Structure theory and stable rank for C*-algebras of finite higher-rank graphs

We study the structure and compute the stable rank of $C^{*}$ -algebras of finite higher-rank graphs. We completely determine the stable rank of the $C^{*}$ -algebra when the $k$ -graph either contains no cycle with an entrance or is cofinal. We also determine exactly which finite, locally convex $k$ -graphs yield unital stably finite $C^{*}$ -algebras. We give several examples to illustrate our results.

The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies ${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and ${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\text{Cu}} (A)$ , and the purely positive part of ${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\text{rc}} (A)> 0$ , ${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$ , and ${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$ .

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 $$\mathcal O_2$$ 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.

C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

We study homeomorphisms of a Cantor set with $k$ ( $k<+\infty$ ) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$ , the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

Self-similar $k$-Graph C$^\ast $-Algebras

In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

A dichotomy for groupoid -algebras

We study the finite versus infinite nature of C$^{\ast }$-algebras arising from étale groupoids. For an ample groupoid$G$, we relate infiniteness of the reduced C$^{\ast }$-algebra$\text{C}_{r}^{\ast }(G)$to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid$S(G)$which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C$^{\ast }$-algebra of$G$in the sense that if$G$is ample, minimal, topologically principal, and$S(G)$is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for$\text{C}_{r}^{\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph$\text{C}^{\ast }$-algebras as well.

Ideal structure and pure infiniteness of ample groupoid -algebras

In this paper, we study the ideal structure of reduced $C^{\ast }$-algebras $C_{r}^{\ast }(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_{r}^{\ast }(G)$ 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 $C_{r}^{\ast }(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_{0}(G^{(0)})$ is properly infinite in $C_{r}^{\ast }(G)$. We also establish a sufficient condition on the ample groupoid $G$ that ensures pure infiniteness of $C_{r}^{\ast }(G)$ in terms of paradoxicality of compact open subsets of the unit space $G^{(0)}$. Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: let $G$ be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then $C_{r}^{\ast }(G)$ is a simple $C^{\ast }$-algebra which is either stably finite or strongly purely infinite.

The Jiang–Su Absorption for Inclusions of Unital C*-algebras

We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras P ⊂ A with index finite, and show that an action α from a finite group G on a simple unital C*- algebra A has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation E: A → AG has the tracial Rokhlin property. Let be a class of infinite dimensional stably finite separable unital C*-algebras that is closed under the following conditions:(1) If A ∊ and B ≅ A, then B ∊ .(2) If A ∊ and n ∊ ℕ, then Mn(A) ∊ .(3) If A ∊ and p ∊ A is a nonzero projection, then pAp ∊ .Suppose that any C*-algebra in is weakly semiprojective. We prove that if A is a local tracial -algebra in the sense of Fan and Fang and a conditional expectation E: A → P is of index-finite type with the tracial Rokhlin property, then P is a unital local tracial -algebra.The main result is that if A is simple, separable, unital nuclear, Jiang–Su absorbing and E: A → P has the tracial Rokhlin property, then P is Jiang–Su absorbing. As an application, when an action α from a finite group G on a simple unital C*-algebra A has the tracial Rokhlin property, then for any subgroup H of G the fixed point algebra AH and the crossed product algebra H is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup W(A) is hereditary to W(P) if A is simple, separable, exact, unital, and E: A → P has the tracial Rokhlin property.