Slice Functional Calculus in Quaternionic Hilbert Spaces

We propose a continuous functional calculus in quaternionic Hilbert spaces. The class of continuous functions considered is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen a generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C^*-algebras and to define, on each of these C^* algebras, a functional calculus for quaternionic normal operators.