Kirchberg Algebras Research Articles

Self-similar graph $C^*$-algebras and partial crossed products

In a recent paper, Pardo and the first named author introduced a class of C*algebras which which are constructed from an action of a group on a graph. This class was shown to include many C*-algebras of interest, including all Kirchberg algebras. In this paper, we study the conditions under which these algebras can be realized as partial crossed products of commutative C*-algebras by groups. In addition, for any n ≥ 2 we present a large class of groups such that for any group H in this class, the Cuntz algebra On is isomorphic to a partial crossed product of a commutative C*-algebra by H.

Purely infinite simple C⁎-algebras that are principal groupoid C⁎-algebras

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.

The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras

We investigate outer symmetries on unital Kirchberg algebras with respect to the Rokhlin property and finite Rokhlin dimension. In stark contrast to the restrictiveness of the Rokhlin property, every such action has Rokhlin dimension at most one. A consequence of these observations is a relationship between the nuclear dimension of an \mathcal O_\infty -absorbing C*-algebra and its \mathcal O_2 -stabilization. We also give a more direct and alternative approach to this result. Several applications of this relationship are discussed to cover a fairly large class of \mathcal O_\infty -absorbing C*-algebras that turn out to have finite nuclear dimension.

One-parameter continuous fields of Kirchberg algebras with rational $K$-theory

We show that separable continuous fields over the unit interval whose fibers are stable Kirchberg algebras that satisfy the universal coefficient theorem in KK-theory (UCT) and have rational K-theory groups are classified up to isomorphism by filtrated K-theory.

Amenable minimal Cantor systems of free groups arising from diagonal actions

Abstract We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

CLASSIFICATION OF CERTAIN CONTINUOUS FIELDS OF KIRCHBERG ALGEBRAS

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

Nuclearity of semigroup [formula omitted]-algebras and the connection to amenability

We study C∗-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C∗-algebras as C∗-algebras of inverse semigroups, groupoid C∗-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C∗-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C∗-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C∗-algebras.

Non-supramenable groups acting on locally compact spaces

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product C^* -algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups.

Unitary groups as a complete invariant

Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. We show that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their K0-ordered groups are isomorphic. Also, using Gong and Dadarlatʼs classification theorem, we prove that such C⁎-algebras are isomorphic if and only if their unitary groups are isomorphic as topological groups. For simple, unital purely infinite C⁎-algebras, we show that two unital Kirchberg algebras are ⁎-isomorphic if and only if their unitary groups are isomorphic as abstract groups.

One-Parameter Continuous Fields of Kirchberg Algebras. II

Abstract Parallel to the first two authors’ earlier classification of separable, unital, one-parameter, continuous fields of Kirchberg algebras with torsion free K-groups supported in one dimension, one-parameter, separable, unital, continuous fields of AF-algebras are classified by their ordered K0-sheaves. Effros-Handelman-Shen type theorems are proved for separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.

The classification of real purely infinite simple C*-algebras

We classify real Kirchberg algebras using united K -theory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that A_{\mathbb{C}} and B_{\mathbb{C}} are also simple. In the stable case, A and B are isomorphic if and only if K^{CRT}(A) \cong K^{CRT}(B) . In the unital case, A and B are isomorphic if and only if (K^{CRT}(A), [1_A]) \cong (K^{CRT}(B), [1_B]) . We also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips[35]. As an application, we find all real forms of the complex Cuntz algebras {\mathcal{O}}_n for 2 \leq n \leq \infty .

The nuclear dimension of [formula omitted]-algebras

We introduce the nuclear dimension of a C ∗ -algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C ∗ -algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.

Formula omitted]-actions on Kirchberg algebras

We classify a large class of Z 2 -actions on the Kirchberg algebras employing the Kasparov group KK 1 as the space of classification invariants.

Continuous fields of C*-algebras over finite dimensional spaces

Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C ( X ) -algebras by C ( X ) -subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra O n , n ∈ { 2 , 3 , … , ∞ } , are locally trivial. They are trivial if n = 2 or n = ∞ . For n ⩾ 3 finite, such a field is trivial if and only if ( n − 1 ) [ 1 A ] = 0 in K 0 ( A ) , where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell type condition.

Fiberwise KK-equivalence of continuous fields of C*-algebras

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

A class of [formula omitted]-algebras generalizing both graph algebras and homeomorphism [formula omitted]-algebras IV, pure infiniteness

This is the final one in the series of papers where we introduce and study the C ∗ -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated C ∗ -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as C ∗ -algebras associated with topological graphs.

A construction of actions on Kirchberg algebras which induce given actions on their K-groups

We prove that every action of a finite group all of whose Sylow subgroups are cyclic on the K-theory of a Kirchberg algebra can be lifted to an action on the Kirchberg algebra. The proof uses a construction of Kirchberg algebras generalizing the one of Cuntz-Krieger algebras, and a result on modules over finite groups. As a corollary, every automorphism of the K-theory of a Kirchberg algebra can be lifted to an automorphism of the Kirchberg algebra with same order.

On Rørdam's classification of certain $C^*$-algebras with one non-trivial ideal, II

In this paper we extend the classification results obtained by Rørdam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we characterize the range in both cases. The invariants are cyclic six term exact sequences together with the class of some unit.

One-Parameter Continuous Fields of Kirchberg Algebras

We prove that all unital separable continuous fields of C*-algebras over [0,1] with fibers isomorphic to the Cuntz algebra $${\mathcal{O}}_n \, (2 \leq n \leq \infty)$$ are trivial. More generally, we show that if A is a separable, unital or stable, continuous field over [0,1] of Kirchberg C*-algebras satisfying the UCT and having finitely generated K-theory groups, then A is isomorphic to a trivial field if and only if the associated K-theory presheaf is trivial. For fixed $$d\in \{0,1\}$$ we also show that, under the additional assumption that the fibers have torsion free K d -group and trivial K d+1-group, the K d -sheaf is a complete invariant for separable stable continuous fields of Kirchberg algebras.

The homotopy groups of the automorphism group of Kirchberg algebras

We compute the homotopy groups of the automorphism group of Kirchberg algebras. More generally, we calculate the homotopy classes [X,Aut(A)] for a Kirchberg algebra A and a path connected metrizable compact space X satisfying a natural continuity property.