A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

Cimprič, Jakob

Canadian mathematical bulletin, Tome 52 (2009) no. 1, pp. 39-52

Résumé

We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$ -algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$ -algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

Détail

DOI : 10.4153/CMB-2009-005-4

Mots-clés : 16W80, 46L05, 46L89, 14P99, Ordered rings with involution, C*-algebras and their representations, noncommutative convexity theory, real algebraic geometry

Bibliographie

[1] [1] Sarymsakov, T. A., Ayupov, S. A., Khadzhiev, D., and Chilin, V. I., Uporyadochennye algebry. “Fan”, Tashkent, 1983. Google Scholar

[2] [2] Berr, R. and Wörmann, T., Positive polynomials on compact sets. Manuscripta Math. 104(2001), no. 2, 135–143. Google Scholar

[3] [3] Bichteler, K., A generalization to the non-separable case of Takesaki's duality theorem for C*-algebras. Invent. Math. 9(1969/1970), 89–98. Google Scholar

[4] [4] Constantinescu, C., C*-algebras. Vol. 2. Banach algebras and compact operators. North-Holland Mathematical Library 59. North-Holland Publishing Co., Amsterdam, 2001. Google Scholar

[5] [5] Constantinescu, C., C*-algebras. Vol. 3. General theory of C*-algebras. North-Holland Mathematical Library 60, North-Holland Publishing Co., Amsterdam, 2001. Google Scholar

[6] [6] Dixmier, J., C*-algebras. North-Holland Mathematical Library 15, North-Holland Publishing Co., Amsterdam–New York–Oxford, 1977. Google Scholar

[7] [7] Dubois, D. W., A note on David Harrison's theory of preprimes. Pacific J. Math. 21(1967), 15–19. Google Scholar

[8] [8] Dubois, D. W., Second note on David Harrison's theory of preprimes. Pacific J.Math. 24(1968), 57–68. Google Scholar

[9] [9] Effros, E. G. and Winkler, S., Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems. J. Funct. Anal. 144(1997), no. 1, 117–152. Google Scholar

[10] [10] Fujimoto, I., CP-duality for C*- and W*-algebras. J. Operator Theory 30(1993), no. 2, 201–215. Google Scholar

[11] [11] Fujimoto, I., Decomposition of completely positive maps. J. Operator Theory 32(1994), no. 2, 273–297. Google Scholar

[12] [12] Fujimoto, I., A Gelfand-Naimark theorem for C*-algebras. Pacific J. Math. 184(1998), no. 1, 95–119. Google Scholar

[13] [13] Jacobi, T., A representation theorem for certain partially ordered commutative rings. Math. Z. 237(2001), no. 2, 259–273. Google Scholar

[14] [14] Jacobi, T. and Prestel, A., Distinguished representations of strictly positive polynomials. J. Reine Angew. Math. 532(2001), 223–235. Google Scholar

[15] [15] Kadison, R. V., A representation theory for commutative topological algebra. Mem. Amer. Math. Soc. 1951(1951), no. 7. Google Scholar

[16] [16] Klep, I., A Kadison-Dubois representation for associative rings. J. Pure Appl. Alg. 189(2004), no. 1–3, 211–220. Google Scholar

[17] [17] Li, B., Real operator algebras. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. Google Scholar

[18] [18] Li, B., Introduction to operator algebras. World Scientific Publishing Co., Inc., River Edge, NJ, 1992. Google Scholar

[19] [19] Marshall, M., Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, American Mathematical Society, Providence, RI, 2008. Google Scholar

[20] [20] Marshall, M., ∗-orderings on a ring with involution. Comm. Algebra 28(2000), no. 3, 1157–1173. Google Scholar

[21] [21] Marshall, M., A general representation theorem for partially ordered commutative rings. Math. Z. 242(2002), no. 2, 217–225. Google Scholar

[22] [22] Palmer, T. W., Real C*-algebras. Pacific J. Math. 35(1970), 195–204. Google Scholar

[23] [23] Pedersen, G. K., Analysis now. Graduate Texts in Mathematics 118, Springer-Verlag, New York, 1989. Google Scholar

[24] [24] Prestel, A., Lectures on formally real fields. Lecture Notes in Mathematics 1093, Springer-Verlag, Berlin, 1984. Google Scholar

[25] [25] Prestel, A., Representations of real commutative rings. Expo. Math. 23(2005), no. 1, 89–98. Google Scholar

[26] [26] Rudin, W., Functional analysis. Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. Google Scholar

[27] [27] Schmüdgen, K., Unbounded operator algebras and representation theory. Operator Theory: Advances and Applications 37, Birkhüser Verlag, Basel, 1990. Google Scholar

[28] [28] Stone, M. H., A general theory of spectra.. Proc. Nat. Acad. Sci. U.S.A. 26(1940), 280–283. Google Scholar

[29] [29] Webster, C. and Winkler, S., The Krein-Milman theorem in operator convexity. Trans. Amer. Math. Soc. 351(1999), no. 1, 307–322. Google Scholar

[30] [30] Wong, Y. C. and Ng, K. F., Partially ordered topological vector spaces. Oxford Mathematical Monographs,. Clarendon Press, Oxford, 1973. Google Scholar