Exercise 24, § 6

Let $A$ be a unital Banach algebra over $\mathbf{C}$. We suppose that $x\mapsto x^*$ and $x\mapsto x^{\circ}$ are involutions on A such that A, furnished with each of them, is a star algebra. We denote by $A_1$ and $A_2$ these two star algebras.

(a) With the notations of exercise 23, now prove that $A'_{1,+}= A'_{2,+}$. We denote this set by $A'_+$.

(b) Let $x\in A_{1,h}$ be an element such that $x^*=x$. Prove that for every $f\in A'_+$, one has $f(i(x-x^{\circ})) = 0$. Deduce that $x\in A_{2,h}$. (Use question b) of exercise 23).

(c) Conclude that $A_1= A_2$, and that a complex Banach algebra admits at most one involution which makes it a star algebra.