A is a Banach $∗$-algebra.

Question

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?

Answer 1

By switching $x $ and $x^*$ you should mention that $x^{**}=x $. Your proof is then complete.