UCT Class Research Articles

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.

On classification of nonunital amenable simple C∗-algebras, III : The range and the reduction

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features. We also show that every stably projectionless separable simple amenable $C^*$-algebra in the UCT class has rationally generalized tracial rank one.

On the Bundle of KMS State Spaces for Flows on a $${\mathcal {Z}}$$-Absorbing C*-Algebra

We obtain three results: 1) Every compact simplex bundle with exactly one point in the fiber over 0 is the KMS bundle of a periodic flow on the Jiang-Su algebra. 2) Let A be a separable unital C*-algebra with a unique trace state. Suppose that A tensorially absorbs the Jiang-Su algebra. The (weak) cocycle-conjugacy classes of flows that are not approximately inner are uncountable. 3) Let B be a separable, simple, unital, purely infinite and nuclear C*-algebra in the UCT class. Assume that the K1 group of B is torsion free. Every proper simplex bundle with empty fiber over 0 is the KMS bundle of a periodic flow on B.

Unitary groups and augmented Cuntz semigroups of separable simple 𝒵-Stable C∗-algebras

Let [Formula: see text] be a separable simple exact [Formula: see text]-stable [Formula: see text]-algebra. We show that the unitary group of [Formula: see text] has the cancellation property. If [Formula: see text] has continuous scale then the Cuntz semigroup of [Formula: see text] has strict comparison property and a weak cancellation property. Let [Formula: see text] be a 1-dimensional noncommutative CW complex with [Formula: see text] Suppose that [Formula: see text] is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms [Formula: see text] such that [Formula: see text] This result leads to the proof that every separable amenable simple [Formula: see text]-algebra[Formula: see text]in the UCT class has rationally generalized tracial rank at most one.

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.

Simple nuclear C*-algebras not equivariantly isomorphic to their opposites

We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.

A new proof of the Tikuisis–White–Winter theorem

Abstract A trace on a C * {\mathrm{C}^{*}} -algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C * {\mathrm{C}^{*}} -algebra is hyperfinite, it is easy to see that traces on nuclear C * {\mathrm{C}^{*}} -algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C * {\mathrm{C}^{*}} -algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C * {\mathrm{C}^{*}} -algebras and, in particular, using a version of the Weyl–von Neumann Theorem due to Elliott and Kucerovsky.

Quasidiagonality of nuclear C^*-algebras

We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.

Quasidiagonal traces on exact C⁎-algebras

Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear C⁎-algebra in the UCT class is quasidiagonal. Building on their work, we generalise the result, and show that any faithful, amenable trace on a separable, exact C⁎-algebra in the UCT class is quasidiagonal. We also prove that any amenable trace on a separable, exact, quasidiagonal C⁎-algebra in the UCT class is quasidiagonal.

Colocalizations of noncommutative spectra and bootstrap categories

We construct a compactly generated and closed symmetric monoidal stable ∞-category NSp′ and show that hNSp′op contains the suspension stable homotopy category of separable C⁎-algebras ΣHoC⁎ constructed by Cuntz–Meyer–Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of NSp′, namely, NSp′[K−1] and NSp′[Z−1], both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov KK-category of separable C⁎-algebras sits inside the homotopy category of KK∞:=NSp′[K−1]op as a fully faithful triangulated subcategory. Hence KK∞ should be viewed as the stable ∞-categorical incarnation of Kasparov KK-category for arbitrary pointed noncommutative spaces (including nonseparable C⁎-algebras). As an application we find that the bootstrap category in hNSp′[K−1] admits a completely algebraic description. We also construct a K-theoretic bootstrap category in hKK∞ that extends the construction of the UCT class by Rosenberg–Schochet. Motivated by the algebraization problem we finally analyze a couple of equivalence relations on separable C⁎-algebras that are introduced via the bootstrap categories in various colocalizations of NSp′.

UCT-Kirchberg algebras have nuclear dimension one

We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial K0 and finite K1 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.

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.

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.

Purely infinite C*-algebras arising from crossed products

AbstractWe 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 a Kirchberg algebra in the UCT class.

A simple C*-algebra with a finite and an infinite projection

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*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers.