Metrics in the sphere of a Hilbert C*-module

Abstract

Given a unital C∗-algebra A and a right C∗-module X over A, we consider the problem of finding short smooth curves in the sphere SX = {x ∈ X :( x, x) = 1}. Curves in SX are measured considering the Finsler metric which consists of the norm of X at each tangent space of SX, The initial x0 ∈ Sx and any tangent vector υ at x0, there exists a curve γ(t)=e^tZ(x0), Z ∈ LA(X), Z*=-Z and ∥Z∥ ≤ π, such that γ(0)=υ, 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 x0, x1 ∈ SX , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by ƒ0 the selfadjoint projection I − x0 ⊗ x0, if the algebra ƒ0LA(X)ƒ0 is finite dimensional, then there exists a curve γ, which is minimizing along its path.