Hansen, H.H. (Helle)

Ossa, C.

Palmigiano, A.

Re, R.

English

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SGF-Quantales and their GroupoidsHelle Hvid Hansen, Catalina Ossa, Alessandra Palmigiano, and Riccardo ReQuantales. The term quantale was introduced by Mulvey as the ‘quantum’ counterpart of the termlocale. Locales can be thought of as pointfree topological spaces, and hence can be considered inthe context of a Gelfand-type duality for commutative C∗-algebras [2]. Quantales were consideredin the context of the quest for a Gelfand-type duality for general C∗-algebras.In the Gelfand duality, the algebra-to-space direction consists in associating any commutativeC∗-algebra with its maximal ideal space. This construction was extended to general C∗-algebrasby considering the spectrum MaxA of any unital C∗-algebra A, i.e. the unital involutive quantaleof closed linear subspaces of A. This gives rise to a functor Max which was extensively studiedfor more than a decade as it was considered the best candidate for the C∗-algebra-to-quantaledirection of a noncommutative Gelfand duality. Remarkably, MaxA is a complete invariant ofA, i.e. if A and A′ are C∗-algebras such that MaxA and MaxA′ are isomorphic, then A and A′are isomorphic. However, there are several problems with Max: 1) it has no adjoint, which is anecessary condition for its providing one direction of a duality; 2) it is not full on isomorphisms,i.e. some isomorphisms of spectra of C∗-algebras do not arise from C∗-algebra morphisms [7]; 3)there is no purely algebraic characterization of the class of quantales isomorphic to quantales oftype MaxA; 4) there is no canonical way of constructing A from MaxA. These difficulties motivatethe quest for alternative ways of linking C∗-algebras and quantales.Besides their interest w.r.t. C∗-algebras, quantales have been extensively studied in logic andtheoretical computer science: not only do they provide the standard algebraic semantics for variousresource-sensitive logics such as linear logic [4,14], they have also been applied to the study ofthe semantics of concurrent systems and their observable behaviour, described in terms of finiteobservations [1]. Finite observations are formalized as semidecidable properties, and can thereforebe identified with open sets of a topological space [13]; however, this perspective does not accountfor those (quantum-theoretic) situations where performing finite observations on a system produceschanges in the system itself. In those cases, the set of the finite observations that can be performedon a system has a natural noncommutative structure of quantale. The basic view on quantales asgeneralized topologies can be retrieved also in this context. In [1], this perspective was applied onquantales to provide a uniform algebraic framework for process semantics and develop a systematicstudy of various notions of observational equivalence between processes.Merging perspectives: the case study of Penrose tilings. Recently, investigation has focused onways to integrate the two perspectives on quantales as noncommutative topologies and as algebrasof experimental observations on computational (or physical) systems, and use them to investigatethe connection between quantales and C∗-algebras. In [8], an important example was studied,which concerns a classification of Penrose tilings using quantales. This classification is alternativeto the one previously introduced by Connes [3] (consisting in associating a certain C∗-algebraAK with the space (K,∼) of Penrose tilings). The classification in [8] is based on a logic of finiteobservations performed on Penrose tilings, the Lindenbaum-Tarski algebra of which is a quantale(denoted by Pen). This classification arises from a canonical representation of Pen as a quantalePen of relations on (K,∼). In [8], the exact connection between Pen and AK was left as anopen problem, but since Pen is not isomorphic to MaxAK , the case study of Penrose tilings wasconsidered pivotal in finding the alternative connection between quantales and C∗-algebras in therestricted but geometrically significant setting in which they both arise from groupoids.E´tale groupoids and their quantales. In [12], Resende generalized the example of Pen to a bijectivecorrespondence between localic e´tale groupoids and certain unital involutive quantales referred toas inverse quantal frames (indeed, their underlying sup-lattice structure is a frame). An importantfeature of inverse quantal frames Q is that, denoting the unit of Q by e, the restriction of theproduct to the subquantale Qe = e↓ coincides with the lattice meet. The groupoid-to-quantaledirection of this correspondence arises from observing that, for every e´tale localic groupoid G =(G0, G1), the groupoid structure-maps induce a structure of unital involutive quantale on the locale2 Helle Hvid Hansen, Catalina Ossa, Alessandra Palmigiano, and Riccardo ReG1, which becomes an inverse quantale frame. Conversely, the e´tale localic groupoid associatedwith an inverse quantal frame Q is based on the locales G0 := Qe and G1 := Q.Towards a non e´tale generalization of Resende’s correspondence. In [9], a unital involutive quantaleis associated with any topological groupoid in a way alternative to Resende’s but compatible withit when the topological groupoid is e´tale. This route makes it possible to account for the connectionbetween the quantale Pen and the C∗-algebra AK , which was left as an open problem in [8]. Thequantale associated with a topological groupoid G is the sub sup-lattice of P(G1) generated bythe inverse semigroup S of the images of the local bisections of G.Spatial SGF-quantales. Building on [9], in [10], a bijective correspondence is established betweencertain unital involutive quantales referred to as spatial SGF-quantales and topological groupoidss.t. G0 is sober. This class of groupoids includes equivalence relations arising from group actions,and significantly extends the class of e´tale topological groupoids. Dually, inverse quantal frames areexactly those SGF-quantales the underlying sup-lattice of which is a frame. The correspondencedefined in [10] extends the theory of [12] to a point-set, non e´tale setting. Interestingly, thiscorrespondence also forms the basis of a representation theorem for SGF-quantales into unitalinvolutive quantales of relations [11], similar to the one for relation algebras in [6].E´tale vs. non e´tale. The comparison between the correspondences in [12] and [10] is facilitated bythe observation that a topological groupoid is e´tale iff the images of its local bisections form abase for the topology on G1. The e´tale topological setting can be shown to be exactly the one inwhich the groupoid-to-quantale routes in [12] and in [10] can be identified.Work in progress. The proposed talk reports on ongoing work [5] aimed at generalizing the cor-respondence of [10] from a topological to a localic setting. This amounts to defining a bijectivecorrespondence between general SGF-quantales and localic (non e´tale) groupoids. For any localicgroupoid G = (G1, G0) satisfying mild additional hypotheses, one defines its local bisections withinthe theory of locales and then, observing that the locale LC(G1) generated by the locally closedsublocales of G1 has a natural quantale structure, one defines the quantale Q(G) as the subquan-tale of LC(G1) generated by the images of the local bisections of G. As to the quantale-to-groupoiddirection, in the point-set case [10], given Q, one defines G0 as the topological space arising fromthe spatial frame Qe, and obtains the points of G1 as equivalence classes of tuples (p, f) such thatp is a point of G0 and f is a partial unit of Q (partial units being the quantale counterparts oflocal bisections), satisfying an algebraic condition which geometrically means that “f is definedon p”. Any such equivalence class can be understood a posteriori as the value f(p) of f at p.The pointfree counterpart of the condition “f is defined on p” is one in which points are replacedby locally closed sublocales of the locale G0 associated with the frame Qe. The main strategyconsists in defining the restrictions of local bisections to locally closed sublocales of G0 in thetheory of SGF-quantales. This makes it possible to extend in a unique way any SGF-quantale Qto a quantale Q˜ s.t. Q˜e is the frame associated with the locale LC(G0), and then derive from theSGF-axiomatization that Q˜ is an inverse quantal frame, which hence gives rise to an e´tale localicgroupoid G˜ = (G˜0, G˜1). Finally, the localic (non-e´tale) groupoid associated with Q will be definedas the push-forward of G˜ induced by the canonical projection LC(G0)→ G0.References1. S. Abramsky and S. Vickers. Quantales, observational logic and process semantics. MathematicalStructures in Computer Science, 3(2):161–227, 1993.2. B. Banaschewski and C. J. Mulvey. A globalisation of the Gelfand duality theorem. Annals of Pureand Applied Logic, 137(1):62–103, 2006.3. A. Connes. Noncommutative geometry. Springer, 1992.4. J.-Y. Girard. Linear logic. Theoret. Comp. Sci., (50):1–102, 1987.5. H. H. Hansen, C. Ossa, A. Palmigiano, and R. Re. SGF-quantales and their groupoids. In prep., 2015.6. B. Jo´nsson and A. Tarski. Boolean algebras with operators, II. Amer. J. of Math, 74:127–162, 1952.7. D. Kruml and P. Resende. On quantales that classify C*-algebras. Cah. top. et ge´ome´. diff.cate´goriques, 45(4):287–296, 2004.8. C. J. Mulvey and P. Resende. A noncommutative theory of Penrose tilings. Int. J. Th. Ph., 44(6):655–689, 2005.SGF-Quantales and their Groupoids 39. A. Palmigiano and R. Re. Topological groupoid quantales. Studia Logica, 95(1-2):125–137, 2010.10. A. Palmigiano and R. Re. Groupoid quantales: A non-e´tale setting. JPAA, 215(8):1945–1957, 2011.11. A. Palmigiano and R. Re. Relational representation of groupoid quantales. Order, 30(1):65–83, 2013.12. P. Resende. E´tale groupoids and their quantales. Advances in Mathematics, 208(1):147–209, 2007.13. S. Vickers. Topology via Logic. Cambridge University Press, 1989.14. D. N. Yetter. Quantales and (noncommutative) linear logic. J. Symb. Logic, 55(01):41–64, 1990.

SGF-Quantales and their Groupoids

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