Definition:Involution on Algebra

Definition

Let $A$ be an algebra over $\C$.

Let $\ast : A \to A$ be a mapping satisfying:

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

We call $\ast$ an involution on $A$.

Sources

• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.1$: $C^\ast$-Algebras