Partial transformation groupoids attached to graphs and semigroups

By Xin Li. 2016

62 citations recorded by the source corpus.

We introduce the notion of continuous orbit equivalence for partial dynamical systems, and give an equivalent characterization in terms of Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and semigroup C*-algebras can be described as C*-algebras attached to partial dynamical systems. As applications, for graphs, we generalize and explain a result of Matsumoto and Matui relating orbit equivalence and Cartan-isomorphism, and for semigroups, we strengthen several structural results for semigroup C*-algebras concerning amenability, nuclearity as well as simplicity of boundary quotients. We also discuss pure infiniteness for partial transformation groupoids arising from graphs and semigroups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2646159220e22f39fc02e0af93a717248250c6b8

How researchers see this paper

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In Geometry and Topology

1. Mikhailo Dokuchaev
2. Xin Li
3. Xin Li

In Mathematical Physics

1. Xin Li
2. Xin Li
3. Mikhailo Dokuchaev
4. J. Cuntz
5. S. Echterhoff
6. Antonio Paques

Citations in the corpus

• Continuous orbit equivalence rigidity (2015, background): “2 can be generalized to partial systems [17].”
• Every classifiable simple C*-algebra has a Cartan subalgebra (2018, background): “rivial pairing between K-theory and traces in [13]. Recently, the notion of Cartan subalgebras in C*-algebras [17, 26] has attracted some attention, due to connections to continuous orbit equivalence [18, 19] and the UCT question [1, 2]. In particular the reformulation of the UCT question in [1, 2] ”
• Recent developments around partial actions (2018, background): “More recent C and topological advances include the groupoid approach to the enveloping C-algebras associated to partial actions of countable discrete groups on (locally) compact spaces in [156], the use of inverse semigroup expansions to treat C-crossed products by twisted partial actions via twiste”