Amenability and coarse geometry of (inverse) semigroups and c*-algebras

• Autores: Luis Diego Martínez Magán
• Directores de la Tesis: Fernando Lledó (dir. tes.)
• Lectura: En la Universidad Carlos III de Madrid ( España ) en 2021
• Idioma: español
• Programa de doctorado: Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid
• Materias:
  • Matemáticas
    • Álgebra
      • Campos, anillos, álgebras
    • Análisis y análisis funcional
      • Álgebra de operadores
      • Espacios de Hilbert
    • Topología
      • Grupos de transformación
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• Resumen

  • The thesis generalizes some classical algebraic and geometric conditions from group theory to the context of inverse semigroups. In particular, it studies Day's definition of amenability for an inverse semigroup, and gives an alternative characterization that does not require taking preimages of a given set. For instance, we prove that Day's amenability is equivalent to two independent conditions, of which only one is dynamical (and hence relevant). Following this path, we characterize the measures of the semigroup that are of this form, and prove that they exactly correspond to the traces of a given C*-algebra associated to the inverse semigroup. Moreover, along the way we introduce a novel Folner sequence for inverse semigroups.

    These results suggest that the inverse semigroup can actually be equipped with a proper and right invariant metric, and this insight is also carried out. We characterize these metrics, and prove exactly when they yield the same C*-algebra as above. In particular, it is of fundamental importance that the local geometry of the inverse semigroup admits a labeling in a finitary manner. Lastly, this condition of admitting a finite labeling is then used to study the metric versions of amenability (as above) and prove that the resulting metric spaces has Yu's property A precisely when the uniform Roe algebra is nuclear.

    The thesis ends initiating the study of when the reduced C*-algebra of the semigroup is quasi-diagonal, that is, we go back to the approximations in trace and try to force these to be in norm. This, as it turns out, is a more restrictive notion, and we give a partial result in this direction.