On extendibility and decomposability of certain *-linear maps into C (X)

TL;DR

This paper investigates the extendibility and decomposability of *-linear maps into commutative C*-algebras of continuous functions, providing extension theorems and minimal decompositions for specific classes of these maps.

Contribution

It introduces new extension results for locally finite *-linear maps and a minimal decomposition method for absolutely continuous *-linear maps into C(X).

Findings

Extension of locally finite *-linear maps into C(X)

Minimal decomposition of absolutely continuous *-linear maps

Characterization of *-linear maps in terms of positivity and continuity

Abstract

We consider *-linear maps into a commutative C*-algebra C (X) of continuous functions on a locally compact Hausdorff space X with certain specified properties and prove two results: (1) an extension result for a class of *-linear maps Y --> C (X) which may be called of locally compact type (locally finite) with respect to an inclusion Y < X of normed vector spaces, and (2) a minimal decomposition for certain *-linear maps into C (X) (absolutely continuous) as a difference of two positive maps.