The Universal Type of Local Algebras
We claim now that the algebra ℜ(K) of a diamond is (as a W*-algebra) isomorphic to a unique mathematical object: the hyperfinite factor of type III1. This means that physical information distinguishing different theories or different sizes of K is not contained in the algebraic structure or topology of an individual algebra ℜ(K). The information comes from the relation between the algebras of different regions, from the net. The universality of ℜ(K) may be seen as analogous to the situation in quantum mechanics where we can associate to each system or subsystem an algebra of type I, i.e. an algebra isomorphic to the set of all bounded operators on a Hilbert space. The change from the materially defined systems in mechanics to “open subsystems” corresponding to sharply defined regions in space-time in a relativistic local theory forces the change in the nature of the algebras from type I to type III1. The fact that there is only one hyperfinite factor of type III1 up to W*-isomorphy has been proved bv Haagerup [Haager 87] based on work by Connes.
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