an element in $\prod_n M_n(\Bbb C)$

Question

I want to find an element $x=(x_n)\in \prod_nM_n(\Bbb C)$ such that $\lim \operatorname{tr}_n(x_n)=0$ but $\lim \operatorname{tr}_n(x_n^*x_n)\not \to 0$,where $tr$ is the unique tracial state on $M_n(\Bbb C)$.But I cannot think of an example for a while.I'll appreciate it anyone can help me.

Answer 1

Consider the bounded sequence $(x_n)_{n=1}^\infty$ given by $$ \begin{cases} x_{2n} := 1_n \oplus (-1_n), \\ x_{2n+1} := 1_{2n+1}. \end{cases} $$ Then $\mathrm{tr}_n(x_n) = 0$ for every even natural nuber $n$. Now, let $\omega$ be a free ultrafilter such that $\omega$ contains the set of all even natural numbers. Then you easily check that $$ \lim_\omega \mathrm{tr}_n(x_n) = 0, $$ but $$ \lim_\omega \mathrm{tr}_n(x_n^*x_n) = 1. $$

Answer 2

You can take $x_n$ to be the shift, that is $x_n=\sum_{j=1}^n E_{j,j+1}$. Then $\operatorname{tr}(x_n)=0$ for all $n$, while $\operatorname{tr}(x_n^*x_n)=1-\tfrac1n$. Then $$\lim_{n\to\infty}\operatorname{tr}(x_n^*x_n)=1,$$ while $$\operatorname{tr}(x_n)=0$$ for all $n$.