On the norm and spectral radius of Hermitian elements

Let A be a complex unital Banach algebra. An element a is an element of A is said to be Hermitian if parallel to exp(ita)parallel to = 1 for all t is an element of R. In the case of the algebra of bounded linear operators in a Hilbert space, this Hermitian property agrees with the ordinary self-adjointness. If a is an element of A is Hermitian, then parallel to a parallel to =vertical bar a vertical bar where a denotes the spectral radius of a. A function F: R -> C is called a universal symbol if parallel to F(a)parallel to = vertical bar F(a)vertical bar for every A and all Hermitian a is an element of A. We characterize universal symbols in terms of positive-definite functions.