Shift invariant preduals of ℓ1(ℤ) Academic Article uri icon

abstract

  • The Banach space ℓ1(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 ℓ1(ℤ) weak*-continuous. This is equivalent to making the natural convolution multiplication on ℓ1(ℤ) separately weak*-continuous and so turning ℓ1(ℤ) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C0(K) (for countable locally compact K) and ℓ1(ℤ) is c0(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c0. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of 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 c0. © 2012 Hebrew University Magnes Press.

altmetric score

  • 2.5

author list (cited authors)

  • Daws, M., Haydon, R., Schlumprecht, T., & White, S.

citation count

  • 4

publication date

  • March 2012