Operator Algebras

Operator algebras are a branch of functional analysis concerned with algebras of bounded linear operators on Hilbert and Banach spaces, most prominently C*-algebras and von Neumann algebras.

Operator algebras provide a unifying framework in which algebraic structures and analytic behavior interact, making it possible to study symmetry, dynamics, and geometry in settings where classical tools are insufficient. Originally motivated by quantum mechanics, the subject has developed deep connections with (abstract) harmonic analysis, ergodic theory, noncommutative geometry, and mathematical physics.

The research in our workgroup revolves around three closely connected topics. A first core direction concerns the structure and classification of C*-algebras and C*-dynamical systems, with an emphasis on understanding how regularity, symmetry, and rigidity properties govern the behavior of algebras and of crossed products arising from group actions. A second, tightly interwoven theme is abstract harmonic analysis and the study of approximation properties for groups and their associated operator and Banach algebras, where questions about amenability, representations, and noncommutative Fourier analysis play a central role and feed directly into structural and dynamical problems. A third direction explores operator-algebraic methods in quantum information theory, where C*-algebras and von Neumann algebras provide a natural language for quantum systems, channels, and correlations, and where structural and approximation phenomena have concrete operational meaning.

Research interests

Structure and classification of C*-algebras
Quantum Information Theory
Abstract harmonic analysis
Topological and C*-dynamics
Approximation properties
Quantum groups

Topics for Master theses

The following are possible topics for a masters thesis within our workgroup:

Commutators of operators (Hannes Thiel). Characterization for commutators of matrices (those matrices with vanishing trace), and the theorem of Wintner-Wielandt (the unit in a Banach algebra is never a commutator) and the theorem of Brown-Pearcy (an operator on a separable, infinite-dimensional Hilbert space is a commutator if and only if it is not a compact perturbation of a nonzero scalar multiple of the identity).
Existence of non-elementary amenable groups (Eusebio Gardella). This project studies amenable groups beyond the elementary class, focusing on explicit constructions that reveal the richness of amenability beyond the “obvious” examples built from finite and abelian groups. After introducing amenability as the existence of an invariant averaging process and reviewing the closure properties that motivate the notion of elementary amenable groups, the project explores landmark counterexamples showing that not all amenable groups are elementary. Central examples include the Grigorchuk group, whose intermediate growth excludes elementary amenability, and amenable simple groups arising as topological full groups of Cantor minimal systems. The aim is to understand both how amenability is proved in these settings and why these groups fall outside the elementary framework, highlighting connections to dynamics, growth, and operator algebras.
Partial actions: restriction and globalization (Eusebio Gardella). This project investigates partial dynamical systems, which arise naturally when transformations are only locally defined, as in the flows of differential equations. After developing the basic theory of partial actions on topological spaces and understanding how they generalize restrictions of global dynamical systems, the project focuses on the fundamental globalization theorem asserting that every partial action admits an enveloping global action. A central goal is to analyze the topological subtleties of this construction, in particular to characterize when the resulting globalization can be chosen to be Hausdorff. Depending on the student’s interests and background, the project may further explore partial actions in algebraic or operator-algebraic settings, such as rings or C*-algebras, highlighting their relevance to modern dynamics and operator theory.
From questions on almost commuting matrices to stability of C*-algebras and groups (Tatiana Shulman).Informally speaking, questions about almost commuting matrices ask the following: Suppose A and B are matrices and their commutator [A, B] is small. Are there any exactly commuting pairs of matrices near A, B? Questions of this kind are classical in Operator Theory and are useful in the theory of C*-algebras and Group Theory. This project starts with two celebrated results: Voiculescu's result that almost commuting unitary matrices need not be close to commuting ones and Lin's Theorem that states that almost commuting self-adjoint matrices are close to commuting ones.We then will see that these questions are in fact naturally related with C*-algebra theory and lead to the general notion of stability of algebraic structures (such as groups and C*-algebras). The project will study obstructions for stability and concrete examples. Then it can be continued in different ways: either to study connection of stability with approximation conjectures in Group Theory or to focus on operator theoretic aspects of stability.
Stone-Weierstrass Theorem and beyond (Tatiana Shulman). This project begins with a proof of the Krein–Milman theorem, an extremely useful result concerning convex compact sets in locally convex topological vector spaces (including Banach spaces). Then this theorem will be applied to prove the Stone–Weierstrass theorem, which generalizes the Weierstrass approximation theorem to arbitrary compact Hausdorff spaces. The next goal is to study the noncommutative Stone–Weierstrass conjecture. A fundamental fact in C*-algebra theory is the duality between compact Hausdorff spaces and commutative C*-algebras. General C*-algebras need not be commutative, so they are often interpreted as "noncommutative spaces." Some statements about topological spaces (or continuous functions on them) can be generalized to the noncommutative setting, some cannot, and for others the answer remains unknown. The noncommutative Stone–Weierstrass conjecture is an example of the latter case. The project will introduce the concepts necessary to understand the noncommutative Stone–Weierstrass conjecture and, as much as time permits, will study some partial results that are known about it.

Seminars

Research Seminar – Thursdays 15:00–16:00

Learning Seminar – Thursdays 13:15–14:15

Upcoming and past activities (since 2020)

June 2024: Conference Noncommutativity in the North - Mikael Fest, Gothenburg. Organizers: Hannes Thiel (Gothenburg), Eduard Vilalta (Gothenburg) and Georg Huppertz (Gothenburg)
November 2022: Swedish Operator Algebra and Noncommutative Geometry Workshop, Gothenburg. Organizers: Lyudmila Turowska (Gothenburg), Magnus Goffeng (Lund) and Sven Raum (Stockholm)
March 2022: Noncommutativity in the North, Gothenburg, Lyudmila Turowska (Gothenburg) Magnus Goffeng (Lund) and Sven Raum (Stockholm). Supported by VR, Marcus Wallenberg Foundation and Wenner-Gren Foundation.
January 2022: Swedish Operator Algebra and Noncommutative Geometry Workshop, Gothenburg. Organizers: Lyudmila Turowska (Gothenburg), Magnus Goffeng (Lund) and Sven Raum (Stockholm)