Algebraic properties of isometries between groups of invertible elements in Banach algebras
Algebraic properties of isometries between groups of invertible elements in Banach algebras
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DOI：
10.1016/j.jmaa.2010.11.027
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发表时间：
2011-04
影响因子：
1.3
通讯作者：
O. Hatori
中科院分区：
文献类型：
--
作者：
O. Hatori
本文证明了单位Banach代数A和B可逆群的开子群之间的等距关系T推广为实线性等距关系，直到这两个Banach代数之间的平移.当单位半单交换Banach代数之间的单位同构不一定是乘法的时，本文证明了：若A是交换的，且A或B是半单的，则[公式：见正文]可推广为从A到B的等距真实的代数同构.特别地，A-1作为度量空间与B-1等距当且仅当它们作为可度量群彼此等距同构当且仅当A作为真实的Banach代数与B等距同构;通过关于非同构Banach代数与同胚同构可逆群的例子将其进行比较。研究了Banach空间上有单位元的闭标准算子代数的可逆群的开子群之间的等距关系，并给出了它们的一般形式。
We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then [Formula: see text] is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1is isometric as a metric space to B−1if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of Żelazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.