Algebraic characterization of symmetric complex Banach manifolds
Algebraic characterization of symmetric complex Banach manifolds
复制标题
对称复 Banach 流形的代数表征
DOI：
10.1007/bf01360772
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发表时间：
1977
影响因子：
1.4
通讯作者：
W. Kaup
中科院分区：
文献类型：
--
作者：
W. Kaup
有限维对称厄米复流形已被E. Cartan [4]利用单复李代数的分类。一个Jordan理论方法是由于Koecher [18]和最近的卢什[25]：对称有界域与厄密Jordan三元系一一对应，其中某个迹形式是正定厄密的。无穷维有界对称域已被许多作者考虑过[7，9，11，14-16，28，32]。哈里斯例如证明了一个大类的复杂Banach空间U（包括所有C*-代数），开放单位球的U是齐次的，因此对称。本文研究对称复Banach流形。一个复Banach流形是一个复流形（可能是无限维的）M，在M的切丛上有一个固定的范数v。我们不要求v对每个切空间T~，x~M的限制是Hilbert范数--否则我们将排除许多有趣的例子，例如Hilbert空间H上所有有界算子的Banach空间L（H）中的开单位球，其中d imH= oo。因此，我们的厄米约当三元系的概念不能依赖于迹形式。本文的主要部分致力于证明以下主要定理。具有基点的单连通对称复Banach流形范畴等价于厄米特Jordan三元系范畴。
The symmetric hermitian complex manifolds (of finite dimension) have been classi fled completely by E. Cartan [4] using the classification of simple complex Lie algebras. A Jordan theoretic approach is due to Koecher [18] and more recently to Loos [25] : The symmetric bounded domains are in a one-to-one correspondence to hermitian Jordan triple systems, for which a certain trace form is positive-definite hermitian. Bounded symmetric domains in infinite dimensions have been considered by various authors [7, 9, 11, 14-16, 28, 32]. Harris for instance proved for a big class of complex Banach spaces U (including all C*-algebras) that the open unit ball of U is homogeneous and hence symmetric. In this paper we study symmetric complex Banach manifolds. A complex Banach manifold is a complex manifold (of possibly infinite dimension) M together with a fixed norm v on the tangent bundle of M. We do not require that the restriction of v to every tangent space T~, x~M, is a Hilbert norm--otherwise we would exclude many interesting examples such as the open unit ball in the Banach space L(H) of all bounded operators on a Hilbert space H with d imH= oo. Therefore our notion of hermitian Jordan triple system cannot rely on trace forms. The major part of the paper is devoted to the proof of the following Main Theorem. The category of simply connected, symmetric, complex Banach manifolds with base point is equivalent to the category of hermitian Jordan triple systems.