Abstract. 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 F-inverse semigroup is strongly Morita equivalent to a cross product of a commutative C ∗-algebra by a group. 1.

Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawvere’s quantale of non-negative real numbers and Walters ’ small quantaloids of closed cribles.

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL. 1

Abstract. Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author’s earlier work on finite inverse semigroups and Paterson’s theorem for the universal C ∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition”. This “finiteness condition ” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent. Contents

My main interests are in geometric group theory, regarded as a subset of the more general field of noncommutative geometry. In my thesis I studied linear group actions on spaces of finite asymptotic dimension. Existence of such actions puts a new light on the result of Higson, Guentner and Weinberger in [GuHW] that linear groups satisfy the Baum-Connes conjecture. Current work also concerns some applications of coarse geometric methods to the theory of étale groupoids and inverse semigroups. 1 Noncommutative Geometry One of the central notions in mathematics is the notion of a space, understood as a collection of points with some structure. Many classical mathematical theories are therefore concerned with the study of the geometry of the underlying space. However, sometimes it is natural to consider not only the space itself, but also the set of functions on it. For example, while classical probability theory treats events as collections of points in the so called set of elementary events, the consideration of measurable functions on this set leads to the notion of a random variable, which is central in the modern theory of stochastic processes. As another example, many classical models in physics appeal to the algebra of smooth

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

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

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

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.

We provide a new characterization of the notion of sheaf on an étale groupoid G, in terms of a particular kind of Hilbert module on the quantale O(G) of the groupoid. All the theory is developed in the context of the more general class of quantales known as stable quantal frames, of which examples are easy to construct because their category is algebraic. The homomorphisms of our Hilbert modules are necessarily adjointable and thus form a strongly self-dual category. By restriction we obtain, for any stable quantal frame, two isomorphic categories of sheaves whose morphisms are related by the duality.