Results 1  10
of
30
Aperiodicity and cofinality for finitely aligned higherrank graphs
, 2009
"... We introduce new formulations of aperiodicity and cofinality for finitely aligned higherrank graphs Λ, and prove that C ∗ (Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms ..."
Abstract

Cited by 21 (12 self)
 Add to MetaCart
(Show Context)
We introduce new formulations of aperiodicity and cofinality for finitely aligned higherrank graphs Λ, and prove that C ∗ (Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C ∗ (Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings, our results are new.
A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY
"... Abstract. 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 Cn associated w ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
Abstract. 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 Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.
Tight representations of semilattices and inverse semigroups”, in preparation
"... By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleannes present in the semilattice of idempotents of S. After observing that the VagnerPreston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E * unitaries.
Noncommutative Stone Duality: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C∗ALGEBRAS
, 2012
"... ..."
STRONG MORITA EQUIVALENCE OF INVERSE SEMIGROUPS
, 2009
"... We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ∗algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing that the C ∗algebra of an Finverse semigroup is strongly Morita equivalent to a cross product of a commutative C ∗algebra by a group.
Inverse semigroup C*algebras associated with left cancellative semigroups, preprint, arXiv:1202.5977v2
 Proc. Edinburgh Math. Soc
"... ar ..."
(Show Context)
Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions
, 2014
"... ..."
(Show Context)
The tiling C∗algebra viewed as a tight inverse semigroup algebra, preprint arXiv:1106.4535v1
"... Abstract We realize Kellendonk´s C*algebra of an aperiodic tiling as the tight C*algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract We realize Kellendonk´s C*algebra of an aperiodic tiling as the tight C*algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.