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Bhishan Jacelon

Publications and source records attributed to Bhishan Jacelon.

16 recordsLinked to original sources
arXiv2026

A $\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem

We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $τ_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $σ(x)=σ(y)$ is convex. Consequently, $d_{U,2}$ equips the set of approximate unitary equivalence classes of contractive normal elements of $\mathcal{M}$ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.

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arXiv2026

Brown-Voiculescu entropy revisited

Aided by the tools and outlook provided by modern classification theory, we take a new look at the Brown-Voiculescu entropy of endomorphisms of nuclear C*-algebras. In particular, we introduce `coloured' versions of noncommutative topological entropy suitable for C*-algebras A of finite nuclear dimension or finite decomposition rank. In the latter case, assuming further that A is simple, separable, unital, satisfies the UCT and has finitely many extremal traces, we prove a variational type principle in terms of quasidiagonal approximations relative to this finite set of traces. Building on work of Kerr, we also show that infinite entropy occurs generically among endomorphisms and automorphisms of certain classifiable C*-algebras that function as noncommutative spaces of observables of topological manifolds.

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arXiv2025

Tracially lyriform $\mathrm{C}^*$-algebras

Quantum metric Choquet simplices are special kinds of compact quantum metric spaces designed for distance measurement in and around the category of stably finite Elliott-classifiable $\mathrm{C}^*$-algebras. The primary objective of this article is to introduce versions of these structures for which the associated tracial metrics need not be induced by Lipschitz seminorms and may induce strictly stronger topologies than the weak$^*$-topology. The resulting category of 'tracially lyriform $\mathrm{C}^*$-algebras' behaves well with respect to sequential inductive limits and accommodates the full family of classical $p$-Wasserstein metrics on probability spaces, including $p=\infty$. Examples of projectionless, classifiable tracial Wasserstein spaces are built as noncommutative spaces of observables of certain compact length spaces, including: fractals like the Sierpiński gasket, the Sierpiński carpet and the Menger sponge; finite-dimensional Alexandrov spaces with two-sided curvature bounds; and metric spaces like simplicial spheres and balls that are Lipschitz equivalent to Riemannian. These simplicial structures are used as building blocks that furnish arbitrary simple inductive limits of prime dimension drop algebras with tracial lyriform structure. Appealing to optimal transport theory, we study the geometry and statistics of the spaces of embeddings of these building blocks and their limits into suitable classifiable $\mathrm{C}^*$-algebras like the Jiang-Su algebra $\mathcal{Z}$ or the universal UHF algebra $\mathcal{Q}$.

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arXiv2025

Applications of $\mathrm{C}^*$-classification

We provide some background on the category of classifiable $\mathrm{C}^*$-algebras, whose objects are infinite-dimensional, simple, separable, unital $\mathrm{C}^*$-algebras that have finite nuclear dimension and satisfy the universal coefficient theorem, and describe some applications of the classification of objects in, and morphisms into, this category.

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arXiv2025

Quantum metric Choquet simplices

Precipitating a notion emerging from recent research, we formalise the study of a special class of compact quantum metric spaces. Abstractly, the additional requirement we impose on the underlying order unit spaces is the Riesz interpolation property. In practice, this means that a 'quantum metric Choquet simplex' arises as a unital $\mathrm{C}^*$-algebra $A$ whose trace space is equipped with a metric inducing the $w^*$-topology, such that tracially Lipschitz elements are dense in $A$. This added structure is designed for measuring distances in and around the category of stably finite classifiable $\mathrm{C}^*$-algebras, and in particular for witnessing metric and statistical properties of the space of approximate unitary equivalence classes of unital embeddings of $A$ into a stably finite classifiable $\mathrm{C}^*$-algebra $B$. As for examples, we recall the construction of classifiable $\mathrm{C}^*$-algebraic quantum metric Bauer simplices that function as noncommutative spaces of observables of compact connected metric spaces $(X,ρ)$. We also explain how to build non-Bauer examples by forming 'tracial quantum crossed products' associated with topological dynamical systems on $(X,ρ)$, and we use classification to show that continuous fields of quantum spaces are obtained by continuously varying either the dynamics or the metric. In the case of deformed isometric actions, we show that equivariant Gromov-Hausdorff continuity implies fibrewise continuity of the quantum structures with respect to Rieffel's quantum Gromov-Hausdorff distance. As an example, we present a field of deformed tracial rotation algebras whose fibres are continuous with respect to a quasimetric that we call the quantum intertwining gap.

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arXiv2025

Operator $K$-theoretic analysis of random adjacency matrices

We appeal to results from combinatorial random matrix theory to deduce that various random graph $\mathrm{C}^*$-algebras are asymptotically almost surely Kirchberg algebras with trivial $K_1$. This in particular implies that, with high probability, the stable isomorphism classes of such algebras are exhausted by variations of Cuntz algebras that we term 'Cuntz polygons'. These probabilistically generic algebras can be assembled into a Fraïssé class whose limit structure $\mathbb{G}$ is consequently relevant to any $K$-theoretic analysis of finite graph $\mathrm{C}^*$-algebras. We also use computer simulations to experimentally verify the behaviour predicted by theory and to estimate the asymptotic probabilities of obtaining stable isomorphism classes represented by actual Cuntz algebras. These probabilities depend on the frequencies with which the Sylow $p$-subgroups of $K_0$ are cyclic and in some cases can be computed from existing theory. For random symmetric $r$-regular multigraphs, current theory can describe these frequencies for finite sets of odd primes $p$ not dividing $r-1$. A novel aspect of the collected data is the observation of new heuristics outside of this case, leading to a conjecture for the asymptotic probability of these graphs yielding $\mathrm{C}^*$-algebras stably isomorphic to Cuntz algebras. For other models of random multigraphs including Bernoulli (di)graphs, the data also allow us to estimate and heuristically explain the (surprisingly high) asymptotic probabilities of exact isomorphism to a Cuntz algebra. Recognising the role played by Cuntz--Krieger algebras in the theory of symbolic dynamics, we also collect supplemental data to estimate (and in some cases, actually compute) the asymptotic probability of a random subshift of finite type being flow equivalent to a full shift.

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arXiv2025

Even the vague specification property implies density of ergodic measures

We prove that if a topological dynamical system $(X,T)$ is surjective and has the vague specification property, then its ergodic measures are dense in the space of all invariant measures. The vague specification property generalises Bowen's classical specification property and encompasses the majority of the extensions of the specification property introduced so far. The proof proceeds by first considering the natural extension $X_T$ of $(X,T)$ as a subsystem of the shift action on the space $X^\mathbb{Z}$ of $X$-valued biinfinite sequences. We then construct a sequence of subsystems of $X^\mathbb{Z}$ that approximate $X_T$ in the Hausdorff metric induced by a metric compatible with the product topology on $X^\mathbb{Z}$. The approximating subsystems consist of $δ$-chains for $δ$ decreasing to $0$. We show that chain mixing implies that each approximating system possesses the classical periodic specification property. Furthermore, we use vague specification to prove that our approximating subsystems of $X^\mathbb{Z}$ converge to $X_T$ in the Hausdorff metric induced the Besicovitch pseudometric. It follows that the simplices of invariant measures of these subsystems of $δ$-chains converge to the simplex of invariant measures of $X_T$ with respect to a generalised version of Ornstein's $\bar{d}$ metric. What is more, the density of ergodic measures is preserved in the limit. The proof concludes by observing that the simplices of invariant measures for $X_T$ and $(X,T)$ coincide. The approximation technique developed in this paper appears to be of independent interest.

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arXiv2023

Metrics on trace spaces

This article continues the investigation of the tracial geometry of classifiable $\mathrm{C}^*$-algebras that have real rank zero and stable rank one. Using the language of optimal transport, we describe several situations in which the distance between unitary orbits of $^*$-homomorphisms into such algebras can be computed in terms of tracial data. The domains we consider are certain (noncommutative) CW complexes, and the measurement is relative to a family of self-adjoint elements that are in a suitable sense tracially Lipschitz. As another application of the utility of this Lipschitz structure, we show how such elements can be repurposed to witness statistical features of endomorphisms in the classifiable category, in particular the tracial version of the (almost-sure) central limit theorem.

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arXiv2023

Chaotic tracial dynamics

The action on the trace space induced by a generic automorphism of a suitable finite classifiable C*-algebra is shown to be chaotic and weakly mixing. Model C*-algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe KK-contractible stably projectionless C*-algebras as crossed products.

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arXiv2023

Random amenable $\mathrm{C}^*$-algebras

What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most $k$ extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz-Krieger algebra is purely infinite and simple, and what can be said about the distribution of its $K$-theory? By constructing $\mathrm{C}^*$-algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.

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arXiv2022

Stably projectionless Fraïssé limits

We realise the algebra $\mathcal W$, the algebra $\mathcal Z_0$ and the algebras $\mathcal Z_0\otimes A$, where $A$ is a unital UHF algebra as Fraïssé limits of suitable classes of structures. In doing so, we show that such algebras are generic objects without the use of any classification result.

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arXiv2021

Optimal transport and unitary orbits in C*-algebras

Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C*-algebras. We combine these two seemingly unrelated disciplines to make progress on a classical problem of Weyl. In particular, we show how results from the Elliott classification programme can be used to translate continuous transport of spectral measures into optimal unitary conjugation in C*-algebras. As a consequence, whenever two normal elements of a sufficiently well-behaved C*-algebra share a spectrum amenable to such continuous transport, and have trivial K_1-class, the distance between their unitary orbits can be computed tracially.

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arXiv2013

Z-stability, finite dimensional tracial boundaries and continuous rank functions

We observe that a recent theorem of Sato, Toms-White-Winter and Kirchberg-Rordam also holds for certain nonunital C*-algebras. Namely, we show that an algebraically simple, separable, nuclear, nonelementary C*-algebra with strict comparison, whose cone of densely finite traces has as a base a Choquet simplex with compact, finite dimensional extreme boundary, and which admits a continuous rank function, tensorially absorbs the Jiang-Su algebra Z.

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arXiv2012

A simple, monotracial, stably projectionless C*-algebra

We construct a simple, nuclear, stably projectionless C*-algebra W which has trivial K-theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C*-algebras as an analogue of the Cuntz algebra O_2. In this context, we show that every nondegenerate endomorphism of W is approximately inner and we construct a trace-preserving embedding of W into the central sequences algebra M(W)_\infty \cap W'. We conjecture that W\otimes W is isomorphic to W and we note some implications of this, for example that W would be absorbed tensorially by a certain class of nuclear, stably projectionless C*-algebras. Finally, we explain why W may play some role in the classification of such algebras.

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arXiv2012

Z is universal

We use order zero maps to express the Jiang-Su algebra Z as a universal C*-algebra on countably many generators and relations, and we show that a natural deformation of these relations yields the stably projectionless algebra W studied by Kishimoto, Kumjian and others. Our presentation is entirely explicit and involves only *-polynomial and order relations.

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