  Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space
$\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$,
$S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators.
Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact
operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the
${}^*$-algebra of all operators $X=A+\lambda I$ with $A\in
S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. We prove that every
operator of $T(\mathcal{M},\tau)$ that is left-invertible in
$T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$. It is a
generalization of Sterling Berberian theorem (1982) on the subalgebra of thin
operators in $\mathcal{B} (\mathcal{H})$. For the singular value function
$\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$ we have $\mu(t; Q)\in
\{0\}\bigcup [1, +\infty)$ for all $t>0$. It gives the positive answer to the
question posed by Daniyar Mushtari in 2010.
