A dynamical uncertainty principle in von Neumann algebras by operator monotone functions

Suppose that A(1),..., A(N) are observables (selfadjoint matrices) and rho is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det{Cov(rho) (A(j), A(k) )}, using the commutators [A(j), A(k)]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [rho, A(j)] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.