to follow

to follow

Recently, Elliott, Li, and Niu proved a classification theorem for Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant that was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of nonisomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.

A $C^*$-diagonal is a certain commutative subalgebra of a $C^∗$ -algebra with a rich structure. Renault and Kumjian showed that finding a $C^*$ -diagonal of a $C^∗$-algebra is equivalent to realizing the $C^*$-algebra via a groupoid. This establishes a close connection between $C^∗$-diagonals and dynamics and allows one to relate the geometric properties of groupoids to the properties of $C^∗$ -diagonals. 

In this talk, I will explore the unique pure state extension property of an Abelian $C^*$-subalgebra of a 1-dim NCCW complex, the approximation of morphisms between two 1-dim NCCW complexes by $C^*$-diagonal preserving morphisms, and the existence of $C^*$-diagonal in inductive limits of certain 1-dim NCCW complexes.