JournalsrmiVol. 35, No. 7pp. 2187–2219

The Poincaré half-space of a C^*-algebra

  • Esteban Andruchow

    Universidad Nacional de General Sarmiento, Los Polvorines, Argentina and Instituto Argentino de Matemática, Buenos Aires
  • Gustavo Corach

    Instituto Argentino de Matemáticas and Universidad de Buenos Aires, Argentina
  • Lázaro Recht

    Universidad Simón Bolívar, Caracas, Venezuela
The Poincaré half-space of a C$^*$-algebra cover
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Let A\mathcal{A} be a unital C^*-algebra. Given a faithful representation AB(L)\mathcal{A}\subset\mathcal B(\mathcal{L}) in a Hilbert space L\mathcal{L}, the set G+AG^+\subset\mathcal{A} of positive invertible elements can be thought of as the set of inner products in L\mathcal{L}, related to A\mathcal{A}, which are equivalent to the original inner product. The set G+G^+ has a rich geometry, it is a homogeneous space of the invertible group GG of A\mathcal{A}, with an invariant Finsler metric. In the present paper we study the tangent bundle TG+TG^+ of G+G^+, as a homogeneous Finsler space of a natural group of invertible matrices in M2(A)M_2(\mathcal{A}), identifying TG+TG^+ with the Poincaré half-space H\mathcal H of A\mathcal{A},

H={hA:Im(h)0,Im(h) invertible}.\mathcal H=\{h\in\mathcal{A}: {\rm Im}(h)\ge 0, {\rm Im}(h) \hbox{ invertible}\}.

We show that HTG+\mathcal H\simeq TG^+ has properties similar to those of a space of non-positive constant curvature.

Cite this article

Esteban Andruchow, Gustavo Corach, Lázaro Recht, The Poincaré half-space of a C^*-algebra. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2187–2219

DOI 10.4171/RMI/1117