Department of Mathematics, King Mongkut’s University of Technology Thonburi (KMUTT)
16:00 Wednesday 29 March 2017, Room 3303, MUIC
The aim of this talk is to give an introduction on the crossed product C*-algebras of C*-dynamical systems by actions of groups of automorphisms. Dynamical systems, as the name suggests, are mathematical formulations of problems in which the situation evolves with time. Traditionally, this evolution is described by a differential, difference or integral equation, or more precisely, by solving initial value problems for this equation.
In qualitative models, the time evolution is given by an action of the additive group of real numbers on the state space of the system. The same mathematical formalism describes the symmetries of the system: the reflections, rotations and translations which do not materially change the system. The allowable symmetries then form a group which should act on the state space in any model of the system.
In classical systems, the groups act by homeomorphisms of a manifold, or equivalently on a commutative algebra of functions on that manifold. The modern subject of non-commutative geometry centers around the idea that one can profitably replace the commutative algebra of functions by a non-commutative algebra, and that many of the basic notions and constructions of geometry and analysis have interesting non-commutative analogues. This idea originally arose in models of quantum mechanics, but has since proved useful in topology, geometry, and the biological sciences, as well as in physics.
Operator algebra provides one way of tackling problems in non-commutative geometry. In operator algebras, objects are studied through their representations as operators on Hilbert spaces, just as one traditionally studies as abstract groups through their realizations as groups of matrices. The different components in dynamical systems are realized by operators on the same Hilbert spaces-the algebra by a representation as unitary operators- and the two are connected by a covariance condition which encodes the action. The main technical tool in their analysis is the crossed product C*-algebras.