The structure of gauge-invariant ideals of labelled graph [formula omitted]-algebras

Abstract

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.