The Bochner measure and holomorphic extension of elementary spherical functions

Localization of elementary systems in the theory of Wigner

Starting from Wigner's theory of elementary systems and following a recent approach of Schroer we define certain subspaces of localized wave functions in the underlying Hilbert space with the help of the theory of modular von-Neumann algebras of Tomita and Takesaki. We characterize the elements of these subspaces as boundary values of holomorphic functions in the sense of distribution theory and show that the corresponding holomorphic functions satisfy the sufficient conditions of the theorems of Paley-Wiener-Schwartz and Hörmander.