Mathematics

1. Banach algebras and C*-algebras, examples from harmonic analysis on locally compact groups and operator algebras 2. Spectrum and resolvent 3. Continuous functional calculus, spectral theorem for bounded normal operators on a Hilbert space, Borel functional calculus 4. Multiplicity-free theory, spectral multiplicity theory 5. Unbounded densely defined operators on Hilbert space, closed, closable, closure, adjoint, differential operators 6. Cayley transform, unbounded spectral theorem, Stone's theorem 7. Positive functionals, Gelfand-Naimark-Segal representation, Gelfand-Naimark theorem 8. Abelian Banach algebras, Gelfand transform, Gelfand theorem 9. Von Neumann algebras, von Neumann bicommutant theorem; abstract definition. 10. Classification of factors into types I, II, and III 11. Elements of modular theory, Tomita's theorem 12 Kubo-Martin-Schwinger states (KMS states) and representations

Measure theory and basic functional analysis on the theory of Banach, Hilbert, and locally convex spaces. Any useful results in the theory of locally convex spaces that are not known to the students will be recalled during the course.

A. Connes: Noncommutative Geometry. J. Dixmier: C∗-algebras. S. Doplicher: Note del corso di Analisi Funzionale, parte I e II S. Doplicher: An invitation to Quantum Mechanics. E. Hewitt, K. Ross: Abstract Harmonic Analysis V.F.R. Jones: Subfactors and knots. AMS G.K. Pedersen: Analysis Now, Springer, G. Pedersen: C*-algebras and their automorphism groups, LMS M. Reed and B. Simon: Functional analysis. W. Rudin: Fourier Analysis on groups S. Sakai: C*-algebras and W*-algebras. Springer. M. Takesaki: Theory of operator algebra, vol. 1, 2, 3.

Participation to lectures is strongly recommended

The exam consists of an interview where the student will be asked questions about the course topics. Students will have the option of exploring a topic of their choosing during the exam. This chosen topic will focus on a list of topics proposed by the teacher during the course.

Two-hour lectures covering the program topics in detail. Some topics will be briefly touched upon and offered to students for further study, optionally, to be presented for the exam.