Linear Hahn Banach Type Extension Operators in Banach Algebras of Operators

TL;DR

This paper investigates the extension of linear Hahn-Banach operators within Banach algebras of operators, providing comprehensive results for both reflexive and certain non-reflexive Banach spaces, including those with few operators.

Contribution

It extends the theory of Hahn-Banach extension operators to Banach modules and algebras of operators, offering new complete characterizations for specific classes of Banach spaces.

Findings

Complete characterization for reflexive Banach spaces.

Extension results for non-reflexive spaces with few operators.

Application to Banach algebras of operators.

Abstract

The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Kozmider and Laustsen(2013), and also Kania and Laustsen (2017), we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon's space with few operators.