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Metrics in the sphere of a C*-module

Esteban Andruchow, Alejandro Varela (2007)

Open Mathematics

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 ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ 𝒜 ( 𝒳 ) , Z* = −Z and ∥Z∥ ≤ π, such...

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