### On a characterization of velocity maps in the space of observables

B. V. Rajarama Bhat
1992 Pacific Journal of Mathematics
Motivated by Heisenberg's picture of quantum dynamics the notion of a velocity map is introduced and its properties are investigated. The main theorem in the present exposition strengthens the well-known result that every derivation on the algebra of all bounded operators on a complex separable Hubert space is inner. A constructive proof leads to an inversion formula for the observables inducing the derivation. Introduction. Let si be a von Neumann algebra. Then a derivation δ on si is a linear
more » ... δ on si is a linear map δ\ s/ -* sf satisfying δ{XY) = Xδ(Y) + δ(X)Y for every X, Y in si . Inner derivations are the derivations of the form δ(X) = [£>, X] for some D in si. It is a well-known result of Sakai and Kadison (cf. , ) that every derivation δ on a von Neumann algebra si is inner. In Heisenberg's picture of quantum dynamics maps of the form δ(X) = i [H 9 X], where H, X are self-adjoint operators, determine the rate of change (or velocity) of observables. However, in this case, we are interested in the action of δ only on the real linear space (9 of observables (self-adjoint elements) of the algebra and not on the full algebra si . Keeping this in mind K. R. Parthasarathy suggested the following "axioms" for a velocity map which measures rate of change of observables: Let (9 be the real linear space of all self-adjoint elements of a von Neumann algebra si . Then a map δ: (9 -> (9 is called a velocity map if it satisfies the following conditions.