Involution on semigroups with identity

Question

I'm trying to understand the following: Let $S$ be a semigroup. By an involution on $S$ we mean a map $* : S \to S$ satisfying for all $a,b\in S$

1. $(ab)^*=b^*a^*$

2. $(a^*)^*=a $

My problem is the following. Why if $S$ has an identity we must have $a^*=a$ for all $a\in S$?

Thank you for your help.

Answer 1

This seems false. Consider the complex numbers with multiplication as binary operation and complex conjugation as involution. Maybe you mean to ask why I*=I, if I denotes the identity? Observe that (I*)a=((a*)I)* =(a*)* =a=(a*)* =(I(a*))* =a(I*).