Regularity properties and K-theory
(K-stability, K_1-injectivity and K_1-surjectivity)
Operator algebraic K-theory encodes intrinsic data of C*-algebras, capturing structural information about the spaces of projections and unitaries in matrix amplifications. It has served as a fundamental invariant in the classification of simple C*-algebras. Among the desirable K-theoretic properties are K_1-injectivity and K_1-surjectivity, which roughly ask whether K-theory can be computed without passing to matrix amplifications. It is well established in the literature that many regularity conditions, such as Z-stability, real rank zero, and stable rank one, imply K_1-injectivity and/or K_1-surjectivity (it has been open of whether real rank zero C*-algebra are K_1-surjective).
I am interested in several open questions surrounding these two notions. One direction is to understand whether milder conditions, such as pureness or proper infiniteness, are still sufficient to guarantee such properties. A more fundamental open question is whether there exists a simple C*-algebra that fails to be K_1-injective; by contrast, a simple example that is not K_1-surjetive has been known since Villadsen's classical examples. Obtaining a full understanding of these properties requires deep insight into the most fundamental internal structure of C*-algebras.
Concrete realizations of C*-algebras
(Groupoid C*-algebras and crossed products)
Following the Elliott classification theorem for a large class of simple, nuclear, separable, and unital C*-algebras, a natural and important direction is to seek concrete models for this class. Existing realizations include ASH inductive-limit models of Elliott's, Spielberg’s groupoid models for Kirchberg algebras, and Li’s twisted groupoid models for stably finite classifiable algebras. Similar questions for crossed product models have been wildly open.
My current research also explores unconventional groupoid models for AF-algebras, with the aim of better understanding their diagonals and symmetries. In the longer term, I hope to develop a more precise correspondence between regularity properties of dynamical or groupoid models and structural properties of their C*-algebras, including comparison, (tracial) Z-stability, nuclear dimension, etc.
Beyond the current scope of classification
(Classification of non-simple C*-algebras, and non-full maps)
I am interested in developing a framework for classifying nonsimple C*-algebras and non-full *-homomorphisms. Such a theory will likely require ideal-related invariants and relative versions of pure largeness, both to formulate suitable regularity hypotheses and to establish uniqueness and classification results. In many cases, even the appropriate definitions remain unclear, leaving substantial scope for foundational development.