Generalization of Wigner's theorem on symmetries in the $C^*$-algebraic approach

On the basis of the abstract algebraic definition of a probability of transition between pure states the following generalization of Wigner's theorem is proved: the $C^*$-algebras of observables $\mathfrak A_1$ and $\mathfrak A_2$ are related by a symmetry transformation if and only if there exists a one-to-one mapping of the set of pure states over $\mathfrak A_1$ onto the set of pure states over $\mathfrak A_2$ that preserves the probability of the transition.