Gruss type inequalities for positive linear maps on C*-algebras

Let A and B be two unital C*- algebras and let for C epsilon A, Gamma(C) = {gamma epsilon C : parallel to C - gamma parallel to = inf(alpha epsilon C) parallel to C - alpha I parallel to}. We prove that if Phi : A -> B is a unital positive linear map, then vertical bar Phi (AB) - Phi (A)Phi(B)vertical bar <= parallel to Phi (vertical bar A* - zeta I vertical bar(2) )parallel to 1/2[Phi(vertical bar B -xi I vertical bar(2) )]1/2 for all A,B epsilon A, zeta epsilon Gamma(A) and xi epsilon Gamma(B). In addition, we show that if (A, tau) is a noncommutative probability space and T epsilon A is a density operator, then vertical bar tau(TAB) - tau(TA) tau (TB)vertical bar <= parallel to A - zeta I parallel to p parallel to B - xi I parallel to q parallel to T parallel to r (p, q >= 4, r >= 2) and vertical bar tau(TAB) - tau(TA) tau (TB)vertical bar <= parallel to A - xi I parallel to p parallel to B - xi I parallel to q parallel to T parallel to (p, q >= 2) for every A, B epsilon A and zeta epsilon Gamma(A), xi epsilon Gamma(B). Our results generalize the corresponding results for matrices to operators on spaces of arbitrary dimension.