TY - BOOK

T1 - Dynamical systems and algebras associated with separated graphs

AU - Lolk, Matias

PY - 2017

Y1 - 2017

N2 - In this thesis, we study partial dynamical systems and graph algebras arising from nitelyseparated graphs. The thesis consists of an introduction followed by three papers, the rst ofwhich is joint work with Pere Ara.In Article [A], we introduce convex subshifts, an abstract generalisation of the partial dynamicalsystems associated with nite separated graphs. We dene notions of a nite and innite typeconvex subshift and show that all such dynamical systems arise from a nite bipartite separatedgraph up to a suitable type of equivalence. We then study various aspects of the ideal structureof the tame separated graph algebras for nite bipartite graphs: We represent the lattice ofinduced ideals by graph-theoretic data, compute all ideals of nite type in the reduced setting,and characterise both simplicity and primitivity.In Article [B], we introduce a generalisation of Condition (K) to nitely separated graphs andshow that it is equivalent to the partial action being essentially free as well as either of thetame algebras having the exchange property. We also demonstrate that Condition (K) is veryrestrictive, and as a consequence, the tame algebras are separative whenever they are exchangerings.Finally, Article [C] completely characterises nuclearity of the tame graph C-algebras in termsof a graph-theoretic property. We also show that the full and reduced tame graph C-algebrascoincide if and only if they are nuclear, and that otherwise the full algebra is in fact non-exact.

AB - In this thesis, we study partial dynamical systems and graph algebras arising from nitelyseparated graphs. The thesis consists of an introduction followed by three papers, the rst ofwhich is joint work with Pere Ara.In Article [A], we introduce convex subshifts, an abstract generalisation of the partial dynamicalsystems associated with nite separated graphs. We dene notions of a nite and innite typeconvex subshift and show that all such dynamical systems arise from a nite bipartite separatedgraph up to a suitable type of equivalence. We then study various aspects of the ideal structureof the tame separated graph algebras for nite bipartite graphs: We represent the lattice ofinduced ideals by graph-theoretic data, compute all ideals of nite type in the reduced setting,and characterise both simplicity and primitivity.In Article [B], we introduce a generalisation of Condition (K) to nitely separated graphs andshow that it is equivalent to the partial action being essentially free as well as either of thetame algebras having the exchange property. We also demonstrate that Condition (K) is veryrestrictive, and as a consequence, the tame algebras are separative whenever they are exchangerings.Finally, Article [C] completely characterises nuclearity of the tame graph C-algebras in termsof a graph-theoretic property. We also show that the full and reduced tame graph C-algebrascoincide if and only if they are nuclear, and that otherwise the full algebra is in fact non-exact.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122927161805763

M3 - Ph.D. thesis

SN - 978-87-7078-925-7

BT - Dynamical systems and algebras associated with separated graphs

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -