Theorem 1   A Banach algebra $A$ with an antilinear involution $*$ such that $\norm{a}^2 \leq \norm{a^* a}$ for all $a \in A$ is a $C^*$ algebra.

Proof. It follows from the product inequality $\norm{ab} \leq \norm{a}\norm{b}$ that $$ \norm{a}^2 \leq \norm{a^* a} \leq \norm{a^*}\norm{a}. $$ Therefore, $\norm{a} \leq \norm{a^*}$ Putting $a^*$ for $a$ we also have $\norm{a^*} \leq \norm{a^{**}} = \norm{a}$ Thus, the involution is an isometry: $\norm{a} = \norm{a^*}$ So now, $$ \norm{a}^2 \leq \norm{a^* a} \leq \norm{a}^2. $$ Hence, $\norm{a^* a} = \norm{a}^2$

Theorem 2   A Banach algebra $A$ with an antilinear involution $*$ such that $\norm{a^* a} = \norm{a^*}\norm{a}$ is a $C^*$ algebra.