Characterization of Inner $*$-Automorphisms of $W^*$-Algebras

For some problems in physics I would like to have a characterization of *-automorphisms of a C*-algebra which can be realized by unitary operators in the enveloping von Neumann algebra. Looking at this problem I realized that one should first treat the same problem, as an "exercise", for PF*-algebras. The simplification is due to the fact that an inner automorphism lies also on a one-parametric group, while, for a permanently ewakly inner automorphism this property is not known. The technique emplyoed here is a further development of a method I have used in a recent paper [5] in order to give a new and constructive proof of the theorem of Kadison [6] and Sakai [9] on derivations and my own result [3] on groups with semibounded spectrum. The same technique as used in [5] has been developed independently by Arveson [1] and also by Pedersens [8]. The advantage of this method is due to the fact that it gives a rather explicit construction of the spectral resolution of the unitary operator we are looking for. The technique is derived from the concept of creationand annihilation-operators which is used in physics. These operators define a shift operation on the spectrum of the unitary operator we are looking for. The aim is, of course to cnostruct a spectral resolution. This means one has to associate to every projection in the center of the invariant elements a subset of the torus. The difficulty is due to the fact that an inner automorphism defines the unitary operator only up to a unitary in the center. This means the mapping from the projections to the