  Let $A$ be a C$*^$-algebra. Given a representation $A\subset B(L)$ in a
Hilbert space $L$, the set $G^+\subset A$ of positive invertible elements can
be thought 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 homogenous Finsler space of a natural group of invertible matrices
in $M_2(A)$, identifying $TG^+$ with the {\it Poincar\'e halfspace} $H$ of $A$,
$$ H=\{h\in A: Im(h)\ge 0, Im(h) \hbox{ invertible}\}. $$ We show that
$\h\simeq TG^+$ has properties similar to those of a space of non-positive
constant curvature.
