Research interests
Research interests
The concept of C*-algebras originally arose with the advent of quantum mechanics. Heisenberg’s matrix mechanics in the 1920s laid the groundwork by using algebraic structures to describe quantum states and observables. In the 1930s, these ideas were further developed by Jordan, Heisenberg and others. They sought a rigorous mathematical framework for quantum mechanics, leading to the consideration of algebras of operators on Hilbert spaces. Murray and von Neumann formalized the theory of operator algebras. Of particular interest were operator norm-closed *-subalgebras of the bounded operators on Hilbert spaces, which are now known as C*-algebras.
Gelfand and Naimark provided an abstract characterization of C*-algebras in 1943, defining them without direct reference to operators on Hilbert spaces. This is the modern definition of a C*-algebra: a Banach algebra with an involution satisfying a specific property which is known as the C*-identity. Basic examples of C*-algebras include the set of square matrices of a fixed size over the complex numbers, (more generally) the set of bounded operators on a Hilbert space, and the set of continuous functions from a locally compact Hausdorff space to the complex numbers. In fact, Gelfand and Naimark proved the remarkable result that every commutative C*-algebra can be realized as such an algebra of continuous functions. In this sense, the theory of C*-algebras can be viewed as non-commutative topology.
This abstract definition broadened the scope and applicability of the theory. In the subsequent decades, the theory of C*-algebras continued to evolve, becoming a rich and expansive field of study in its own right. Its relevance is exemplified by its intricate ties to numerous different fields in mathematics, from topology and geometry to algebra and group theory.
The general principle of classification is that one describes complex objects by certain simpler, better understood objects associated to them, in the sense that the former is completely determined by the latter. The simpler objects are known as invariants. In the setting of C*-algebras, the combined effort of many researchers over the past three decades resulted in the classification of a broad collection of C*-algebras (simple, separable, nuclear C*-algebras which absorb the Jiang-Su algebra tensorially and satisfy the universal coefficient theorem of Rosenberg and Schochet). The invariant, known as the Elliott invariant, consists of a pair of abelian groups (the K-theory, a generalization of topological K-theory), a Choquet simplex (the space of traces) and a map between these two (known as the pairing map).
Purely infinite C*-algebras (the analogue of type III von Neumann algebras) are a specific class of C*-algebras I am interested in. Aside from their numerous striking properties, they are also of interest because their classification theory becomes particularly elegant. In fact, the classification result described earlier was already obtained in the 90s for purely infinite C*-algebras, by Kirchberg and Phillips independently. One of the crucial ingredients for this result are the so-called Kirchberg-Phillips absorption theorems, one of which I gave a more efficient and modern proof for in my Master's thesis.
In recent years, Carrión, Gabe, Schafhauser, Tikuisis and White developed a new, more abstract approach to classification, which connects more explicitly to the von Neumann algebraic classification results. For classification in the purely infinite setting, however, they refer to the original result obtained by Kirchberg and Phillips. The main goal of my PhD project was to use similar techniques to reprove the purely infinite classification theorem. This goal has now been achieved; the resulting paper is currently available as a preprint on arXiv (arXiv:2412.15968).