Definition:C*-Algebra

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Definition

Let $A$ be a Banach algebra over $\C$.

Let $A$ have a conjugate-linear anti-automorphic involution $*$ satisfying the following axioms:



\((\text C^* 1)\)   $:$     \(\ds \forall x \in A:\)    \(\ds x^{**} \)   \(\ds = \)   \(\ds x \)      
\((\text C^* 2)\)   $:$     \(\ds \forall x \in A:\)    \(\ds x^* + y^* \)   \(\ds = \)   \(\ds \paren {x + y}^* \)      
\((\text C^* 3)\)   $:$     \(\ds \forall x, y \in A:\)    \(\ds x^* y^* \)   \(\ds = \)   \(\ds \paren {y x}^* \)      
\((\text C^* 4)\)   $:$     \(\ds \forall x \in A, c \in \C:\)    \(\ds \paren {c x}^* \)   \(\ds = \)   \(\ds \overline c \paren x^* \)      
\((\text C^* 5)\)   $:$     \(\ds \forall x \in A:\)    \(\ds \norm {x x^*} \)   \(\ds = \)   \(\ds \norm x^2 \)      

Then $A$ is referred to as a $\text C^*$-algebra.


Also known as

An older term for $C^*$-algebra is $B^*$-algebra.


Sources