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Hossein Larki

Publications and source records attributed to Hossein Larki.

13 recordsLinked to original sources
arXiv2025

Minimality and effectiveness of the groupoid associated to a self-similar ultragraph

The notion of a self-similar ultragraph $(G,\mathcal{U},\varphi)$ and its $C^*$-algebra $\mathcal{O}_{G,\mathcal{U}}$ were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such $C^*$-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph $(G,\mathcal{U},\varphi)$. In particular, we obtain a result for simplicity of the $C^*$-algebras $\mathcal{O}_{G,\mathcal{U}}$ in a certain case.

math.OA↗
arXiv2023

Exel-Pardo algebras of self-similar $k$-graphs

We introduce the Exel-Pardo $*$-algebra $\mathrm{EP}_R(G,Λ)$ associated to a self-similar $k$-graph $(G,Λ,φ)$. We prove the $\mathbb{Z}^k$-graded and Cuntz-Krieger uniqueness theorems for such algebras and investigate their ideal structure. In particular, we modify the graded uniqueness theorem for self-similar 1-graphs, and then apply it to present $\mathrm{EP}_R(G,Λ)$ as a Steinberg algebra and to study the ideal structure.

math.RA↗
arXiv2020

A dichotomy for simple self-similar graph $C^\ast$-algebras

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo $C^*$-algebras $\mathcal{O}_{G,E}$ for countable self-similar graphs $(G,E,φ)$. In particular, we associate a specific ordinary graph $\widetilde{E}$ to $(G,E,φ)$ such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph $C^*$-algebra $C^*(\widetilde{E})$ imply that of $\mathcal{O}_{G,E}$. Among others, this follows a dichotomy for simple $\mathcal{O}_{G,E}$: if $(G,E,φ)$ contains no $G$-circuits, then $\mathcal{O}_{G,E}$ is stably finite; otherwise, $\mathcal{O}_{G,E}$ is purely infinite. Furthermore, Li and Yang recently introduced self-similar $k$-graph $C^*$-algebras $\mathcal{O}_{G,Λ}$. We also show that when $|Λ^0|<\infty$ and $\mathcal{O}_{G,Λ}$ is simple, then it is purely infinite.

math.OA↗
arXiv2019

Non-simple purely infinite Steinberg Algebras with applications to Kumjian-Pask algebras

In this paper, we characterize properly purely infinite Steinberg algebras $A_K(\mathcal{G})$ for strongly effective, ample Hausdorff groupoids $\mathcal{G}$. As an application, when $Λ$ is a strongly aperiodic $k$-graph, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian-Pask algebra $\text{KP}_K(Λ)$, which may be determined by the proper infiniteness of vertex idempotents. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of $\text{KP}_K(Λ)$. Furthermore, since the complex Steinberg algebra $A_\mathbb{C}(\mathcal{G})$ is a dense subalgebra of the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$, we focus on the problem that "when does the proper pure infiniteness of $A_\mathbb{C}(\mathcal{G})$ imply that of $C^*_r(\mathcal{G})$ in the $C^*$-sense?". In particular, we show that if the Kumjian-Pask algebra $\mathrm{KP}_{\mathbb{C}}(Λ)$ is purely infinite, then so is $C^*(Λ)$ in the sense of Kirchberg-Rørdam.

math.RA↗
arXiv2018

Primitive ideal space of Higher-rank graph $C^*$-algebras and decomposability

In this paper, we describe primitive ideal space of the $C^*$-algebra $C^*(Λ)$ associated to any locally convex row-finite $k$-graph $Λ$. To do this, we will apply the Farthing's desourcifying method on a recent result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain maximal ideals of $C^*(Λ)$. Furthermore, we study the decomposability of $C^*(Λ)$. We apply the description of primitive ideals to show that if $I$ is a direct summand of $C^*(Λ)$, then it is gauge-invariant and isomorphic to a certain $k$-graph $C^*$-algebra. So, we may characterize decomposable higher-rank $C^*$-algebras by giving necessary and sufficient conditions for the underlying $k$-graphs. Moreover, we determine all such $C^*$-algebras which can be decomposed into a direct sum of finitely many indecomposable $C^*$-algebras.

math.OA↗
arXiv2017

Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras

Let $\mathcal{G}$ be an ultragraph and let $C^*(\mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(\mathcal{G})$, we approach the quotient $C^*$-algebra $C^*(\mathcal{G})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-R$ø$rdam) via Fell bundles.

math.OA↗
arXiv2017

Quotients of Ultragraph C*-Algebras

Let $\mathcal{G}$ be an ultragraph and let $C^*(\mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(\mathcal{G})$, we analyze the structure of the quotient $C^*$-algebra $C^*(\mathcal{G})/I_{(H,B)}$. For simplicity's sake, we first introduce the notion of quotient ultragraph $\mathcal{G}/(H,B)$ and an associated $C^*$-algebra $C^*(\mathcal{G}/(H,B))$ such that $C^*(\mathcal{G}/(H,B))\cong C^*(\mathcal{G})/I_{(H,B)}$. We then prove the gauge invariant and the Cuntz-Krieger uniqueness theorems for $C^*(\mathcal{G}/(H,B))$ and describe primitive gauge invariant ideals of $C^*(\mathcal{G}/(H,B))$.

math.OA↗
arXiv2017

Prime and primitive Kumjian-Pask algebras

In this paper, prime as well as primitive Kumjian-Pask algebras $\mathrm{KP}_R(Λ)$ of a row-finite $k$-graph $Λ$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when $Λ$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ are determined.

math.RA↗
arXiv2017

Purely infinite simple Kumjian-Pask algebras

Given any finitely aligned higher-rank graph $Λ$ and any unital commutative ring $R$, the Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ is known as the higher-rank generalization of Leavitt path algebras. After characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela (and others), we focus in this article on the purely infinite simple ones. Briefly, we show that if $\mathrm{KP}_R(Λ)$ is simple and every vertex of $Λ$ is reached from a generalized cycle with an entrance, then $\mathrm{KP}_R(Λ)$ is purely infinite. We next prove a standard dichotomy for simple Kumjian-Pask algebras: in the case that each vertex of $Λ$ is reached only from finitely many vertices and $\mathrm{KP}_R(Λ)$ is simple, then $\mathrm{KP}_R(Λ)$ is either purely infinite or locally matritial. This result covers all unital simple Kumjian-Pask algebras.

math.RA↗
arXiv2012

Ideal structure of Leavitt path algebras with coefficients in a unital commutative ring

Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set of prime ideals and the set of primitive ideals of $\lr$ coincide.

math.RA↗
arXiv2012

Stable rank of Leavitt path algebras of arbitrary graphs

The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this for an arbitrary directed graph. In some parts, we proceed our computation as the row-finite case while in some parts we use the knowledge about row-finite setting by applying the desingularizing method due to Drinen and Tomforde. In particular, we characterize purely infinite simple quotients of a Leavitt path algebra.

math.RA↗