The ideal structure of the C*-algebras of infinite graphs

On the Lie ideals of C*-algebras

Various questions on Lie ideals of C*-algebras are investigated. They fall roughly under the following topics: relation of Lie ideals to closed two-sided ideals; Lie ideals spanned by special classes of elements such as commutators, nilpotents, and the range of polynomials; characterization of Lie ideals as similarity invariant subspaces.

On Invariants of $\text{C}^*$-algebras with the ideal property

In this paper, we consider $\text{C}^*$-algebras with the ideal property (the ideal property unifies the simple and real rank zero cases). We define two categories related the invariants of the $\text{C}^*$-algebras with the ideal property. And we showed that these two categories are in fact isomorphic. As a consequence, the Elliott's Invariant and the Stevens' Invariant are isomorphic for $\text{C}^*$-algebras with the ideal property.

Ideals in Operator Space Projective Tensor Product of $C^*$-algebras

For $C^*$-algebras $A$ and $B$, we prove the slice map conjecture for ideals in the operator space projective tensor product $A \hat\otimes B$. As an application, a characterization of prime ideals in the Banach $\ast$-algebra $A\hat\otimes B$ is obtained. Further, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\hat\otimes B$. It is also shown that the Banach $\ast$-algebra $A\hat\otimes B$ possesses Wiener property; and that, for a subhomogenous $C^*$-algebra $A$, $A\hat\otimes B$ is symmetric.

The Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras

We describe the Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras under certain conditions.

On the Primitive Ideals of Nest Algebras

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the Continuum Hypothesis). This provides for the first time concerete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras we show how primitive ideals can be categorized by their behaviour on the diagonal, and provide concrete examples of all types.

The Maximal Two-Sided Ideals of Nest Algebras

We give a necessary and sufficient criterion for an operator in a nest algebra to belong to a proper two-sided ideal of that algebra. Using this result, we describe the strong radical of a nest algebra, and give a general description of the maximal two-sided ideals.

Hausdorffifized algebraic $K_1$ group and invariants for $C^*$-algebras with the ideal property

A $C^*$-algebra $A$ is said to have the ideal property if each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. $C^*$-algebras with the ideal property are generalization and unification of real rank zero $C^*$-algebras and unital simple $C^*$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see the introduction), consisting of scaled ordered total $K$-group $(\underline{K}(A), \underline{K}(A)^{+},ΣA)_Λ$ (used in the real rank zero case), the tracial …

The Stable Ideals of a Continuous Nest Algebra II

We continue the study of the rich family of norm-closed, automorphism invariant ideals of a continuous nest algebra. First we present a unified framework which captures all stable ideals as the kernels of limits of diagonal compressions. We then characterize when two such limits give rise to the same ideal, and we obtain detailed information of the structure of sums and intersections of ideals.

Classification of graph C*-algebras with no more than four primitive ideals

We describe the status quo of the classification problem of graph C*-algebras with four primitive ideals or less.

Ideal structure of C*-algebras of singly generated dynamical systems

In this paper, we show that the set of all ideals of the C*-algebras of a singly generated dynamical system corresponds bijectively to the set of all subsets of the product of the space of the system and the circle satisfying three conditions.

Graph $C^\ast$-algebras with a $T_1$ primitive ideal space

We give necessary and sufficient conditions which a graph should satisfy in order for its associated $C^\ast$-algebra to have a $T_1$ primitive ideal space. We give a description of which one-point sets in such a primitive ideal space are open, and use this to prove that any purely infinite graph $C^\ast$-algebra with a $T_1$ (in particular Hausdorff) primitive ideal space, is a $c_0$-direct sum of Kirchberg algebras. Moreover, we show that graph $C^\ast$-algebras with a $T_1$ primitive ideal space canonically may be given the structure of a $C(\widetilde{\mathbb N})$-algebra, and that isomorp…

Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras

In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample gr…

The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.

Aperiodicity and the primitive ideal space of a row-finite $k$-graph $C^*$-algebra

We describe the primitive ideal space of the $C^{\ast}$-algebra of a row-finite $k$-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the $k$-graph in question.

一致 Roe 代数中几何理想的刚性

Rigidity for geometric ideals in uniform Roe algebras

本文研究有界几何离散度量空间的一致 Roe 代数中几何理想的刚性问题，证明了稳定同构的几何理想对应粗等价的粗空间，并探讨了 ghostly ideals 情形与开放问题。

A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we intr…

真无限C*-代数中的李理想

Lie ideals in properly infinite C*-algebras

该研究证明含单位元的真无限C*-代数中李理想与唯一双边理想交换子等价，得到实秩零无特征标C*-代数的类似结果，并应用于von Neumann代数解决了相关公开问题。

Ideals in Toeplitz Algebras

We determine the ideal structure of the Toeplitz C*-algebra on the bidisk.

A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras III, ideal structures

We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C^*-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph such that the associated C^*-algebra is prime but not primitive.

Ideals and structure of operator algebras

We continue the study of r-ideals, l-ideals, and HSA's in operator algebras. Some applications are made to the structure of operator algebras, including Wedderburn type theorems for a class of operator algebras. We also consider the one-sided $M$-ideal structure of certain tensor products of operator algebras.

Partial and Total Ideals of Von Neumann Algebras

A notion of partial ideal for an operator algebra is a weakening the notion of ideal where the defining algebraic conditions are enforced only in the commutative subalgebras. We show that, in a von Neumann algebra, the ultraweakly closed two-sided ideals, which we call total ideals, correspond to the unitarily invariant partial ideals. The result also admits an equivalent formulation in terms of central projections. We place this result in the context of an investigation into notions of spectrum of noncommutative $C^*$-algebras.

Ideals in transformation-group $C^*$-algebras

We characterize the ideal of continuous-trace elements in a separable transformation-group $C^{*}$-algebra $C_0(X)\times G$. In addition, we identify the largest Fell ideal, the largest liminal ideal and the largest postliminal ideal.

Reduction to dimension two of local spectrum for $AH$ algebra with ideal property

A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.

Every finitely generated ideal of a Leavitt path algebra is a principal ideal

Let E be an arbitrary graph and K be any field. For every non-graded ideal I of the Leavitt path algebra L_{K}(E), we give an explicit description of the generators of I. Using this, we show that every finitely generated ideal of L_{K}(E) must be principal. In particular, if E is a finite graph, then every ideal of L_{K}(E) must be principal ideal.

Ideal and weak Amenability of Fréchet locally $C^*$-algebra

The notion of Fréchet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fréchet locally $C^*$-algebra is ideally amenable.

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.

One-sided ideals and approximate identities in operator algebras

A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in $C^*$-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study left ideals from the perspective of `Hamana theory' and using the multiplier algebras introduced by the author. More generally, we develop some general theory f…

The ideal envelope of an operator algebra

A left ideal of any C*-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in a `pre-quel' to this paper [B], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C*-algebra. We study operator algebras and their multiplier algebras from the perspective of `Hamana theory' and using the multiplier algebras introduced by the first author.

Regular Ideals of Locally-Convex Higher-Rank Graph Algebras

We give a vertex set description for basic, graded, regular ideals of locally-convex Kumjian-Pask Algebras. We also show that Condition (B) is preserved when taking the quotient by a basic, graded, regular ideal. We further show that when a locally-convex, row-finite k-graph satisfies Condition (B), all regular ideals are graded. We then show the same things hold for gauge-invariant, regular ideals in locally-convex k-graph C*-algebras.

On the ideals of ultragraph Leavitt path algebras

In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen…

The topology of ideals in some triangular AF algebras

A set of meet-irreducible ideals is described for a class of maximal triangular almost finite algebras. This set forms a topological space under the hull-kernel closure, and there is a one-to-one correspondence between closed sets in this space and ideals in the AF algebra. Some connections with nest representations and nest-primitive ideals are also described.

The Ideal Structure of Steinberg Algebras

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it a…

On Bounded Approximate Identities and Existence of Dense Ideals in Real Locally C*- and Locally JB-Algebras

It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras).

Prime and Primitive Ideals of Ultragraph Leavitt Path Algebras

Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals. We show that the non-graded prime ideals of $L_K(\mathcal G)$ are always primitive.

Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions

We study some topological properties of maximal ideal spaces of certain algebras of almost periodic functions. Our main result is that such spaces are contractible. We present certain corollaries of this result.

Meet irreducible ideals in direct limit algebras

We study the meet irreducible ideals in certain direct limit algebras, namely the strongly maximal triangular subalgebras of AF C*-algebras. These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C*-envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. Completely meet irreducible ideals are considered, and a distance formula is presented.

Dense nuclear Fréchet ideals in $C^\star$-algebras

We show that a $C^\star$-algebra $B$ contains a dense left or right Fréchet ideal $A$, with $A$ a nuclear locally convex space, if and only if the primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is finite dimensional for each $I \in $ Prim$(B)$. We show the forward implication holds for a general Banach algebra $B$, if the ideal is assumed two-sided. For $C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining a set of matrix-valued Schwartz functions on the countable discrete space Prim$(B)$.

Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all effective. For such a groupoid, we completely describe the ideal lattice of the associated Steinberg algebra over any commutative ring with identity. Our results are new even for the special case of Leavitt path algebras; so we describe explicitly what they say in this context, and give two concrete examples.

Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras

For a locally compact Hausdorff space $X$ and a $C^*$-algebra $A$ with only finitely many closed ideals, we discuss a characterization of closed ideals of $C_0(X,A) $ in terms of closed ideals of $A$ and certain (compatible) closed subspaces of $X$. We further use this result to prove that a closed ideal of $C_0(X) \otimes^{\min} A$ is a finite sum of product ideals. We also establish that for a unital $C^*$-algebra $A$, $C_0(X,A)$ has centre-quotient property if and only if $A$ has centre-quotient property. As an application, we characterize the closed Lie ideals of $C_0(X,A)$ and identify al…

Ideal structures and rigidity of uniform Roe algebras

In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.

The ideal structure of algebraic partial crossed products

Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant, compactly supported functions on $X$, taking values in a given field $K$. We then study the ideal structure of the algebraic partial crossed product $L_c(X)\rtimes G$. After developping a theory of induced ideals, we show that every ideal in $L_c(X)\rtimes G$ may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic ver…

Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C*-algebra is isomorphic to a graph C*-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C*-algebra (respectively, a Leavitt path algebra).

C*-代数中的半素理想

Semiprime ideals in C*-algebras

本研究证明C*-代数中未必闭的理想是半素理想的充要条件为幂等或对正元平方根封闭，素与半素理想自动自伴；通过提出并研究Dixmier理想及其幂根理论，利用其对C*-代数理想的夹逼性质完成证明。

Primitive Ideals of Labelled Graph $C^*$-algebras

Given a directed graph $E$ and a labeling $\mathcal{L}$, one forms the labelled graph $C^*$-algebra by taking a weakly left--resolving labelled space $(E, \mathcal{L}, \mathcal{B})$ and considering a universal generating family of partial isometries and projections. In this paper we provide characterization for primitive ideals of labelled graph $C^*$-algebras.