Metrics in the sphere of a Hilbert C*-module
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 an...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/101032 |
| Acceso en línea: | http://hdl.handle.net/11336/101032 |
| Access Level: | acceso abierto |
| Palabra clave: | C*MODULES SPHERES GEODESICS https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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. |
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