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Dilian Yang

Publications and source records attributed to Dilian Yang.

At least 19 recordsLinked to original sources
arXiv2026

Endomorphisms of the 2-adic ring $C^*$-algebra and its Weyl group

In this paper, we establish a one-to-one correspondence between a natural monoid and the endomorphisms of the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. As a consequence, we classify endomorphisms of $\mathcal{Q}_2$ with prescribed images and derive several criteria for the uniqueness of extensions of endomorphisms of a canonical copy of the Cuntz algebra $\mathcal{O}_2$ inside $\mathcal{Q}_2$ to endomorphisms of $\mathcal{Q}_2$. Moreover, we construct an example of an extendable automorphism of $\mathcal{O}_2$ that is not a composition of the flip-flop automorphism, the gauge automorphisms, and inner automorphisms, thereby providing negative answers to certain open questions. Using this explicit construction, we also show that the canonical image of $\operatorname{Aut}(\mathcal{Q}_2,\mathcal{O}_2)$ in the outer automorphism group $\operatorname{Out}(\mathcal{Q}_2)$ is non-abelian. Finally, we characterize the automorphisms of $\mathcal{Q}_2$ that globally preserve $C^*(u)$, and completely describe the Weyl group $\mathcal{W}(\mathcal{Q}_2, C^*(u))$. Consequently, several related open questions are answered affirmatively.

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arXiv2026

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

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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arXiv2025

Maximality and symmetry related to the \(2\)-adic ring \(C^*\)-algebra

The 2-adic ring $C^*$-algebra $\mathcal{Q}_2$ is the universal $C^*$-algebra generated by a unitary and an isometry satisfying certain relations. It contains a canonical copy of the Cuntz algebra $\mathcal{O}_2$. We show that $\mathcal{O}_2$ is a maximal $C^*$-subalgebra of $\mathcal{Q}_2$. Furthermore, we examine the structure of the fixed-point algebra under a periodic \(^*\)-automorphism $\sigma$ of $\mathcal{Q}_2$, which is extended from the flip-flop \(^*\)-automorphism of $\mathcal{O}_2$. We show that the maximality of $\mathcal{O}_2$ in $\mathcal{Q}_2$ extends to the crossed product $\mathcal{O}_2 \rtimes_{\sigma} \mathbb{Z}_2$ in $\mathcal{Q}_2 \rtimes_{\sigma} \mathbb{Z}_2$, and to the fixed-point algebra $\mathcal{O}_2^\sigma$ in $\mathcal{Q}_2^\sigma$. As a consequences of our main results, a few open questions concerning $\mathcal{Q}_2$ are resolved.

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arXiv2024

Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups

In this paper, we initiate the study of higher rank Baumslag-Solitar semigroups and their related C*-algebras. We focus on two extreme, but interesting, classes - one is related to products of odometers and the other is related to Furstenberg's $\times p,, \times q$ conjecture. For the former class, whose C*-algebras are studied by H. Li and the second author, we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting which encompasses (single-vertex) generalized Baumslag-Solitar semigroups. One of our main tools is from self-similar higher rank graphs and their C*-algebras.

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arXiv2024

Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs

We study the Wold decomposition for representations of a self-similar semigroup $P$ action on a directed graph $E$. We then apply this decomposition to the case where $P=\mathbb{N}$ to study the C*-envelope of the associated universal non-selfadjoint operator algebra $\mathcal{A}_{\mathbb{N}, E}$ by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of $\mathcal{A}_{\mathbb{N}, E}$ coincides with the self-similar C*-algebra $\mathcal{O}_{\mathbb{Z}, E}$.

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arXiv2024

Representing topological full groups in Steinberg algebras and C*-algebras

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.

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arXiv2021

Higman-Thompson Like Groups of Higher Rank Graph C*-Algebras

Let $Λ$ be a row-finite and source-free higher rank graph with finitely many vertices. In this paper, we define the Higman-Thompson like group $\Lht$ of the graph C*-algebra $\mathcal{O}_Λ$ to be a special subgroup of the unitary group in $Ø_Λ$. It is shown that $\Lht$ is closely related to the topological full groups of the groupoid associated with $Λ$. Some properties of $\Lht$ are also investigated. We show that its commutator group $\DLht$ is simple and that $\DLht$ has only one nontrivial uniformly recurrent subgroup if $Λ$ is aperiodic and strongly connected. Furthermore, if $Λ$ is single-vertex, then we prove that $\Lht$ is C*-simple and also provide an explicit description on the stabilizer uniformly recurrent subgroup of $\Lht$ under a natural action on the infinite path space of $Λ$.

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arXiv2020

Zappa-Szép actions of groups on product systems

Let $G$ be a group and $X$ be a product system over a semigroup $P$. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Szép product $P\bowtie G$. We define a Zappa-Szép action of $G$ on $X$ to be a collection of functions on $X$ that are compatible with both actions from $P\bowtie G$ in a certain sense. Given a Zappa-Szép action of $G$ on $X$, we construct a new product system $X\bowtie G$ over $P\bowtie G$, called the Zappa-Szép product of $X$ by $G$. We then associate to $X\bowtie G$ several universal C*-algebras and prove their respective Hao-Ng type isomorphisms. A special case of interest is when a Zappa-Szép action is homogeneous. This case naturally generalizes group actions on product systems in the literature. For this case, besides the Zappa-Szép product system $X\bowtie G$, one can also construct a new type of Zappa-Szép product $X \widetilde\bowtie G$ over $P$. Some essential differences arise between these two types of Zappa-Szép product systems and their associated C*-algebras.

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arXiv2020

Nuclearity of semigroup C*-algebras

We study the semigroup C*-algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P\cap P^{-1}=\{e\} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C*-algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group. Here we give a new definition of a controlled map and discuss examples from different sources. We apply our main result to establish nuclearity for semigroup C*-algebras of a class of one-relator semigroups, motivated by a recent work of Li, Omland and Spielberg. This includes all the Baumslag--Solitar semigroups. We also analyse semidirect products of weakly quasi-lattice ordered groups and use our theorem in examples to prove nuclearity of the semigroup C*-algebra. Moreover, we prove that the graph product of weak quasi-lattices is again a weak quasi-lattice, and show that the corresponding semigroup C*-algebra is nuclear when the underlying groups are amenable.

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arXiv2019

The ideal structures of self-similar $k$-graph C*-algebras

Let $(G, Λ)$ be a self-similar $k$-graph with a possibly infinite vertex set $Λ^0$. We associate a universal C*-algebra $\mathcal{O}_{G,Λ}$ to $(G,Λ)$. The main purpose of this paper is to investigate the ideal structures of $\mathcal{O}_{G,Λ}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $Λ^0$ and the set of all gauge-invariant and diagonal-invariant ideals of $\mathcal{O}_{G,Λ}$. Under some conditions, we characterize all primitive ideas of $\mathcal{O}_{G,Λ}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$-graph C*-algebras in depth.

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arXiv2018

KMS States of Self-Similar $k$-Graph C*-Algebras

Let $G$ be a countable discrete amenable group, and $Λ$ be a strongly connected finite $k$-graph. If $(G,Λ)$ is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra $Ø_{G,Λ}$ associated to $(G,Λ)$ is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group $\Per_{G,Λ}$ of $(G,Λ)$, depending on whether the Perron-Frobenius eigenvector of $Λ$ preserves the $G$-action. As applications of our main results, we also exhibit several classes of important examples.

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arXiv2018

Self-Similar $k$-Graph C*-Algebras

In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $Λ$, and associate it a universal C*-algebra $Ø_{G,Λ}$. We prove that $Ø_{G,Λ}$ can be realized as the Cuntz-Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then $Ø_{G,Λ}$ is shown to be isomorphic to a "path-like" groupoid C*-algebra. This facilitates studying the properties of $Ø_{G,Λ}$. We show that $Ø_{G,Λ}$ is always nuclear and satisfies the Universal Coefficient Theorem; we characterize the simplicity of $Ø_{G,Λ}$ in terms of the underlying action; and we prove that, whenever $Ø_{G,Λ}$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $Λ$ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

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arXiv2017

Boundary Quotient C*-algebras of Products of Odometers

In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of $k$ odometers over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph C*-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the boundary quotient C*-algebra. Some relations between the boundary quotient C*-algebra and the C*-algebra $\Q_\bN$ introduced by Cuntz are also investigated. As an easy consequence of our main results, it settles a boundary quotient C*-algebra constructed by Brownlowe-Ramagge-Robertson-Whittaker.

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arXiv2016

Cartan Subalgebras of Topological Graph Algebras and k-Graph C*-algebras

In this paper, two sufficient and necessary conditions are given. The first one characterizes when the boundary path groupoid of a topological graph without singular vertices has closed interior of its isotropy group bundle, and the second one characterizes when the path groupoid of a row-finite k-graph without sources has closed interior of its isotropy group bundle. It follows that the associated topological graph algebra and the associated k-graph C*-algebra have Cartan subalgebras due to a result of Brown-Nagy-Reznikoff-Sims-Williams.

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arXiv2016

Affine Actions and the Yang-Baxter Equation

In this paper, the relations between the Yang-Baxter equation and affine actions are explored in detail. In particular, we classify solutions of the Yang-Baxter equations in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the C*-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained.

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arXiv2015

Factoriality and type classification of \textsf{k}-graph von Neumann algebras

Let $\Fth$ be a single vertex \textsf{k}-graph, and $π_ω(Ø_θ)"$ be the von Neumann algebra induced from the GNS representation of a distinguished state $ω$ of its $\textsf{k}$-graph C*-algebra $Ø_θ$. In this paper, we prove the factoriality of $π_ω(Ø_θ)"$ and further determine its type, when either $\Fth$ has the little pull-back property, or the intrinsic group of $\Fth$ has rank $0$. The key step to achieve this is to show that the fixed point algebra of the modular action corresponding to $ω$ has a unique tracial state.

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arXiv2015

The interplay between $k$-graphs and the Yang-Baxter equation

In this paper, we initiate the study of the interplay between $k$-graphs and the Yang-Baxter equation. For this, we provide two very different perspectives. One one hand, we show that the set of all set-theoretic solutions of the Yang-Baxter equation is a special class of single-vertex $k$-graphs. As a consequence, we construct an infinite family of large solutions of the Yang-Baxter equation from an arbitrarily given one. On the other hand, we prove that all single-vertex $k$-graphs are YB-semigroups of square-free, involutive solutions of the Yang-Baxter equation. Other various connections are also investigated.

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