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Existence of normal linear positive functionals on a von Neumann algebra invariant with respect to a semigroup of contractions Hsieh, Tsu-Teh

Abstract

Let A be a von Neumann algebra of linear operators on the Hilbert space H . A linear operator T (resp. a linear bounded. functional ϕ ) on A is said to be normal if for any increasing net [formula omitted] of positive elements in A with least upper bound B , T(B) is the least upper bound of [formula omitted]. Two linear positive functionals ψ1 and ψ2 on A are said to be equivalent if ψ1 (B) = 0 ψ2 (B) = 0 for any positive element B in A. Let ϕ0 be a positive normal linear functional on A . Let S be a semigroup and, {T(s) : s ε S} an antirepresentation of S as normal positive linear contraction operators on A . We find in this thesis equivalent conditions for the existence of a positive normal linear functional ϕ on A which is equivalent to ϕ0 and invariant under the semigroup {T(s) : s ε S} (i.e. ϕ(T(s)B) = ϕ(B) for all B in A and s ε S ). We also extend the concept of weakly-wandering sets, which was first introduced by Hajian-Kakutani, to weakly-wandering projections in A. We give a relation between the non-existence of weakly-wandering projections in A and the existence of positive normal linear functionals on A, invariant with respect to an antirepresentation {T(s) : s ε S} of normal *-homomorphisms on A . Finally we investigate the existence of a complete set of positive normal linear functionals on A which are invariant under the semigroup {T(s) : s ε S}.

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