Trace and norm bounded sequence of positive elements has convergent subsequence in hyperfinite $II_1$ factor

Question

Let $A$ is a hyperfinite $\operatorname{II_1}$ factor and $x_n \in A$ is some sequence of positive elements such that $||x_n||$ convergent and $\operatorname{Tr}(x_n^2) = 1$ (where $\operatorname{Tr}$ - is a unique faithful normal tracial state in $A$). Is it true that $x_n$ has convergent subsequence in norm topology?

Answer 1

In norm, it is clear that no. All II$_1$-factors contain a copy of $L^\infty[0,1]$, where the trace corresponds with integration against Lebesgue measure. You can take $x_n=\sqrt2\cdot 1_{[1/n,1/n+1/2]}$. Then $\|x_n\|=\sqrt2$, $\text{Tr}\,(x_n^2)=1$ for all $n$, and $\|x_n-x_m\|=1$ for all $n\ne m$.