Convexity of the geodesic distance on spaces of positive operators (Q1320963)

Let \(A\) be a \(C^*\)-algebra with 1 and denote by \(A^ +\) the set of positive invertible elements of \(A\). The set \(A^ +\) has a rich differential structure as a reductive homogeneous space [the authors, Integral Equations Oper. Theory 16, No. 3, 333-359 (1993; Zbl 0786.58006); Int. J. Math. 4, No. 2, 193-202 (1993)] and the first author [Banach Cent. Publ. 30, 101-115 (1994)]. The main result of this paper is the convexity of the function \(d(\gamma(t),\delta(t))\) where \(\gamma\) and \(\delta\) are two geodesics of \(A^ +\). This extends a result of [the authors, Linear Algebr. Appl. 179, 271-275 (1993)] and generalizes to infinite dimensions a previous theorem by \textit{R. R. Coifman} and \textit{S. Semmes} [Am. J. Math. 115, No. 2, 243-278 (1993; Zbl 0789.46021)].