The structure of gauge-invariant ideals of labelled graph C⁎-algebras

In this paper we consider the gauge-invariant ideal structure of a C⁎-algebra C⁎(E,L,B) associated to a set-finite, receiver set-finite and weakly left-resolving labelled space (E,L,B), where L is a labelling map assigning an alphabet to each edge of the directed graph E with no sinks. It is obtained that if an accommodating set B is closed under relative complements, there is a one-to-one correspondence between the set of all hereditary saturated subsets of B and the gauge-invariant ideals of C⁎(E,L,B). For this, we introduce a quotient labelled space (E,L,[B]R) arising from an equivalence relation ∼R on B and show the existence of the C⁎-algebra C⁎(E,L,[B]R) generated by a universal representation of (E,L,[B]R). Finally we give necessary and sufficient conditions for simplicity of certain labelled graph C⁎-algebras.