Suppose that $A$ is a Banach algebra with an involution $x \to x^∗$ that satisfies $\|x\|^2 \leq \|x^∗x\|$. Then show that A is a Banach $∗$-algebra.
I.E. we have to show $\|x^*\| = \|x\|$.
We know $\|x\|^2 \leq \|x^∗x\| \leq \|x^∗\| \|x\| \implies \|x\| \leq \|x^∗\|$
and switching $x$ and $x^*$ we have $\|x^*\| \leq \|x\|$. Thus we have $\|x^*\| = \|x\|$.
Is the proof correct?