# Why $\|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ in the definition of $C^*$ algebra?

I read the definition of $C^*$ algebra in Wikipedia where it says $\|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ but I do not know why. Can you show me how to derive $\|xx^*\| = \|x\|^2$ from $\|x^* x \| = \|x\|\|x^*\|$?

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For one direction, if we start by assuming $\|xx^*\| = \|x\|^2$ then as $C^*$-algebras are Banach algebras we get $\|x\|^2 = \|xx^*\| \le \|x\|\|x^*\|$ so that $\|x\| \le \|x^*\|$. Then $\|x\| \le \|x^*\| \le \| (x^*)^*\| = \|x\|$ so that $\|x\| = \|x^*\|$. It then follows that $\|xx^*\| = \|x\|^2 = \|x\|\|x^*\|$.