2002/04/15 by Hellmut Baumgärtel, Matthias Jurke, Fernando Lledó · 24 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Algebraic structures and combinatorial models #Direct proof #Duality (order theory) #Fock space #Hilbert space #Linear subspace #Modular design #Operator (biology) #Random Matrices and Applications #Simple (philosophy) #math-ph #math.FA #math.MP #math.OA #msc:46L10 #msc:47A05 #msc:81T05
paper · pdf · open access · doi:10.1063/1.1483376
published in Journal of Mathematical Physics 43(8), 4158-4179 (American Institute of Physics) · 32 pages, Latex2e, to appear in Journal of Mathematical Physics
arxiv created 2002/04/15 · openalex publication_date 2002/08/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We give a complete proof of the twisted duality property M(q)′=Z̃M(q⊥)Z̃* 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.