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Gilles G. de Castro

Publications and source records attributed to Gilles G. de Castro.

27 records · Page 2Linked to original sources
arXiv2020

Infinite sum relations on universal C*-algebras

We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one still wants to avoid using relations with respect to the strong operator topology.

math.OA↗
arXiv2020

Coverages on Inverse Semigroups

First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exel's notions of tight filters and tight groupoids.

math.RA↗
arXiv2019

Haar systems, KMS states on von Neumann algebras and $C^*$-algebras on dynamically defined groupoids and Noncommutative Integration

We analyse certain Haar systems associated to groupoids obtained by certain natural equivalence relations of dynamical nature on sets like $\{1,2,...,d\}^\mathbb{Z}$, $\{1,2,...,d\}^\mathbb{N}$, $S^1\times S^1$, or $(S^1)^\mathbb{N}$, where $S^1$ is the unitary circle. We also describe properties of transverse functions, quasi-invariant probabilities and KMS states for some examples of von Neumann algebras (and also $C^*$-Algebras) associated to these groupoids. We relate some of these KMS states with Gibbs states of Thermodynamic Formalism. While presenting new results, we will also describe in detail several examples and basic results on the above topics. In other words it is also a survey paper. Some known results on non-commutative integration are presented, more precisely, the relation of transverse measures, cocycles and quasi-invariant probabilities. We describe the results in a language which is more familiar to people in Dynamical Systems. Our intention is to study Haar systems, quasi-invariant probabilities and von Neumann algebras as a topic on measure theory (intersected with ergodic theory) avoiding questions of algebraic nature (which, of course, are also extremely important).

math.DS↗
arXiv2018

Groupoid models for the C*-algebra of labelled spaces

We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.

math.OA↗
arXiv2018

Recovering the boundary path space of a topological graph using pointless topology

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_1$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

math.OA↗
arXiv2016

C*-algebras of labelled spaces and their diagonal C*-subalgebras

Motivated by Exel's inverse semigroup approach to combinatorial C*-algebras, in a previous work the authors defined an inverse semigroup associated with a labelled space. We construct a representation of the C*-algebra of a labelled space, inspired by how one might cut or glue labelled paths together, that proves that non-zero elements in the inverse semigroup correspond to non-zero elements in the C*-algebra. We also show that the spectrum of its diagonal C*-subalgebra is homeomorphic to the tight spectrum of the inverse semigroup associated with the labelled space.

math.OA↗
arXiv2008

KMS States, Entropy and a Variational Principle for Pressure

We want to relate the concepts of entropy and pressure to that of KMS states for $C^*$-Algebras. Several different definitions of entropy are known in our days. The one we describe here is quite natural and extends the usual one for Dynamical Systems in Thermodynamic Formalism Theory. It has the advantage of been very easy to be introduced. It is basically obtained from transfer operators (also called Ruelle operators). Later we introduce the concept of pressure as a min-max principle. Finally, we consider the concept of a KMS state as an equilibrium state for a potential (in the context of $C^*$-Algebras) and we show that there is a relation between KMS states for certain algebras coming from a continuous transformation and equilibrium measures.

math.OA↗
arXiv2008

C*-Algebras Associated with Iterated Function Systems

We review Kajiwara and Watatani's construction of a C*-algebra from an iterated function system (IFS). If the IFS satisfies the finite branch condition or the open set condition, we build an injective homomorphism from Kajiwara-Watatani algebras to the Cuntz algebra, which can be thought as the algebra of the lifted system, and we give the description of its image. Finally, if the IFS admits a left inverse we show that the Kajiwara-Watatani algebra is isomorphic to an Exel's crossed product.

math.OA↗