Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is implemented by an automorphism of A of order two. As a consequence, we prove that every countable Z/2Z-graded module over the representation ring of Z/2Z is isomorphic to the equivariant K-theory for some action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra.
Along the way, we prove that every not necessarily finitely generated module over the group ring of Z/2Z which is free as an abelian group has a direct sum decomposition with only three kinds of summands, namely the group ring itself and Z on which the nontrivial element of Z/2Z acts either trivially or by multiplication by -1.
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.
We give a construction of Kirchberg algebras from graphs. By using product graphs in the construction we are able to provide models for general (UCT) Kirchberg algebras while maintaining the explicit generators and relations of the underlying graphs.
We show that a Kirchberg algebra is semiprojective if and only if it is KK-semiprojective. In particular, this shows that a Kirchberg algebra in the UCT-class is semiprojective if and only if its K-theory is finitely generated, thereby giving a positive solution to a longstanding conjecture by Blackadar.
In this note we show that every Kirchberg algebra in the UCT class is the $C^{\ast}$-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg's graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg's construction.
We give a partial answer to the question for the precise value of the nuclear dimension of UCT-Kirchberg algebras raised by W. Winter and J. Zacharias. It is shown that every Kirchberg algebra in the UCT-class with torsion free $K_1$-group has nuclear dimension 1.
Toward the complete classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly-$\mathbb{Z}$ groups of Hirsch length not greater than three on unital Kirchberg algebras up to $KK$-trivial cocycle conjugacy.
We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra.
We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial $K_0$ and finite $K_1$ have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.
For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier--Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles.
We investigate 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 outer action has Rokhlin dimension at most 1. 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. A more direct and alternative approach to this is given as well. Several applications of this relationship are discussed to cover a fairly large class of $\mathcal{O}_\infty$-absorbing…
In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup an…
In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a prec…
We give a framework to produce C*-algebra inclusions with extreme properties. This gives the first constructive nuclear minimal ambient C*-algebras. We further obtain a purely infinite analogue of Dadarlat's modeling theorem on AF-algebras: Every Kirchberg algebra is rigidly and KK-equivalently sandwiched by non-nuclear C*-algebras without intermediate C*-algebras. Finally we reveal a novel property of Kirchberg algebras: They embed into arbitrarily wild C*-algebras as rigid maximal C*-subalgebras.
A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.
This is the second part of our serial work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. Based on technical results obtained in our previous work, we completely reduce the problem to the classification of continuous fields of Kirchberg algebras over the classifying spaces. As an application, we determine the number of cocycle conjugacy classes of outer $\mathbb{Z}^n$-actions on the Cuntz algebras.
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 and have rational K-theory groups are classified up to isomorphism by filtrated K-theory.
A universal coefficient theorem is proved for C*-algebras over an arbitrary finite T_0-space X which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero Kirchberg X-algebras with intermediate cancellation. Range results are obtained for (unital) purely infinite graph C*-algebras with intermediate cancellation and Cuntz-Krieger algebras with intermediate cancellation. Permanence results for extensions of these classes follow.
We prove the following theorem: let $A$ be a UCT Kirchberg algebra, and let $α$ be a prime-order automorphism of $K_*(A)$, with $α([1_A])=[1_A]$ in case $A$ is unital. Then $α$ is induced from an automorphism of $A$ having the same order as $α$. This result is extended to certain instances of an equivariant inclusion of Kirchberg algebras. As a crucial ingredient we prove the following result in representation theory: every module over the integral group ring of a cyclic group of prime order has a natural presentation by generalized lattices with no cyclotomic summands.
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.
We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group $Γ$ and any infinite group $Λ$, we construct an outer action of $Λ$ on the Cuntz algebra $\mathcal{O}_2$ whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}^\ast_{\rm r}(Γ)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.
In this note we show that a combinatorial model of Kirchberg algebras in the UCT, namely the Katsura algebras O_{AB}, can be expressed both as groupoid C*-algebras and as inverse semigroup crossed products. We use this picture to obtain results about simplicity, pure infiniteness and nuclearity of O_{AB} using different methods than those used by Katsura.
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 define continuous C*-algebras over a topological space X and establish some basic results. If X is a locally compact Hausdorff space, continuous C*-algebras over X are equivalent to ordinary continuous C_0(X)-algebras. The main purpose of our study is to prove that every continuous, full, separable, nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra over X. (Here a Kirchberg algebra over X is a separable, nuclear, and strongly purely infinite C*-algebra over X with primitive ideal space homeomorphic to X.) In the case that X is a one-point space, this result is know…
The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if $\mathcal{U}_{nc}(2) \otimes_{\min} \mathcal{U}_{nc}(2)=\mathcal{U}_{nc}(2) \otimes_{\max} \mathcal{U}_{nc}(2)$. Our results follow from properties of the finite-dimensional operator system $\mathcal{V}_n$ spanned by $1$ and the generators of $\mathcal{U}_{nc}(n)$. We show that $\mathcal{V}_n$ is an operator system quotient of $M_{2n}$ and has…
Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C$^*$ algebras. Along the way, we establish a connection between existentially closed C$^*$ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper c…
The theory of exact C*-algebras was introduced by Kirchberg and has been influential in recent development of C*-algebras. A fundamental result on exact C*-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality.
In the paper, we provide some examples of MF algebras by considering minimal or maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C$^*$-algebras, whose BDF extension semigroup is not group. These examples include, for example, $ C_r^*(F_n)\otimes_{max} C^*(F_n) $, $ C_r^*(F_n)\otimes_{min} C^*(F_n) $, $C_r^*(H_1\ast H_2)$ with $2\le |H_1|<\infty$ and $3\le |H_2|<\infty$ where $|H_1|$, $|H_2|$ are the orders of thr groups $H1$, or $H_2$ respectively, and many others.
We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.
The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using the quantum complexity result MIP*=RE, we establish two quantum complexity consequences of a positive solution to KEP. Both results involve almost-commuting strategies to nonlocal games.
We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_Λ (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results o…
We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.
I present a proof of Kirchberg's classification theorem: two separable, nuclear, $\mathcal O_\infty$-stable $C^\ast$-algebras are stably isomorphic if and only if they are ideal-related $KK$-equivalent. In particular, this provides a more elementary proof of the Kirchberg--Phillips theorem which is isolated in the paper to increase readability of this important special case.
We extend in this paper several results of E. Kirchberg, S. Wassermann and the author dealing with continuous fields of C*--algebras to the semi-continuous case. We provide a new characterisation of separable lower semi-continuity C*--bundles and we present exactness criteria for C(X)-algebras and unital bisimplifiable Hopf \cst-algebras.
We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$-theory. Our main results are $\mathbb{T}$-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK^\mathbb{T}$-equivalent to a Rokhlin action on a Kirchberg algebra; any Rokhlin action on an exact separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique Rokhlin action); and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK^\mathbb{T}$-…
Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an ex…
We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C*-algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C*-algebras.
We generalize Kirchberg's weak exactness to inclusions of C*-algebras in von Neumann algebras and study some characterizations and permanence properties which are similar to those of exact groups. We then consider a similar condition to Ozawa's condition (AO) with our weak exactness and generalize Ozawa's theorem for bi-exact groups. As a corollary, we give new examples of prime factors.
We extend in this paper the characterisation of a separable nuclear \cst-algebra given by Kirchberg proving that given a unital separable continuous field of nuclear C*-algebras A over a compact metrizable space X, the C(X)-algebra A is isomorphic to a unital C(X)-subalgebra of the trivial continuous field C(X;O_2), image of C(X;O_2) by a norm one projection.
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.
