Research interests
Research interests
My main interests are C*-algebras and their classification, Cartan subalgebras and C*-diagonals in C*-algebras, the structure of generalized inductive systems, and K-theory of operator 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. But 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. That is why I am particularly interested in explicit examples of C*-diagonals in prominent C*-algebras. 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. In particular, it is an interesting question how we can access the K-theory of a C*-algebra presented as a limit of such a generalized inductive system. This question has natural connections to the recent notion of K-theory for operator systems by van Suijlekom. Therefore, I would also like to understand interesting examples for computing the K-theory of operator systems that are not C*-algebras.
If you are interested, you may consider: