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Operator Algebras (B-KUL-G0B07A)

6 ECTSEnglish40 Second term
Szabo Gabor
This course is taught this academic year, but not next year. This course is taught this academic year, but not next year.
POC Wiskunde

Aims

After following this course, the student

(1) knows the notion of spectrum in several contexts; in simple cases, he/she can compute the spectrum,

(2) has acquired insight in the elementary theory of operator algebras, in particular C*-algebras and von Neumann algebras,

(3) can deal with functions of operators,

(4) can illustrate the various concepts and results treated in this course with relevant examples,

(5) has gained intuition about linear mappings between infinite-dimensional Hilbert spaces and is able to verify intuitive conjectures by giving either rigorous proofs or counterexamples,

(6) is able to explore some problems, examples, applications or extensions related to the course, independently using the literature.

Previous knowledge

The student should be fully familiar with (rigorous) analysis and linear algebra on bachelor level. More specifically, concepts as norm, scalar product, Hilbert space, analytic function, matrices, linear mapping, eigenvalues, ... should be very well understood. Basic knowledge of topology is needed. The course G0P55A Topologie amply provides that basic knowledge. But notions from (metric) topology, for example treated in bachelor courses G0N30A Analyse I and G0N86A Analyse II, can suffice initially, provided the student has the maturity to brush up his/her knowledge of topology independently. Previous knowledge of some measure theory is definitely useful. A course such as G0P63B Probability and Measure certainly gives sufficient measure theoretical background. But one can also manage with the basic measure theoretical notions and results as treated in the bachelor course G0N86A Analyse II. It is strongly recommended to have followed G0B03A Functional Analysis. Indeed, some fundamental theorems of Functional Analysis are invoked at times. Whoever hasn't studied the relevant concepts and results will have to acquire independently the insight to understand and use them at least at the level of a "black box".

Is included in these courses of study

Activities

2 ects. Operator Algebras: Exercises (B-KUL-G00J6a)

2 ECTSEnglishFormat: Practical20 Second term
Szabo Gabor
POC Wiskunde

Content

see G0B07a

Format: more information

Discussion

Weekly exercise sessions integrated with the lectures, in which the students further develop the topics of the course and apply the material in different situations.

4 ects. Operator Algebras (B-KUL-G0B07a)

4 ECTSEnglishFormat: Lecture20 Second term
Szabo Gabor
POC Wiskunde

Content

Below a general overview of possible themes and subjects is described. According to the specific background and interests of the students, emphasis can be modulated and possibly extra topics or applications might be covered.

 

Spectral theory in Banach algebras

  • Banach algebras: definition, examples, basic properties
  • The spectrum of an element in a unital Banach algebra: definition, examples, general properties of the spectrum, spectral radius

Gelfand's theory of commutative Banach algebras and C*-algebras

  • The Gelfand transform for commutative Banach algebras
  • C*-algebras: definition, examples, special elements (unitary, self-adjoint, normal) and their spectrum
  • The continuous functional calculus for normal elements in a C*-algebra
  • Gelfand-Naimark theorem

C*-algebras

  • Positivity for elements and functionals
  • Non-unital C*-algebras; approximate units
  • Universal C*-algebras from generators and relations
  • States and representations; GNS construction
  • Pure states and irreducible representations
  • Construction and study of special C*-algebras (e.g. group C*-algebra, irrational rotation algebra)
  • Inductive limits

von Neumann algebras

  • the weak, s−weak, strong and s−strong topologies on the bounded operators on a Hilbert space
  • Defintion of von Neumann algebras, elementary examples
  • Bicommutant theorem
  • Kaplansky density theorem
  • enveloping von Neumann algebras
  • Borel functional calculus
  • Construction and study of special examples (e.g. group von Neumann algebra)

Course material

Concise lecture notes are provided by the lecturer. Those notes have to be elaborated by the student using the literature.

Evaluation

Evaluation: Operator Algebras (B-KUL-G2B07a)

Type : Exam during the examination period
Description of evaluation : Written
Type of questions : Open questions
Learning material : Course material