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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  
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}.

Genre  
Type  
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  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
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.