1,147 research outputs found
Induced and reduced unbounded operator algebras
The induction and reduction precesses of an O*-vector space \M obtained by
means of a projection taken, respectively, in \M itself or in its weak
bounded commutant \M'_\w are studied. In the case where \M is a partial
GW*-algebra, sufficient conditions are given for the induced and the reduced
spaces to be partial GW*-algebras again
Quantum Probability Theory
The mathematics of classical probability theory was subsumed into classical
measure theory by Kolmogorov in 1933. Quantum theory as nonclassical
probability theory was incorporated into the beginnings of noncommutative
measure theory by von Neumann in the early thirties, as well. To precisely this
end, von Neumann initiated the study of what are now called von Neumann
algebras and, with Murray, made a first classification of such algebras into
three types. The nonrelativistic quantum theory of systems with finitely many
degrees of freedom deals exclusively with type I algebras. However, for the
description of further quantum systems, the other types of von Neumann algebras
are indispensable. The paper reviews quantum probability theory in terms of
general von Neumann algebras, stressing the similarity of the conceptual
structure of classical and noncommutative probability theories and emphasizing
the correspondence between the classical and quantum concepts, though also
indicating the nonclassical nature of quantum probabilistic predictions. In
addition, differences between the probability theories in the type I, II and
III settings are explained. A brief description is given of quantum systems for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for
clarity and accuracy made. This is the version to appear in Studies in the
History and Philosophy of Modern Physic
An introduction to quantum filtering
This paper provides an introduction to quantum filtering theory. An
introduction to quantum probability theory is given, focusing on the spectral
theorem and the conditional expectation as a least squares estimate, and
culminating in the construction of Wiener and Poisson processes on the Fock
space. We describe the quantum It\^o calculus and its use in the modelling of
physical systems. We use both reference probability and innovations methods to
obtain quantum filtering equations for system-probe models from quantum optics.Comment: 41 pages, 1 figur
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Dirac Triples for Unital AF Algebras
For a unital AF algebra A, we construct a family of triples (A, H, D) where A is represented faithfully on the Hilbert space H and D is an unbounded self-adjoint operator on H. These triples have the same properties as spectral triples except for the compact resolvent condition, so we call them Dirac triples. They serve as a generalization of Pearson-Bellissard spectral triples for an ultrametric Cantor set corresponding to choice functions. Pearson and Bellissard showed that the underlying ultrametric can be recovered by considering spectral triples associated to all choice functions. We obtain an analogue for unital AF algebras: the supremum of the Connes spectral distances induced by a large family of Dirac triples from our construction coincides with a generalized version of the Aguilar seminorm, which is a Leibniz Lip-norm for a unital AF algebra. Moreover, the convergence result of Aguilar is retained: equipped with the generalized Aguilar seminorm, a unital AF algebra is the limit of its defining finite-dimensional subalgebras for the quantum Gromov-Hausdorff propinquity
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