Robert Neagu - Research

In the attempt to produce a mathematical formulation of quantum mechanics, Murray and von Neumann introduced operator algebras in the 1930's as a description of the algebra of observables in a quantum system. With time, operator algebras proved to be an interesting field in its own right, having deep interactions with functional analysis, algebraic topology, group theory, or quantum information.

A C*-algebra is a *- subalgebra of the collection of bounded linear operators on a Hilbert space which is closed in the norm topology. Early in the theory, Gelfand and Naimark remarkably showed that any commutative C*-algebra can be realised as an algebra of continuous functions on a locally compact Hausdorff space. Then, as even more C*-properties turned out to have a strong topological flavour, an important philosophy is that studying C*-algebras is somehow in the realm of non-commutative topology.

Classification is a common theme in mathematics, encompassing the idea that quite complicated objects are completely determined by some simpler objects associated to them, called invariants. To a C*-algebra, one can associate a pair of abelian groups (its K-theory, generalising the topological K-theory) and a Choquet simplex (its space of traces). This so-called Elliott invariant cannot distinguish all C*-algebras, but due to remarkable work by many researchers over the last three decades, it completely determines simple, separable, nuclear C*-algebras which have enough regularity (technically, they tensorially absorb an infinite-dimensional version of the complex numbers called the Jiang-Su algebra and satisfy the universal coefficient theorem of Rosenberg and Schochet).

My research interests span across operator algebras and their dynamics. I am interested in using tools from the classification theory of C*-algebras to study inclusions of C*-algebras and develop regularity properties for them. Another important direction of my research is the study of classical and quantum symmetries of C*-algebras. More precisely, I am interested in studying actions of the real numbers on C*-algebras via their induced KMS states as well as in the classification of actions of tensor categories on C*-algebras in the spirit of subfactor theory. For a glimpse into this last topic, you can have a look at this poster presenting joint work with Sergio Giron Pacheco.