Operational calculus on a complex semisimple Lie group

For every complex semisimple Lie algebra $\mathfrak g$ we construct a so-called operational calculus, which consists in the isomorphic embedding of $\mathfrak g$ along with its associative hull $\mathfrak G$ into a certain algebra of operator polynomials. We investigate the image of $\mathfrak G$ under this embedding; the resulting theorems comprise the algebraic analog of the functional duality theorems of harmonic analysis (theorems of Paley–Wiener type).