PhD Defense Matias Lolk

Title: Algebras and dynamical systems associated with separated graphs

Abstract:

In this thesis, we study partial dynamical systems and graph algebras arising from finitelyseparated graphs. The thesis consists of an introduction followed by three papers, the first of which is joint work with Pere Ara.

In Article [A], we introduce convex subshifts, an abstract generalisation of the partial dynamical systems associated with finite separated graphs. We define notions of a finite and infinite type convex subshift and show that all such dynamical systems arise from a finite bipartite separated graph up to a suitable type of equivalence. We then study various aspects of the ideal structure of the tame separated graph algebras for finite bipartite graphs: We represent the lattice of induced ideals by graph-theoretic data, compute all ideals of finite type in the reduced setting, and characterise both simplicity and primitivity.

In Article [B], we introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to the partial action being essentially free as well as either of the tame algebras having the exchange property. We also demonstrate that Condition (K) is very restrictive, and as a consequence, the tame algebras are separative whenever they are exchange rings.

Finally, Article [C] completely characterises nuclearity of the tame graph C*-algebras in terms of a graph-theoretic property. We also show that the full and reduced tame graph C*-algebras coincide if and only if they are nuclear, and that otherwise the full algebra is in fact non-exact.

Supervisors:  

Prof. Mikael Rørdam, University of Copenhagen

Assessment committee:

Prof. Søren Eilers (Chairman), University of Copenhagen

Prof. Wojciech Szymanski, University of Southern Denmark

Senior Lecturer, Xin Li, Queen Mary University of London