Esteban Andruchow, Alejandro Varela
December 1, 2007

### Abstract

Given a unital C*-algebra $$\mathcal{A}$$ and a right C*-module $$\mathcal{X}$$ over $$\mathcal{A}$$ , we consider the problem of finding short smooth curves in the sphere $$\mathcal{S}_\mathcal{X} $$ = {x ∈ $$\mathcal{X}$$ : 〈x, x〉 = 1}. Curves in $$\mathcal{S}_\mathcal{X} $$ are measured considering the Finsler metric which consists of the norm of $$\mathcal{X}$$ at each tangent space of $$\mathcal{S}_\mathcal{X} $$ . The initial value problem is solved, for the case when $$\mathcal{A}$$ is a von Neumann algebra and $$\mathcal{X}$$ is selfdual: for any element x 0 ∈ $$\mathcal{S}_\mathcal{X} $$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $$\dot \gamma $$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $$\mathcal{S}_\mathcal{X} $$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.