There are no prerequisites.

# Introduction to C* algebras

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 .

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.

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

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.

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

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]