Inverse semigroups and combinatorial C*-algebras

By R. Exel. 2007

329 citations recorded by the source corpus.

Abstract.We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc6f6cfca1c2b93a730c73f13c229357bff982e7

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Simplicity of algebras associated to non-Hausdorff groupoids (2018, methodology): “ra. So neither AK(G⋊α Z) nor C∗ r(G ⋊α Z) is simple by Theorem 3.14 and Theorem 4.10 respectively. 5.2. Inverse semigroup actions and their groupoids. For any unreferenced claims in this section, see [11] and [12]. We will use the notation Y ⊆ ﬁn X to indicate that Y is a ﬁnite subset of X. An inver”
Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids (2015, background): “ction in G, also known as a G-set, is a set U ⊂ Gsuch that r,srestrict to homeomorphisms on U. An important feature of ´etale groupoids is that they have plenty of open bisections: Proposition 3.5 of [9] together with local compactness implies that the topology on an ´etale groupoid has a basis cons”
Tight representations of semilattices and inverse semigroups (2007, background): “the last equality following from the assumption that N0 = G. One may similarly prove that [x,g][0,1] = [0,1], which means that [0, 1 ] is a zero element for S. Moreover”
A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY (2009, background): “This work has been developed in a number of directions [4, 5, 6, 13, 14, 22, 24, 26, 27, 28], to name but a few.”
\'Etale groupoids and their $C^*$-algebras (2017, background): “Exel [18] and Paterson [31] have both given excellent discussions of étale groupoids—particularly as they relate to inverse semigroups—but in the non-Hausdorff setting, where the details are a little trickier.”
Partial Dynamical Systems, Fell Bundles and Applications (2015, background): “The relationship between partial actions and inverse semigroups is further discussed in [99], [49], [54], [60], [21], [22] and [23].”
Partial transformation groupoids attached to graphs and semigroups (2016, unknown): “The boundary quotient in Exel’s sense (see [9, 10, 11, 13]) is given by ∂C λ(S) := C ∗ λ(S)/ 〈I〉.”
A Groupoid Approach to Discrete Inverse Semigroup Algebras (2009, methodology): “For basic properties of étale groupoids (also called r-discrete groupoids), we refer to the treatises [6, 17, 19].”
ON INVERSE SEMIGROUP C ∗ -ALGEBRAS AND CROSSED PRODUCTS (2011, methodology): “S/ is the groupoid of germs of the action ˇ [22], [4], [26].”
Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups (2012, methodology): “on of Lif this πis an isomorphism. The fact that πisn’t always an isomorphism is related to the problem of ﬁnding the right way to map semilattices into Boolean algebras which is discussed by Exel in [11, 12]. Before we investigate this we want to discuss ﬁlters, which is a concept that is important”
Boundary quotients of C*-algebras of right LCM semigroups (2014, methodology): “π must be a commutative C*-algebra. The set Bπ = {e∈ Dπ | e2 = e} is a Boolean algebra when given the operations e∧ f= ef e∨ f= e+f− ef ¬e= 1− e We will recall a subclass of representations deﬁned in [Exe08]. Let S be an inverse semigroup and let F⊂ Z⊂ E(S). We say that Fis a cover of Zif for every ”
A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras (2016, methodology): “ generated by open invariant subsets of the unit space. In the last two sections, we give two more classes of examples. Both are classes whose C∗-algebras have been constructed using the machinery of [14]. In Section 6 we consider groups acting on graphs, as deﬁned in [17]; we describe the Steinberg”
Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras (2014, methodology): “See [9, 19] for details.”
Recent developments around partial actions (2018, background): “Moreover, the tight spectrum Êtight (see [154]) is invariant under the action of S, and it is shown in [235] that the corresponding groupoid of germs Gtight(S) is also isomorphic to a transformation groupoid of a partial action of G.”
Pseudogroups and their étale groupoids (2011, background): “7, 16] we then have that (L(S),·) is a groupoid and is in fact the groupoid underlying Paterson’s universal groupoid of an inverse semigroup S [20]. Filters also play an important role in Exel’s work [2, 3] on relating inverse semigroups and C∗-algebras. A ﬁlter is said to be proper if it does not c”