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Positive elements in the algebra of the quantum moment problem

Authors:	Palle E T Jorgensen Robert T Powers

Institution:	(1) Department of Mathematics, University of Iowa, 52242 Iowa City, IA, USA;(2) University of Pennsylvania, 19104 Philadelphia, PA, USA

Abstract:	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=B _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 $\mathfrak{A}_0$ .Work supported in part by the NSF

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