A certain limit of a state is zero

Question

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ of A: $\lim\limits_{n\to\infty}\varphi(b^*(1-u_n)^2b)=0$.
Could anybody explain me how to prove this? Regards

Answer 1

This is not true. For example you could take $A$ unital, $a=\frac12\,1$. Then $$ (1-u_n)^2=\left(1-\frac 1{\frac12+\frac1n}\right)^2\to 1. $$ So $\varphi(b(1-u_n)^2b^*)\to\varphi(bb^*)$ for any $b$.