The Poincaré half-space of a C${}^{\ast }$-algebra

Abstract

Let $\mathcal{A}$ be a unital C${}^{\ast }$-algebra. Given a faithful representation $\mathcal{A}\subset \mathcal{B}(\mathcal{L})$ in a Hilbert space $\mathcal{L}$, the set $G^{+}\subset \mathcal{A}$ of positive invertible elements can be thought of as the set of inner products in $\mathcal{L}$, related to $\mathcal{A}$, 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 $\mathcal{A}$, with an invariant Finsler metric. In the present paper we study the tangent bundle $T{G}^{+}$ of $G^{+}$, as a homogeneous Finsler space of a natural group of invertible matrices in $M_2(\mathcal{A})$, identifying $T{G}^{+}$ with the Poincaré half-space $\mathcal{H}$ of $\mathcal{A}$,

We show that $\mathcal{H}\simeq T{G}^{+}$ has properties similar to those of a space of non-positive constant curvature.