Positive definite operator sequences
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- by T. Bisgaard PDF
- Proc. Amer. Math. Soc. 121 (1994), 1185-1191 Request permission
Abstract:
An example is given of a linear mapping from ${\mathbf {C}}[x]$ to ${{\mathbf {M}}_2}({\mathbf {C}})$ which is positive but not completely positive. It is shown that a positive linear mapping from ${\mathbf {C}}[x]$ to ${\mathbf {B}}(\mathcal {H})$ is completely positive if certain scalar moment sequences associated with it are determinate.References
-
N. I. Akhiezer, The classical moment problem, Oliver & Boyd, Edinburgh, 1965.
- William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
- Ch. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, vi, 99–114 (English, with French summary). MR 638619
- Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR 747302, DOI 10.1007/978-1-4612-1128-0
- David E. Evans, Quantum dynamical semigroups, symmetry groups, and locality, Acta Appl. Math. 2 (1984), no. 3-4, 333–352. MR 753699, DOI 10.1007/BF02280858
- D. E. Evans and J. T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Communications Dublin Inst. Advanced Studies. Ser. A 24 (1977), v+104. MR 489494
- J. S. MacNerney, Hermitian moment sequences, Trans. Amer. Math. Soc. 103 (1962), 45–81. MR 150550, DOI 10.1090/S0002-9947-1962-0150550-1
- Włodzimierz Mlak, Dilations of Hilbert space operators (general theory), Dissertationes Math. (Rozprawy Mat.) 153 (1978), 61. MR 496046
- Konrad Schmüdgen, On a generalization of the classical moment problem, J. Math. Anal. Appl. 125 (1987), no. 2, 461–470. MR 896176, DOI 10.1016/0022-247X(87)90101-6
- J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. I, American Mathematical Society, New York, 1943. MR 0008438
- W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
- Béla Sz.-Nagy, A moment problem for self-adjoint operators, Acta Math. Acad. Sci. Hungar. 3 (1952), 285–293 (1953) (English, with Russian summary). MR 55583, DOI 10.1007/BF02027827 —, Extension of linear transformations in Hilbert space which extend beyond this space, Ungar, New York, 1960.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 1185-1191
- MSC: Primary 43A35; Secondary 43A65, 44A60, 46L05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1197531-3
- MathSciNet review: 1197531