Extension of $c_0(I)$-valued operators on spaces of continuous functions on compact lines

TL;DR

This paper studies conditions under which bounded $c_0(I)$-valued operators defined on subalgebras of continuous functions on compact lines can be extended to the whole space, generalizing previous results and analyzing specific classes of compact lines.

Contribution

It extends existing results on $c_0$-valued operator extensions to $c_0(I)$-valued operators on spaces of continuous functions on compact lines, providing bounds and characterizations.

Findings

Extension can be achieved with norm at most twice the original

Characterization of compact lines where extension properties are equivalent

Generalization of previous $c_0$-operator extension results

Abstract

We investigate the problem of existence of a bounded extension to $C(K)$ of a bounded $c_0(I)$-valued operator $T$ defined on the subalgebra of $C(K)$ induced by a continuous increasing surjection $\phi:K\to L$, where $K$ and $L$ are compact lines. Generalizations of some of the results of [6] about extension of $c_0$-valued operators are obtained. For instance, we prove that when a bounded extension of $T$ exists then an extension can be obtained with norm at most twice the norm of $T$. Moreover, the class of compact lines $L$ for which the $c_0$-extension property is equivalent to the $c_0(I)$-extension property for any continuous increasing surjection $\phi:K\to L$ is studied.