# `Unsolved problems in mathematics': J. von Neumann's address to the International Congress of

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Local properties of a quantum system are defined as the expectation values of its observables in a microstate of some complete set of commuting observables. An equation for the time evolution of local properties is obtained for any system whose statistical operator (density matrix) evolves by...

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The properties of quantum fields generating local nets of von Neumann algebras are discussed. For fields that satisfy certain energy bounds, necessary and sufficient conditions for the existence of such a net and for the existence of product states are given in terms of the vacuum expectation...

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In this paper, we introduce a new class of non-amenable groups denoted by NC1âˆ©Quot(Crss) which give rise to prime von Neumann algebras. This means that, for every Î“âˆˆNC1âˆ©Quot(Crss), its group von Neumann algebra L(Î“) cannot be decomposed as a tensor product of diffuse von...

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According to a procedure previously introduced from Y. Ilamed and N. Salingaros, we start giving proof of two existing Clifford algebras, the Si that has isomorphism with that one of Pauli matrices and the N i,Â±1 where N i stands for the dihedral Clifford algebra. The salient feature is that...

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We prove that each element of the von Neumann algebra without a direct abelian summand is representable as a finite sum of products of at most three projections in the algebra. In a properly infinite algebra the number of product terms is at most two. Our result gives a new proof of equivalence...

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We show that for F âˆŠ GL(2;â„‚), the von Neumann algebra associated to the universal compact quantum group Au(F) is a free Araki-Woods factor.

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Discusses the use of the concepts of von Neumann lattices and tight frames for defining discrete quantum mechanical transforms in the phase lane. Construction of the discrete Weyl-Heisenberg transform; Bargmann representation in quantum mechanics; Fourier transform in signal processing.

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We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for...