arXiv subjects

Adam Dor-On

Publications and source records attributed to Adam Dor-On.

At least 19 recordsLinked to original sources

arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗

arXiv2026

Obstructions to the full Hao-Ng isomorphism

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

math.OA↗

arXiv2026

Space-time boundaries for random walks and their application to operator algebras

We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(\Gamma, \mu)$ with spectral radius $\rho$ and relate it to several classical compactifications of $\Gamma$. Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the $0$-Martin boundary, which governs the behaviour of $\infty$-harmonic functions, and show that the $0$-Martin kernels arise as rescaled limits of $\lambda$-Martin kernels as $\lambda\rightarrow 0$. For symmetric random walks on hyperbolic groups, the $0$-Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal $\lambda$-Martin boundaries over $\lambda\in [0, \rho^{-1}]$ with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk $(\Gamma, \mu)$ coincides with its Toeplitz $C^*$-algebra.

math.PR↗

arXiv2025

Products and factorization in operator systems

We study unital operator spaces endowed with a partially defined product. We give a matrix-norm characterization of such products that allows for a representation theorem where the partial product is realized as composition of operators on Hilbert space. We study product-respecting C*-covers, including a universal product C*-cover, and product quotients. We show that for the Haagerup tensor product of unital operator spaces remains injective, while projectivity holds relative to product quotients. Moreover, we identify the commuting tensor product as a complete product quotient of the Haagerup tensor product. Our framework yields new factorization norm formulas for a variety of product structures, as well as an intrinsic trace-extension criterion that resolves a question posed by Sinclair. Our work unifies and extends tensor products for operator systems, with applications in quantum information theory.

math.OA↗

arXiv2025

On almost commuting unitary matrices

A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately commuting unitary matrices to be close, in a dimension-independent way, to genuinely commuting unitary matrices. In this paper, under the vanishing of the said obstruction, we obtain effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices.

math.OA↗

arXiv2025

The Hao-Ng isomorphism theorem for reduced crossed products

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate $\mathrm{C}^*$-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction. This is done by proving that the reduced crossed product of an operator algebra commutes with the C*-envelope, which relies on refined W*-dynamical covers of C*-dynamical systems, unitary implementation of W*-dynamical systems, and an operator-valued extension of Maharam's lifting theorem.

math.OA↗

arXiv2024

Shift equivalence relations through the lens of C*-correspondences

We continue the study of shift equivalence relations from the perspective of C*-bimodule theory. We study emerging shift equivalence relations following work of the second-named author with Carlsen and Eilers, both in terms of adjacency matrices and in terms of their C*-correspondences, and orient them when possible. In particular, we show that if two regular C*-correspondences are strong shift equivalent, then the intermediary C*-correspondences realizing the equivalence may be chosen to be regular. This result provides the final missing piece in answering a question of Muhly, Pask and Tomforde, and is used to confirm a conjecture of Kakariadis and Katsoulis on shift equivalence of C*-correspondences.

math.OA↗

arXiv2024

Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras

We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Va{\v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Corti\~nas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.

math.RA↗

arXiv2024

Equivariant homotopy classification of graph C*-algebras

We show that shift equivalence of essential adjacency matrices coincides with gauge-equivariant homotopy equivalence of their stabilized graph C*-algebras. This provide the first equivalent formulation of shift equivalence of essential matrices in terms of gauge actions on graph C*-algebras. Our proof uses bicategory theory for C*-bimodules developed by Meyer and Sehnem, allowing us to avoid the use of K-theory classification of C*-algebras.

math.OA↗

arXiv2024

Arveson's hyperrigidity conjecture is false

Arveson's hyperrigidity conjecture predicts that if the non-commutative Choquet boundary of a separable operator system $\mathcal{S}$ is the entire spectrum of its generated C*-algebra $\mathcal{B}$ then $\mathcal{S}$ is hyperrigid in $\mathcal{B}$. We provide a counterexample to the conjecture with a C*-algebra $\mathcal{B}$ of type I generated by a single operator.

math.OA↗

arXiv2023

Normal coactions extend to the C*-envelope

We show that a normal coaction of a discrete group on an operator algebra extends to a normal coaction on the C*-envelope. This resolves an open problem attempted by several experts in the area, and provides a more direct proof of a prominent result of Sehnem. As an application, we resolve a question of X. Li, where we identify the C*-envelopes of the operator algebras of groupoid-embeddable categories and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable.

math.OA↗

arXiv2023

Ratio-limit boundaries for random walks on relatively hyperbolic groups

We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We extend, as well as determine significant limitations of, a strategy employed by Woess for computing ratio-limit boundaries for the class of hyperbolic groups. On the one hand we employ results of the second and third authors to adapt this strategy to spectrally non-degenerate random walks, and show that the closure of minimal points in $R$-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. On the other hand we show that the general strategy can fail when the random walk is spectrally degenerate and adapted on a free product. Using our results, we are able to extend a theorem of the first author beyond the hyperbolic case and establish the existence of a co-universal quotient for Toeplitz C*-algebras arising from random walks which are spectrally non-degenerate on relatively hyperbolic groups. Finally, we exhibit an example of a relatively hyperbolic group carrying two random walks such that the ratio limit boundaries are not equivariantly homeomorphic and no two equivariant quotients of their respective Toeplitz C*-algebras are equivariantly $*$-isomorphic.

math.GR↗

arXiv2021

Toeplitz quotient C*-algebras and ratio limits for random walks

We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*-algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique smallest symmetry-equivariant quotient C*-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C*-algebras of subproduct systems.

math.OA↗

arXiv2021

Finite-dimensional approximations and semigroup coactions for operator algebras

The residual finite-dimensionality of a $\mathrm{C}^*$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of whether residual finite-dimensionality of an operator algebra is inherited by its maximal $\mathrm{C}^*$-cover, which we resolve in many cases of interest.

math.OA↗

arXiv2020

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

math.OA↗

arXiv2020

Shift equivalences through the lens of Cuntz-Krieger algebras

Motivated by Williams' problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper.

math.OA↗

arXiv2019

Doob equivalence and non-commutative peaking for Markov chains

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

math.OA↗

arXiv2019

Classification of irreversible and reversible Pimsner operator algebras

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^*$-algebras with additional $C^*$-algebraic structure. Our approach naturally applies to algebras arising from $C^*$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

math.OA↗