Examples of homogeneous manifolds with uniformly bounded metric projection
Let M be a finite von Neumann algebra with a faithful normal trace τ. Denote by Lp(M)sh the skew-Hermitian part of the non-commutative Lp space associated with (M, τ). Let 1 < p < ∞, z ∈ Lp(M)sh and S be a real closed subspace of Lp(M)sh. The metric projection Q : Lp(M)sh −→ S is defined for e...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2012 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/18935 |
| Acceso en línea: | http://hdl.handle.net/11336/18935 |
| Access Level: | acceso abierto |
| Palabra clave: | finite von Neumann algebra metric projection homogeneous space https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | Let M be a finite von Neumann algebra with a faithful normal trace τ. Denote by Lp(M)sh the skew-Hermitian part of the non-commutative Lp space associated with (M, τ). Let 1 < p < ∞, z ∈ Lp(M)sh and S be a real closed subspace of Lp(M)sh. The metric projection Q : Lp(M)sh −→ S is defined for every z ∈ Lp(M)sh as the unique operator Q(z) ∈ S such that kz − Q(z)kp = miny∈ S kz − ykp. We show the relation between metric projection and metric geometry of homogeneous spaces of the unitary group UM of M, endowed with a Finsler quotient metric induced by the p-norms of τ, kxkp = τ(|x| p) 1/p, p an even integer. The problem of finding minimal curves in such homogeneous spaces leads to the notion of uniformly bounded metric projection. Then we show examples of metric projections of this type. |
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