CONICET | Buscador de Institutos y Recursos Humanos

The space $g^+$ of postive invertible operators of a C$^*$-algebra $al$, with the appropriate Finsler metric, behaves like a (non positively curved) symmetric space. Among the characteristic properties of such spaces, one has that two selfadjoint elements $x,yinal$ (regarded as tangent vectors at $ain g^+$) verify that$$|x-y|_ale d(exp_a(x),exp_a(y)).$$In this paper we investigate the ocurrence of the equality $$|x-y|_a=d(exp_a(x),exp_a(y)).$$If $al$ has a trace, and the trace is used to measure tangent vectors then - as in the finite dimensional classical setting - this equality is equivalent to the fact that $x$ and $y$ commute. In arbitrary C$^*$-algebras, when the usual C$^*$-norm is used, the equality is equivalent to a weaker condition. We introduce in $g^+$ an analogous of the sectional curvature for pairs of selfadjoint operators, and study the vanishing of this invariant.