We classify the gauge-invariant ideals in the C * -algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C * -algebras of arbitrary infinite graphs.

This paper explores the effect of various graphical constructions upon the associated graph C * -algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that outsplittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C * -algebras. We generalise the notion of a delay as defined in [D] to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C * -algebras. We provide examples which suggest that our results are the most general possible in the setting of the C * -algebras of arbitrary directed graphs.

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz-Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as Prim C * (E) are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph E from E such that C * ( E) is an AF-algebra and Prim C * (E) and Prim C * ( E) are homeomorphic.

We prove simplicity and pure infiniteness results for a certain class of labelled graph C * -algebras. We show, by example, that this class of unital labelled graph C * -algebras is strictly larger than the class of unital graph C * -algebras.

In this paper we give a formula for the K-theory of the C * -algebra of a weakly left-resolving labelled space. This is done by realising the C * -algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner algebra of a C * -correspondence. As a corollary we get a gauge invariant uniqueness theorem for the C * -algebra of any weakly left-resolving labelled space. In order to achieve this we must modify the definition of the C * -algebra of a weakly left-resolving labelled space.We also establish strong connections between the various classes of C * -algebras which are associated with shift spaces and labelled graph algebras. Hence, by computing the K-theory of a labelled graph algebra we are providing a common framework for computing the K-theory of graph algebras, ultragraph algebras, Exel-Laca algebras, Matsumoto algebras and the C *algebras of Carlsen.We provide an inductive limit approach for computing the K-groups of an important class of labelled graph algebras, and give examples.C * -ALGEBRAS OF LABELLED GRAPHS III -K-THEORY COMPUTATIONS 2 see Subsection 2) in the main part of the paper, but in the appendix we show how to deal with the general case.The motivation for our K-theory computations is to investigate the relationship between certain dynamical invariants of shift spaces and the K-theoretical invariants of C * -algebras associated to these shift spaces. This connection was first brought to light in the work of Cuntz and Krieger in [12] where they showed how to associate a C * -algebra O A to a finite 0-1 matrix A with no zero rows or columns, provided that the matrix satisfied a certain condition called (I). In [13, Proposition 3.1] it was shown that the K-groups of a Cuntz-Krieger algebra are isomorphic to the Bowen-Franks groups of the shift of finite type X A associated to A (see [7]). Thus, a deep connection was established between the combinatorially-defined C * -algebra O A and the (one-sided) shift of finite type X A . Several generalisations of Cuntz-Krieger algebras have now been widely studied.Combining the universal algebra approach of [1] and the graphical approach to Cuntz-Krieger algebras begun in [15], graph algebras were introduced in [25]. Graph algebras were originally defined for graphs satisfying a finiteness condition -the need for this condition was removed by Fowler and Raeburn in [18] (see also [17]). Using a different approach, Exel and Laca showed how to associate a C * -algebra to an infinite 0-1 matrix with no zero rows or columns in [16]. A link between graph algebras and Exel-Laca algebras was provided by the ultragraph algebras introduced by Tomforde (see [39,40]).Motivated by the symbolic dynamical data contained in a Cuntz-Krieger algebra, Matsumoto provided a generalisation of Cuntz-Krieger algebras by associating to an arbitrary two-sided shift space Λ over a finite alphabet a C * -algebra O Λ (see [27,28,29,30,32,33]). Later Carlsen and Matsumoto modified this construction (see [9,35]), and Carlsen further modified the definition of O Λ and extended it...