Vol. 46, No. 1, 1973

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The upper envelope of invariant functionals majorized by an invariant weight

Alfons Van Daele

Vol. 46 (1973), No. 1, 283–302

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 σweakly 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 p and representation of G in p

Finally two examples are discussed.

Mathematical Subject Classification 2000
Primary: 46L05
Received: 6 January 1972
Published: 1 May 1973
Alfons Van Daele
Department of Mathematics
Katholieke Universiteit Leuven
3030 Leuven