Metrics in the sphere of a C*-module

Esteban Andruchow; Alejandro Varela

Open Mathematics, Tome 5 (2007), p. 639-653 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ∈ $ℒ_{𝒜} (𝒳)$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $\dot{γ}$ (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 ∈ $𝒮_{𝒳}$ , 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.

Publié le : 2007-01-01

Zbl 1149.46045

EUDML-ID : urn:eudml:doc:269531

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     author = {Esteban Andruchow and Alejandro Varela},
     title = {Metrics in the sphere of a C*-module},
     journal = {Open Mathematics},
     volume = {5},
     year = {2007},
     pages = {639-653},
     zbl = {1149.46045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0025-1}
}

Esteban Andruchow; Alejandro Varela. Metrics in the sphere of a C*-module. Open Mathematics, Tome 5 (2007) pp. 639-653. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0025-1/

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