Approximate Unit of a $C^*$-subalgebra

Question

Here is part of a proof I am working through. Note that $A$ is a $C^*$-algebra, $I$ is a closed two-sided ideal, and $C^*(a)$ denotes the $C^*$-subalgebra generated by $1$ and $a$.

Let $b \in I$, and observe that $C^*(b^*b) \subseteq I$. By 2.5.2, we can choose an approximate unit $\{u_{\lambda}\}$ for $C^*(b^*b)$. Since $$||b(1-u_{\lambda})||^2 = ||(1-u_{\lambda})b^*b(1-u_{\lambda})|| \le ||b^*b(1-u_{\lambda})||$$ we have $b u_{\lambda} \to b$ in norm.

I'm having trouble seeing why the inequality holds.

Answer 1

Because the elements in the approximate unit satisfy $0\leq u_\lambda\leq 1$, you have $0\leq 1-u_\lambda\leq1$. This implies $\|1-u_\lambda\|\leq1$. Then $$ \|(1-u_\lambda)b^*b(1-u_\lambda)\|\leq\|1-u_\lambda\|\,\|b^*b(1-u_\lambda)\|\leq\|b^*b(1-u_\lambda)\|. $$