Graph C*-algebras and stability

Abstract

In this thesis, we present three results on C∗-algebras. The first result provides, for certain graphs, a decomposition of the corresponding graph C∗-algebras into an amalgamated free product. This decomposition is used to study two properties of graph C∗-algebras: residually finite dimensionality and a stability property. For a graph with finitely many vertices, its C∗-algebra is residually finite-dimensional if and only if no cycle has an entry (here we use Raeburn’s convention for associating a C∗-algebra to a graph). Moreover, for a graph with finitely many vertices, its C∗-algebra is matricially semiprojective if and only if ˜G, a certain subgraph of G, is finite. The second result characterises residually finite-dimensional C∗-algebras associated with directed graphs. The C∗-algebra of a directed graph is residually finite-dimensional if and only if the graph satisfies four conditions: it has no infinite receivers, it has no cycles with an exit, it has no infinite backward chains, and for every vertex there exists a finite path from that vertex to a sink, a cycle, or an infinite emitter (here we use the classical convention for associating a C∗-algebra to a graph, according to the Tomforde notation). The proof relies on a characterisation of the residually finite-dimensional property for groupoid C∗-algebras given in [1], and on a groupoid constructed from a graph, whose C∗-algebra is isomorphic to the C∗-algebra of the graph. The third result concerns a stability property of C∗-algebras, namely matricial semiprojectivity. We study soft deformations of C∗-algebras, and in particular the soft deformation of the crystallographic group P2. Using an idea of D. Enders from on the soft torus, we prove that the soft deformation of the C∗-algebra of the group P2 is not matricially semiprojective.