Almost periodic functionals on Banach algebras (Q1209548)

This paper discusses the almost periodic functionals \(ap(A)\) and the weakly almost periodic functionals \(wap(A)\) on a Banach algebra \(A\). A pleasing and surprising result is an early key lemma: if the set \(\Phi_ A\) of multiplicative functionals separates the points of \(A\), and if \(a\in A\) is such that left translation by \(a\) is weakly compact on \(A\), then \(fa\) is almost periodic for every \(f\in A^*\) (indeed, \(fa\in\overline{\text{span}} \Phi A\)). As a corollary, if \(A\) has a bounded left approximate identity, is a right ideal in its second dual, and \(\Phi_ A\) separates the points of \(A\), then \(wap(A) = ap(A) = A^*A\). The conclusion \(wap(A) = ap(A)\) is also obtained if \(A\) has a bounded left approximate identity, as a right ideal in its second dual, and has the Dunford-Pettis property. The authors determine \(ap(A)\) for some classical Banach algebras, for example \(ap(\ell^ 1) = c_ 0\), \(ap(K(X)) = \{0\}\) where \(K(X)\) is the algebra of compact operators on any infinite dimensional Banach space \(X\) with the approximation property, and \(ap(C(K))\) (for a compact Hausdorff \(K\)) is the space of atomic measures on \(K\). Finally, the behaviour of \(ap(A)\) under standard constructions is investigated.