Kirchberg Algebras Research Articles

C⁎-algebras associated to directed graphs of groups, and models of Kirchberg algebras

We introduce C⁎-algebras associated to directed graphs of groups. In particular, we associate a combinatorial C⁎-algebra to each row-finite directed graph of groups with no sources, and show that this C⁎-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these C⁎-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.

The reciprocal Kirchberg algebras

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.

Classifiability of crossed products by nonamenable groups

Abstract We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {\widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.

Non-amenable tight squeezes by Kirchberg algebras

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.

Equivariant KK-Theory and the Continuous Rokhlin Property

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb {T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb {T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb {T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb {T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

On the classification of group actions on C*-algebras up to equivariant KK-equivalence

We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of Izumi is equivalent to an equivalence between cocycle conjugacy and equivariant KK-equivalence for actions of torsion-free amenable groups on Kirchberg algebras. Let G be a cyclic group of prime order. We describe its actions up to equivariant KK-equivalence, based on previous work by Manuel K\"ohler. In particular, we classify actions of G on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.

A weak homotopy equivalence type result related to Kirchberg algebras

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.

Poly-$${\mathbb {Z}}$$ group actions on Kirchberg algebras II

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.

Cartan subalgebras and the UCT problem, II

We study the connection between the UCT problem and Cartan subalgebras in C*-algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra mathcal O_2) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.

The Cuntz–Toeplitz algebras have nuclear dimension one

We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.

On pathological properties of fixed point algebras in Kirchberg algebras

AbstractWe 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 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

Every classifiable simple C*-algebra has a Cartan subalgebra

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.

Poly-ℤ Group Actions on Kirchberg Algebras I

AbstractToward 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.

All classifiable Kirchberg algebras are [formula omitted]-algebras of ample groupoids

In this note we show that every Kirchberg algebra in the UCT class is the C∗-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.

THE UNITAL EXT-GROUPS AND CLASSIFICATION OFC*-ALGEBRAS

AbstractThe semigroups of unital extensions of separableC*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtainK-theoretic classification of both unital and non-unital extensions ofC*-algebras, and in particular we obtain a completeK-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions

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 precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C*-dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.

$K$-theory for the crossed products by certain actions of $\mathbb Z^2$

We investigate the K -theory of crossed product C^* -algebras by actions of \mathbb Z^2 . Given a \mathbb Z^2 -action, we associate to it a homomorphism between certain subquotients of the K -theory of the underlying C^* -algebra, which we call the obstruction homomorphism. This homomorphism together with the K -theory of the underlying algebra and the induced action in K -theory determine the K -theory of the associated crossed product C^* -algebra up to group extension problems. A concrete description of this obstruction homomorphism is provided as well. We give examples of \mathbb Z^2 -actions, where the associated obstruction homomorphisms are non-trivial. One class of examples comprises certain outer \mathbb Z^2 -actions on Kirchberg algebras, which act trivially on KK -theory. This relies on a classification result by Izumi and Matui. A second class of examples consists of certain pointwise inner \mathbb Z^2 -actions. One instance is given as a natural action on the group C^* -algebra of the discrete Heisenberg group. We also compute the K -theory of the corresponding crossed product. A general and concrete construction yields various examples of pointwise inner \mathbb Z^2 -actions on amalgamated free product C^* -algebras with non-trivial obstruction homomorphisms. Among these, there are actions that are universal, in a suitable sense, for pointwise inner \mathbb Z^2 -actions with non-trivial obstruction homomorphisms. We also compute the K -theory of the crossed products associated with these universal C^* -dynamical systems.

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

On the -theory of -algebras arising from integral dynamics

We investigate the$K$-theory of unital UCT Kirchberg algebras${\mathcal{Q}}_{S}$arising from families$S$of relatively prime numbers. It is shown that$K_{\ast }({\mathcal{Q}}_{S})$is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct$C^{\ast }$-algebra naturally associated to$S$. The$C^{\ast }$-algebra representing the torsion part is identified with a natural subalgebra${\mathcal{A}}_{S}$of${\mathcal{Q}}_{S}$. For the$K$-theory of${\mathcal{Q}}_{S}$, the cardinality of$S$determines the free part and is also relevant for the torsion part, for which the greatest common divisor$g_{S}$of$\{p-1:p\in S\}$plays a central role as well. In the case where$|S|\leq 2$or$g_{S}=1$we obtain a complete classification for${\mathcal{Q}}_{S}$. Our results support the conjecture that${\mathcal{A}}_{S}$coincides with$\otimes _{p\in S}{\mathcal{O}}_{p}$. This would lead to a complete classification of${\mathcal{Q}}_{S}$, and is related to a conjecture about$k$-graphs.

The general linear group as a complete invariant for $C^*$-algebras

In 1955 Dye proved that two von Neumann factors not of type I2n are isomorphic (via a linear or a conjugate linear u -isomorphism) if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for C u -algebras. We show that the topological gen- eral linear group is a classifying invariant for simple, unital AH-algebras of slow dimension growth and of real rank zero, and the abstract gen- eral linear group is a classifying invariant for unital Kirchberg algebras in UCT.