On simplicity of the C⁎-algebras associated with λ-graph systems

λ-graph systems are generalizations of finite directed labeled graphs. We introduce the notion of λ-minimality of λ-graph system and prove that it is equivalent to the minimality of the associated étale groupoid. The λ-minimality is also equivalent to the irreducibility of the λ-graph system. Hence the C⁎-algebra OL associated with λ-graph system L is simple if and only if the λ-graph system is irreducible. We also introduce the notion of locally contracting λ-graph system, and prove that it yields pure infiniteness of the simple C⁎-algebras OL.