On uniformly continuous operators and some weight-hyperbolic function Banach algebra

We consider an abelian non-unitary Banach algebra $\mathfrak{A}$, ruled by an hyperbolic weight, defined on certain space of Lebesgue measurable complex valued functions on the positive axis. Since the non-convolution Banach algebra $\mathfrak{A}$ has its own interest by its applications to the representation theory of some Lie groups, we search on various of its properties. As a Banach space, $\mathfrak{A}$ does not have the Radon–Nikodým property. So, it could be exist not representable linear bounded operators on $\mathfrak{A}$ (cf. [6]). However, we prove that the class of locally absolutely continuous bounded operators are representable and we determine their kernels.