Séminaire Physique mathématique ICJ

Noncommutative dynamical systems and crossed products

by Dr Jonathan Taylor

Fokko du Cloux (Bâtiment Braconnier)

Fokko du Cloux

Bâtiment Braconnier


Gelfand duality gives the (contravariant) equivalence of categories between locally compact Hausdorff (lcH) spaces and commutative C*-algebras. An action of a group G on a lcH space X induces an action of G on the continuous functions on X via pullback. The analogue for the orbit space X/G is given by the crossed product of C(X) by G, and *-homomorphisms from this crossed product correspond to representations of the dynamical system. 

The shift to noncommutative dynamics is much easier to phrase in the setting of C*-algebras (rather than topological spaces), and one may still construct crossed products by group actions. A new question arises: what is the correct notion of a `non-trivial' action on a C*-algebra? Going further, what properties of the inclusion of the algebra C(X) into the crossed product generalise to the noncommutative setting? The aim of this talk is to introduce noncommutative dynamical systems and show how certain properties in the classical setting generalise. Time permitting, we shall discuss some of the convenient properties of sufficiently `non-trivial' NC dynamical systems and their crossed products.