A noncommutative moment problem
HTML articles powered by AMS MathViewer
- by Don Hadwin PDF
- Proc. Amer. Math. Soc. 129 (2001), 1785-1791 Request permission
Abstract:
We prove a noncommutative moment theorem and relate it to Connes’ problem of embedding finite factor von Neumann algebras into an ultraproduct of the hyperfinite $\mathrm {II}_1$ factor. We include a linear-algebraic equivalent of Connes’ problem, which asks for a characterization of all noncommutative polynomials which have positive trace when the variables are replaced by contractive hermitian $n\times n$ matrices.References
- J. Alcántara-Bode and J. Yngvason, Algebraic quantum field theory and noncommutative moment problems. I, Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 2, 147–159. MR 952659
- Arlen Brown and Carl Pearcy, Commutators in factors of type III, Canadian J. Math. 18 (1966), 1152–1160. MR 201987, DOI 10.4153/CJM-1966-115-2
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- A. Yu Daletskiui, Yu. S. Samouilenko, A noncommutative moment problem, (Russian) Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 72–73.
- Michel Dubois-Violette, Topics on a noncommutative moment problem, Proceedings of the International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics (Leipzig, 1977) Teubner, Leipzig, 1978, pp. 241–254. MR 528281
- L. Ge and D. Hadwin, Ultraproducts of $\mathrm {C}^*$-algebras, preprint.
- Th. Fack and P. de la Harpe, Sommes de commutateurs dans les algèbres de von Neumann finies continues, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 49–73 (French). MR 597017
- P. R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215–223. MR 231233
- Herbert Halpern, Commutators in properly infinite von Neumann algebras, Trans. Amer. Math. Soc. 139 (1969), 55–73. MR 251546, DOI 10.1090/S0002-9947-1969-0251546-8
- R. A. Kunze, $L_{p}$ Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519–540. MR 100235, DOI 10.1090/S0002-9947-1958-0100235-1
- Carl Pearcy and David Topping, Commutators and certain $\textrm {II}_{1}$-factors, J. Functional Analysis 3 (1969), 69–78. MR 0239432, DOI 10.1016/0022-1236(69)90051-2
- K. Schmüdgen, Commutative and noncommutative moment problems, Operator algebras and operator theory (Craiova, 1989) Pitman Res. Notes Math. Ser., vol. 271, Longman Sci. Tech., Harlow, 1992, pp. 184–197. MR 1189174
- Konrad Schmüdgen, Noncommutative moment problems, Math. Z. 206 (1991), no. 4, 623–649. MR 1100846, DOI 10.1007/BF02571369
- J. Yngvason, Algebraic quantum field theory and noncommutative moment problems. II, Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 2, 161–173. MR 952660
Additional Information
- Don Hadwin
- Affiliation: Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824
- Email: don@math.unh.edu
- Received by editor(s): June 2, 1999
- Received by editor(s) in revised form: October 11, 1999
- Published electronically: January 23, 2001
- Communicated by: David R. Larson
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1785-1791
- MSC (2000): Primary 44A60, 46L89; Secondary 46L50, 47A57
- DOI: https://doi.org/10.1090/S0002-9939-01-05772-0
- MathSciNet review: 1814111
Dedicated: Dedicated to the memory of my brother, Gary, whose cheerful spirit still fills my heart