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Étale groupoids and their quantales
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
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Cited by 16 (7 self)
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We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal
A Noncommutative Theory of Penrose Tilings
 International Journal of Theoretical Physics 44: 655689, 2005. [3] A. Palmigiano and
"... Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible ..."
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Cited by 8 (6 self)
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Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.
On quantales that classify C*algebras
"... The functor Max of Mulvey assigns to each unital C*algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*algebras up to ∗isomorphism. In this paper we provide a proof of this and of the stronger fact that for every iso ..."
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Cited by 4 (4 self)
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The functor Max of Mulvey assigns to each unital C*algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*algebras up to ∗isomorphism. In this paper we provide a proof of this and of the stronger fact that for every isomorphism u: Max A → Max B of unital involutive quantales there is a ∗isomorphism �u: A → B such that Max �u coincides with u when restricted to the leftsided elements of Max A. But we also show that isomorphisms u: Max A → Max B may exist for which no isomorphism v: A → B is such that Max v = u. Keywords: quantale, C*algebra, noncommutative topology.
1 Research supported in part by FCT through the Program POCI2010/FEDER
, 2006
"... Groupoids and inverse semigroups are two generalizations of the notion of group. Both provide a handle on more general kinds of symmetry than groups do, in particular symmetries of a local nature, and applications of them crop up almost everywhere in mathematics — evidence of this is the number of ..."
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Groupoids and inverse semigroups are two generalizations of the notion of group. Both provide a handle on more general kinds of symmetry than groups do, in particular symmetries of a local nature, and applications of them crop up almost everywhere in mathematics — evidence of this is the number of
Quantales as geometric objects: symmetry beyond groupoids?
, 2005
"... Modern mathematics has become pervaded by the idea that in order to cater for certain notions of symmetry, in particular of a local nature, one needs to go beyond group theory, replacing groups by ..."
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Modern mathematics has become pervaded by the idea that in order to cater for certain notions of symmetry, in particular of a local nature, one needs to go beyond group theory, replacing groups by
Quantum triads: an algebraic approach
, 2008
"... frame, quantum triad. A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frame ..."
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frame, quantum triad. A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices. 1
Quantale Modules, with Applications to Logic and Image Processing
, 2007
"... Università degli Studi di SalernoTo the memory of my father, Francesco. To my newborn nephew and godson, wishing he will share with his grandad much more than the bare name. Acknowledgements Most of the people who read a doctoral dissertation are academics. Then, as it often understandably happens, ..."
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Università degli Studi di SalernoTo the memory of my father, Francesco. To my newborn nephew and godson, wishing he will share with his grandad much more than the bare name. Acknowledgements Most of the people who read a doctoral dissertation are academics. Then, as it often understandably happens, they may underestimate its importance for the author. A Ph.D. thesis represents a sort of finishing line, of a run begun more than twenty years before. So there is no reason for being sparing of thanks and gratitude. A few years ago, while writing my degree thesis, I read a booklet by Umberto Eco, entitled “Come si fa una tesi di laurea ” (that is “How to make a degree thesis”). One of the first hints he gives in that book is that it is inelegant to thank your advisor in the acknowledgements of your thesis, because he’s simply doing his job, nothing more, and in most cases your acknowledgement would be nothing but an act of obedience. It is probably true in many