FMA340F

Temporary

Introduction to Operator Algebras

7.5

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

2026-01-15

/Jonas Johansson

English

If sufficient demand

The theory of operator algebras provides essential theoretical foundations for advanced studies in functional analysis, non-commutative geometry, and quantum physics. The theory of C*-algebras and von Neumann algebras offers a powerful framework for analyzing operators on Hilbert space and for generalizing the topology and measure theory of classical spaces. The course introduces key concepts such as spectral theory, the GNS construction, and continuous functional calculus, enabling the study of abstract algebraic structures through their concrete representations. These tools are indispensable in modern research and of importance to students aiming to specialize in pure mathematics, particularly analysis and geometry, or in mathematical physics.

This course provides a rigorous foundation in the theory of operator algebras, progressing from the general theory of Banach algebras to the rich structure of C*-algebras and von Neumann algebras. The course begins with the fundamentals of Banach algebras and Gelfand's powerful theory for the commutative case. We then explore the core of the subject: the structure of C*-algebras, including the spectral theory of special elements (normal, self-adjoint, positive) and the pivotal continuous functional calculus. A central highlight is the Gelfand-Naimark theorem, which establishes that every C*-algebra is isometrically isomorphic to an algebra of bounded operators on a Hilbert space. This representation theory is further developed through the study of states and the profound Gelfand-Naimark-Segal (GNS) construction. The course culminates in the theory of von Neumann algebras, analyzed through the lens of the bicommutant theorem and Kaplansky density theorem.

be able to explain the notion of the spectrum in various contexts and compute it in simple cases.

be able to demonstrate knowledge of the elementary theory of operator algebras, specifically regarding C*-algebras and von Neumann algebras.

be able to illustrate central concepts and theorems from the course with concrete examples.

be able to analyze linear mappings on infinite-dimensional Hilbert spaces and test intuitive conjectures by constructing rigorous proofs or counterexamples.

be able to independently investigate problems, examples, or applications related to the course content using relevant literature.

be able to justify functional analytic and spectral properties of operators using key tools such as the continuous functional calculus, positive linear functionals, and the GNS construction.

be able to calculate spectra of operators and utilize the techniques of the continuous functional calculus.

be able to apply the GNS construction to build Hilbert space representations from states and analyze their properties.

be able to construct many examples of C*-algebras via standard operations.

Lectures

Oral exam

Written assignments

Failed, pass

Measure theory, Topology and Functional analysis.

 

Course literature

Murphy, Gerard J. C*-Algebras and Operator Theory. Academic Press, 1990.

Davidson, Kenneth R. C*-Algebras by Example. American Mathematical Society, 1996.

Douglas, Ronald G. Banach Algebra Techniques in Operator Theory. 2nd ed., Springer, 1998.

Szabó, Gábor. Lecture Notes on Operator Algebras, 2024.

FMA340F

2026-01-15

/Jonas Johansson