A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: Algorithms. Issue 5 (3rd December 2012)

• Record Type: Journal Article
• Authors: Cimprič, Jakob; Helton, J. William; McCullough, Scott; Nelson, Christopher
• Is Part Of: Proceedings of the London Mathematical Society. Volume 106:Issue 5(2012:May)
• Page Start: 1060
• Page End: 1086
• Publication Date: 2012-12-03
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pds060
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract : This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free *‐algebras ℝ 〈 x, x * 〉 with x =( x 1, …, x n ). First, we introduce notions of the (noncommutative) zero set of a left ideal and of a real left ideal. We prove that every element from ℝ 〈 x, x * 〉 whose zero set contains the intersection of zero sets of elements from a finite subset S of ℝ 〈 x, x * 〉 belongs to the smallest real left ideal containing S . Next, we give an implementable algorithm, which for every finite S ⊂ℝ 〈 x, x * 〉, computes the smallest real left ideal containing S, and prove that the algorithm succeeds in a finite number of steps. Our definitions and some of our results also work for other *‐algebras. As an example, we treat real left ideals in M n (ℝ[ x 1 ]).