Research interests
Research interests
My main interests are C*-algebras and their classification, Cartan subalgebras and C*-diagonals in C*-algebras, and the structure of generalized inductive systems.
Cartan pairs form an intriguing bridge between the realms of C*-algebras and topological dynamics as they arise as groupoid C*-algebras, fully remembering the underlying groupoid. This allows us to translate techniques back and forth, resulting in a fruitful interplay between the two fields. For example, it is known that any classifiable C*-algebra contains a Cartan subalgebra, and in the stably finite case, one even gets a C*-diagonal. However, the study of the structure of Cartans respective C*-diagonals in C*-algebras, e.g., towards the long-term goal of classification of certain classes of Cartans and C*-diagonals in interesting C*-algebras, is still in its beginnings. To have a chance to understand the entirety of the underlying dynamics in a given classifiable C*-algebra, we still need more explicit constructions of examples of C*-diagonals, in particular in prominent C*-algebras. This is one of my main interests. Moreover, I am interested in regularity conditions for diagonal pairs like diagonal dimension, diagonal comparison, or uniform property Gamma for diagonal pairs, and their interplay.
If you are interested, you may consider:
Xin Li, Every classifiable simple C*-algebra has a Cartan subalgebra,
Li, Liao, Winter, The diagonal dimension of sub-C*-algebras,
Kopsacheilis, Liao, Tikuisis, Vaccaro, Uniform property Γ and the small boundary property,
Kopsacheilis, Winter, Diagonal comparison of ample C*-diagonals,
and other interesting work in this area.
The notion of generalized inductive limits goes back to work of Blackadar and Kirchberg in which they modeled quasidiagonal C*-algebras as limits of inductive systems of C*-algebras with c.p.c. connecting maps. This was extended by Kristin Courtney and Wilhelm Winter to a class of generalized inductive systems of finite dimensional C*-algebras accessing any separable, nuclear C*-algebra. Writing C*-algebras as the limit of finite dimensional ones in this sense is an intriguing point of view, and I am working on exploring the possibilities of this notion further. The overarching goal is to identify structural properties of limits of these generalized inductive systems in terms of intrinsic data of the system. In particular, it is an interesting question how we can access the K-theory a C*-algebra presented as a limit of a generalized inductive system. This question has natural connections to the recent notion of K-theory for operator systems by van Suijlekom.
If you are interested, you may consider: