CLASSICAL SYMMETRIES OF NONCOMMUTATIVE SPACES: A GUIDED TOUR THROUGH THE CUNTZ ALGEBRA CASE

Multiplets of isometries with orthogonal ranges summing up to 1 were systematically used by Doplicher and Roberts in the early 70's  in their study of the superselection structure of Quantum Field Theory. Nowadays, the C*-algebra generated by any such multiplet is known as the Cuntz algebra O_n, where n is the cardinality of the given multiplet. Since their introduction, Cuntz algebras have been the subject of endless investigations from many different points of view. They are perhaps the most studied/used class of C*-algebras ever. Notably, the study of their automorphisms has many challenging facets, where operator algebras meet Lie theory, dynamical systems and combinatorics. We will present an overview of old and new results along with some open questions.