The alternative Dunford-Pettis property on projective tensor products
A Banach space X has the Dunford–Pettis property (DPP) if and only if whenever (xn) and (pn) are weakly null sequences in X and X*, respectively, we have pn(xn)→ 0. Freedman introduced a stricly weaker version of the DPP called the alternative Dunford–Pettis property (DP1). A Banach space X has the...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2006 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/49452 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/49452 |
| Access Level: | acceso abierto |
| Palabra clave: | 517 Randon-Nikodym property JB*-triples Jordan triples Banach-spaces Star-triples Algebras Análisis matemático 1202 Análisis y Análisis Funcional |
| Sumario: | A Banach space X has the Dunford–Pettis property (DPP) if and only if whenever (xn) and (pn) are weakly null sequences in X and X*, respectively, we have pn(xn)→ 0. Freedman introduced a stricly weaker version of the DPP called the alternative Dunford–Pettis property (DP1). A Banach space X has the DP1 if whenever xn ! x weakly in X, with kxnk = kxk, and (xn) is weakly null in X*, we have that xn(xn)→ 0. The authors study the DP1 on projective tensor products of C*-algebras and JB*-triples. Their main result, Theorem 3.5, states that if X and Y are Banach spaces such that X contains an isometric copy of c0 and Y contains an isometric copy of C[0, 1], then Xˆ_Y , the projective tensor product of X and Y , does not have the DP1. As a corollary, they get that if X and Y are JB*-triples such that X is not reflexive and Y contains `1, then Xˆ_Y does not have the DP1. Furthermore, if A and B are infinite-dimensional C*-algebras, then Aˆ_B has the DPP if and only if Aˆ_B has the DP1 if and only if both A and B have the DPP and do not contain `1. |
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