Abstract
We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ∗-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime ∗-ideal, that every maximal proper quadratic module in a Noetherian ∗-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen’s Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra \(\mathcal{W}(d)\) which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements \(r_1,\ldots,r_k \in \mathcal{W}(d)\) such that ∑ j=1 k r j c r j ∗ is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound k ≤d.
Similar content being viewed by others
References
Ambrozie, C.-G., Vasilescu, F.-H.: Operator-theoretic Positivstellensätze. Z. Anal. Anwendungen 22(2), 299–314 (2003)
Cimprič, J.: A representation theorem for archimedean quadratic modules on ∗-rings. Canad. Math. Bull. (2007) (in press)
Cohn, P.M.: Skew Fields, Theory of General Division Rings. Cambridge University Press, Cambridge (1995), ISBN 0-521-43217-0
McConnell, J.C., Robson, J.C.: Noncommutative noetherian rings. In: Graduate Studies in Mathematics (Rev. edn.), vol. 30 (xx+636 pp.). American Mathematical Society, Providence, RI (2001), ISBN: 0-8218-2169-5
McCullough, S., Putinar, M.: Noncommutative sums of squares. Pacific J. Math. 218(1), 167–171 (2005)
Domokos, M.: Goldie’s theorems for involution rings. Comm. Algebra 22(2), 371–380 (1994)
Handelman, D.: Rings with involution as partially ordered abelian groups. Rocky Mountain J. Math. 11(3), 337–381 (1981)
Helton, J.W.: Positive noncommutative polynomials are sums of squares. Ann. of Math. 156(2), 675–694 (2002)
Jacobi, T.: Über die Darstellung positiver Polynome auf semi-algebraischen Kompakta. Ph.D. thesis, University of Konstanz (1999)
Klep, I., Schweighofer, M.: A Nichtnegativstellensatz for polynomials in noncommuting variables. Israel J. Math. (in press)
Lam, T.Y.: A First course in noncommutative rings. In: Graduate Texts in Mathematics (2nd edn.), vol. 131 (xx+385 pp.). Springer, New York (2001), ISBN: 0-387-95183-0
Lam, T.Y.: Lectures on modules and rings. In: Graduate Texts in Mathematics, vol. 189 (xxiv+557 pp.). Springer, New York (1999), ISBN: 0-387-98428-3
Marshall, M.: *-orderings on a ring with involution. Comm. Algebra 28(3), 1157–1173 (2000)
Putinar, M., Vasilescu, F.-H.: Solving moment problems by dimensional extension. Ann. of Math. 149(3), 1087–1107 (1999)
Schmüdgen, K.: Unbounded operator algebras and representation theory. In: Operator Theory: Advances and Applications, vol. 37 (380 pp.). Birkhäuser, Basel (1990), ISBN: 3-7643-2321-3
Schmüdgen, K.: A strict Positivstellensatz for the Weyl algebra. Math. Ann. 331(4), 779–794 (2005)
Schmüdgen, K.: A strict Positivstellensatz for enveloping algebras. Math. Z. 254(3), 641–653 (2006)
Schwartz, N.: About Schmüdgen’s theorem. Preprint at ihp-raag.org (2007)
Walter, L.J.: Orders and signatures of higher level on a commutative ring. Ph.D. thesis, University of Saskatchewan (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cimprič, J. Maximal Quadratic Modules on ∗-rings. Algebr Represent Theor 11, 83–91 (2008). https://doi.org/10.1007/s10468-007-9076-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-007-9076-z