibly unbounded twists, we prove that fuzzy tori equipped with their natural discrete calculus converge to the standard flat Dirac triple on the torus, while the underlying twists converge to the identity.

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math-ph 2026-07-02

The authors introduce a classification scheme for topological phases using the topology…

by Giuseppe De Nittis, Santiago G. Rendel

A scheme for topological phases of the Weyl C^*-algebra

A classification scheme for topological phases is defined via homotopy classes of sections of pure-state fiber bundles over the Weyl…

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In this work, we introduce a classification scheme for topological phases of matter based on the topology of the space of pure states of a model $C^*$-algebra. Under it, topological phases are described by homotopy classes of sections of certain fiber bundles of (pure) states. Applying this classification procedure on states of the Weyl $C^*$-algebra that are invariant under translations by a lattice, we recover the $K$-theoretic classification of gapped spectral projectors for topological insulators of types A and AI, thus essentially generalizing this notion.

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math.OA 2026-07-02

Graph conditions make ultragraph algebras residual finite-dimensional

by Daniel Gonçalves, Danilo Royer

Residual finite-dimensionality of ultragraph algebras via branching systems

For RFUM2 ultragraphs the combinatorial conditions become equivalent to RFD of both Leavitt path algebras and C*-algebras.

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We study residual finite-dimensionality for ultragraph algebras, both in the algebraic and in the C-star-algebraic settings. We introduce graph-theoretic RFD conditions for ultragraphs, extending the conditions that characterize RFD graph C-star-algebras. Using the boundary ultrapath branching system, we construct finite-dimensional branching-system representations associated to terminal boundary sets and no-exit cycles. These representations are used to prove that, whenever an ultragraph satisfies the graph-theoretic RFD conditions, its ultragraph Leavitt path algebra LK(G) is RFD, for every field K, and its ultragraph C-star-algebra RFD. For ultragraphs satisfying Condition (RFUM2), we prove converses in both settings. The analytic converse uses the groupoid model and the density of periodic points, while the algebraic converse is proved directly by finite-dimensional linear algebra. Thus, for RFUM2 ultragraphs, RFD of LK(G), RFD of C(G), and the graph-theoretic RFD conditions are equivalent. This gives, in particular, a common combinatorial description linking the algebraic and analytic theories, recovers the graph C-start-algebra characterization, and yields an algebraic characterization for Leavitt path algebras of graphs. We also construct an RFD ultragraph algebra which is genuinely outside the graph-algebra class in both settings.

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math.OA 2026-07-02

State-space yields Davis-Wielandt shell bounds in C*-algebras

by Debarati Bhattacharya, Fuad Kittaneh +2 more

On the geometry of the algebraic Davis--Wielandt shell and norm-parallelism in C^*-algebra

Geometric properties established and radii of sums bounded, with links to norm-parallelism.

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This article is devoted to the study of the Davis--Wielandt shell and the Davis--Wielandt radii of elements in a $C^*$-algebra. Utilizing a state-space approach, several geometric properties of the algebraic Davis--Wielandt shell are established. Upper and lower bounds for the algebraic Davis--Wielandt radii are obtained including the Davis--Wielandt radius of the sum of $k$ elements. We also explore the relationship between norm-parallelism and the Davis--Wielandt radii of elements.

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math.OA 2026-07-02

HK conjecture holds for poly-Z actions up to Hirsch length four

by Hiroki Matui

HK and GL

Sequence comparisons prove the conjecture for Z, Z2 and Klein bottle groups while gap-labelling works up to factor two for length three grou

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We study the HK conjecture and the gap-labelling problem for transformation groupoids associated with free actions of poly-$\Z$ groups on Cantor sets. The main tool is a comparison of the long exact sequences in groupoid homology and cohomology with the Pimsner--Voiculescu exact sequence for crossed products by $\Z$. In addition to the canonical homology comparison maps $\mu_0$ and $\mu_1$, we introduce cohomology comparison maps associated with suitable $K$-theory classes of the acting group. Together with Poincar\'e duality, these maps detect the higher homology terms occurring in the HK conjecture. We apply this method to free actions of poly-$\Z$ groups of small Hirsch length. For actions of $\Z$, $\Z^2$, and the Klein bottle group, we recover HK and gap-labelling. For several classes of groups of Hirsch length three and four, we either prove HK or obtain explicit exact sequences describing the $K$-groups in terms of groupoid homology and cohomology. For gap-labelling, we combine the de la Harpe--Skandalis determinant, the trace formula for the Pimsner--Voiculescu boundary map, and transposition decompositions in topological full groups. This gives gap-labelling up to a factor of two for all free actions of poly-$\Z$ groups of Hirsch length three and for certain groups of Hirsch length four, including $\Z^4$. We also recover gap-labelling for $\Z^3$-actions and prove gap-labelling up to a factor of two for $\Z^5$-actions by using cohomology comparison maps for mapping tori.

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math.KT 2026-07-02

Explicit unitaries realize real K-theory generators on involuted spheres

by Jeffrey L. Boersema

The real K-theory of the sphere with an arbitrary involution

Matrices for dimensions up to 3 and a recipe for all higher dimensions turn abstract classes into concrete operators.

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We complete the investigation begun in a previous paper to find unitary representations of the non-trivial real $K$-theory elements for the sphere $S^d$ with an involution. Here we consider all involutions except the antipodal involutions. We write down explicit unitaries representing the generators in all cases for $d \leq 3$, and for $d > 0$ we describe a recipe for generating such unitaries.

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math.OA 2026-07-01

Graph products admit L_p-bounded Hilbert transforms

by Xiao-Qi Lu, Runlian Xia

Hilbert transforms on graph products of finite von Neumann algebras

A length-dependent Cotlar identity plus a Haagerup-type inequality extends free-product boundedness to graph products and settles an Ozawa c

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We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interests on their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality and show, in particular, that it is equivalent to the graph product being generated by finite-dimensional von Neumann algebras with uniformly bounded dimensions. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide positive answers to a compactness problem posed by Ozawa in the setting of graph products of finite groups and right-angled Hecke von Neumann algebras.

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math.OA 2026-07-01

Nuclearity of operator systems equals approximate factorization

by Roy Araiza, Larissa Kroell +2 more

Approximate factorization properties for operator systems

Standard conditions on the tensor product hierarchy match factorization through finite-dimensional systems.

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We show that many of the standard nuclearity properties considered in the literature for the hierarchy of operator system tensor products can be expressed as approximate factorization properties, generalizing the well-known Completely Positive Approximation Property for nuclear C*-algebras due to Choi and Effros and its generalization to nuclear operator systems due to Han and Paulsen.

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math.OA 2026-07-01

II1 factors without crossed product decompositions found

by Adriana Fernández Quero, Adrian Ioana +1 more

A class of II₁ factors without non-trivial crossed product decompositions

New separable examples restrict embeddings into their tensor squares to only the canonical maps.

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We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, for any trace preserving action $G\curvearrowright^\sigma (B,\tau)$ of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. These provide the first examples of II$_1$ factors that do not arise as crossed products of noncommutative dynamical systems. Our approach relies on a novel construction of separable II$_1$ factors $M$ whose embeddings into their tensor product square $M\overline{\otimes}M$ all arise from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.

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math.PR 2026-07-01

Free SDEs gain global well-posedness under local Lipschitz conditions

by Jiaxin Wei, Zhi Yin

Well-posedness and stationary distribution of free stochastic differential equations

Local operator Lipschitz and Lyapunov conditions ensure unique solutions and stationary distributions in noncommutative probability spaces.

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This paper studies free stochastic differential equations driven by free Brownian motion. Under local operator Lipschitz and Lyapunov-type conditions on the coefficients, we prove the global well-posedness of solutions in the noncommutative probability setting using free It\^o calculus. We further establish the existence and uniqueness of stationary solutions under appropriate dissipativity conditions. Our results extend classical theory to the free probability framework.

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math.OA 2026-07-01

Real Robbin-Salamon theorem equates spectral flow to index

by Chris Bourne, Alan L. Carey +2 more

Analytic index theory and spectral flow in real Hilbert C^*-modules

Spectral flow of real Fredholm operators on Hilbert C*-modules equals a Fredholm index via Van Daele K-theory.

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We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules. Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $\sigma$-unital $C^*$-algebra. We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing. We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators. Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module. We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices. Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index. Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.

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math-ph 2026-06-30

Any conformal net yields Cardy CFTs on Minkowski space

by Bin Gui

Minkowskian open/closed conformal field theory possibly without vacuum: the Cardy case

The axioms alone produce closed and open string theories plus three duality relations that realize modular invariance and the Cardy conditio

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For any conformal net, not necessarily rational, we construct the associated Cardy-type conformal field theory on the Minkowski spacetimes $(\mathbb R/2\pi\mathbb Z)\times\mathbb R$ for closed strings and $[0,\pi]\times\mathbb R$ for open strings within the framework of algebraic quantum field theory. In addition to verifying some of their basic properties, we prove three forms of Haag duality for multi-double-cones and boundary intervals, interpreted respectively as the Minkowskian versions of modular invariance, the Cardy consistency condition, and the Morita equivalence of boundary field algebras.

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math.OA 2026-06-30

Noncommutative Wiener-Wintner theorem for amenable groups

by Panchugopal Bikram, Sudipta Kundu +1 more

Non-Commutative Wiener-Wintner theorem for amenable group actions

Decomposition into almost periodic and weakly mixing parts yields the result on finite von Neumann algebras.

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Let $G$ be a locally compact second countable amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau)$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this action, obtaining a decomposition of $\mathcal{M}$ into its almost periodic and weakly mixing components. As an application, we prove a noncommutative Wiener--Wintner theorem for amenable group actions on finite von Neumann algebras.

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math.OA 2026-06-30

Heat semigroup on free orthogonal quantum groups hypercontracts at optimal time

by Haonan Zhang

Sharp hypercontractivity for free orthogonal quantum groups of Kac type

Result gives sharp contraction rate to equilibrium for O_F^+ of Kac type in noncommutative L^p spaces.

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We prove that the heat semigroup on the free orthogonal quantum group $O_F^+$ of Kac type satisfies hypercontractivity with the optimal time.

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math.FA 2026-06-29

Bijections on C*-algebra cones satisfying Fischer-Muszély equation extend to Jordan isomor

by Daisuke Hirota, Jyamira Oppekepenguin

On the Fischer-Musz\'ely equation for the positive cones of C^*-algebras

Maps on positive semidefinite cones extend via Jordan *-isomorphism and positive multiplier on both sides.

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We study the Fischer-Musz\'ely functional equation for the positive semidefinite and the positive definite cones of unital $C^*$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Musz\'ely equality extends to a Jordan $*$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^*$-algebras.

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math.OA 2026-06-29

Fermionic matrices fail strong convergence to semicircle

by Dimitri Shlyakhtenko

Failure of Strong Convergence of Matrices with Fermionic Entries

Norms of individual real parts approach the limit, but joint operator space structure does not converge in free probability.

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Let $Q^{(k)}_N$ be an $N\times N$ matrices with entries satisfying CAR, normalized to have variance $1/\sqrt{N}$ with respect to the trace of the CAR algebra. We show that, although the operator norm of the real part of an individual matrix $Q^{(k)}_N$ converges as $N\to\infty$ to the semicircular limit, the family of matrices does not converge to the free probability limit strongly. In fact, even the operator space structure of the linear spans of the real and imaginary parts of $Q^{(k)}_N$'s, $k=1,\dots,M$, does not converge to the semicircular limit.

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quant-ph 2026-06-29

Quantum instruments compose by integrating channel-valued functions

by Robert I. Booth, Dominik Leichtle +2 more

Composing Quantum Instruments

The Okamura-Ozawa extension supplies monad multiplication, identifying quantum Markov kernels as the Kleisli category.

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We study the composition of classically-controlled quantum instruments--the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.

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math.FA 2026-06-29

Maximal symbol subalgebra forms complete locally convex algebra

by Miguel Angel Rodriguez Rodriguez

Commutative topological algebras on translation-invariant reproducing kernel Hilbert spaces

Transporting from symbols to operators on translation-invariant kernel spaces produces commutative topological algebras of operators and ker

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We study commutative topological algebras naturally associated with translation-invariant reproducing kernel Hilbert spaces whose direct integral decomposition has one-dimensional fibers. Starting from the bounded algebra of translation-invariant operators, we pass to a common dense domain generated by reproducing kernels and identify the corresponding diagonalizable operators with multiplication by symbols in an intersection of weighted $L^2$-spaces. On the symbol side this gives a canonical space $\mathcal F_0$ and a maximal multiplicative subalgebra $\mathcal F_M$, which is a complete locally convex $*$-algebra. Transporting the structure back yields corresponding algebras of operators and integral kernels. We also discuss when the inclusions $L^\infty(\Omega)=\mathcal F_\infty\subset \mathcal F_M\subset \mathcal F_0$ are strict, and illustrate the results with vertical and radial operators on classical Bergman and Fock spaces.

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math.GR 2026-06-29

Thick building groups satisfy Howe-Moore property

by Andreas Thom

On the Howe--Moore property for automorphism groups of buildings

Weakly mixing unitary representations vanish at infinity for large thickness, implying character rigidity for associated lattices.

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Let $G<Aut(X)$ be a totally disconnected locally compact group acting strongly transitively on a locally finite building $X$ of finite-rank and minimal non-spherical type. For sufficiently large thickness, every weakly mixing strongly continuous unitary representation of $G$ is $C_0$. Consequently, if $G$ has no non-trivial finite-dimensional unitary representations, then $G$ has the Howe--Moore property. More concretely, this applies to rank-three compact-hyperbolic crystallographic types of thickness $q+1$ for $q\geq 19379$, if there are no compact quotients. As an application, we prove that the corresponding Caprace--R\'emy Kac--Moody lattices in these types, which are known to be finitely presented simple and Kazhdan, are character-rigid: their extremal characters are only the regular and the trivial character. Consequently they also have no non-trivial invariant random subgroups.

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math.OA 2026-06-29

Free analog of Bobkov isoperimetry inequality proved

by Dimitri Shlyakhtenko

A Free Analog of Bobkov's Gaussian Isoperimetry Inequality

One-variable bound on L1 norm of difference quotients gives non-local isoperimetric statement in free probability.

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We prove a one-variable functional inequality which is the free probability analog of Bobkov's isoperimetry inequality. The inequality involves the $L^1$ norm of the difference quotient of a function $f$ and can be viewed as a non-local isoperimetric inequality. We also prove related inequalities for subsets of an interval as well as for subsets of roots of Hermite polynomials. This paper is also an experiment in AI-based exploration of free analogs of classical probability statements.

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math.FA 2026-06-26

Property A implies p-nuclearity of ℓ^p uniform Roe algebras

by Yeong Chyuan Chung

On some p-approximation properties of exact discrete groups and ell^p uniform Roe algebras

For discrete spaces with bounded geometry this links coarse geometry to algebraic approximation for every p in (1, ∞).

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We study $p$-approximation properties of $\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For a discrete metric space $X$ with bounded geometry, we prove that property A implies $p$-nuclearity of the $\ell^p$ uniform Roe algebra $B^p_u(X)$ for every $p\in(1,\infty)$, while $B^1_u(X)$ is always 1-nuclear. We introduce the $p$-invariant translation approximation property ($p$-ITAP) for discrete groups, generalizing the 2-ITAP of Roe. We also introduce the $p$-operator ITAP. For exact groups, we show that the $p$-operator ITAP is equivalent to the $p$-approximation property of An-Lee-Ruan. We also characterize exactness of discrete groups in terms of their $\ell^p$ uniform Roe algebras with coefficients in $p$-operator spaces.

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math.OA 2026-06-26

Kleppner's condition decides selflessness of twisted C*-algebras

by Tron Omland

Selflessness for twisted group C*-algebras of amenable groups and their inclusions

For amenable virtually nilpotent groups the algebra is selfless exactly when the pair meets the condition; same for inclusions under the rel

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For a discrete amenable group $G$ with a two-cocycle $\sigma$ we first record a few results on when the twisted group $C^*$-algebra $C^*_r(G,\sigma)$ is selfless, in the sense of Robert. In particular, for an infinite finitely generated virtually nilpotent $G$, this holds exactly when $(G,\sigma)$ satisfies Kleppner's condition. For the larger class of FC-hypercentral groups the same holds modulo $\mathcal{Z}$-stability, equivalently finite nuclear dimension. Further, using the relative Kleppner condition we obtain corresponding selflessness results for inclusions $C^*_r(H,\sigma')\subseteq C^*_r(G,\sigma)$, when $H$ is a normal subgroup of $G$. For amenable $G$ such an inclusion is selfless precisely when $C^*_r(H,\sigma')$ is selfless and $(H\leq G,\sigma)$ satisfies the relative Kleppner condition. Thus, for an infinite finitely generated virtually nilpotent $G$, selflessness of the inclusion $C^*_r(H,\sigma')\subseteq C^*_r(G,\sigma)$ is equivalent to the relative Kleppner condition.

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math.CT 2026-06-26

Semantics fixes the geometry of categorified spectra

by Shih-Yu Chang

Intrinsic Geometry of Categorified Spectral Objects

Tangent complex, singular locus, inertia stack and curvature class arise canonically from duality and reconstruction.

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This paper develops the intrinsic geometry of the categorified spectral object $\mathfrak{Spec}(A)$ associated with an admissible operator-semantic system $A$ in the Categorified Spectral Duality (CSD) framework. We prove that the tangent complex, singular locus, inertia stack, and contextual curvature class are canonically determined by the duality adjunction between $\mathfrak{Spec}$ and the global sections functor, together with the reconstruction theorem identifying $A$ with the global sections of $\mathfrak{Spec}(A)$. The Canonical Geometry Theorem establishes that any CSD-compatible geometric structure is induced by the canonical datum consisting of $\mathfrak{Spec}(A)$, its tangent complex, its singular locus, its inertia stack, and its contextual curvature class; hence the geometry of $\mathfrak{Spec}(A)$ is intrinsic to the semantic structure of $A$. We prove that the tangent complex controls the deformation theory of $\mathfrak{Spec}(A)$ and satisfies a Hochschild realization, establishing a direct bridge between geometry and algebra. The assignment sending $A$ to its canonical geometric datum is functorial and Morita invariant, with explicit computations for the complex numbers and matrix algebras demonstrating that noncommutativity, detected by the inertia stack, is distinct from contextuality, which requires additional structures. Thus semantics determines geometry, and the intrinsic geometry of $\mathfrak{Spec}(A)$ provides a canonical geometric encoding of $A$ up to Morita equivalence.

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math.OA 2026-06-25

Subhomogeneous system duals are quotients of subhomogeneous systems

by Markus Dannemüller, Tim Netzer

On Subhomogeneous Operator Systems

Even when the dual is not itself subhomogeneous, it arises as a quotient from one that is, in the finite-dimensional case.

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We study subhomogeneity for finite-dimensional operator systems, and collect and extend characterizations in terms of the $C^*$-envelope, $d$-maximality, complete positivity, dual $d$-minimality, and non-commutative boundary conditions. We then show that the dual of a subhomogeneous operator system, while not necessarily subhomogeneous itself, is always a quotient of a subhomogeneous system. We complement these characterizations with examples and counterexamples, including minimal and maximal systems over certain polyhedral cones.

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math.OA 2026-06-25

Operator inequalities yield Wold decomposition for covariant representations

by Dimple Saini, Azad Rohilla

Wold-type decomposition and Beurling-Type Theorem for Covariant Representations

Every nonzero invariant subspace is then uniquely recovered from its wandering subspace.

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Using operator inequalities, we study a Wold-type decomposition of covariant representations. Building on this decomposition, we prove a Beurling-type theorem showing that every nonzero invariant subspace is uniquely determined by its wandering subspace. Our results extend classical theorems of Beurling and subsequent developments for left-invertible operators to the setting of covariant representations of $C^*$-correspondences, providing a unified framework for invariant subspace theory under operator inequalities.

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math.FA 2026-06-24

Bargmann transform makes every translation invariant operator a Toeplitz operator

by Raul Quiroga-Barranco

A Bargmann transform for translation invariant operators on weighted Bergman spaces of the complex half-plane

On weighted Bergman spaces over the half-plane the correspondence is a *-isomorphism realized by a commutative algebra of symbols.

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Let us denote with $\mathcal{A}^2_\lambda(\mathbb{C}_+)$ ($\lambda > -1$) a weighted Bergman space over the right half-plane $\mathbb{C}_+$, which admits a unitary representation of $\mathbb{R}$ given by the (imaginary) translations of $\mathbb{C}_+$. We study the von Neumann algebra $\mathfrak{T}\big(\mathcal{A}^2_\lambda(\mathbb{C}_+)\big)$ of bounded translation invariant operators. We prove that every element of $\mathfrak{T}\big(\mathcal{A}^2_\lambda(\mathbb{C}_+)\big)$ is a Toeplit operator $T^{(\lambda)}_A$ for some translation invariant operator $A$ of the ambient $L^2$-space of $\mathcal{A}^2_\lambda(\mathbb{C}_+)$. Furthermore, we prove that this can be achieved through a commutative von Neumann algebra $\mathfrak{A}_\lambda$ that yields an assignment $A \mapsto T^{(\lambda)}_A$ that turns out to be a $*$-isomorphism of $*$-algebras. Our main tool is a Bargmann transform $\mathcal{B}_\lambda$ for which we establish several operator and representation theoretic properties. We also describe the translation invariant subspaces of $\mathcal{A}^2_\lambda(\mathbb{C}_+)$ and obtain formulas for the diagonalizing spectral functions for translation invariant Toeplitz operators whose symbols are translation invariant operators. The latter generalize previously known results for function symbols.

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math.OA 2026-06-24

Group growth sets metric dimension of twisted groupoid C*-algebras

by Arnab Chattopadhyay, Soumalya Joardar

Metric dimension of C^(ast)-algebras of cocycle twisted transformation groupoids: Growth and dynamical complexity

Polynomial growth keeps the dimension finite; exponential growth makes it generically infinite, even after cocycle twists.

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We consider a natural CQMS structure on a twisted transformation groupoid $C^{\ast}$-algebra coming from stratified $_{\text {c}}$Lip-norm introduced by Austad. We obtain upper bounds of metric dimension of reduced $C^{\ast}$-algebra of a transformation groupoid $\Gamma\rtimes X$ and its cocycle twist for a suitably chosen CQMS structure, provided $(X,d)$ is a compact metric space of finite Kolmogorov dimension and $\Gamma$ is a discrete group of polynomial growth. When $\Gamma$ has exponential growth, we prove that the dimension is generically $+\infty$ proving that the dichotomy between polynomial growth and exponential growth of groups survive even after considering cocycle twists of transformation groupoids.

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math.FA 2026-06-24

Infinite doubly stochastic matrices keep norm 1 only with large almost-stochastic blocks

by Ludovick Bouthat, Javad Mashreghi +1 more

Norm of infinite doubly stochastic matrices

The condition Θ(D^*D)=1 gives the exact combinatorial test for when the ℓ^p operator norm remains 1.

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In finite dimensions, every doubly stochastic matrix has the $\ell^p$-operator norm equal to $1$ for all $1 \le p \le \infty$. However, in the infinite-dimensional setting, this property may fail since the norm can be strictly smaller than $1$ when $1<p<\infty$. In this paper, a complete characterization of infinite doubly stochastic matrices for which the norm remains equal to $1$ is obtained. More precisely, for $1<p<\infty$, it is shown that $$ \|D\|_{\ell^p(I)\to\ell^p(I)}=1 \quad\iff\quad \Theta(D^*D)=1, $$ where $\Theta$ measures the maximal average mass of a finite square submatrix. Thus, the norm is equal to $1$ precisely when the matrix contains arbitrarily large finite regions in which it behaves almost like a finite doubly stochastic matrix. The proof uses a Cheeger-type argument, highlighting a natural connection with ideas from spectral graph theory.

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math.OA 2026-06-24

Baum-Connes assembly map extended to inverse semigroup Fell bundles

by Diego Martínez

A Baum-Connes assembly map for essential semigroup crossed products

Functoriality of cross-sectional algebras allows the map for Cartan pairs and non-Hausdorff groupoids.

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We construct an equivariant E-theory and a Baum-Connes assembly map at the level of Fell bundles of inverse semigroups over separable C*-algebras. This generalizes previous work of several authors, and allows to discuss E-theoretic matters in the context of Cartan pairs; maximal and essential C*-algebras of non-Hausdorff groupoids; and Fell bundles over discrete groups and \'etale groupoids, among others. In order to do this we establish several functoriality properties for maximal, reduced and essential cross-sectional C*-algebras associated with a (saturated) Fell bundle of an inverse semigroup. This allows to discuss when these algebras give rise to short exact sequences, generalizing the classical case of discrete groups. We also introduce the adequate notion of ``proper'' Fell bundle, or ``proper'' action of an inverse semigroup, and prove a weak containment property for these. Using these functoriality properties and these proper actions we then introduce (maximal, reduced and/or essential) equivariant E-theory by means of adequately equivariant asymptotic morphisms, and construct a Baum-Connes assembly map that is both natural and reasonably well-behaved.

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hep-th 2026-06-23

Gaussian states with VEVs excite only if mean differences stay bounded

by Jacqueline Caminiti, Federico Capeccia +1 more

Excitability of Gaussian states with VEVs

The bound on averages plus the correlation criteria equate bulk and boundary excitation problems in anti-de Sitter space.

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In arXiv:2604.19861, we gave general criteria for when one zero-mean Gaussian state can be excited out of another in a (generalized) free field theory. Here we extend this analysis to the case of nonzero mean, i.e., to Gaussian states with vacuum expectation values (VEVs). We prove that excitability is possible exactly when (i) the connected two-point functions satisfy criteria like those in arXiv:2604.19861, and (ii) the difference of the VEVs is bounded relative to the two-point functions. As an application, we give an explicit computation showing that in anti-de Sitter spacetime, a VEV shift can be excited from the Klein-Gordon vacuum if and only if its boundary extrapolation can be excited from the vacuum of the dual conformal field theory.

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math.OA 2026-06-23

Bicentralizer flow ergodicity yields MASAs in type III1 subfactors

by Amine Marrakchi

Ergodicity of the bicentralizer flow and Kadison's problem

This finishes Kadison's 1967 problem for irreducible subfactors with expectation in factors with separable predual.

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We show that the relative bicentralizer flow of a type $\mathrm{III}_1$ irreducible subfactor with expectation is always ergodic. As a consequence, every irreducible subfactor with expectation in a factor with separable predual contains a maximal abelian subalgebra. This completes the solution to Kadison's problem on maximal abelian subalgebras from 1967.

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math.GR 2026-06-23

Infinite one-relator groups have cost |S| minus 1/m

by Antoine Poulin, Konrad Wróbel

Cost of one-relator groups

The formula also sets the first L2 Betti number to cost minus one, for presentations where the relator is a proper power.

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For any infinite one-relator group $\Gamma=\langle S \mid w^m\rangle$, we prove that $\mathrm{cost}(\Gamma)=|S|-\frac{1}{m}$. For such groups, this gives $\beta^{(2)}_1(\Gamma)=\mathrm{cost}(\Gamma) - 1$, answering a special case of Gaboriau's question on the relationship between cost and first $\ell^2$-Betti number.

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math.OA 2026-06-22

Pure contractive rep of N^2_0 product system dilates to isometric

by Dimple Saini

Completely contractive covariant representations of product system over mathbb N²₀

Explicit construction shows the dilation theorem extends to two-parameter product systems while keeping purity.

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A pure completely contractive covariant representations of a $C^*$-correspondence dilate to a pure isometric covariant representations due to Muhly-Solel. More specifically, we are curious about the following question for pairs: Does a pure completely contractive covariant representations of a product system dilate to a pure isometric covariant representations of a product system? The goal of this study is to find a pure completely contractive covariant representations of a product system that provide an affirmative answer for the previous question.

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math.NT 2026-06-22

Zeta functions over F1 satisfy Weil conjectures except RH analog

by Igor V. Nikolaev

Geometry of F₁ and Cuntz-Krieger algebras

K-theory of Cuntz-Krieger algebras gives Frobenius action and point counts for varieties over the field with one element.

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We study a natural map between projective varieties $V(\mathbf{F}_{1})$ over the field with one element and the Cuntz-Krieger algebras $O_A$. Using the $K$-theory of $O_A$, we calculate the Frobenius action and cardinality of the set $V(\mathbf{F}_{1^r})$. It is proved that the zeta function of $V(\mathbf{F}_{1})$ satisfies all Weil's Conjectures except for an analog of the Riemann hypothesis. We use the crossed product structure of $O_A$ to establish a morphism of the schemes $\operatorname{Spec} ~(\mathbf{Z})\to \operatorname{Spec} ~(\mathbf{F}_{1})\simeq \{\operatorname{pt}\}$.

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math.KT 2026-06-22

Discretisation equates independent groupoids to discrete ones

by Xin Li, Alistair Miller

Discretisation and independent resolutions of ample groupoids

Independent resolutions then reduce homology and K-theory computations for ample groupoids to the discrete case.

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We develop a general framework for understanding and computing both the groupoid homology of an ample groupoid and the topological K-theory of its reduced C*-algebra, based on two main ideas: discretisation and independent resolutions. Discretisation shows that a special class of ample groupoids we term independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. We introduce the notion of a resolution by independent groupoids and provide a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory. In order to illustrate our general ideas and methods, we work out several concrete examples and applications. Garside categories provide a wide range of examples, including higher rank graphs, self-similar groups and spherical Artin-Tits groups. We also present an application to the homology of Stein's groups.

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math.OA 2026-06-22

Path indicators defined for finite quantum graphs

by Sourav Khatua, Sutanu Roy

On the path correspondences of quantum graphs

The notions extend classical paths and receive explicit formulas in the classical, trivial, and complete cases.

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We introduce notions of path indicators and path correspondences for finite quantum graphs, study their basic properties, and compute them explicitly for classical graphs, trivial graphs, and complete quantum graphs.

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math.FA 2026-06-22

Tensor inequalities keep matrix constants but their slack depends on the transform

by Shih-Yu Chang, Michael K. Ng

Operator Inequalities in Φ-Product Tensor Algebras: Invariance and Transform Sensitivity

Defect vanishes under DFT yet grows linearly in p under DCT for explicit pairs; no transform is best for all tensors.

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We study classical operator inequalities in $\Phi$-product tensor algebras (a transformation $\Phi$-based generalization of the $t$-product framework) for third-order tensors. Although these algebras are algebraically isomorphic under different unitary transforms $\Phi$, we show that their quantitative behavior is not invariant. We prove that fundamental inequalities, including Golden--Thompson, Jensen, Klein, and Lieb, extend to the $\Phi$-product setting with the same constants as in the matrix case. However, the associated defect -- the slack between the two sides of the inequality -- depends explicitly on transform-domain noncommutativity. In particular, we establish a sharp characterization of the defect in terms of slice-wise commutators, revealing that inequality tightness is governed by transform-induced noncommutativity. We further demonstrate strong transform sensitivity by constructing explicit tensor pairs for which the defect vanishes under one transform (e.g., discrete Fourier transform) but grows linearly with the tensor depth under another transform (e.g., discrete Cosine transform), yielding an $\Omega(p)$ separation where $p$ is the matrix dimension of $\Phi$. Moreover, we prove that no transform is universally optimal: for any pair of transforms, there exist tensors for which each is strictly better than the other. These results show that the choice of transform defines a coordinate system in which commutativity is measured, inducing a nontrivial geometry of inequality tightness. Consequently, optimal transform selection is inherently data-dependent and can be formulated as an optimization problem over the unitary group.

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quant-ph 2026-06-22

Quantum phase operators are trace-class perturbations of Susskind-Glogower

by Bogdan D. Djordjevic, Nikolay A. Ivanov

Variants of the Quantum Phase Operator for the Harmonic Oscillator

The variants for the harmonic oscillator differ by trace-class terms, motivated by the two-phase case.

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We introduce and study quantum phase operators associated with the Quantum Harmonic Oscillator (QHO). We show that these operators are trace-class perturbations of the Susskind-Glogower operators and examine their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

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math.PR 2026-06-22

Positive coeffs bound GUE edges within 9 sqrt(nN/M) of Lehner rho

by Benoit Collins, Yuta Yamagishi

A Sudakov--Fernique proof of Lehner-type edge bounds for matrix-valued GUE sums

The finite-M expectation of the operator norm stays controlled by the free edge plus a term vanishing when M grows faster than N.

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Let $A_0,A_1,\ldots,A_n\in M_N(\mathbb{C})$ be Hermitian matrices and let $G_1,\ldots,G_n$ be independent $M\times M$ GUE matrices normalized so that $\|M^{-1/2}G_i\|\to 2$ almost surely as $M\to\infty$. We study the spectral edges and operator norm of $H_M = A_0\otimes I_M + \frac{1}{\sqrt{M}}\sum_{i=1}^n A_i\otimes G_i$. Lehner's formula identifies the right and left edges of the corresponding free semicircular operator as $\rho_+ = \inf_{Z\succ 0}\lambda_{\max}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$ and $\rho_- = \sup_{Z\prec 0}\lambda_{\min}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$. Assuming $A_i\succeq 0$ for $i\ge 1$ and $M\ge N$, we prove via concentration and minimax duality the finite-dimensional bounds $\mathbb{E}\lambda_{\max}(H_M)\le \rho_+ + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$ and $\mathbb{E}\lambda_{\min}(H_M)\ge \rho_- - 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. With $\rho_* = \max\{\rho_+,-\rho_-\}$, this yields $\mathbb{E}\|H_M\|_{\mathrm{op}}\le \rho_* + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. For uniformly bounded positive coefficients, bounded $n$, and $N=o(M)$, one obtains $\limsup_{M\to\infty}\mathbb{E}\|H_M\|_{\mathrm{op}}\le\rho$ whenever $\rho_{*,M}\to\rho$. The proof is a matrix-coefficient extension of classical Sudakov--Fernique comparison, combined with a Davidson--Szarek-type singular-value estimate and dual variational formulas for Lehner's edge quantities over density matrices. We also explain why this approach does not extend sharply to signed Hermitian coefficients.

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math.GR 2026-06-19

Lacunary hyperbolic groups are selfless under fast radius growth

by Goulnara Arzhantseva, Martin Finn-Sell

Lacunary hyperbolic groups with fast injectivity radius growth and enough loxodromic elements are selfless

The condition delta (log r)^7 = o(r) suffices for groups with enough generics, separating selfless from C*-selfless properties.

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We prove that a lacunary hyperbolic group $G = \varinjlim G_i$ with sufficient generics is selfless in the sense of Amrutam--Gao--Kunnawalkam Elayavalli--Patchell, provided the hyperbolicity constants $\delta_i$ and injectivity radii $r_i$ satisfy $\delta_i(\log r_i)^{7} = o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $\delta$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^*$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson's mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^*$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^*$-algebra with strict comparison.

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math.OA 2026-06-19

Canonical KK map is isomorphism for UCT algebras and infinite products

by Diego Martínez

On the relation between the product of KK-groups and the KK-group of the product

The comparison between KK of a product and the product of KK groups becomes bijective when targets are unital simple purely infinite.

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We observe that the canonical map $KK(A, \prod_{n \in \mathbb{N}} B_n) \to \prod_{n \in \mathbb{N}} KK(A,B_n)$ is an isomorphism of abelian groups whenever $A$ enjoys the Universal Coefficient Theorem and $B_n$ are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat--Eilers and Tikuisis--White--Winter.

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math.FA 2026-06-19

Full Gabor frames exist iff lower density exceeds 1

by Rui Liu, Xin Ma +1 more

Full Gabor frames, its existence problem, and a non-uniform Balian-Low type theorem

The equivalence for Schwartz windows holds on significant Delone sets in any dimension and yields a non-uniform Balian-Low theorem

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For a broad class of Delone sets in $\mathbb{R}^n$ that are of significance in both mathematics and physics, we prove a non-uniform Balian-Low type theorem and settle the converse problem on the existence of Gabor frames, for arbitrary dimension $n$. To this end, we introduce a class of Gabor frames, termed full Gabor frames, and prove that the existence of such a frame on the Delone set with Schwartz window functions is equivalent to the condition that the lower Beurling density be strictly greater than one. In fact, the usual Balian-Low direction using window functions from the Feichtinger's algebra can be proven for arbitrary point sets, thereby improving an earlier density theorem by Christensen, Deng, and Heil. The corresponding dual result for Riesz sequences is also obtained. The main technical tools employed in this paper are tiling groupoid constructions and $C^*$-algebraic methods. As a byproduct, we resolve an open question from Ito's thesis concerning the bounded dynamical asymptotic dimension of tiling groupoids. Furthermore, this result allows us to extend the classification theorem of Ito, Whittaker, and Zacharias to the twisted case.

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math.AT 2026-06-18

K-theory spectrum obstructs QCA linearization over any field

by Mattie Ji, Bowen Yang

K-Theoretic Obstructions to Linearizing QCA Representations

Homotopy types of QCA spaces produce universal classes that block linearization under locality constraints

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Projective representations arise naturally in physics and representation theory, and determining whether they can be linearized has been a fundamental problem. In this work, we study the analogous problem for quantum cellular automata (QCA) representations, which incorporate locality constraints imposed by a metric space $X$. Over an arbitrary field $\mathbb{F}$, we develop an obstruction theory for the linearization of QCA representations, using the algebraic $K$-theory spectrum of QCA constructed in previous work of the authors. The resulting obstructions are governed by the homotopy type of the QCA spaces, from which we extract universal obstruction classes to linearization. In the complex algebraic and unitary case, we also fully compute the homotopy types of the QCA spaces over a point, a line, and a plane.

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math.OA 2026-06-18

q-Gaussians and free Gibbs laws carry compact quantum metric spaces

by David Jekel, Therese Basa Landry

Compact quantum metric spaces from free probability

Generator-based Lip-norms regularized by semigroups induce the weak-* topology on states and transfer via free transport.

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We study quantum metric space structures on operator algebras arising from free probability, namely those associated to $q$-Gaussians and free Gibbs laws for convex potentials. We note that even for free semicirculars, Voiculescu's dual system does not produce a quantum metric space structure that recovers the weak-$*$ topology on the state space. However, for $q$-Gaussians, we can define a compact quantum metric space using length-like functions by the same method as has already been used for hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Next, motivated by the free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. We show using semigroup regularization that this Lip-norm defines a quantum metric space structure for $q$-Gaussians, and then transfer this property to free Gibbs laws for convex potentials using free transport.

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math.GR 2026-06-18

Existential embeddings force amenable intersections in bi-exact groups

by Connor MacMahon

Existential Inclusions of Bi-exact Groups are Conjugacy Representation Rigid

The weak equivalence class of the quasi-regular representation then determines the subgroup up to conjugacy among self-commensurating ones.

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If $\Lambda$ is a non-amenable bi-exact group and $\Lambda \hookrightarrow \Gamma$ is an existential embedding, then each of the intersections $\Lambda \cap g \Lambda g^{-1}$ for $g$ a member of $\Gamma \backslash \Lambda$ is amenable. This in conjunction with work of Bekka and Kalantar demonstrates that in this situation, the weak equivalence class of the quasi-regular representation $\lambda_{\Gamma/\Lambda}$ determines $\Lambda$ up to conjugacy among the self-commensurating subgroups of $\Gamma$.

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math.PR 2026-06-18

Polynomials solve truncated perpetuity equation and converge to free laws

by Julia Le Bihan, Bartosz Ko{l}odziejek

Finite free perpetuities

Degree-n monic solutions to the affine recursion yield polynomial models whose roots approach free-probability distributions.

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We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation \[ p(z) = \mathbb{E}\!\left[ A^{n}\,p\!\left(\frac{z-B}{A}\right)\mathbf{1}_{\{A\neq0\}} \right] + \mathbb{E}\!\left[ (z-B)^n\mathbf{1}_{\{A=0\}} \right], \] where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetuities between classical perpetuities and free-probabilistic fixed-point laws. We prove existence and uniqueness under weak conditions, and we identify a broad class of admissible pairs $(A,B)$ for which the resulting polynomial has only real, nonnegative zeros. Our approach uses finite free additive and multiplicative convolutions together with a probabilistic representation via the $U$-transform. As a motivating example, we exhibit an explicit family of finite free perpetuities expressed in terms of Jacobi polynomials and show that their empirical root distributions converge to a free-beta-prime law. More generally, for admissible sequences of parameters, we prove weak convergence of the empirical root distributions of finite free perpetuities to the law of a free perpetuity characterized by the corresponding free fixed-point equation. This yields a finite-degree polynomial model approximating free perpetuities and clarifies the connection between classical affine recursions, finite free convolutions, and free probability.

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math.OA 2026-06-17

Joint radius bounds detect amenable traces on C*-algebras

by Vern I. Paulsen, Mizanur Rahaman +1 more

Amenable traces and the joint numerical radius

Necessary and sufficient conditions on tuples of unitaries and isometries also obstruct several lifting properties.

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We provide necessary and sufficient characterizations of the existence of an amenable trace on a C$^*$-algebra in terms of the joint free numerical radius of tuples of unitaries, isometries, and partial isometries in the algebra. We apply these results to obtain new obstructions to various lifting properties.

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math.OA 2026-06-17

Symmetric cycline subgroupoid yields Cartan subalgebra for Z-actions on graphs

by Dawn Archey, Anna Duwenig +4 more

Cartan subalgebras in self-similar graph C^*-algebras

For many self-similar graphs with integer action the distinguished subgroupoid is maximal among open abelian ones and closed, inducing a Car

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For a self-similar graph $(G, E)$, we find a distinguished subgroupoid of the associated path groupoid $\mathcal{G}_{G,E}$ -- the symmetric cycline subgroupoid $\mathcal{S}_{\text{sym}}$. If the acting group $G$ is abelian, we show that $\mathcal{S}_{\text{sym}}$ is open, abelian, and normal. For $G=\mathbb{Z}$, we describe the dual bundle $\hat{\mathcal{S}}_{\text{sym}}$ of $\mathcal{S}_{\text{sym}}$ which can be used to provide a different groupoid model for the self-similar graph $C^*$-algebra $\mathcal{O}_{\mathbb{Z}, E}\cong C^*_r(\mathcal{G}_{\mathbb{Z},E})$. For a large class of self-similar graphs $(\mathbb{Z}, E)$, we further prove that $\mathcal{S}_{\text{sym}}$ is maximal among open abelian subgroupoids of $\mathrm{Iso}(\mathcal{G}_{\mathbb{Z},E})^{\circ}$ and closed in $\mathcal{G}_{\mathbb{Z},E}$, so that it gives rise to a Cartan subalgebra of $\mathcal{O}_{\mathbb{Z}, E}$. This result seems new even for genuine actions. Our proofs heavily rely on careful studies of dynamical behaviours of cycline triples of $(\mathbb{Z}, E)$ and on a dynamical-flavour classification for the vertices of $E$. Some results hold in more general settings and may be of independent interest.

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math.OA 2026-06-17

2-periodic Fock space position operators form type II1 factor

by Vitonofrio Crismale, Yun Gang Lu +1 more

Periodicity, type II₁ factors and free Poisson laws in interacting Fock spaces

Squared operators match Marchenko-Pastur law, giving concrete free Poisson realization.

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We show that the von Neumann algebra generated by position operators in a 2-periodic interacting Fock space is a type $II_1$ factor. On the probabilistic side, we prove that the squared position operators have a Marchenko-Pastur distribution with respect to the vacuum state, yielding a natural realization of free Poisson laws within this framework.

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math.OA 2026-06-17

Strongly reflexive masa-bimodules form a Boolean lattice

by Rupert H. Levene, Ying-Fen Lin +1 more

Lattices of strongly reflexive masa-bimodules

Natural operations create the lattice while support determines density of positive rank one subspaces, with topological views for group case

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We characterise the density of the positive rank one subspace of a masa-bimodule in terms of its support. We prove that strongly reflexive masa-bimodules form a Boolean lattice under naturally defined operations. We examine the lattice-theoretic properties of the class of strongly reflexive masa-bimodules that are also operator systems and some natural subclasses thereof, and provide a topological description of the lattice operations in the case the masa-bimodules arise from closed subsets of a locally compact group.

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quant-ph 2026-06-17

Log-Sobolev constants of quantum products stay within 0.96 of single-site value

by Yangjing Dong, Li Gao +3 more

Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups

The dimension-free bound holds for any n and any local dimension under the PODS condition on depolarizing semigroups

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We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes an exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. As an application of the same method, we also obtain sharp $(q,2)$-hypercontractivity estimates for qubit depolarizing channels.

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math.OA 2026-06-12

Restricted Fell bundle algebra embeds isometrically into groupoid C*-algebra

by Md Amir Hossain

Inclusions of Fell bundles C^*-algebras and coaction crossed products

The embedding induces a coaction of the discrete group whose crossed product is the C*-algebra of a bundle built from the cocycle data.

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Let $p \colon \mathcal{A} \to G$ be a Fell bundle over a locally compact Hausdorff second countable groupoid $G$ equipped with a Haar system, and let $\Gamma$ be a discrete group. Given a continuous $1$-cocycle $c \colon G \to \Gamma$, we show that the $\mathrm{C}^*$-algebra of the restricted Fell bundle $\mathcal{A}|_{G_e}$ embeds isometrically into $\mathrm{C}^*(G;\mathcal{A})$, where $G_e = c^{-1}(e)$ is the clopen subgroupoid corresponding to the identity element. We exploit this embedding to show that $\mathrm{C}^*(G;\mathcal{A})$ admits a natural structure of a topologically graded $\mathrm{C}^*$-algebra in the sense of Exel. As a consequence, we obtain a canonical coaction $\delta$ of $\Gamma$ on $\mathrm{C}^*(G; \mathcal{A})$. We further show that the associated coaction crossed product $\mathrm{C}^*(G; \mathcal{A})\rtimes_\delta \Gamma$ is naturally isomorphic to the $\mathrm{C}^*$-algebra of a Fell bundle constructed from the cocycle data.

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math.OA 2026-06-12

B-splines capture statistics of semigroup AF algebras

by Konrad Aguilar, Stephan Ramon Garcia +3 more

Rapidly growing AF algebras

Bratteli diagrams from numerical semigroups yield ensembles open to probabilistic analysis.

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We introduce certain families of AF algebras associated to Bratteli diagrams arising from numerical semigroup theory, a branch of combinatorics. Curry-Schoenberg B-splines, staples of computer-aided design, provide insight into the statistical properties of these algebras. This permits us to consider certain ensembles of "rapidly growing" AF algebras from a probabilistic viewpoint.

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math.FA 2026-06-11

Contractions dilate to normals with bound 2/√φ

by Ian Thompson

On the universal commuting dilation constant

Universal commuting dilation constant C₂ now satisfies 1.5438 ≲ C₂ ≲ 1.5724.

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The universal commuting dilation constant $C_d$ is the smallest constant $\alpha$ such that every $d$-tuple of contractions dilates to a commuting $d$-tuple of normal operators with norm at most $\alpha$. The work of several authors shows that $1.5438 \lesssim C_2 \leq 2$, and it has been asked on a few accounts whether $C_2 < 2$. We provide a positive answer that, in fact, produces a near optimal upper bound of $C_2 \leq \frac{2}{\sqrt{\phi}}$ where $\phi$ is the golden ratio. This tightens the gap on the universal commuting dilation constant to $1.5438 \lesssim C_2 \lesssim 1.5724$. We also tighten the known upper and lower bounds on $C_d$ for arbitrary $d$-tuples.

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math.OA 2026-06-11

Schubert calculus blocks uniform property Γ in a nuclear C*-algebra

by Andrew S. Toms

Schubert Calculus and uniform property Gamma

A quadratic obstruction from degeneracy loci persists through inductive limits and prevents trace comparison of projections.

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We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.

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math.OA 2026-06-11

Non-locally trivial W*-bundle exists with fixed II1 factor fibres

by Kiefer Mommaerts

A non-locally trivial W^*-bundle with fixed factorial fibres

Absence of uniform spectral gap obstructs local triviality even over zero-dimensional bases.

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In this paper we construct the first example of a non-locally trivial $\mathrm{W}^*$-bundle whose fibres are all isomorphic to some fixed $\mathrm{II}_1$ factor. This is achieved by introducing a notion of uniformly having spectral gap for $\mathrm{W}^*$-bundles. For bundles with fixed factorial fibres, the negation of having this uniform spectral gap property provides an obstruction for being locally trivial. This results in seemingly elementary examples of $\mathrm{W}^*$-bundles whose fibres are all isomorphic to some fixed factor but that are not locally trivial, even over spaces with covering dimension equal to zero.

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math.FA 2026-06-11

BPB property holds for low-density domains into C(K) on extremally disconnected K

by Tattwamasi Amrutam, Priyadarshi Dey +2 more

The Bishop--Phelps--Bollob\'as Property for Extremally Disconnected Ranges: Separable and Low-Density Domains

The result applies whenever domain density is below the Baire number, covers all separable spaces, and supplies an explicit quadratic modulu

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We prove a Bishop--Phelps--Bollob\'as theorem for operators into spaces of continuous scalar-valued functions on extremally disconnected compact Hausdorff spaces over both the real and complex scalar fields. The main result applies whenever the density character of the domain is strictly smaller than the Baire number of the underlying compact space. The proof also yields an explicit quadratic Bishop--Phelps--Bollob\'as modulus. In particular, every separable Banach space paired with such a function space has the Bishop--Phelps--Bollob\'as property for operators.

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math.GT 2026-06-11

Anti-tori require bi-reversible Mealy automata and aperiodic graphs

by David Pask

Full Mealy automata, complete square complexes, and anti-tori

The square complex Y_A from a full automaton contains an anti-torus exactly when both independent conditions hold.

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To a full $m\times n$ Mealy automaton $A$ we associate a bijection $\theta_A$, a one-vertex rank-two graph $F_{\theta_A}$, and a one-vertex $VH$-square complex $Y_A$ tiled by $mn$ Wang tiles. We prove that $Y_A$ contains an anti-torus if and only if $A$ is bi-reversible and $F_{\theta_A}$ is aperiodic. The two hypotheses are independent and play disjoint roles: bi-reversibility is exactly what makes $Y_A$ a complete square complex, so that its universal cover splits as a product of two trees and anti-tori can be discussed at all; and, within that setting, an anti-torus is precisely a period-free configuration in the two-sided path space of $F_{\theta_A}$, whose existence is the aperiodicity condition. Working at the level of configurations removes any appeal to the geometry of products of trees from the main equivalence; the geometric (loop-spanned) form of Wise is shown to be strictly stronger, the lamplighter being aperiodic with no loop-spanned anti-torus.

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math.OA 2026-06-11

Mixing upgrades relative biexactness to full biexactness

by Srivatsav Kunnawalkam Elayavalli, Zhiyuan Yang

Relative biexactness and mixing in von Neumann algebras

When subalgebras are both mixing and biexact, relative biexactness of the ambient algebra implies the algebra is biexact, producing new exam

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We develop a new technique to upgrade relative biexactness in general von Neumann algebras: suppose that $\{N_i\}_{i\in I}\subset M$ are mixing and biexact subalgebras of a separable von Neumann algebra with expectation, and if $M$ is biexact relative to $\{N_i\}_{i\in I}$, then $M$ is biexact. This result yields several new examples of biexact von Neumann algebras, notably including amalgamated free products. By generalizing the relative biexactness results of Hoshino to the von Neumann algebra setting and applying our result above along with certain bimodule computations, we in fact obtain, as an application, a new classification result for biexactness for graph products of finite dimensional von Neumann algebras. This yields significant extensions of prior works of Caspers-Borst, and Blufstein-Goldman-Oyakawa.

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math.QA 2026-06-10

O(2) subgroups label distinct 2-categorical Hilbert spaces

by Giovanni Ferrer, Lukas Müller +2 more

The many faces of higher Hilbert spaces

Fixed points under an O(2) action on 2-vector spaces recover the module categories of C*, W*, and H*-algebras via different choices of G.

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Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.

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math.CO 2026-06-10

Graph Laplacian pencils belong to finite free convolution class LC_n

by Thomas Sinclair

Finite free convolution via reproducing kernels and squarefree algebras

Their higher-order cumulants equal Hamiltonian cycle counts in induced subgraphs under the squarefree algebra model.

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We give a structural account of the finite free convolutions of Marcus, Spielman, and Srivastava in terms of reproducing kernel inner products on polynomial spaces and a multilinear model over the squarefree algebra. In this model, additive convolution becomes algebra multiplication, and the nilpotent logarithm linearizes it, recovering the finite free cumulants of Arizmendi and Perales. This perspective leads to a class $\mathcal{LC}_n$ of multilinear polynomials characterized by nonpositivity of higher-order cumulants, closed under additive convolution and satisfying several key permanence properties associated with negatively dependent measures. We show that every graph Laplacian pencil belongs to this class, with higher-order cumulants given by Hamiltonian cycle counts in induced subgraphs.

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math.OA 2026-06-10

q-Gaussian C*-algebras are simple with unique trace

by Tattwamasi Amrutam, David Jekel +1 more

Simplicity of q-Gaussian C*-algebras

Rapid decay and spectral gap estimates from free probability establish the Dixmier averaging property for q in (-1,1).

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We show that q-Gaussian $C^*$-algebras for $q \in (-1, 1)$ have the Dixmier averaging property, and hence are simple with a unique trace. We argue by combining rapid decay and spectral gap estimates for the commutators with the generators, which are obtained from free probability.

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math.OA 2026-06-09

Hyperbolic groups force amenable subalgebras inside maximal ones

by Juan Felipe Ariza Mejia, Ionut Chifan +2 more

Amenable absorption in von Neumann algebras of hyperbolic groups

Any amenable Q intersecting L(H) diffusely must sit inside L(H) for maximal amenable H.

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We prove that the von Neumann algebra $\cL(G)$ associated with any hyperbolic group $G$ satisfies the following \emph{amenable absorption property}: for any infinite maximal amenable subgroup $H \leqslant G$ and any amenable von Neumann subalgebra $\mathcal{Q} \subset \cL(G)$ with diffuse intersection with $\cL(H)$, one must have $\mathcal{Q} \subset \cL(H)$. This strengthens a result of Boutonnet and Carderi \cite{BC2}. We also establish similar amenable absorption results for the broader class of acylindrically hyperbolic groups, including relatively hyperbolic groups, mapping class groups, and limit groups.

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math.OA 2026-06-09

C*-simple groups have exactly one amenable recurrent subalgebra

by Tattwamasi Amrutam, Pierre Fima +1 more

Uniformly recurrent subalgebras in finite von Neumann algebras

In crossed products with amenable coefficients, a group is C*-simple precisely when the only amenable uniformly recurrent subalgebra contain

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We introduce the notion of a uniformly recurrent subalgebra (URA) for a trace-preserving action of a countable discrete group $\Gamma$ on a finite von Neumann algebra $M$, providing an operator-algebraic counterpart to the theory of uniformly recurrent subgroups (URS). We also show that the Effros-Mar\'echal space $\text{Sub}(M)$ is compact if and only if $M$ lacks a diffuse direct summand. Leveraging this, we show that URAs can exhibit arbitrary topological complexity and construct exotic URAs homeomorphic to any prescribed minimal Polish space. In the context of crossed products $M \rtimes \Gamma$ with amenable coefficients, we utilize URAs to formulate a new characterization of C*-simplicity, proving that $\Gamma$ is C*-simple if and only if the only amenable URA of the crossed product containing $M$ is $\{M\}$. Finally, to bypass the failure of compactness in $\text{Sub}(M)$, we develop a generalized state-space machinery using Baire-category methods on the weak-* compact space of trace-extending states. This construction captures compact, discrete, and exotic URAs, while recovering the classical URS framework as a special case.

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math.OA 2026-06-09

Selfless C*-algebras are either purely infinite or stably finite

by Miles Gould

The Selfless Dichotomy

Nonfaithful cases prove purely infinite and simple, so all such algebras are pure.

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The purpose of this note is to address the gap in the stably finite/purely infinite dichotomy of selfless $C^*$-probability spaces. In particular, we show that nonfaithful selfless $C^*$-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless $C^*$-algebra is either purely infinite or stably finite. Notably, this shows that every selfless $C^*$-algebra is pure. To accomplish this, we show that infinite reduced free products of $C^*$-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having resolved the selfless dichotomy, we improve existing permanence properties of selfless $C^*$-probability spaces, make progress on a conjecture of Choda and Dykema, and produce several new isomorphisms arising from reduced free products.

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math.CT 2026-06-09

Centre comonad empties state spaces of non-commutative algebras

by Joey Woo

The Degeneracy of the Centre Comonad Model and the Precomposition Obstruction for Quantum Modalities on Presheaf Topoi

It sends their representables to the empty presheaf, makes Day convolution cartesian, and collapses linear logic to classical logic.

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The centre comonad model provided the first concrete cohesive linear $\infty$-topos, settling an open problem of Schreiber. However, the model is degenerate: the quantum modality annihilates all non-commutative algebras, and the associated linear logic collapses to classical cartesian logic. In this paper we give a complete mathematical diagnosis of this degeneracy. We prove that the centre comonad sends the representable sheaf of a simple non-commutative algebra to the empty presheaf, and that the state space of any such algebra is empty. We then prove that the Day convolution on the classical core is cartesian, forcing the Seely isomorphism to hold trivially and collapsing the linear logic. We isolate the structural reason behind this collapse: whenever the opposite of the classical core is monoidally equivalent to a cartesian monoidal category, any coreflective precomposition comonad will exhibit the same degeneracy. We conclude that a non-degenerate quantum modality must be constructed without precomposition, and we briefly discuss possible directions.

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math.OA 2026-06-09

Equivariant KK-theory computes homotopy groups of automorphism groups

by Hiroro Kamikawa

The homotopy groups of the equivariant automorphism group of Kirchberg algebras with compact group actions and equivariant Dadarlat-Pennig theory

For isometrically shift-absorbing compact group actions on Kirchberg algebras, this extends Dadarlat's result to the equivariant setting.

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In the first half of this paper, we describe the homotopy groups of the equivariant automorphism group of Kirchberg algebras with isometrically shift-absorbing actions of compact groups in terms of equivariant KK-theory. This provides an equivariant version of Dadarlat's result. In the second half, we present a unified treatment of the equivariant Dadarlat-Pennig theory for strongly self-absorbing actions.

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math-ph 2026-06-09

Hard-core Markov measure yields hyperfinite factor with type fixed by κ

by Farrukh Mukhamedov, Yuri Suhov

Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model

The GNS algebra is II_1 when κ=1 and III_λ with λ=min{κ,κ^{-1}} otherwise, κ=q/p² from the transition matrix.

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We study a quasi-product state associated with the one-dimensional hard-core Gibbs measure. After coding the model by the topological Markov chain, we construct the standard path $AF$-algebra of admissible hard-core words and show that the stationary Markov measure induces on it a faithful diagonal state in the sense of Evans. We then analyze the von Neumann algebra generated by the corresponding GNS representation. The resulting algebra is a hyperfinite factor, and its type is determined by the single parameter $\kappa=q/p^2,$ where $\begin{pmatrix}p&q\\ 1&0\end{pmatrix}$ is the transition matrix of the Markov chain. More precisely, the factor is of type $\mathrm{II}_1$ when $\kappa =1$, and of type $\mathrm{III}_{\l}$ with $ \l=\min\{\kappa,\kappa^{-1}\}$ for $\k\neq 1$. We also specify the centralizer and the weight flow for the resulting factor.

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math.AT 2026-06-08

Finite-type equivariant bundles extend across compactifications

by Alexandru Chirvasitu

Equivariant compactifications, trivial embeddability and finite type

They become finite-type after structure group extension along Lie group embeddings and satisfy local or K-theoretic conditions under isotrop

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We characterize finite-type $\mathbb{G}$-principal $\mathbb{U}$-equivariant bundles on normal $\mathbb{U}$-spaces for compact Lie groups $\mathbb{U}$ and $\mathbb{G}$, in several ways, including (a) their extensibility across the $\mathbb{U}$-equivariant compactification $\beta_{\mathbb{U}}X$ and (b) their becoming finite-type upon extending the structure group along at least one $\mathbb{U}$-equivariant compact-Lie-group embedding $\mathbb{G}\le \mathbb{K}$. This generalizes non-equivariant results of Phillips and the author's characterizing finite-type matrix-algebra bundles, upon specializing $\mathbb{G}$ to projective unitary groups. When the $\mathbb{U}$-action on $X$ has virtually abelian isotropy, matrix-algebra equivariant bundles are also finite-type precisely when, locally over a finite open $\mathbb{U}$-cover, they are tensor factors of trivial matrix bundles. In a $K$-theoretic offshoot we prove that for $\mathbb{U}$-actions with finite isotropy groups on compact Hausdorff spaces $X$ equivariant vector bundles $\mathcal{E}\to X$ are factors of trivial bundles $K$-theoretically: there is a class $a\in K_{\mathbb{U}}(X)$ with $[\mathcal{E}]a$ the class of a bundle induced by a $\mathbb{U}$-representation (which furthermore can be chosen so as to restrict to isotropy groups to multiples of the regular representations). This generalizes a result of Donovan and Karoubi.

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math.OA 2026-06-08

Hyperfinite II1 factor is Ulam stable in trace norm

by Vadim Alekseev, Andreas Thom

The hyperfinite II₁-factor is Ulam stable

Sufficiently multiplicative maps become genuine homomorphisms after small target amplification and the factor is isolated among II1 factors.

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We prove Ulam stability of the hyperfinite II$_1$-factor with respect to the trace norm on the operator-norm unit ball. More precisely, every sufficiently additive, multiplicative, unital, $*$-preserving map from the hyperfinite II$_1$-factor-factor into a II$_1$-factor-factor von Neumann algebra is uniformly close, after passing to a small amplification of the target, to a genuine unital $*$-homomorphism. As a key finite-dimensional ingredient, we establish a dimension-free stability theorem for matrix algebras in the same trace-norm setting. As an application, we show that the hyperfinite II$_1$-factor is isolated among II$_1$-factors with respect to sufficiently accurate approximate $*$-isomorphisms.

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math.FA 2026-06-05

Weighted and unweighted multiplier algebras coincide on amenable groups

by Mahmood Alaghmandan, Olof Giselsson +2 more

Multipliers of Beurling-Fourier algebras

The reduced Beurling-Fourier-Stieltjes algebra equals the cb-multipliers of the weighted Fourier algebra precisely when G is amenable; embed

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For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.

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math.QA 2026-06-05

Equivalence equates twisted reps of net to fixed-point reps

by Adrià Marín-Salvador

Balanced tensor categories of representations of fixed-points conformal nets

G-equivariantization of G-crossed category matches Rep of A^G as balanced W*-tensor categories for any finite faithful action.

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Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let $\text{Rep}^G(\mathcal{A})$ be the $G$-crossed balanced $\mathrm{W}^*$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced $\mathrm{W}^*$-tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ between the $G$-equivariantization of $\text{Rep}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net $\mathcal{A}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

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math.CT 2026-06-05

Planar higher-rank trees have rank at most four

by David Pask

Finite connected singly connected locally convex non-degenerate cases cannot exceed rank four, via K5 subdivision

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We prove that a finite, connected, singly connected, locally convex higher-rank tree whose $1$-skeleton is planar and which is \emph{non-degenerate}, in the sense that every edge of each colour forms a commuting square with every other colour, has rank at most four. Under these hypotheses this establishes the planarity conjecture stated in \cite{Pask}. The obstruction side of the argument uses only the non-planarity of $K_5$; it makes no appeal to the four-colour theorem. The engine is a monotonicity property of the set of colours emitted at a vertex (``backward propagation''), which forces, in any finite singly connected non-degenerate $k$-graph, a single vertex emitting all $k$ colours; once $k\ge 5$, local convexity manufactures a subdivision of $K_5$ at such a vertex.

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math.OA 2026-06-05

Operator algebras with commutative diagonals are rigid under stable isomorphism

by Elias G. Katsoulis, Feifei Miao +2 more

Rigidity for Isomorphisms between Operator Algebras with Commutative Diagonals

Isometric isomorphism coincides with stable isomorphism for multiplicity-free CSL algebras and semicrossed products of commutative C*-algebr

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We show that two families of operator algebras, the CSL algebras of multiplicity free CSLs and the semicrossed products of commutative C$^*$-algebras, demonstrate a strong form of rigidity with respect to isometric isomorphisms. Specifically, the isomorphism class of any such algebra remains unchanged within its family, even if we allow for isomorphism after tensoring with operator algebras containing the compact operators. For semicrossed products of commutative C$^*$-algebras, the same conclusion holds even when tensoring with operator algebras whose diagonals are irreducibly acting. Collectively, these results imply rigidity with respect to stable isomorphisms: two algebras are isometrically isomorphic if and only if they are stably isomorphic.

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math.CT 2026-06-05

Three invariants fix every spectral propagation rule

by Shih-Yu Chang

A Universal Theory of Spectral Propagation for Compositional Operator Networks

Operadic spectra, derivatives and residues alone determine how spectra combine in any compositional system.

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Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum circuits. This paper develops a universal theory governed by three invariants: the operadic spectrum (local spectral data), spectral derivatives (perturbation sensitivity), and interaction residue (emergent interface-generated content). We prove three main theorems: the Spectral Propagation Theorem decomposes global output into propagated local spectra, residues, and derivative corrections; the Stability Theorem introduces the SOC stability radius and condition number; and the Universality Theorem shows any reasonable propagation rule is uniquely determined by the three invariants. These results provide a coordinate-free, representation-invariant language for spectral analysis of compositional operator systems.

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