Shift invariant preduals of ℓ 1(ℤ)

Abstract

The Banach space 1 (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 1 (Z) weak * -continuous. This is equivalent to making the natural convolution multiplication on 1 (Z) separately weak * -continuous and so turning 1 (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 C 0 (K) (for countable locally compact K) and 1 (Z) is c 0 (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 c 0 . 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 c 0 .

Second duals of measure algebras

“…However this does not give a counter-example to our conjecture. For further study of preduals of 1 (Z), see [23].…”

The bidual of a radical operator algebra can be semisimple

“…The lengthy paper of Dales and Lau [3] refers to Gulick's paper [5], but does not use the false theorem 4.6; private communication with my colleague Garth Dales reveals a history of previous suspicion about that result, but no actual counterexamples as presented here. The paper of Daws, Haydon, Schlumprecht and White [2] refers to (the proof of) Theorem 3.3 of [5], which we believe to be completely correct. Likewise the paper of Bouziad and Filali [1] quotes the proof, given by Gulick in [5] (Lemma 5.2), that the radical of L ∞ (G) * is nonseparable for any nondiscrete locally compact group G. This proof also is perfectly valid.…”

“…For n ∈ N, let T n : H → H be the rank 1 operator with (2) T n e i = e i+1 , if i = 2n; 0, otherwise.…”

Connes-amenability of Fourier–Stieltjes algebras

“…1. The seemingly pedantic formulation of Definition 1.1 is necessary because the predual space A * need not be unique: in [DHSW12], the authors construct a continuum of different preduals of the convolution algebra ℓ 1 (Z), each of which is isometrically isomorphic to c 0 (Z), and each of which turns ℓ 1 (Z) into a dual Banach algebra.…”