Vol. 52, No. 1, 1974

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Automorphism groups of operator algebras

Richard Howard Herman

Vol. 52 (1974), No. 1, 91–99

The general setting of this paper is that of a von Neumann algebra, M, with weight, φ, and a group, G, of automorphisms which commute with the modular automorphism group associated with this weight.

The first section is devoted to the question of when the given weight is invariant under the action of G. Should G leave the center of M elementwise fixed results have been obtained by Pedersen, Størmer, Takesaki, and the present author. If, alternatively, it is assumed that φ is invariant under a subgroup, H, of G, then by requiring an ergodic action of H on the center of M it is shown here that φ must be (semi-) invariant under the action of G. This is done with the aid of some technical assumptions H. Less demanding hypothesis are shown to lead to a G-invariant weight bicommuting with the given weight.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 29 October 1973
Published: 1 May 1974
Richard Howard Herman