2
$\begingroup$

I saw the definition of involution: An involution on an algebra $A$ is a map $a→ a^*$ of A onto itself such that

  1. $(a^*)^* = a$,

  2. $(ab)^* = b^*a^*$ and

  3. $(a + λb)^* = a^* + \barλb^*$ for all $a, b \in A$ and $λ\in\mathbb{C}$.

But what if the space which are over a field that is not $\mathbb{C}$? Indeed, on wikipedia, the definition just contains (1),(2). I am confused with the definition, what's the formal definition?

  • 0
    If the field is $\Bbb R$ then (c) is replaced by $(a+\lambda b)^*=a^*+\lambda b^*$. I don't know so many fields all that well, but if you have something like a conjugation on the field, the most natural extension of (c) is to have $^*$ to be anti-linear wrt this conjugation. For example if you are considering $\Bbb H$ algebras (not a field but a skew-field), then (c) is replaced by the same expression using the conjugation of $\Bbb H$. – 2017-02-05

0 Answers 0