Semi Étale Groupoids and Applications

The main purpose of this paper is to develop new tools for the investigation of C-algebras which have been constructed from shift spaces in a series of papers by K. Matsumoto and T. Carlsen, cf. [Ma1]-[Ma5], [C], [CM]. The main results about the structure of these algebras which we obtain here give necessary and sufficient conditions for the algebras to be simple, and show that they all contain a Cartan subalgebra in the sense introduced by J. Renault in [Re3]. Previous results on simplicity of the C-algebras defined from subshifts are all due to Matsumoto and give only sufficient conditions under various additional assumptions on the subshift. As a step on the way we show that the algebras are all the crossed product in the spirit of Paschke, [P], arising from a full corner endomorphism of an AF-algebra. The methods we employ are useful beyond the study of C-algebras of subshifts because they extend the applicability of locally compact groupoids to the construction and study of C-algebras. The use of groupoids in relation to C-algebras was initiated by the pioneering work of J. Renault in [Re1]. After a relatively slow beginning during the eighties the last two decades has witnessed an increasing recognition of the importance of groupoids as a tool to encode various mathematical structures in a C-algebra. Of particular importance in this respect are the socalled étale groupoids which have been used in many different contexts, for example in connection with graph algebras and dynamical systems. In an étale groupoid the range and source maps are local homeomorphisms, and in particular open as they must be if there is a Haar-system in the sense of Renault, cf. [Re1]. But in the locally compact groupoid which is naturally associated to a dynamical system by the construction of Renault, Deacuno and Anantharaman-Delaroche, cf. [Re1], [De] and [A], the range and source maps are only open if the map of the dynamical system is also open, and this is a serious limitation which for example prevents the method from being used on subshifts which are not of finite type. For this reason we propose here a construction of a C-algebra from a more general class of locally compact Hausdorff groupoids which differ from the étale groupoids in that the range and source maps are locally injective, but not necessarily open. This class of groupoids is not new; it coincides with the locally compact Hausdorff groupoids which were called r-discrete by Renault in [Re1] and they are equipped with a (continuous) Haar-system if and only if they are étale. In many influential places in the litterature on the C-algebras of groupoids, such [A] or [Pa] for example, an r-discrete groupoid is assumed to have a continuous Haar-system and hence to be étale in the terminology which is now generally accepted. In order to avoid any misunderstanding we therefore propose the name semi étale for the class of locally compact groupoids where the range map is locally injective, but not necessarily open.