Maximal Quadratic Modules on ∗-rings

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 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 , . . . , r k ∈ W(d) such that k j=1 r j cr * 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.

Strong majorization in a free ✱-algebra

“…Proofs and statements similar to (2) and (3) above appear in [7] and more generally in [3]. We mention that in [7] the evaluation matrices X j are all symmetric, while here we remain consistent with the rest of this note and leave them unconstrained.…”

A Nichtnegativstellensatz for polynomials in noncommuting variables

“…We are indebted to Prof. Dr. J. Cimprič who motivated us to look at this problem. After having completed this work, we have received his work [C2] containing a different and abstract approach to the same problem (see [C2,Theorem 5]). …”

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“…In the noncommutative case the second assertion is not true, for the first we have the following theorem due to J. Cimpric [2]. …”