分数代数与 *-代数的严格正性
Algebras of Fractions and Strict Positivstellensätze for *-Algebras
摘要：
本文研究了与单位复 *-代数$\cA$相关联的分数的 *-代数$\cX$.利用代数$\cX$及其希尔伯特空间表示证明了$\cA$的抽象非交换严格正性.研究了$\cA$的多重分次问题，作为验证该定理假设的技术工具。作为应用，我们得到了实直线的仿射群的Weyl代数和李代数$\cg$的新的严格正性定理.利用生成元的预解式刻画了李代数$\cg$的可积表示，并给出了$\cg$表示的一个新的可积判据.
英文摘要：
In this paper we investigate a *-algebra $\cX$ of fractions associated with a unital complex *-algebra $\cA$. The algebra $\cX$ and its Hilbert space representations are used to prove abstract noncommutative strict Positivstellensätze for $\cA$. Multi-grading of $\cA$ are studied as technical tools to verify the assumptions of this theorem. As applications we obtain new strict Positivstellensätze for the Weyl algebra and for the Lie algebra $\cg$ of the affine group of the real line. We characterize integrable representations of the Lie algebra $\cg$ in terms of resolvents of the generators and derive a new integrability criterion for representations of $\cg$.
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