On the norm and spectral radius of Hermitian elements

Let be a complex unital Banach algebra. An element a ∈ is said to be Hermitian if ‖exp(ita)‖ = 1 for all t ∈ 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 Hermitian, then ‖a‖ = |a|, where |a| denotes the spectral radius of a. A function F: ℝ → ℂ is called a universal symbol if ‖F(a)‖ = | F(a)| for every and all Hermitian a ∈ . We characterize universal symbols in terms of positive-definite functions.