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Positive elements in the algebra of the quantum moment problem
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  • Published: June 1991

Positive elements in the algebra of the quantum moment problem

  • Palle E. T. Jorgensen1 &
  • Robert T. Powers2 

Probability Theory and Related Fields volume 89, pages 131–139 (1991)Cite this article

Summary

Let\(\mathfrak{A}\) denote the extended Weyl algebra,\(\mathfrak{A}_0 \subset \mathfrak{A}\), the Weyl algebra. It is well known that every element of\(\mathfrak{A}\) of the formA=ΣB * k B k is positive. We prove that the converse implication also holds: Every positive elementA in\(\mathfrak{A}\) has a quadratic sum factorization for some finite set of elements (B k ) in\(\mathfrak{A}\). The corresponding result is not true for the subalgebra\(\mathfrak{A}_0 \). We identify states on\(\mathfrak{A}_0 \) which do not extend to states on\(\mathfrak{A}\). It follows from a result of Powers (and Arveson) that such states on\(\mathfrak{A}_0 \) cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on\(\mathfrak{A}\), and this property is not in general shared by representations generated by states defined only on the subalgebra\(\mathfrak{A}_0 \).

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, 52242, Iowa City, IA, USA

    Palle E. T. Jorgensen

  2. University of Pennsylvania, 19104, Philadelphia, PA, USA

    Robert T. Powers

Authors
  1. Palle E. T. Jorgensen
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  2. Robert T. Powers
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Work supported in part by the NSF

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Jorgensen, P.E.T., Powers, R.T. Positive elements in the algebra of the quantum moment problem. Probab. Th. Rel. Fields 89, 131–139 (1991). https://doi.org/10.1007/BF01366901

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  • Received: 16 April 1990

  • Revised: 24 January 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01366901

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Regularity Property
  • Positive Element
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