Shift invariant preduals of l1(Z), and isomorphisms with c0(Z)

[摘要] The relation between shift-invariant preduals of the space of summable sequences $\ell_{1}(\mathbb Z)$ and the dual Banach algebra $\ell_{1}(\mathbb Z)$ equipped with the convolution product have resulted in recent development of research on preduality of this space. According to the survey paper entitled 'Shift Invariant Preduals of $\ell_{1}(\mathbb Z)$', written by Matthew Daws, Richard Hadon, Thomas Schlumprecht and Stuart White, we know that there exists an uncountable family $\left\{F^{(\lambda)}\right\}_{\lambda\in \mathbb C}$ of shift-invariant preduals of $\ell_{1}(\mathbb Z)$ and all these preduals $F^{(\lambda)}$ constructed in the above paper are isomorphic to $c_{0}(\mathbb Z)$, the space of sequences converging to zero. This conclusion is based on an abstract theory of the Szlenk index, without stating the explicit form of that isomorphism. This thesis will make an attempt to define this sort ofisomorphism. In other words, I will form an isomorphism between $c_{0}(\mathbb Z)$ and $F^{(\lambda)}_{+}$, which is a subspace of $F^{(\lambda)}$.

[发布机构] University:University of Glasgow;Department:School of Mathematics and Statistics

[关键词] Functional analysis, Banach space, dual Banach algebra, duality, preduality, isomorphism.