TY - THES
AU - Hsieh, Tsu-Teh
PY - 1971
TI - Existence of normal linear positive functionals on a von Neumann algebra invariant with respect to a semigroup of contractions
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - 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}.
N2 - 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}.
UR - https://open.library.ubc.ca/collections/831/items/1.0080522
ER - End of Reference