Metrics in the sphere of a C*-module

✍ Scribed by Esteban Andruchow; Alejandro Varela

• Book ID: 111488404
• Publisher: SP Versita
• Year: 2007
• Tongue: English
• Weight: 237 KB
• Volume: 5
• Category: Article
• ISSN: 1895-1074
• DOI: 10.2478/s11533-007-0025-1

✦ Synopsis

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 S X = {x ∈ X : x, x = 1}. Curves in S X are measured considering the Finsler metric which consists of the norm of X at each tangent space of S X . The initial value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any element x 0 ∈ S X and any tangent vector v at x 0 , there exists a curve γ(t) = e tZ (x 0 ), Z ∈ L A (X ), Z * = -Z and Z ≤ π, such that γ(0) = x 0 and γ(0) = v, 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 ∈ S 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 L A (X )f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1 , which is minimizing along its path.