Non-commutative Analysis Seminar Ical Atom

In 1975, Powers proved that the free group on two generators is a $C^{\star}$-simple group. The key insight in Powers’s proof of the $C^\star$-simplicity is that the left regular representation of $\mathbb{F}_2$ satisfies Dixmier type averaging property. Using the pioneering work of Kalantar-Kennedy, it was shown by Haagerup and Kennedy independently that the $C^\star$-simplicity of the group $\Gamma$ is equivalent to the group having Powers’ averaging property. In this talk, we introduce a generalized version of Powers’ averaging property for commutative crossed products. Using the notion of generalized Furstenberg boundary introduced by Kawabe and Naghavi (independently), we show that the simplicity of the commutative crossed products $C(X)\rtimes_r\Gamma$ (for minimal $\Gamma$-spaces $X$) is equivalent to the crossed product having generalized Powers’ averaging. As an application, we will show that every intermediate $C^\star$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. This is a joint work with Dan Ursu.

In a seminal 1973 paper, Williams recast conjugacy and eventual conjugacy for subshifts of finite type purely in terms of equivalence relations between adjacency matrices of the directed graphs. Williams expected these two notions to be the same, but after around 20 years the last hope for a positive answer, even under the most restrictive conditions, was extinguished by Kim and Roush.

In this talk, we will discuss operator algebras associated with adjacency matrices / directed graphs, which are naturally $\mathbb{Z}$-graded algebras. These operator algebras were first introduced by Cuntz and Krieger in tandem with early attacks on Williams’ problem, and manifest several natural properties of subshifts through their classification up to various kinds of isomorphisms.

The works on Cuntz-Krieger algebras later inspired a systematic study of purely algebraic versions called Leavitt path algebras, promoting new interactions between pure algebra and analysis. A well-known conjecture of Hazrat claims that two Leavitt path algebras are graded isomorphic if and only if their unital graded Grothendieck K0 groups are isomorphic. The topological version of this problem asks for a characterization of graded (stable) isomorphisms between Cuntz-Krieger algebras in terms of equivariant K-theory.

A solution to these problems has been sought after by many, and although substantial progress has been made, a proof is still missing in general. In joint work with Carlsen and Eilers we were able to discover subtle obstructions to certain algebraic methods of proof for the latter conjecture, by building on the counterexamples of Kim and Roush

Earlier this year, we discovered a new class of zero-dimensional dynamical systems, which we call “fiberwise essentially minimal”, that are of importance to operator algebras because of the nice properties, in particular K-theoretic classification, of the crossed product. Today, we discuss the Bratteli diagrams associated to these systems, and extend the K-theoretic classification to include a dynamical condition called “strong orbit equivalence”, extending the existing result in the minimal case due to Giordano-Putnam-Skau.

Earlier this year, we discovered a new class of zero-dimensional dynamical systems, which we call “fiberwise essentially minimal”, that are of importance to operator algebras because of the nice properties, in particular K-theoretic classification, of the crossed product. Today, we discuss the Bratteli diagrams associated to these systems, and extend the K-theoretic classification to include a dynamical condition called “strong orbit equivalence”, extending the existing result in the minimal case due to Giordano-Putnam-Skau.

TBA