Polarization identities

We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an involution, and polarization is viewed as an averaging operation over a compact multiplicative subgroup of the scalars. Using this we prove a general form of the Jordan-von Neumann theorem on characterizing inner product spaces among normed linear spaces, when the scalars are taken in an associative algebra.