K-theory of C*-algebras
K-theory; K-stability; K1-injectivity and K1-surjectivity
K-theory provides a fundamental invariant for C*-algebras, defined in terms of equivalence classes of projections and unitaries in matrix amplifications of the algebra. The resulting K-groups can be identified with higher homotopy groups of unitary groups after stabilization. In certain cases, however, matrix amplification is unnecessary—for instance, when the C*-algebra is stable, i.e. tensorially absorbs the compact operators. Understanding which regularity properties guarantee that such stabilization is automatic remains an intriguing question.
Classification program of C*-algebras
Elliott classification program; Z-stability; K-theory and traces
The Elliott classification program, motivated by the cornerstone classification results of separable hyperfinite von Neumann factors, aims to classify unital, simple, separable and nuclear C*-algebras by K-theory and traces. It turns out that a regularity property called Z-stability is necessary, which is a tensorial absorption property by the Jiang-Su algebra.
Graph C*-algebras
Directed graphs; higher rank / self-similar graph C*-algebras
From combinatorial data, such as finite directed graphs or group actions on graphs, one can construct analytic objects in the form of C*-algebras. These C*-algebras admit canonical groupoid models, whose properties are often more accessible and easier to analyze. Establishing regularity properties and investigating substructures of these C*-algebras constitute an active and ongoing area of research.