非对易实理想对应的非对易实零点空间及其算法
A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms
摘要：
本文将Dubois和Risler的经典实零点定理推广到 *-代数A中的左理想。在介绍了非交换零集和实理想的概念之后，我们发展了与我们的基本问题相关的三个主题：A的一个元素p，其零集包含A的一个有限集S的元素的零集的交集，是否属于包含S的最小实理想？首先，我们构造了一些一般理论，证明了如果某些对象的规范拓扑闭包是允许的，那么答案是肯定的，而在纯代数水平上则是否定的。其次，对于g不定多项式及其形式伴随的自由 *-代数R<x，x*>的每个有限子集S，本文给出了一个计算包含S的最小实理想的可实现算法，并证明了该算法在有限步内是成功的。最后，我们提供了非交换实理想的例子，其中一个纯粹的代数非交换实Nullstellenbrands持有。例如，这包括由n乘n矩阵的 *-代数中的有限集合S生成的实（左）理想，其元素是单变量多项式。进一步，给出了R<x，x*>中左理想的显式充分条件，这些条件覆盖了我们所知道的这类理想的所有例子，并且显著地更多.
英文摘要：
This article takes up the challenge of extending the classical Real Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After introducing the notions of non-commutative zero sets and real ideals, we develop three themes related to our basic question: does an element p of A having zero set containing the intersection of zero sets of elements from a finite set S of A belong to the smallest real ideal containing S? Firstly, we construct some general theory which shows that if a canonical topological closure of certain objects are permitted, then the answer is yes, while at the purely algebraic level it is no. Secondly for every finite subset S of the free *-algebra R<x,x*> of polynomials in g indeterminates and their formal adjoints, we give an implementable algorithm which computes the smallest real ideal containing S and prove that the algorithm succeeds in a finite number of steps. Lastly we provide examples of noncommutative real ideals for which a purely algebraic non-commutative real Nullstellensatz holds. For instance, this includes the real (left) ideals generated by a finite sets S in the *-algebra of n by n matrices whose entries are polynomials in one-variable. Further, explicit sufficient conditions on a left ideal in R<x,x*> are given which cover all the examples of such ideals of which we are aware and significantly more.
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