Instituto Superior Técnico – Ruy Exel (Univ. Fed. de Santa Catarina, Brasil)
January 2013, 21-25, 14:30, Lisboa, Portugal
21, 23, 25 - sala 3.10 ; 22, 24 - sala 4.35
21, 23, 25 - sala 3.10 ; 22, 24 - sala 4.35
The main goal of this series of five lectures will be to describe several generalizations of the classical notion of C*-crossed products to situations in which groups act on C*-algebras in non-classical ways, such as via partial actions, endomorphisms/transfer operators or via interactions.
The central example which motivated the introduction of these new theories is the Cuntz–Krieger algebra OA, where A is a 0–1 matrix. We will therefore begin by carefully defining and studying OA, hoping to provide the necessary motivation for the notion of partial actions.
The next step will be to define partial actions in general, followed by its accompanying notion of crossed product. Once this is done we will introduce the notion of covariant representations, which will be a central tool in the description of OA as the crossed product associated to a natural partial action of a finitely generated free group on Markov’s space.
The appearance of Markov’s space, here denoted by ΩA, brings to mind the all important Markov’s shift which, once viewed as an endomorphism of the commutative C*-algebra C(ΩA), suggests the need of a notion of crossed-product by an endomorphism.
After surveying some proposals found in the literature, we will introduce the notion of transfer operators and we will show how these appear naturally in the context of the Cuntz–Krieger algebras and, most importantly, how these suggest a new notion of crossed product by an endomorphism which, as it turn out, is the most suitable such notion to deal with endomorphisms arising from irreversible dynamical systems.
In the special case of group actions (partial or endomorphic) on commutative C*-algebras, the resulting crossed product may often be described as the groupoid C*-algebra for anétale groupoid. Arguably, the theory of groupoid C*-algebras is among the best tools available to study C*-algebras, most notably those bearing some relationship with dynamical systems. However a full course on groupoids and their C*-algebras would certainly take longer than the time allotted to this mini course, but I plan to do my best to include some basics on groupoids, and at least to describe the relationship between crossed products of abelian C*-algebras by partial group actions on the one hand, and groupoids on the other hand.
Since last September, when I was invited to deliver this series of lectures, and motivated by recent papers by Bastos, Fernandes and Karlovich, I have been thinking about the question of characterizing invertibility for elements in a groupoid C*-algebra in terms of a given family of representations. Time permitting, I will therefore attempt to describe some ideas in that direction. This will require the study of induced representations in the context of groupoids, as developed by Renault, Ionescu and Williams which, time permitting, we plan to briefly describe.
Given the vast literature on the rich interplay between dynamical systems and C*-algebras, many other topics could be included depending on the specific interests of the audience, such as: twisted partial actions; partial representations of groups; generalizations to inverse semigroups; topological freeness, ideals and simplicity criteria; reduced crossed products and amenability; partial group algebras subject to a set of relations; K- theory and generalized Pimsner-Voiculescu exact sequences; trace-scaling automorphisms and KMS states; envelopping actions; partial actions and Fell bundles; algebraic issues: associativity, etc. Further alternatives would be to discuss some examples in depth, such as partial group C*-algebras, AF algebras, graph algebras, Cuntz-Krieger algebras for infinite matrices or Cuntz-Li algebras.