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53
Tiling Semigroups
 11TH ICALP, LECTURE NOTES IN COMPUTER SCIENCE 199
, 1999
"... It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without fixed points on ..."
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Cited by 52 (13 self)
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It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without fixed points on an inverse category associated with the tiling.
A homeomorphism invariant for substitution tiling spaces
, 2000
"... We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed dire ..."
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Cited by 26 (14 self)
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We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheellike tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.
Patternequivariant functions and cohomology
 J. Phys. A
"... Abstract. The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion ..."
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Cited by 26 (8 self)
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Abstract. The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity. 1.
Topological Equivalence of Tilings
 J.MATH.PHYS., VOL
, 1997
"... We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomor ..."
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Cited by 24 (7 self)
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We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.
Delone dynamical systems and associated random operators
 Proc. OAMP
, 2003
"... ABSTRACT. We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via noncommutative integration theory t ..."
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Cited by 23 (15 self)
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ABSTRACT. We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via noncommutative integration theory these representations give rise to von Neumann algebras of random operators. Features of these algebras and operators are discussed. Restricting our attention to a certain subalgebra of tight binding operators, we then discuss a Shubin trace formula.
Crossed products of the Cantor set by free, minimal actions of Z d
 Comm. Math. Phys
"... Abstract. Let d be a positive integer, let X be the Cantor set, and let Z d act freely and minimally on X. We prove that the crossed product C ∗ (Z d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K0(C ∗ (Z d, X)) is determined by traces. We obtain th ..."
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Cited by 20 (2 self)
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Abstract. Let d be a positive integer, let X be the Cantor set, and let Z d act freely and minimally on X. We prove that the crossed product C ∗ (Z d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K0(C ∗ (Z d, X)) is determined by traces. We obtain the same conclusion for the C*algebras of various kinds of aperiodic tilings. In [35], Putnam considered the C*algebra A associated with a substitution tiling system satisfying certain additional conditions, and proved that the order on K0(A) is determined by the unique tracial state τ on A. That is, if η ∈ K0(A) satisfies τ∗(η)> 0, then there is a projection p ∈ M∞(A) = ⋃∞ n=1 Mn(A) such that η = [p]. In this paper, we strengthen Putnam’s theorem, obtaining Blackadar’s Second Fundamental Comparability Question ([7], 1.3.1) for A, namely that if p, q ∈ M∞(A) are projections such that τ(p) < τ(q) for every tracial state τ on A, then p � q, that is, that p is Murrayvon Neumann equivalent to a subprojection of q. We further prove that the C*algebra A has real rank zero [10] and stable rank one [37].
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 ..."
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Cited by 16 (8 self)
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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.
Noncommutative Stone Duality: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C∗ALGEBRAS
, 2012
"... ..."
Tilings; C ∗ algebras, and Ktheory
 In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Provicence, RI
, 2000
"... Abstract. We describe the construction of C ∗algebras from tilings. We describe the Ktheory of such C ∗algebras and discuss applications of these ideas in physics. We do not assume any familiarity with C ∗algebras or Ktheory. 1. ..."
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Cited by 12 (0 self)
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Abstract. We describe the construction of C ∗algebras from tilings. We describe the Ktheory of such C ∗algebras and discuss applications of these ideas in physics. We do not assume any familiarity with C ∗algebras or Ktheory. 1.