Local Quantum Physics pp 267-269 | Cite as

# The Universal Type of Local Algebras

## Abstract

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 III_{1}. 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 III_{1}. The fact that there is only one hyperfinite factor of type III_{1} up to W*-isomorphy has been proved bv Haagerup [Haager 87] based on work by Connes.

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