Characterizations of the trace (Q1112920)

Extending a result of \textit{L. T. Gardner} [Can. J. Math. 31, 1322-1328 (1979; Zbl 0394.47059)] the authors prove the following: a linear functional on M(n,\({\mathbb{C}})\) is a (nonnegative) scalar multiple of the trace iff \(f(A^ k)=f(| A|^ k)\) for all \(A\in M(n,{\mathbb{C}})\) and some fixed k. Similar characterizations for more general operator algebras, in particular in terms of projections, are obtained.