C*-代数

C*-代数（或读作“C星代数”）是数学分支中泛函分析的重要研究对象。C*-代数的典型例子是满足以下两个性质的希尔伯特空间线性算子的代数A

1943年前后，伊斯拉埃尔·盖尔范德和马可·奈马克对C*-代数建立了不依赖于算子的抽象刻画。

抽象刻画

C*-代数A是复数域上的巴拿赫代数以及映射* : AA（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

• A中任意的两个元素xy
${\displaystyle (x+y)^{*}=x^{*}+y^{*}}$
${\displaystyle (xy)^{*}=y^{*}x^{*}}$
• C中任意复数${\displaystyle \lambda }$以及A中任一元素x
${\displaystyle (\lambda x)^{*}={\overline {\lambda }}x^{*}}$
• A中任一元素x
${\displaystyle (x^{*})^{*}=x}$
• C*–恒等映射A中任一元素x成立：
${\displaystyle \|x^{*}x\|=\|x\|\|x^{*}\|}$
C*–恒等映射性质等价于
${\displaystyle \|xx^{*}\|=\|x\|\|x^{*}\|}$

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${\displaystyle \}}$

• A中任意的两个元素xy
${\displaystyle \pi (xy)=\pi (x)\pi (y)}$
• A中任一元素x
${\displaystyle \pi (x^{*})=\pi (x)^{*}}$

参考来源

• Arveson, W., An Invitation to C*-Algebra, 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, Non-commutative geometry (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, Les C*-algèbres et leurs représentations, 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., Characterizations of C*-algebras: The Gelfand-Naimark Theorems, CRC Press, 1986, ISBN 9780824775698.
• Emch, G., Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972, ISBN 0-471-23900-3. Mathematically rigorous reference which provides extensive physics background.
• A.I. Shtern, C* algebra, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Sakai, S., C*-algebras and W*-algebras, Springer, 1971, ISBN 3-540-63633-1.
• Segal, Irving, Irreducible representations of operator algebras, Bulletin of the American Mathematical Society, 1947, 53 (2): 73–88, doi:10.1090/S0002-9904-1947-08742-5.

C*-algebra