Introduction to C* algebras

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Banach algebras: ideals, quotients, holomorphic functional calculus, weak topology, Banach Alaoglu's theorem, Gelfand's transform.
C
-algebras: order, approximate units, ideals,
quotients, the characterization of commutative C*-algebras, continuous functional calculus, states and representations, the universal representation.
Operator topologies and approximation theorems: von Neumann's bicommutation theorem, Kaplansky's density theorem and Kadison's transitivity theorem.
The spectral theorem for bounded normal operators: the Borel functional calculus, commutative von Neumann algebras, the group algebra .

Readings

G. K. Pedersen: Analysis Now, Springer, Berlin, 1989.
J. B. Conway: A Course in Functional Analysis, Springer, Berlin, 1978.
J. B. Conway: A Course in Operator Theory, GSM 91, Amer. Math. Soc., 2000.
R. V. Kadison in J. R. Ringrose: Fundamentals of theTtheory of Operator Algebras I, II, Graduate Studies in Math. 15, 16, Amer. Math. Soc., 1997.
I. Vidav: Banachove algebre, DMFA-založništvo, Ljubljana, 1982.
I. Vidav: Uvod v teorijo C*-algeber, DMFA-založništvo, Ljubljana, 1982.
N. Weaver: Mathematical Quantization, Chapman & Hall/CRC, London, 2001.

Objectives and competences

To master basic tools of spectral theory and their use in C*-algebras.

Intended learning outcomes

Knowledge and understanding: the basic knowledge on C-algebras may be useful also outside of mathematics, for example, it may facilitate the understanding of quantum physics.
Application: The acquired knowledge is applicable elsewhere in mathematics and mathematical physics.
Reflection: C
-algebras are one of the basic active fields of modern mathematics.
Transferable skills:
An approach to problems using abstract methods.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Matej Brešar:
BREŠAR, Matej, KISSIN, Edward, SHULMAN, Victor S. Lie ideals: from pure algebra to C[star]-algebras. Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, vol. 2008, b. 623, str. 73-121. [COBISS-SI-ID 14931289]
BREŠAR, Matej, ŠPENKO, Špela. Determining elements in Banach algebras through spectral properties. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2012, vol. 393, iss. 1, str. 144-150. [COBISS-SI-ID 16287833]
BREŠAR, Matej, MAGAJNA, Bojan, ŠPENKO, Špela. Identifying derivations through the spectra of their values. Integral equations and operator theory, ISSN 0378-620X, 2012, vol. 73, no. 3, str. 395-411. [COBISS-SI-ID 16339289]