# 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代数A

1943年前后，伊斯拉埃爾·蓋爾范德马可·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
$(x + y)^* = x^* + y^*$
$(x y)^* = y^* x^*$
• C中任意复数$\lambda$以及A中任一元素x
$(\lambda x)^* = \overline{\lambda} x^*$
• A中任一元素x
$(x^*)^* = x$
• C*–恒等映射A中任一元素x成立：
$\|x^* x \| = \|x\|\|x^*\|$
C*–恒等映射性质等价于
$\|x x^* \| = \|x\|\|x^*\|$

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：

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

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

## 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.
• . 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// (编) Hazewinkel, Michiel, 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