Holder and Young inequalities for the trace of operators

If A(1),..., A(m) are positive semidefinite n x n matrices, and if p(1),..., p(m) are positive real numbers such that 1/p(1) + ... + 1/p(m) = 1, then [GRAPHICS] where vertical bar X vertical bar denotes (X*X)(1/2) and tr (X) denotes the trace of X. Moreover, equality holds in either of these inequalities if and only if A p(1)(p1) = ... = A(m)(pm). This result will be shown to hold as well in unital C*-algebras that have a faithful tracial state.