On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall Isomorphisms
On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall Isomorphisms
复制标题
关于 Kakutani-Itô-Segal-Gross 和 Segal-Bargmann-Hall 同构
DOI：
--
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
B. Driver
中科院分区：
文献类型：
--
作者：
B. Driver
最近Gross证明了Kakutani-Ito-Segal同构定理从向量空间上的Gaussian测度的设置推广到紧致型单连通李群（G）上的“热核”测度（pt）。同构将L2（pt）与g= Lie（G）的泛包络代数的某种完备化联系起来。格罗斯使用角谷-伊藤-西格尔定理和与G值布朗运动相关的无穷维微积分证明了这一结果。Hijab大大简化和澄清了格罗斯的证明。Hiiab的证明避免了原证明中的大部分（但不是全部）“无穷维”分析。本文在Hijab证明的基础上给出了Gross同构定理的一个完全“有限维”的非概率证明.本文的证明在很大程度上依赖于Hall将Segal-Bargmann变换漂亮地“推广”到紧李群的情形。将L2（pt）与复化李群GC上的某个全纯函数的L2-空间（L2（GC）<$H）联系起来的定理也将推广到紧型李群。在这个过程中，它示出了如何特征，在GC中的单位元的所有导数的可和性条件，这些全纯函数是在L2（GC）。
Recently, Gross has shown that the Kakutani-Ito-Segal isomorphism theorem has an extension from the setting of Gaussian measure on a vector space to "heat kernel" measure (pt) on a simply connected Lie group (G) of compact type. The isomorphism relates L2(pt) to a certain completion of the universal enveloping algebra of g= Lie(G). Gross proves this result using the Kakutani-Ito-Segal theorem and an infinite dimensional calculus associated to G-valued Brownian motion. Hijab has greatly simplified and clarified Gross′ proof. Hiiab′s proof avoids most, but not all, of the "infinite dimensional" analysis in the original proof. In this paper, we will build on Hijab′s proof to give a completely "finite dimensional" non-probabilistic proof of Gross′ isomorphism theorem. The proof given here relies heavily on Hall′s beautiful "extension" of the Segal-Bargmann transformation to the setting of compact Lie groups. This theorem relating L2(pt) to a certain L2-space of holomorphic functions (L2(GC) ∩ H) on the complexified Lie group GC will also be generalized to Lie groups of compact type. In the process, it is shown how to characterize, in terms of summability conditions on all the derivatives at the identity in GC, those holomorphic functions which are in L2(GC).