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Existence of normal linear positive functionals on a von Neumann algebra invariant with respect to a semigroup of contractions Hsieh, TsuTeh
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 weaklywandering sets, which was first introduced by HajianKakutani, to weaklywandering projections in A. We give a relation between the nonexistence of weaklywandering 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}.
Item Metadata
Title  Existence of normal linear positive functionals on a von Neumann algebra invariant with respect to a semigroup of contractions 
Creator  Hsieh, TsuTeh 
Publisher  University of British Columbia 
Date Issued  1971 
Description 
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 weaklywandering sets, which was
first introduced by HajianKakutani, to weaklywandering projections in A.
We give a relation between the nonexistence of weaklywandering 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}.

Subject  Von Neumann algebras; Linear algebraic groups 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110419 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080522 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.