## Introduction to C* algebras

### Dr. Gherardo Piacitelli

Number of cycles: 1

First semester: Tuesdays 2-4pm, Wednesdays 4-6pm (starting October 3)

### Programme

**First part (8 hours): Basics about C*-algebras and their representations.**

Definition of C*-algebras, Gelfand transform, functional calculus,
representation theory, GNS construction, universal representation,
geometry of states. Operator algebras: weak topologies, bicommutant
theorem. Tomita-Takesaki theorem (statement only)

**Second Part (12 hours): selected topics in Quantum Physics.**

Axioms for Quantum Physics (Copenhagen interpretation), generalised
Heisenberg uncertainty ``principle'' (a theorem, actually).
Localisability and Weyl relations. Von Neumann uniqueness of the
Schr\"odinger representation, Weyl quantisation. The quantum harmonic
oscillator (sketchy). Rotations. Elements of local (relativistic) quantum physics.
Superselection theory as a duality problem for compact groups.