The Radon Nikodym property and multipliers of \(\mathcal{HK}\)-integrable functions (Q2188792)

The aim of the paper is the study of multipliers of Henstock-Kurzweil (shortly, HK) integrable functions in a commutative Banach algebra \(X\) with identity of norm one. It is proved that if \(X\) has the Radon-Nikodym property, then any \(X\)-valued function of strong bounded variation is a multiplier mapping the space \(\mathcal{SHK}\) of strongly HK-integrable functions into itself. It is then seen that, under the same assumptions on \(X\), if \(f\) is HK-integrable and \(g\) has strong bounded variation (both \(X\)-valued), then there exists \(h\in \mathcal{SHK}\) such that \[ \tau(fg)=\tau(h) \] for every multiplicative linear functional \(\tau\) of the Banach algebra. Moreover, when \(X\) has the weak Radon-Nikodym property, the same is available with \(h\) Henstock-Kurzweil-Pettis integrable.