795 research outputs found
Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras
We study a non-commutative generalization of Stone duality that connects a
class of inverse semigroups, called Boolean inverse -semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse -semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
-semigroup is pre-Boolean if and only if every tight filter is an
ultrafilter, where the definition of a tight filter is obtained by combining
work of both Exel and Lenz. A simple necessary condition for a semigroup to be
pre-Boolean is derived and a variety of examples of inverse semigroups are
shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees
matrix semigroups over the polycyclics, are pre-Boolean and it is proved that
the groups of units of their completions are precisely the Thompson-Higman
groups . The inverse semigroups arising from suitable directed graphs
are also pre-Boolean and the topological groupoids arising from these graph
inverse semigroups under our non-commutative Stone duality are the groupoids
that arise from the Cuntz-Krieger -algebras.Comment: The presentation has been sharpened up and some minor errors
correcte
The Booleanization of an inverse semigroup
We prove that the forgetful functor from the category of Boolean inverse
semigroups to inverse semigroups with zero has a left adjoint. This left
adjoint is what we term the `Booleanization'. We establish the exact connection
between the Booleanization of an inverse semigroup and Paterson's universal
groupoid of the inverse semigroup and we explicitly compute the Booleanization
of the polycyclic inverse monoid and demonstrate its affiliation with
the Cuntz-Toeplitz algebra.Comment: This is an updated version of the previous paper. Typos where found
have been corrected and a new section added that shows how to construct the
Booleanization directly from an arbitrary inverse semigroup with zero
(without having to use its distributive completion
A non-commutative generalization of Stone duality
We prove that the category of boolean inverse monoids is dually equivalent to
the category of boolean groupoids. This generalizes the classical Stone duality
between boolean algebras and boolean spaces. As an instance of this duality, we
show that the boolean inverse monoid associated with the Cuntz groupoid is the
strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its
group of units is a Thompson group
A perspective on non-commutative frame theory
This paper extends the fundamental results of frame theory to a
non-commutative setting where the role of locales is taken over by \'etale
localic categories. This involves ideas from quantale theory and from semigroup
theory, specifically Ehresmann semigroups, restriction semigroups and inverse
semigroups. We establish a duality between the category of complete restriction
monoids and the category of \'etale localic categories. The relationship
between monoids and categories is mediated by a class of quantales called
restriction quantal frames. This result builds on the work of Pedro Resende on
the connection between pseudogroups and \'etale localic groupoids but in the
process we both generalize and simplify: for example, we do not require
involutions and, in addition, we render his result functorial. We also project
down to topological spaces and, as a result, extend the classical adjunction
between locales and topological spaces to an adjunction between \'etale localic
categories and \'etale topological categories. In fact, varying morphisms, we
obtain several adjunctions. Just as in the commutative case, we restrict these
adjunctions to spatial-sober and coherent-spectral equivalences. The classical
equivalence between coherent frames and distributive lattices is extended to an
equivalence between coherent complete restriction monoids and distributive
restriction semigroups. Consequently, we deduce several dualities between
distributive restriction semigroups and spectral \'etale topological
categories. We also specialize these dualities for the setting where the
topological categories are cancellative or are groupoids. Our approach thus
links, unifies and extends the approaches taken in the work by Lawson and Lenz
and by Resende.Comment: 69 page
Graph inverse semigroups: their characterization and completion
Graph inverse semigroups generalize the polycyclic inverse monoids and play
an important role in the theory of C*-algebras. This paper has two main goals:
first, to provide an abstract characterization of graph inverse semigroups; and
second, to show how they may be completed, under suitable conditions, to form
what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the
ample semigroup of a topological groupoid associated with the graph, and the
semigroup analogue of the Leavitt path algebra of the graph.Comment: Some minor corrections made and tangential material remove
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