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}. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/33824 |
| Date | January 1971 |
| Creators | Hsieh, Tsu-Teh |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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