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

Pierzchala, Tomasz (2014) Shift invariant preduals of l1(Z), and isomorphisms with c0(Z). MSc(R) thesis, University of Glasgow.

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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 of isomorphism. In other words, I will form an isomorphism between $c_{0}(\mathbb Z)$ and $F^{(\lambda)}_{+}$, which is a subspace of $F^{(\lambda)}$.