Twisted duality of the CAR-Algebra

Details Twisted duality of the CAR-Algebra Twisted duality of the CAR-Algebra

2002-04-15

arxiv.org/abs/math-ph/0204029v1

J.Math.Phys. 43 (2002) 4158-4179

Physics

Mathematical Physics

32 pages, Latex2e, to appear in Journal of Mathematical Physics

Scientific paper

10.1063/1.1483376

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.

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