Banach algebras and Bott periodicity
Banach algebras and Bott periodicity
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Banach代数和Bott周期
DOI：
10.1016/0040-9383(66)90035-8
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发表时间：
1966
期刊：
影响因子：
--
通讯作者：
R. Wood
中科院分区：
文献类型：
--
作者：
R. Wood
最近 Atiyah 和 Bott 给出了无限复一般线性群周期性定理的新证明 [l]。他们的证明是在向量丛和 K 理论的框架中给出的。在本工作中，[l]的两个主要思想，即用多项式逼近单位圆上的Banach值函数和将多项式映射变形为标准形式，应用于Banach代数的背景下，以获得具有对合6的实数或复数Banach代数A的单个定理。通过适当选择A和u，可以推导出Bott[3]的定理。该材料呈现在同质空间的框架中。没有K理论。这种遗漏并不是出于愿望，而是因为在撰写本文时缺乏对真正的 K 理论的周期性定理中的“一级步骤”的可行描述。
RECENTLY Atiyah and Bott have given a new proof of the periodicity theorem for the infinite complex general linear group [l]. Their proof is given in the framework of vector bundles and K-theory. In the present work, the two principal ideas of [l], namely the approximation of Banach valued functions on the unit circle by polynomials and the deformation of polynomial maps into standard form are applied in the context of Banach algebras to obtain a single theorem for a real or complex Banach algebra A with involution 6. By suitably choosing A and u the theorems of Bott [3] can be deduced. The material is presented in the framework of homogeneous spaces. There is no K-theory. This omission is not by desire but rather by the lack, at the time of writing, of a workable description of the “one stage steps” in the periodicity theorem for the real K-theory.