Isometric direct limits of bidual Banach spaces

TL;DR

This paper studies sequences of bidual Banach spaces called tower systems, introduces functors to construct new Banach spaces from them, and explores their properties and applications in uniform algebras.

Contribution

It introduces the functors $ extrm{Dir}$ and $ extrm{Inv}$ for Banach spaces, analyzing their effects on spectra, decompositions, and applications in function algebras.

Findings

The functor $ extrm{Dir}$ preserves direct sum decompositions.

The functor $ extrm{Dir}$ preserves spectra and Fredholm properties.

Application to locating supports of representing measures in function algebras.

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.