## The Radon Nikodym property and multipliers of $$\mathcal{HK}$$-integrable functions.(English)Zbl 1451.28010

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.

### MSC:

 28B05 Vector-valued set functions, measures and integrals 26A39 Denjoy and Perron integrals, other special integrals
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### References:

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