The Philosophical Significance of Unitarily Inequivalent Representations in Quantum Field Theory

counterpart , these new observables in A∈A ( ) ′′ A A ω π A have no abstract counterpart in . However, they will have an abstract counterpart in . There is also a natural embedding of A into ∗∗ ∗∗ ∗∗ ⊆ A A A such that . The most common way to construct a von Neumann algebra is to start with a C*-algebra and an abstract state A ω , construct a representation ( , ω ) π ω π H ( via the GNS theorem, and then close ) ω π A ( ) ω π A ′′ ( ) ω π A ′′ ∗∗ A in the weak topology .10F63 However, there is an equivalent alternative way to build using the bidual . Since C*-algebras and W*-algebras are both *-algebras, the GNS theorem can be used to construct Hilbert spaces from both algebras. In order for ∗∗ A ω to be a state for it must be extended to be a normal state ω ∗∗ A on .11F64 If the GNS construction is done using ∗ ω and ∗ A , then a von ( ) ω π _ 63 Though the notation for a von Neumann algebra closed in the weak topology is A , a von Neumann algebra is usually symbolized as ( ) ω π ( ) π A A ′′ A ′′ even if it is being discussed as being closed in the weak topology. This is because . ( ) ω ω π = _ 64 A linear functional ∗ ρ on a W*-algebra ∗ A ( ) ) sup sup T T α α ρ ρ = Tα} ∗∗ A is said to be normal if and only if for every uniformly bounded increasing directed set { of positive elements of . ( α α