Sectional curvature and commutation of pairs of selfadjoint operators

Abstract

The space G^+ of postive invertible operators of a C*-algebra A, 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, y ∈ A (regarded as tangent vectors at a ∈ G^+)verify that ∥x − y∥a ≤ 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 A 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 *-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 pairsof selfadjoint operators, and study the vanishing of this invariant.