ON UNIFORMLY CONTINUOUS OPERATORS AND SOME WEIGHT-HYPERBOLIC FUNCTION BANACH ALGEBRA

We consider an abelian non-unitary Banach algebra 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 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, A does not have the Radon-Nikodym property. So, it could be exist not representable linear bounded operators on A (cf. [6]). However, we prove that the class of locally absolutely continuous bounded operators are representable and we determine their kernels.