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

The aim of this course is to acquire the basic elements of the theory of C*-algebras and von Neumann algebras as a unified language for spectral theory and non commutative probability. 

AIMS AND LEARNING OUTCOMES

The goal is for the student 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 be able to:

-understand the mathematical language of C*-algebras and von Neumann algebras,

- reproduce and comment on constructions and proofs of relevant objects in the theory of C*-algebras and von Neumann algebras,

- work with functionals and normal and positive operators, analysing their spectral properties.

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.

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

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: Exposition of a topic covered during teaching with relevant proofs.

Students with disabilities or specific learning disorders (SLDs) are reminded that in order to request adaptations in the exam, they must first enter their certification on the University website at servizionline.unige.it in the “Students” section. The documentation will be verified by the University's Services for the Inclusion of Students with Disabilities and DSA Sector.

Subsequently, significantly in advance (at least 7 days) of the examination date, an e-mail must be sent to the teacher with whom the examination will be taken, including in the knowledge copy both the School's Teacher Referent for the Inclusion of Students with Disabilities and with DSA (sergio.didomizio@unige.it) and the above-mentioned Sector. The e-mail should specify:

- the name of the teaching

- the date of the roll call

- the student's last name, first name and roll number

- the compensatory tools and dispensatory measures deemed functional and required.

The contact person will confirm to the teacher that the applicant has the right to request adaptations in the examination and that these adaptations must be agreed upon with the teacher. The lecturer will respond by informing whether it is possible to use the requested adaptations.

Requests should be sent at least 7 days before the date of the call in order to allow the lecturer to assess the content. In particular, in the case of intending to make use of concept maps for the exam (which must be much more concise than the maps used for study) if the submission does not meet the deadline there will be no technical time to make any adjustments.

For more information regarding the request for services and adaptations see the document: Guidelines for requesting services, compensatory tools and/or dispensatory measures and specific aids.

ASSESSMENT METHODS

Verification of learning is by oral examination only and will focus on topics covered in class.

The following will be evaluated:

 -the correctness of mathematical language and formalism,

-the knowledge of mathematical objects and results introduced,

-the ability to use these tools.

FURTHER INFORMATION

Contact the lecturer for further information not included in the teaching sheet.