An exercise (about positive elements) in C*-algebra

Question

Let $A$ be a C*-algebra, $a\in A$ be a positive element and $b\in A$ be an arbitary element in $A$. Can we verify that $$b^{*}ab\leq \|b\|^{2}a~~?$$

Answer 1

I might be missing something trivial- but this result seems to be false.

Let's take the $C^*$ algebra to be $M_2(\mathbb{C})$, $a=\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix}$, $b=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$.

Then $b^*b=I$, and $b^*ab=\begin{pmatrix} 2 & 0\\ 0 & 1 \end{pmatrix}$, and it's not true that $b^*ab \leq ||b||^2a$