Master's Thesis Seminar

Ehsaan Hossain, Pure Mathematics, University of Waterloo

"The Algebraic Kirchberg--Phillips Conjecture"

To a directed graph $E$, one can associate two algebras: the C*-algebra $C^*(E)$, and the Leavitt path algebra $L(E)$. Remarkably many algebraic properties of $L(E)$ mirror C*-algebraic properties of $C^*(E)$ --- in fact $L(E)$ is a dense subalgebra of $C^*(E)$. The famous Kirchberg--Phillips Theorem from C*-algebras says that certain purely infinite simple C*-algebras can be determined up to isomorphism from their K-theory; this has nice implications for graph C*-algebras because if $C^*(E)$ is simple then it is often purely infinite (as long as $E$ has at least one cycle). Since Leavitt path algebras mimic C*-algebras in many ways, one may ask if they also satisfy a Kirchberg--Phillips classification: if $L(E)$ is purely infinite simple, can we recover its isomorphism class from its K-theory? This remains an unresolved question, but many clues from symbolic dynamics hint that at a positive answer. The aim of this presentation is to give an overview of the state-of-affairs regarding the algebraic Kirchberg--Phillips question.