On a characterization of velocity maps in the space of observables (Q1183102)

Motivated by Heisenberg's picture of quantum dynamics, the author introduces the notion of a velocity map. A map taking the real linear space of all selfadjoint elements of a von Neumann algebra into itself is called a velocity map if it is (real) homogeneous, and if it is additive and satisfies the derivation-type condition on couples of commuting elements. It is noted that a norm-continuous mapping satisfying the derivation-type condition (on commuting elements) and vanishing on (real) multiples of identity is automatically a velocity map. The main result is that a (real) linear velocity map on the algebra of all bounded operators on a separable Hilbert space is necessarily inner. Thus, it extends uniquely to a derivation on the algebra. The selfadjoint operator \(H\) for which the velocity map equals \(ad(iH)\) is given explicitly.