Abstract
This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.
Highlights
We explain the philosophy behind this paper and provide the context for the two theorems (Theorem 1.3 and Theorem 1.4) that we prove; any undefined terms will be defined later in this paper.The theory of C∗-algebras is the theory of non-commutative spaces
It is often the case that theetale groupoids that occur in constructing such C∗-algebras are those whose spaces of identities are locally compact Boolean spaces — by which we mean 0-dimensional, locally compact Hausdorff spaces
We are interested in topological groupoids, that is groupoids which carry a topology with respect to which multiplication and inversion are continuous, but those topological groupoids which are alsoetale, meaning that the domain and range maps are local homeomorphisms
Summary
We explain the philosophy behind this paper and provide the context for the two theorems (Theorem 1.3 and Theorem 1.4) that we prove; any undefined terms will be defined later in this paper. Our first new result is an application of Theorem 1.1 and uses the theory of ideals of Boolean inverse semigroups described in [45]. It is based on two ideas: that of a cover of an element and that of a cover-to-join map. Our second new result, which is the main theorem of this paper, is a description of the Stone groupoid of the Exel completion of S This involves what is termed the tight groupoid Gt(S) of an inverse semigroup S, introduced in [7]; it will be explicitly defined at the beginning of Section 9. The Stone groupoid of the Exel completion T(S) of S is the tight groupoid Gt(S)
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