$C^*$-alg\`ebres gradu\'ees par un semi-treillis

TL;DR

This thesis provides a comprehensive analysis of graded $C^*$-algebras by semi-lattices, detailing their structure, invariance properties, K-theory, and examples, advancing the mathematical understanding of these algebras.

Contribution

It systematically characterizes graded $C^*$-algebras by semi-lattices, including their invariance under operations and their K-theory, building on prior foundational work.

Findings

Complete description via homogeneous subalgebras

Invariance under tensor and crossed products

Explicit K-theory calculations

Abstract

Graded $C^*$-algebras by a semi lattice were introduced and studied by Anne Boutet de Monvel, Vladimir Georgescu and their collaborators in relation with the quantum N body problem. This thesis is devoted to a systematic study of these algebras and their properties. In particular, we show that they are completely described by their homogenous subalgebras and that they are invariant under several operations such as tensor products, crossed products (by actions of locally compact groups that respect the graduation). We give the $K$-theory of graded $C^*$-algebras and finally, we study some examples of such algebras.