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Notes and Remarks
CAlgebras and von Neumann Algebras
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abelian algebra analytic application argument associated assume automorphisms Banach space boundary conditions bounded C*-algebra calculation CCR algebra closed commutation complete condition consequence consider contains continuous convergence Corollary corresponding defined definition denote dense density derived described determined discussion dynamics elements energy entropy equilibrium equivalent established estimates examine Example exists extension extremal fact field finite Fock follows function Gibbs given gives ground hence Hilbert space identity implies inequality integral interaction introduce invariant latter Lemma limit linear matrices mean measure Moreover norm normal Note Observation obtained operator particles particular perturbation phase positive possible proof properties Proposition prove quantum relation Remarks representation respect result satisfies selfadjoint sense separating spin systems stability statement strongly temperature Theorem theory thermodynamic transformations unique unitary vector