Some applications of simultaneous continuous functional calculus

We define and study a continuous functional calculus for a commutative family of normal elements of a C*-algebra. We obtain that this functional calculus is unique, continuous and satisfies the spectral mapping theorem. We also provide two applications. The first one concerns the existence of a particular orthonormal basis on a locally Hilbert space. The second for multi-dimensional continuous versions of N. Wiener and P. Le vy theorems.