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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.
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