Isometric direct limits of bidual Banach spaces

Abstract

Sequences of $n$-th order bidual Banach spaces, called tower systems and their direct and inverse limits are considered. Motivated by recent applications in uniform algebras, we introduce two functors: $\textrm{Dir}$ and $\textrm{Inv}$ assigning to Banach spaces (and to bounded linear operators) some new Banach spaces and operators. Of particular interest is the enormous "tower space" built over the space of continuous functions. We prove that the action of $\textrm{Dir}$ preserves direct sum decompositions. This functor preserves also spectra of operators, their Fredholmness and compactness properties. An application of these functors to the problem of location of supports of representing measures for function algebras is outlined in the last section.