Labelled space C⁎-algebras as partial crossed products and a simplicity characterization
Introduction
The interplay between Symbolic Dynamics and -algebras started with the work of Cuntz and Krieger [16]. They associated a -algebra, known as a Cuntz-Krieger algebra, to a finite square matrix of 0's and 1's representing what is known as a subshift of finite type. These subshifts can also be described using finite graphs [31]. Cuntz-Krieger algebras were generalized to a general subshift by Matsumoto and Carlsen [32], [14], [12], to graph -algebras by considering infinite graphs in [29], and to Exel-Laca algebras by considering infinite matrices in [22]. In [41] Tomforde introduces ultragraphs and their -algebras as a common framework for Exel-Laca algebras and graph -algebras.
From the point of view of Symbolic Dynamics, a broader class of subshifts, called sofic subshifts, can be defined by using labelled graphs [31]. With labelled graphs and Tomforde's definition of an ultragraph in mind, Bates and Pask defined the notion of a labelled space and associated a -algebra to it [4]. The class of labelled space -algebras generalizes all the classes of algebras mentioned above, including -algebras associated to general subshifts, not necessarily sofic. There were some problems with Bates and Pask's original definition of a labelled space -algebra, and a new one was proposed, independently, in [3] and by the first named author together with Boava and Mortari in [7].
In some instances there is more than one model for an associated -algebra. For example, it is well known that a directed graph has a groupoid model such the associated -algebras are isomorphic [29], [33]. In [13] it is shown that a directed graph also has a partial dynamical system associated with it such that the graph -algebra and the crossed product -algebra are isomorphic. In [8], the first named author in collaboration with Boava and Mortari gave a groupoid model to labelled space -algebras.
In this paper we give a new description of a labelled space -algebra by associating a partial action with a labelled space. We prove that the labelled space -algebra is isomorphic to the partial crossed product of this partial action by giving an explicit isomorphism. In addition, we show that the partial action groupoid (as in [1]) is isomorphic to the labelled space groupoid as given in [8], which generalizes the various models for graphs described above to labelled spaces. Unlike graphs, the case of labelled spaces brings new technical difficulties that need to be dealt with. Specifically, our partial action is defined on the tight spectrum, which is a set of filters in an inverse semigroup associated with the labelled space [6]. And, each tight filter is characterized by a pair consisting of a path and family of filters in the underlying vertex set. Therefore, unlike graphs where one only considers paths, here we also need to consider these families of filters in conjunction with paths.
A particular property of -algebras associated to some underlying object that has been well studied is simplicity. Simplicity of graph -algebras is characterized in [29], [38], [33], [18], for partial actions associated to graphs in [24] and for étale groupoids in [9]. Simplicity of ultragraphs -algebras is studied in [40].
We use our new description of labelled space -algebras as partial crossed product as well as description as a groupoid -algebra to characterize simplicity of labelled space -algebras in terms of the tight spectrum of the labelled space and in terms of the labelled space itself. Sufficient conditions for simplicity of labelled space -algebras are given in [5], and a converse is given in [26], [28]. However, in both instances it is assumed that the labelled space is set-finite and receiver set-finite, the underlying graph has no sinks and no sources, and that the accommodating family of the labelled space is the smallest such family. We do not make any of these assumptions in this paper. In [15] Boolean dynamical systems are studied, which generalize labelled spaces. A characterization of simplicity for a -algebra associated with a Boolean dynamical system is also given in [15]. In this instance, though, the Boolean dynamical system is assumed to be countable with a certain domain condition. We do not assume countability or any domain conditions in our simplicity characterization. In particular, we remove the domain condition in [28, Theorem 3.7]. In the process we recover some of the simplicity results of [15].
In order to describe our results on the simplicity of labelled spaces -algebras, we recall three conditions found to describe the simplicity of -algebras associated to an arbitrary graph [18, Corollary 2.15]. The first condition is called condition (L), which says that every cycle on the graph has an exit. The second one, called cofinality, says that every infinite path on the graph can be reached in a sense from finite paths, and the third condition deals with singular vertices. Also, in [18], the authors study the ideal structure of graph -algebras using the concept of hereditary and saturated sets of vertices. In broad terms, hereditary means if a vertex is in the set then all vertices that can be reached by this one are also in the set, and saturated means all vertexes that reaches a certain vertex are in the set, then the reached vertex is in the set as well. There are versions of condition (L), called condition (), and hereditary saturated subsets for labelled spaces as well (Definition 6.12, Definition 6.14 respectively). We prove, under some mild hypothesis, that a labelled space -algebra is simple if and only if the labelled space satisfies condition () and the only hereditary saturated sets are trivial (Theorem 6.16). We also point out that, in some cases, hereditary saturated sets can be used to describe ideals in labelled spaces -algebras [27], [19], [2].
In [15], the simplicity characterization for the -algebra of a Boolean dynamical system is applied to the Cuntz-Pimsner algebra of a one-sided subshift. It is stated that the -algebra of the subshift is simple if and only if there is no cyclic point isolated in past equivalence and the subshift is cofinal in past equivalence, [15, Example 11.4]. However, a different simplicity characterization of these algebras is given in [17]. We apply our results to -algebras of one-sided subshifts and recover the results in [17]. Then we give an example where the subshift has no cyclic points isolated in past equivalence and is cofinal in past equivalence, but the associated algebra is not simple, which shows that a stronger condition than cofinality in past equivalence is needed in [15, Example 11.4].
This paper is structured as follows. Section 2 contains some preliminaries and notation on labelled spaces and their -algebras which are used throughout this paper. In Section 3 we define a partial action on the tight spectrum of a labelled space. Section 4 contains our first main result, Theorem 4.8. Here we show that the partial crossed product, obtained from the partial action of the previous section, is isomorphic to the labelled space -algebra. In section 5 we show that the groupoid associated with a labelled space (as in [8]) is the same groupoid that is obtained from the partial action (as in [1]). In Section 6 we apply our results on the partial action and its associated groupoid to characterize simplicity of labelled space -algebras in terms of the tight spectrum (Theorem 6.7, Theorem 6.16). Finally, in Section 7, by applying our results to subshifts, we characterize simplicity of certain -algebras associated with subshifts, and recover some of the results in [17]. We also give an example to show that a stronger condition is needed for simplicity of -algebras associated with subshifts than that given in [15, Example 11.4].
Acknowledgments: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The majority of this paper was completed while the second author worked at Universidade Federal de Santa Catarina under the guidance of the first author. He thanks him for his guidance and warm hospitality. The authors would also like to thank the referee for several suggestions that helped improve the clarity of the exposition.
Section snippets
Filters and characters
A filter in a partially ordered set P with least element 0 is a subset ξ of P such that
- (i)
;
- (ii)
if and , then ;
- (iii)
if , there exists such that and .
For , we define and for subsets of P define and ; the sets , , and are defined analogously.
Lemma 2.1 Let
A partial action on the tight filters of
In this section we define a partial action of the free group generated by on the tight spectrum of a labelled space. Our construction is in the same spirit as that of graphs [13]. However, as opposed to graphs where we only need to consider paths, filters in are in one-to-one correspondence with pairs consisting of a labelled path and a family of filters (see Theorem 2.4), which adds an extra layer of complexity that needs to dealt with.
We recall the definition of a topological
Labelled space -algebra as a crossed product
In this section we show that the partial crossed product -algebra obtained from the partial action in Section 3 is isomorphic to the labelled space -algebra .
We begin by describing the partial crossed product -algebra . Let be a weakly left-resolving labelled space and let be the partial action associated with in Proposition 3.12. For every and , the subset is compact and open, with the exception of which might
Partial action and labelled space groupoids are isomorphic
Let be a normal labelled space. In this section we show that the groupoid associated with a partial action (as defined in [1]) is isomorphic to the groupoid associated with a labelled space (as defined in [8]).
For the definition of a groupoid we refer the reader to [34]. A topological groupoid is a groupoid with a topology such that multiplication and involution are continuous. A locally compact groupoid is étale if the range and source maps are local homeomorphisms. An open set in a
Simplicity of labelled spaces C*-algebras
In this section we characterize simplicity of in terms of the tight spectrum . We do this by using known simplicity results for groupoid -algebras and partial crossed product C*-algebras.
Definition 6.1 Let be a locally compact, Hausdorff groupoid. A unit has trivial isotropy if the set contains only u. A subset is invariant if for all , when then . We say that is topologically principal if the set of units with trivial isotropy is dense in
Cuntz-Pimsner algebras associated to subshifts
In [12] (see also [11]) a -algebra is associated with a one-sided subshift. These algebras were studied as a groupoid C*-algebras and as partial crossed products in [11], [17], [36], [39]. There is a topological space that serves both as the unit space of the groupoid and the space on which the free group acts. In [3, Example 4] it is shown that the -algebra of a one-sided subshift may be realized as the -algebra of a normal labelled space. In this section we describe how the topological
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Current address: Department of Mathematics, Dartmouth College, Hanover, NH 03755.