Maximal Quadratic Modules on ∗-rings


DOI: 10.1007/s10468-007-9076-z

Cite this article as:
Cimprič, J. Algebr Represent Theor (2008) 11: 83. doi:10.1007/s10468-007-9076-z


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=1krjcrj is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound kd.


Rings with involution Ordered structures Noncommutative real algebraic geometry 

Mathematics Subject Classifications (2000)

Primary: 16W80 13J30 Secondary: 14P10 12D15 

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia

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