Title: A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: Algorithms
Abstract: This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free *-algebras ℝ 〈 x, x* 〉 with x=(x1, …, xn). 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 Mn(ℝ[x1]).