Shift invariant preduals of ${\ell }_1(\mathbb{Z})$

Abstract

The Banach space ${\ell }_1(\mathbb{Z})$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ${\ell }_1(\mathbb{Z})$ weak${}^{\ast }$-continuous. This is equivalent to making the natural convolution multiplication on ${\ell }_1(\mathbb{Z})$ separately weak*-continuous and so turning ${\ell }_1(\mathbb{Z})$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_0(K)$ (for countable locally compact $K$) and ${\ell }_1(\mathbb{Z})$ is $c_0(\mathbb{Z})$. We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak${}^{\ast }$-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to $c_0$. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of $\mathbb{Z}$. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to $c_0$.

31 pages, minor typos corrected, to appear in Israel Journal of Mathematics

Cited by in corpus (2)

• Connes-amenability of Fourier--Stieltjes algebras
• The Bidual of a Radical Operator algebra can be Semisimple