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