Extremal invariant states with a spectrum condition and Borchers' theorem
✍ Scribed by Akitaka Kishimoto
- Publisher
- Springer
- Year
- 1979
- Tongue
- English
- Weight
- 191 KB
- Volume
- 3
- Category
- Article
- ISSN
- 0377-9017
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✦ Synopsis
If the energy spectrum of an extremal invariant state w is not the whole real line, it is shown that co is either pure or uniquely decomposed into nmtually disjoint pure states in the way that co = X-~fXo~oatdt where 4~ is a pure state satisfying q~oax = 4~ with X > 0. Next we give a slightly generalized version of Borchers' theorem [1 ] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.
Let (a, R, a) be a C*-dynamicaI system, i.e. a is a C*-algebra and a is a homomorphism of R into the automorphism group of a such that t -~ at(x) is continuous for any x E a. Let ~ be an extremal invariant state of a under a. On the GNS triple 0r, Jr, ~2) associated with co we have the unitary representation of R on ~ such that Ut~2 = ~2, UtTr(x)~2 = 7roat(x)I2, for t ~R , x E a. Let Sp(U) be the spectrum of U (i.e. the spectrum of the infinitesimal generator of Ut) and PSp(U) the point spectrum of U.