Definition 30.1.1 (Involution).label Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an involution if:

1. (C1)

   For each $\lambda \in \complex$, $(\lambda x)^{*} = \ol \lambda x^{*}$.

2. (C2)

   For each $x \in A$, $x^{**}= x$.

3. (I)

   For every $x, y \in A$, $(xy)^{*} = y^{*}x^{*}$.

The space $A$ equipped with an involution is an involutive algebra over $\complex$.