Research

My research focuses on C*-algebras and their applications. Such algebras are defined abstractly, so I like to see them as an abstraction of the algebra of n by n matrices: it has a product, an involution (the transpose conjugate), and its complete with respect to a norm (the operator norm). Moreover, there are few properties linking the norm with the operations:

This subject can be seen as a combination of analysis and algebra, in the sense that we can apply analytic techniques to prove results stated in algebraic terms. Additionally, this area is highly interdisciplinary, as it allows us, in many cases, to consider a mathematical object and then define a C*-algebra encoding its structure. So, there are C*-algebras constructions for graphs, groups, dynamical systems, etc. A common research problem is to try to understand if a property in the original mathematical object provides an insight about the corresponding C*-algebra. There are also several applications in mathematical physics, including statistical mechanics and quantum field theory.

Most of the main classes of C*-algebras are examples of groupoid C*-algebras. Roughly speaking, a groupoid is a generalisation of groups, in which not every pair of elements can be multiplied.