OVERVIEW

The teaching presents the theory of C*-algebras and von Neumann algebras, which are the basis for the study of Quantum Mechanics and Quantum Probability.

AIMS AND CONTENT

LEARNING OUTCOMES

Introduction to Quantum Probability, C*-algebras, von Neumann algebras and Quantum Markov Semigroups.

AIMS AND LEARNING OUTCOMES

The goal is to learn the theory of C*-algebras and von Neumann algebras, which are the natural spaces on which to define quantum evolutions. 

Specifically, upon completion of the teaching, the student will:

-have learned the mathematical language of quantum probability,

-know how to work with normal and positive functionals and operators, and study their spectral properties,

-gain mastery of the topics covered so that he/she can then continue the study independently on related and more advanced topics.

PREREQUISITES

Theory of linear bounded and compact operators, strong and weak operator topology, Banach algebras.

TEACHING METHODS

Teaching is done in the traditional way, with lectures held at the blackboard. 

Attendance is not mandatory but strongly recommended.

SYLLABUS/CONTENT

- C*-algebras: positive elements, positive functionals, GNS representation.

-Bounded operators on Hilbert spaces: sesquilinear forms, projections, partial isometries and polar decomposition theorem.

- Trace class and Hilbert-Schmidt operators. Spectral theorem.

- Weak, strong and sigma-weak topologies on the algebra of bounded operators on a Hilbert space.

- Von Neumann algebras: normal states and predual, tensor product of von Neumann algebras, type I factors and representation theorem.

-Completely positive maps. 

-Introduction to Quantum Markov Semigroups.

Teaching contributes to the achievement of Goals 4 (provide quality, equitable and inclusive education and learning opportunities for all) and 5 (achieve gender equality and empower all women and girls) of Sustainable Development of the UN 2030 Agenda.

RECOMMENDED READING/BIBLIOGRAPHY

- Bratteli, Robinson: Operator algebras and quantum statistical mechanics 1

- Conway: A course in functional analysis

- Murphy: C*-algebras and operator theory

- Sakai: C*-algebras and W*-algebras

- Takesaki: Theory of operator algebras I

- Dixmier: Von Neumann algebras

TEACHERS AND EXAM BOARD

Exam Board

VERONICA UMANITA' (President)

DAMIANO POLETTI

EMANUELA SASSO (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral test.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

ASSESSMENT METHODS

Verification of learning is by oral examination only and will focus on topics covered in class. The student will be expected to show correctness in mathematical language and formalism, be well acquainted with the mathematical objects and results of the course, and be able to use them naturally.

Exam schedule