Classification of Amenable C*-algebras

We will discuss recent progresses in the Elliott program of classification of simple amenable C*-algebras, including non-unital cases

A C*-algebra is said to have ideal property if each of its ideal is generated by projections inside the ideal. This class of C*-algebras is a common generalization of unital simple C*algebras and real rank zero C*-algebras. In this talk, we will present a classification of AH algebras with ideal property. The invariant involves scaled ordered total K-theory, tracial state spaces of cutting down algebras and new ingredient of Hausdorffized algebraic $K_1$ of cutting down algebras with certain compatibility. The talk is based on two joint papers of Gong-Jiang-Li-Pasnicu and Gong-Jiang-Li.

Abstract: Let $A$ be a unital $C^*$-algebra, and let $CU(A)$ denote the closure of the set of all commutators of the unitary group of $A$. Let $cel_{CU}(A)$ denote supremum of exponential lengths of all $u\in CU(A)$. Huaxin Lin proved that if $A$ is a TAI algebra, then $cel_{CU}(A)\leq 2\pi$. Lin also proved that for each countable ordered weakly unperforated Riesz group $(G, G_+)$ and each countable group $H$, there is a simple AH algebra of tracial rank one such that $(K_0(A), K_0(A)_+, K_1(A))= (G, G_+, H) $ and $cel_{CU}(A)>\pi$. In this talk, I will present the following theorem: for any simple AH algebra $A$ of traicial rank one, $cel_{CU}(A)=2\pi$. This is a joint work with Chunguang Li and Ivan Valesques.

We shall discuss results on classifying categories of real C*-algebras and actions on them using K-theoretic invariants.

Given an action $\alpha$ of an inverse semigroup $\mathcal{S}$ on a associative ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_\alpha \mathcal{S}$. We assume that $\mathcal{A}$ is commutative and we define a certain commutative subring $\mathcal{T}$ of $\mathcal{A} \rtimes_\alpha S$ which coincides with the embedding of $\mathcal{A}$ in $\mathcal{A} \rtimes_\alpha S$ whenever $S$ is unital. Our main result asserts that $\mathcal{A} \rtimes_\alpha S$ is a simple ring if, and only if, $\mathcal{T}$ is a maximal commutative subring of $\mathcal{A} \rtimes_\alpha S$ and $\mathcal{A}$ is $\mathcal{S}$-simple. As an application of our result we present a new proof of the simplicity criterion for a Steinberg algebra $A_R(\mathcal{G})$ associated with a Hausdorff and ample groupoid $\mathcal{G}$.

Spatial $L^p$ AF algebras were introduced by Phillips and Viola, and shown to be completely classifiable by their scaled preordered $K_0$ group. In this talk we describe the structure of ideals of a spatial $L^p$ AF algebra. We also show that any spatial $L^p$ AF algebra is residually incompressible and completely residually incompressible. We conclude by discussing some properties of the automorphisms of a spatial $L^p$ AF algebras.

I call "Dixmier set" a closed convex subset of a C*-algebra that is invariant under unitary conjugation. Similar sets can be defined in the dual of a C*-algebra (now using the weak* topology). I will talk about several natural questions on these sets and the answers that we know. Some of this work is joint with Ng and Skoufranis, some joint with Archbold and Tikuisis, some is still in progress.