Gauge-invariant Ideals Research Articles

Ideal Structure of Nica-Toeplitz Algebras

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\mathcal {T}(X)$$\\end{document} of a product system (A, X) over Nn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}^n$$\\end{document}. We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} to A. The main result is a classification of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.

Weighted Cuntz–Krieger Algebras

Let E be a finite directed graph with no sources or sinks and write \(X_E\) for the graph correspondence. We study the \(C^*\)-algebra \(C^*(E,Z):=\mathcal {T}(X_E,Z)/\mathcal {K}\) where \(\mathcal {T}(X_E,Z)\) is the \(C^*\)-algebra generated by weighted shifts on the Fock correspondence \(\mathcal {F}(X_E)\) given by a weight sequence \(\{Z_k\}\) of operators \(Z_k\in \mathcal {L}(X_{E^{k}})\) and \(\mathcal {K}\) is the algebra of compact operators on the Fock correspondence. If \(Z_k=I\) for every k, \(C^*(E,Z)\) is the Cuntz–Krieger algebra associated with the graph E. We show that \(C^*(E,Z)\) can be realized as a Cuntz–Pimsner algebra and use a result of Schweizer to find conditions for the algebra \(C^*(E,Z)\) to be simple. We also analyse the gauge-invariant ideals of \(C^*(E,Z)\) using a result of Katsura and conditions that generalize the conditions of subsets of \(E^0\) (the vertices of E) to be hereditary or saturated. As an example, we discuss in some details the case where E is a cycle.

Annihilator ideals of graph algebras

If I is a (two-sided) ideal of a ring R, we let $${\text {ann}}_l(I)=\{r\in R\mid rI=0\},$$ $${\text {ann}}_r(I)=\{r\in R\mid Ir=0\},$$ and $${\text {ann}}(I)={\text {ann}}_l(I)\cap {\text {ann}}_r(I)$$ be the left, the right and the double annihilators. An ideal I is said to be an annihilator ideal if $$I={\text {ann}}(J)$$ for some ideal J (equivalently, $${\text {ann}}({\text {ann}}(I))=I$$ ). We study annihilator ideals of Leavitt path algebras and graph $$C^*$$ -algebras. Let $$L_K(E)$$ be the Leavitt path algebra of a graph E over a field K. If I is an ideal of $$L_K(E),$$ it has recently been shown that $${\text {ann}}(I)$$ is a graded ideal (with respect to the natural grading of $$L_K(E)$$ by $$\mathbb Z$$ ). We note that $${\text {ann}}_l(I)$$ and $${\text {ann}}_r(I)$$ are also graded. If I is graded, we show that $${\text {ann}}_l(I)={\text {ann}}_r(I)={\text {ann}}(I)$$ and describe $${\text {ann}}(I)$$ in terms of the properties of a pair of sets of vertices of E, known as an admissible pair, which naturally corresponds to I. Using such a description, we present properties of E which are equivalent with the requirement that each graded ideal of $$L_K(E)$$ is an annihilator ideal. We show that the same properties of E are also equivalent with each of the following conditions: (1) the lattice of graded ideals of $$L_K(E)$$ is a Boolean algebra; (2) each closed gauge-invariant ideal of $$C^*(E)$$ is an annihilator ideal; (3) the lattice of closed gauge-invariant ideals of $$C^*(E)$$ is a Boolean algebra. In addition, we present properties of E which are equivalent with each of the following conditions: (1) each ideal of $$L_K(E)$$ is an annihilator ideal; (2) the lattice of ideals of $$L_K(E)$$ is a Boolean algebra; (3) each closed ideal of $$C^*(E)$$ is an annihilator ideal; (4) the lattice of closed ideals of $$C^*(E)$$ is a Boolean algebra.

$$C^*$$-algebras of generalized Boolean dynamical systems as partial crossed products

In this paper, we realize $$C^*$$ -algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a $$C^*$$ -algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of $$C^*$$ -algebras of generalized Boolean dynamical systems are themselves $$C^*$$ -algebras of generalized Boolean dynamical systems.

Left regular representations of Garside categories I. C*-algebras and groupoids

AbstractWe initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

The simplicity of the C*-algebras associated to arbitrary labeled spaces

In this paper, we consider the simplicity of the C*-algebra associated to an arbitrary weakly left-resolving labeled space (E, L, E), where E is the smallest non-degenerate accommodating set. We classify all gauge-invariant ideals of C*(E, L, E) and characterize minimality of (E, L, E) in terms of ideal structure of C*(E, L, E). Using these results, we prove simplicity results for C*-algebras associated to arbitrary labeled spaces.

Regular ideals of graph algebras

Let $C^*(E)$ be the graph C$^*$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^*(E)$. It is shown that when $E$ satisfies Condition (L) the regular ideals $C^*(E)$ are a class of gauge-invariant ideals which preserve Condition (L) under quotients. That is, we show that if $E$ satisfies Condition (L) then a regular ideal $J \unlhd C^*(E)$ is necessarily gauge-invariant. Further, if $J \unlhd C^*(E)$ is a regular ideal, it is shown that $C^*(E)/J \simeq C^*(F)$ where $F$ satisfies Condition (L).

Gauge-invariant ideals of C⁎-algebras of Boolean dynamical systems

We enlarge the class of C⁎-algebras of Boolean dynamical systems in order to include all weakly left-resolving normal labelled space C⁎-algebras in it. We prove a gauge-invariant uniqueness theorem and classify all gauge-invariant ideals of these C⁎-algebras of generalized Boolean dynamical systems and describe the corresponding quotients as C⁎-algebras of relative generalized Boolean dynamical systems.

C∗-Algebras for partial product systems over N

We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C∗-algebras and relative Cuntz-Pimsner algebras for them and show that the section C∗-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C∗-algebra.

C⁎-algebras associated to Boolean dynamical systems

The goal of these notes is to present the C⁎-algebra C⁎(B,L,θ) of a Boolean dynamical system (B,L,θ), that generalizes the C⁎-algebra associated to labelled graphs introduced by Bates and Pask, and to determine its simplicity, its gauge invariant ideals, as well as compute its K-Theory.

Ideal structure of crossed products by endomorphisms via reversible extensions of C*-dynamical systems

We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)⊥, and if J = ( ker α)⊥it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K -theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.

The structure of gauge-invariant ideals of labelled graph [formula omitted]-algebras

In this paper we consider the gauge-invariant ideal structure of a C ⁎ -algebra C ⁎ ( E , L , B ) associated to a set-finite, receiver set-finite and weakly left-resolving labelled space ( E , L , B ) , where L is a labelling map assigning an alphabet to each edge of the directed graph E with no sinks. It is obtained that if an accommodating set B is closed under relative complements, there is a one-to-one correspondence between the set of all hereditary saturated subsets of B and the gauge-invariant ideals of C ⁎ ( E , L , B ) . For this, we introduce a quotient labelled space ( E , L , [ B ] R ) arising from an equivalence relation ∼ R on B and show the existence of the C ⁎ -algebra C ⁎ ( E , L , [ B ] R ) generated by a universal representation of ( E , L , [ B ] R ) . Finally we give necessary and sufficient conditions for simplicity of certain labelled graph C ⁎ -algebras.

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

We prove that the classes of graph algebras, Exel-Laca algebras, and ultra- graph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C � (G) is isomorphic to a full corner of C � (E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

Ideal structure of C∗-algebras associated with C∗-correspondences

We study the ideal structure of C*-algebras arising from C*-correspondences. We prove that gauge-invariant ideals of our C*-algebras are parameterized by certain pairs of ideals of original C*-algebras. We show that our C*-algebras have a nice property that should be possessed by a generalization of crossed products. Applications to crossed products by Hilbert C*-bimodules and relative Cuntz?Pimsner algebras are also discussed.

Gauge-Invariant Ideals in theC*-Algebras of Finitely Aligned Higher-Rank Graphs

AbstractWe produce a complete description of the lattice of gauge-invariant ideals inC*(Λ) for a finitely alignedk-graph Λ. We provide a condition on Λ under which every ideal is gauge-invariant. We give conditions on Λ under whichC*(Λ) satisfies the hypotheses of the Kirchberg–Phillips classification theorem.

The ideal structure of the $C\sp *$-algebras of infinite graphs

We classify the gauge-invariant ideals in the $C^*$-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the $K$-theory of the $C^*$-algebras of arbitrary infinite graphs.