Let A be a C∗-algebra, G a
group of ∗-automorphisms of A and φ a G-invariant weigh t. Assume that φ takes
finite values on a dense subset of A+. It is shown that there is a largest element
among the G-invariant weights ψ0 maiorized by φ and weakly adherent to the set of
G-invariant continuous positive linear functionals majorized by ψ0. Moreover this
weight majorizes every G-invariant continuous positive linear functional majorized by
φ. If A is a von Neumann algebra it is sufficient to assume that φ takes finite
values on a σ− weakly dense subset of A+ to get a similar result for normal
functionals. Further characterisations of this weight are given in terms of
the representation associated with φ. This relation is then used to prove
that if φ is lower semicontinuous, the existence of G-invariant continuous
positive linear functionals majorized by φ is equivalent to the existence of
fixed points in the associated Hilbert space ℋp and representation of G in
ℋp
