We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\infty tensorially. Thus one can reach an O_\infty-absorbing C*-algebra as an inductive limit of the finite and elementary C*-algebras C_0([0,1),M_k).
As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class of C*-algebras can be embedded.
We investigate crossed products of Cuntz algebras by quasi-free actions of abelian groups. We prove that our algebras are AF-embeddable when actions satisfy a certain condition. We also give a necessary and sufficient condition that our algebras become simple and purely infinite, and consequently our algebras are either purely infinite or AF-embeddable when they are simple.
It is proved that classifiable simple separable nuclear purely infinite C*-algebras having finitely generated K-theory and torsion-free K_1 are semiprojective. This is accomplished by exhibiting these algebras as C*-algebras of infinite directed graphs.
Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
We prove that a simple, separable, nuclear, purely infinite classifiable $C^*$-algebra is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.
We show that separable, nuclear and strongly purely infinite C*-algebras have finite nuclear dimension. In fact, the value is at most three. This exploits a deep structural result of Kirchberg and Rørdam on strongly purely infinite C*-algebras that are homotopic to zero in an ideal-system preserving way.
In the reduced free product of C*-algebras (A,phi)=(A_1,phi_1)*(A_2,phi_2), A is shown to be purely infinite and simple under the hypothesis that A_1 is the crossed product of a C*-algebra by a discrete infinite group, phi_1 is well behaved with respect to this crossed product and A_2 is not one dimensional.
Given a separated graph $(E,C)$, there are two different C*-algebras associated to it, the full graph C*-algebra $C^*(E,C)$, and the reduced one $C^*_{\text{red}} (E,C)$. For a large class of separated graphs $(E,C)$, we prove that $C^*_{\text{red}} (E,C)$ either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C*-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.
Assume that A is a purely infinite simple C*-algebra whose K_0 is a torsion group, namely, contains no free element. Then a positive element a in A can be written as a finite sum of projections in A if and only if either a is a projection or the norm ||a||>1.
We give a classification theorem for a class of C*-algebras which are direct limits of extensions of circle algebras by purely infinite C*-algebras. The invariant consists of the following: (1) the set of Murray-von Neumann equivalence classes of projections; (2) the set of homotopy classes of hyponormal partial isometries; (3) a map d; and (4) total K-theory.
Certain reduced free products of C*-algebras, (A,phi)=(A_1,phi_1)*(A_2,ϕ_2), taken with respect to faithful states, at least one of which is not a trace, are shown to be purely infinite and simple. It is assumed that one of the A_i contain a partial isometry in the spectral subspace of phi_i corresponding to a positive number not equal to one. For example, if A_1 and A_2 are copies of the two-by-two complex matrices and if phi_1 and phi_2 are not unitarily conjugate, it is shown that A is simple and purely infinite.
In this paper, we consider pure infiniteness of generalized Cuntz-Krieger algebras associated to labeled spaces $(E,\mathcal{L},\mathcal{E})$. It is shown that a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{E})$ is purely infinite in the sense that every nonzero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E, \mathcal{L},\mathcal{E})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^*(E,\mathcal{L},\mathcal{E})=C^*(p_A, s_a)$ is purely infinite in the sense of Kirchberg and …
We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann algebras, on purely infinite simple C*-algebras, and on their associated multiplier algebras.
Let (A,G,α) be a partial dynamical system. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A xr G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition---automatic when A is abelian---we show that A xr G is purely infinite if and only if the positive nonzero elements in A are properly infinite in A xr G. As an application we verify pure infiniteness of various partial crossed products, including realisations of the Cuntz algebras O_n, O_A, O_N, and O_Z as partial cr…
From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised as the C*-algebras of amenable principal groupoids.
We study the crossed product $C^*$-algebra associated to injective endomorphisms, which turns out to be equivalent to study the crossed product by the dilated autormorphism. We prove that the dilation of the Bernoulli $p$-shift endomorphism is topologically free. As a consequence, we have a way to twist any endomorphism of a $\D$-absorbing $C^*$-algebra into one whose dilated automorphism is essentially free and have the same $K$-theory map than the original one. This allows us to construct purely infinite crossed products $C^*$-algebras with diverse ideal structures.
Let $0\longrightarrow \B\stackrel{j}{\longrightarrow}E\stackrelπ{\longrightarrow}\A\longrightarrow 0$ be an extension of $\A$ by $\B$, where $\A$ is a unital simple purely infinite $C^{*}$--algebra. When $\B$ is a simple separable essential ideal of the unital $C^{*}$--algebra $E$ with $\RR(\B)=0$ and {\rm(PC)}, $K_{0}(E)=\{[p]\mid p$ is a projection in $E\setminus B\}$; When $B$ is a stable $C^{*}$--algebra, $\U(C(X,E))/\U_0(C(X,E))\cong K_1(C(X,E))$ for any compact Hausdorff space $X$.
Counterexamples to classification of purely infinite, nuclear, separable C*-algebras (in the ideal-related bootstrap class) and with primitive ideal space X using ideal-related K-theory occur for infinitely many finite primitive ideal spaces X, the smallest of which having four points. Ideal-related K-theory is known to be strongly complete for such C*-algebras if they have real rank zero and X has at most four points for all but two exceptional spaces: the pseudo-circle and the diamond space. In this article, we close these two remaining cases. We show that ideal-related K-theory is strongly …
In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a *-isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.
Let $n$ be a positive integer. We introduce a concept, which we call the $n$-filling property, for an action of a group on a separable unital $C^*$-algebra $A$. If $A=C(Ω)$ is a commutative unital $C^*$-algebra and the action is induced by a group of homeomorphisms of $Ω$ then the $n$-filling property reduces to a weak version of hyperbolicity. The $n$-filling property is used to prove that certain crossed product $C^*$-algebras are purely infinite and simple. A variety of group actions on boundaries of symmetric spaces and buildings have the $n$-filling property. An explicit example is the ac…
We show that a separable purely infinite C*-algebra is of real rank zero if and only if its primitive ideal space has a basis consisting of compact-open sets and the natural map K_0(I) -> K_0(I/J) is surjective for all closed two-sided ideals J contained in I in the C*-algebra. It follows in particular that if A is any separable C*-algebra, then A tensor O_2 is of real rank zero if and only if the primitive ideal space of A has a basis of compact-open sets, which again happens if and only if A tensor O_2 has the ideal property, also known as property (IP).
We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0…
We introduce filling families with matrix diagonalization as a refinement of the work by Rørdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly purely infinite $C^*$-algebra and a exact $C^*$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $A\rtimes_φ\mathbb{N}$ by an endomorphism $φ$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harn…
Let A be a simple, sigma-unital, non-unital C*-algebra, with metrizable tracial simplex T(A), which is projection-surjective and injective and has strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) The multiplier algebra M(A) has strict comparison of positive elements by traces; (iii) The coronal algebra M(A)/A is purely infinite; (iii') The quotient M(A)/Imin is purely infinite; (iv) M(A) has finitely many ideals; (v) Imin=Ifin. If furthermore algebra of n by n matrices of A is projection-surjective and injective for ever…
We continue our investigation, from \cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a "Cuntz-like" Banach $*$-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algeb…
We show a noncommutative Rohlin type theorem for automorphisms of a certain class of purely infinite simple $C^{\ast}$-algebras. This class consists of the purely infinite unital simple $C^{\ast}$-algebras which are in the bootstrap category ${\cal N}$ and have trivial $K_{1}$-groups.
A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $*$-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets $U$ of the unit space of the g…
Examples of simple, separable, unital, purely infinite $C^*$--algebras are constructed, including: (1) some that are not approximately divisible; (2) those that arise as crossed products of any of a certain class of $C^*$--algebras by any of a certain class of non--unital endomorphisms; (3) those that arise as reduced free products of pairs of $C^*$--algebras with respect to any from a certain class of states.
We will introduce a notion of normal subshifts. A subshift $(Λ,σ)$ is said to be normal if it satisfies a certain synchronizing property called $λ$-synchronizing and is infinite as a set. We have lots of purely infinite simple $C^*$-algebras from normal subshifts including irreducible infinite sofic shifts, Dyck shifts, $β$-shifts, and so on. Eventual conjugacy of one-sided normal subshifts and topological conjugacy of two-sided normal subshifts are characterized in terms of the associated $C^*$-algebras and the associated stabilized $C^*$-algebras with its diagonals and gauge actions, respect…
We augment Restorff's classification of purely infinite Cuntz-Krieger algebras by describing the range of his invariant on purely infinite Cuntz-Krieger algebras. We also describe its range on purely infinite graph C*-algebras with finitely many ideals, and provide 'unital' range results for purely infinite Cuntz-Krieger algebras and unital purely infinite graph C*-algebras.
Given an n x n integer matrix A whose eigenvalues are strictly greater than 1 in absolute value, let σ_A be the transformation of the n-torus T^n=R^n/Z^n defined by σ_A(e^{2πix})=e^{2πiAx} for x\in R^n. We study the associated crossed-product C*-algebra, which is defined using a certain transfer operator for σ_A, proving it to be simple and purely infinite and computing its K-theory groups.
It was shown by Rordam and the second named author that a countable group G admits an action on a compact space such that the crossed product is a Kirchberg algebra if, and only if, G is exact and non-amenable. This construction allows a certain amount of choice. We show that different choices can lead to different algebras, at least with the free group.
We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is necessary and sufficient for the associated $C^*$-algebra to have topological dimension zero. We prove that a purely infinite $2$-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the $2$-graph. We also show that a $k$-graph $C^*$-algebra with topological dimension zero i…
In the given article the notion of infinite norm decomposition of a C$^*$-algebra is investigated. The norm decomposition is some generalization of Peirce decomposition. It is proved that the infinite norm decomposition of any C$^*$-algebra is a C$^*$-algebra. C$^*$-factors with an infinite and a nonzero finite projection and simple purely infinite C$^*$-algebras are constructed.
We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is…
In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke-Li and Rainone-Sims. We a…
We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C*-algebras. An operator theoretic proof will be given to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K_1-group. Some upper and lower bounds for the distance between unitary orbits will be obtained based on these methods. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infin…
Using Kirchberg KK_X-classification of purely infinite, separable, stable, nuclear C*-algebras with finite primitive ideal space, Bentmann showed that filtrated K-theory classifies purely infinite, separable, stable, nuclear C*-algebras that satisfy that all simple subquotients are in the bootstrap class and that the primitive ideal space is finite and of a certain type, referred to as accordion spaces. This result generalizes the results of Meyer-Nest involving finite linearly ordered spaces. Examples have been provided, for any finite non-accordion space, that isomorphic filtrated K-theory d…
For $\MvN$ a separable, purely infinite von Neumann algebra with almost periodic weight $ϕ$, a decomposition of $\MvN$ as a crossed product of a semifinite von Neumann algebra by a trace--scaling action of a countable abelian group is given. Then Takasaki's continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki's notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one--parameter trace--scaling action of $\Real_+$ on the injective II$_\infty$ factor. Finally, another model o…
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…
Let ${\cal O}_{{\cal H}^{A,B}_κ}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^κ$ is transitive, the $C^*$-algebra ${\cal O}_{{\cal H}^{A,B}_κ}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra ${\cal O}_{{\cal H}^{[N],[M]}_κ}$ are computed by using the Euclidean algorithm.
Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A is isomorphic to A if in addition A is purely infinite, we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C*-algebras. It follows that if A and B are unital separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_* (B) which preserves the class of the identity, then A is isomorphic to B.…
Pure infiniteness (in sense of E.Kirchberg and M.Rørdam) is considered for C*-algebras arising from singly generated dynamical systems. In particular, Cuntz-Krieger algebras and their generalizations, i.e., graph-algebras and O_A of an infinite matrix A, admit characterizations of pure infiniteness. As a consequence, these generalized Cuntz-Krieger algebras are traceless if and only if they are purely infinite. Also, a characterization of AF-algebras among these C*-algebras is given. In the case of graph-algebras of locally finite graphs, characterizations of stability are obtained.
One of the main tools to classify \cst-algebras is the study of its projections and its unitaries. It was proved by Cuntz in \cite{Cu81} that if $A$ is a \textit{purely infinite} simple \cst-algebra, then the kernel of the natural map for the unitary group $\U(A)$ to the $K$-theory group $K_1(A)$ is reduced to the connected component $\U^0(A)$, i.e. $A$ is \textit{$K_1$-injective} (see §3). We study in this note a finitely generated \cst-algebra, the $K_1$-injectivity of which would imply the $K_1$-injectivity of all unital \textit{properly infinite} \cst-algebras.
A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a uniq…
An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-alg…
Given an arbitrary infinite 0--1 matrix A having no identically zero rows, we define an algebra OA as the universal C*-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize OA as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. OA is always nuclear and we obtai…
We prove that the Bowen-Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where $2$ is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are $K$-theoretic, and we prove several results about (Hermitian, bivariant) $K$-theory of Leavitt path a…
In this paper we address the classification problem for purely infinite simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph $E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open question which asks whether the pair $(K_0(L(E)), [1_{L(E)}])$, consisting of the Grothendieck group together with the class $[1_{L(E)}]$ of the identity, is a complete invariant for the classification, up to algebra isomorphism, of those Leavitt path algebras of finite graphs which are purely infinite simple. We show that $(K_0(L(E)), [1_{L(E)}])$ is a complete invariant f…
