Sectional curvature and commutation of pairs of selfadjoint operators
The space G^+ of postive invertible operators of a C*-algebra A, with the appropriate Finsler metric, behaves like a (non positively curved)symmetric space. Among the characteristic properties of such spaces, one has that two selfadjoint elements x, y ∈ A (regarded as tangent vectors at a ∈ G^+)veri...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2006 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositório: | CONICET Digital (CONICET) |
| Idioma: | inglês |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/109699 |
| Acesso em linha: | http://hdl.handle.net/11336/109699 |
| Access Level: | Acceso aberto |
| Palavra-chave: | POSITIVE OPERATOR SELFADJOINT OPERATOR SECTIONAL CURVATURE https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | The space G^+ of postive invertible operators of a C*-algebra A, with the appropriate Finsler metric, behaves like a (non positively curved)symmetric space. Among the characteristic properties of such spaces, one has that two selfadjoint elements x, y ∈ A (regarded as tangent vectors at a ∈ G^+)verify that ∥x − y∥a ≤ d(exp_a(x), exp_a(y)). In this paper we investigate the ocurrence of the equality ∥x − y∥a = d(exp_a(x), exp_a(y)). If A has a trace, and the trace is used to measure tangent vectors then, as in the finite dimensional classical setting, this equality is equivalent to the fact that x and y commute. In arbitrary *-algebras, when the usual C*-norm is used, the equality is equivalent to a weaker condition. We introduce in G^+ an analogous of the sectional curvature for pairsof selfadjoint operators, and study the vanishing of this invariant. |
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