Smooth paths of conditional expectations
Let A be a von Neumann algebra with a finite trace , represented in H = L2(A, ), and let Bt ⊂ A be sub- algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique -preserving conditional expectation. We say that the path t 7→ Et is smooth if for every a ∈ A and ∈ H, the map I ∋ t 7→ Et(...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2011 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/20219 |
| Acceso en línea: | http://hdl.handle.net/11336/20219 |
| Access Level: | acceso abierto |
| Palabra clave: | conditional expectations finite von Neumann algebras systems of projections https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | Let A be a von Neumann algebra with a finite trace , represented in H = L2(A, ), and let Bt ⊂ A be sub- algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique -preserving conditional expectation. We say that the path t 7→ Et is smooth if for every a ∈ A and ∈ H, the map I ∋ t 7→ Et(a) ∈ H is continuously differentiable. This condition implies the existence of the derivative operator dEt(a) : H → H, dEt(a) = d dt Et(a). If this operator satifies the additional boundedness condition, ZJ kdEt(a)k2 2dt ≤ CJ kak2 2, for any closed bounded sub-interval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras Bt are ∗-isomorphic. More precisely, there exists a curve Gt : A → A, t ∈ I of unital, ∗-preserving linear isomorphisms which intertwine the expectations, Gt ◦ E0 = Et ◦ Gt. The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps B0 onto Bt. We show that this restriction is a multiplicative isomorphism. |
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