Spectrally bounded traces on C*-algebras

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M greater than or equal to 0 such that r(Tx) less than or equal to Mr(x) for all x is an element of E, where r((.)) denotes the spectral radius. We establish the equivalence of the following properties of a unital linear mapping T from a unital C*-algebra A into its centre: (a) T is spectrally bounded; (b) T is a spectrally bounded trace; (c) T is a bounded trace.