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A \(C^{*}\)-algebraic model for locally noncommutative spacetimes. (English) Zbl 1127.53075
Authors’ summary: Locally noncommutative space-times provide a refined notion of noncommutative space-times where the noncommutativity is present only for small distances. Here the authors discuss a non-perturbative approach based on Rieffel’s strict deformation quantization. To this end, the authors extend the usual \(C^{\ast}\)-algebraic results to a pro-\(C^{\ast}\)-algebraic framework.

MSC:
53D55 Deformation quantization, star products
46L65 Quantizations, deformations for selfadjoint operator algebras
46L87 Noncommutative differential geometry
81R60 Noncommutative geometry in quantum theory
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[1] Bahns D., Doplicher S., Fredenhagen K., Piacitelli G. (2003). Ultraviolet finite quantum field theory on quantum spacetime. Commun. Math. Phys. 237: 221–241 · Zbl 1046.81093
[2] Bahns, D., Waldmann, S.: Locally Noncommutative Space-Times. Preprint math. QA/0607745, 28 pages (To appear in Rev. Math. Phys.) (2006) · Zbl 1127.81027
[3] Connes A. (1994). Noncommutative Geometry. Academic, San Diego · Zbl 0818.46076
[4] Connes A., Douglas M.R., Schwarz A. (1998). Noncommutative geometry and matrix theory: compactification on tori. J. High Energy Phys. 02: 003 · Zbl 1018.81052 · doi:10.1088/1126-6708/1998/02/003
[5] Doplicher S., Fredenhagen K., Roberts J.E. (1995). The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172: 187–220 · Zbl 0847.53051 · doi:10.1007/BF02104515
[6] Jurčo B., Möller L., Schraml S., Schupp P., Wess J. (2001). Construction of non-Abelian gauge theories on noncommutative spaces. Eur. Phys. J. C21: 383–388 · Zbl 1099.81524
[7] Jurčo B., Schupp P., Wess J. (2000). Noncommutative gauge theory for Poisson manifolds. Nucl. Phys. B584: 784–794 · Zbl 0984.81167
[8] Lam T.Y. (1999). Lectures on modules and rings, vol. 189. Graduate Texts in Mathematics. Springer, Berlin · Zbl 0911.16001
[9] Lance E.C. (1995). Hilbert C *-modules. A toolkit for operator algebraists, vol. 210. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge · Zbl 0822.46080
[10] Phillips N.C. (1988). Inverse limits of C *-algebras. J. Operator Theory 19: 159–195 · Zbl 0662.46063
[11] Phillips, NC.: Inverse limits of C *-algebras and applications. In: Evans, D.E., Takesaki, M. (eds.) Operator Algebras and Applications, vol. 1. London Mathematical Society Lecture Note Series, vol. 135, pp. 127–185. Cambridge University Press, Cambridge (1988)
[12] Rieffel M.A. (1993). Deformation quantization for actions of \({\mathbb{R}}^d\) . Mem. Am. Math. Soc. 106(506): 93 · Zbl 0798.46053
[13] Rieffel, M.A.: On the operator algebra for the space-time uncertainty relations. In: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.): Operator Algebras and Quantum Field Theory, pp. 374–382. International Press, Cambridge (1997) In: Proceedings of the conference held in Rome, 1–6 July (1996)
[14] Schweitzer L.B. (1993). Dense m-convex Fréchet subalgebras of operator algebra crossed products by Lie groups. Int. J. Math. 4: 601–673 · Zbl 0802.46040 · doi:10.1142/S0129167X93000315
[15] Weinstein, A.: Commuting vector fields with compact support. In: Private communication during the Poisson 2006 conference (2006)
[16] Wells R.O. (1980). Differential analysis on complex manifolds, vol. 65. Graduate Texts in Mathematics. Springer, New York · Zbl 0435.32004
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