Operators preserving $$\ell _\infty $$

Abstract

Let \(Y\) be a Banach space, let the space \(\ell _\infty \) be real, let \(W\) denote the Banach space \(\ell _\infty /c_0,\) and let \(Q\) denote the quotient map \(\ell _\infty \rightarrow W.\) In 1981, Partington proved there is a topological embedding \(J\) of \(\ell _\infty \) into \(W\) such that the composition \(QJ\) is an isometry; in particular, \(Q\) preserves \(\ell _\infty .\) In this paper we prove that if the kernel of an operator \(T:\ell _\infty /c_0 \rightarrow Y\) does not contain an isometric copy of \(c_0\) (in particular, if \(T\) is injective), then \(T\) preserves \(\ell _\infty ,\) and hence \(T\) is non-weakly compact. This, in turn, allows us to extend Partington’s theorem: we show that natural quotient mappings of some real function spaces preserve \(\ell _\infty .\) We also remark that our results apply to some quotients of both Orlicz and Marcinkiewicz spaces.