C*-代数

此处给出蓋爾范德和奈马克在1943年给出的定义。

C^*-代数A是复数域上的巴拿赫代数以及映射* : A → A（称为对合映射）的组合。A中元素x关于对合映射* 的像写作x^*。对合映射拥有下列性质

• 对A中任意的两个元素x、y：

${\displaystyle (x+y)^{*}=x^{*}+y^{*}}$

${\displaystyle (xy)^{*}=y^{*}x^{*}}$

• 对C中任意复数${\displaystyle \lambda }$以及A中任一元素x：

${\displaystyle (\lambda x)^{*}={\overline {\lambda }}x^{*}}$

• 对A中任一元素x：

${\displaystyle (x^{*})^{*}=x}$

• C^*–恒等映射对A中任一元素x成立：

${\displaystyle \|x^{*}x\|=\|x\|\|x^{*}\|}$

C^*–恒等映射性质等价于

${\displaystyle \|xx^{*}\|=\|x\|\|x^{*}\|}$

这个关系等价于${\displaystyle \|xx^{*}\|=\|x\|^{2}}$，有时亦称为B^*–恒等映射

C^*–恒等映射是一个很强的约束条件。举例来说，C^*–恒等映射和谱半径公式可以推出C^*–范数由以下代数结构唯一确定：

${\displaystyle \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda |:x^{*}x-\lambda \,1}$不可逆${\displaystyle \}}$

给定的从C^*-代数A射到B的有界线性算子π : A → B被称为*-同态，如果

• 对A中任意的两个元素x、y，

${\displaystyle \pi (xy)=\pi (x)\pi (y)}$

• 对A中任一元素x，

${\displaystyle \pi (x^{*})=\pi (x)^{*}}$

• Arveson, W., , Springer-Verlag, 1976, ISBN 0-387-90176-0. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.
• Connes, Alain, (PDF), [2012-01-16], ISBN 0-12-185860-X, （原始内容存档 (PDF)于2011-09-27）. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
• Dixmier, Jacques, , Gauthier-Villars, 1969, ISBN 0-7204-0762-1. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
• Doran, Robert S.; Belfi, Victor A., , CRC Press, 1986, ISBN 9780824775698.
• Emch, G., , Wiley-Interscience, 1972, ISBN 0-471-23900-3. Mathematically rigorous reference which provides extensive physics background.
• A.I. Shtern, , Hazewinkel, Michiel (编), , Springer, 2001, ISBN 978-1-55608-010-4
• Sakai, S., , Springer, 1971, ISBN 3-540-63633-1.
• Segal, Irving, , Bulletin of the American Mathematical Society, 1947, 53 (2): 73–88, doi:10.1090/S0002-9904-1947-08742-5.

C*-algebra