Abstract
In this thesis, we study partial dynamical systems and graph algebras arising from nitely
separated graphs. The thesis consists of an introduction followed by three papers, the rst 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 nite separated graphs. We dene notions of a nite and innite type
convex subshift and show that all such dynamical systems arise from a nite 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 nite bipartite graphs: We represent the lattice of
induced 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 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.
separated graphs. The thesis consists of an introduction followed by three papers, the rst 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 nite separated graphs. We dene notions of a nite and innite type
convex subshift and show that all such dynamical systems arise from a nite 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 nite bipartite graphs: We represent the lattice of
induced 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 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.
| Originalsprog | Engelsk |
|---|
| Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
|---|---|
| Antal sider | 160 |
| ISBN (Trykt) | 978-87-7078-925-7 |
| Status | Udgivet - 2017 |