Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space , with , define , and let be the unique regular Borel positive measure invariant under the action of the unitary operators in , with . We prove that if a complex frame function satisfies , then it verifies Gleason's statement: there is a unique linear operator such that for every is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.