An identity in C* algebra

Question

Let $A$ be a C* algebra and $a,x \in A$. Is the identity $x^*a^*ax \leq \|a\|^2 x^*x$ true ? How to prove it?

Answer 1

HINT

Here are two useful propositions for proving this:

1. For any positive element $a$ in a unital $C^*$-algebra $A$, we have $a\leq \|a\|$
2. If $a,b\in A$ and $a$ is positive, then $b^*ab$ is positive.