The Poincaré half-space of a C∗-algebra

• Esteban Andruchow ^[1] ; Gustavo Corach ^[2] ; Lázaro Recht ^[3]
  1. [1] Universidad Nacional de General Sarmiento

     Universidad Nacional de General Sarmiento

     Argentina

     
  2. [2] Instituto Argentino de Matemáticas
  3. [3] Universidad Simón Bolívar, Caracas

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 35, Nº 7, 2019, págs. 2187-2219
• Idioma: inglés
• DOI: 10.4171/rmi/1117
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let A be a unital C∗-algebra. Given a faithful representation A⊂B(L) in a Hilbert space L, the set G+⊂A of positive invertible elements can be thought of as the set of inner products in L, related to A, which are equivalent to the original inner product. The set G+ has a rich geometry, it is a homogeneous space of the invertible group G of A, with an invariant Finsler metric. In the present paper we study the tangent bundle TG+ of G+, as a homogeneous Finsler space of a natural group of invertible matrices in M2(A), identifying TG+ with the Poincaré half-space H of A, H={h∈A:Im(h)≥0,Im(h) invertible}.

    We show that H≃TG+ has properties similar to those of a space of non-positive constant curvature.