The algebra of thin $\tau$-measurable operators is directly finite

Let $\mathscr{M}$ be a von Neumann algebra of operators on a Hilbert space $\mathscr{H}$ , $\mathscr{P (M)}$ be the lattice of projections in $\mathscr{M}$, $I$ and be the unit of $\mathscr{M}$. Let $\tau$ be a faithful normal semifinite trace on $\mathscr{M}$, $S(\mathscr{M},\tau)$ be the *-algebra of all $\tau$-measurable operators. Let $S_0(\mathscr{M},\tau)$ be the *-algebra of all $\tau$-compact operators and $T(\mathscr{M},\tau) = S_0(\mathscr{M},\tau)+\mathbb{C}I$ be the *-algebra of all operators $X = A +\lambda I$ with $A \in S_0(\mathscr{M},\tau)$ and a complex number $\lambda$.
For every operator $X \in S(\mathscr{M},\tau)$ the range projection $\mathrm{r}(X)$ is the projection onto the closure of its range $\mathrm{Ran}(X)$ ; it lies in $\mathscr{P (M)}$. Consider $F_0(\mathscr{M},\tau) = {A \in S_0(\mathscr{M},\tau) : \tau(\mathrm{r}(A)) < +\infty}$ and $A (\mathscr{M},\tau) = F_0(\mathscr{M},\tau)+\mathbb{C}I$. Then $A (\mathscr{M},\tau)$ is a *-subalgebra of $T (\mathscr{M},\tau)$.
We obtain the following extension of the S. Berberian’s result [1] on direct finiteness of the algebra of thin operators on an infinite-dimensional Hilbert space to the I. Segal’s non-commutative integration setting.
Theorem 1. If $X,Y \in T (\mathscr{M},\tau)$ such that $X Y = I$, then $Y X = I$.
The generalized singular value function $\mu(X) : t \to \mu(t;X)$ of the operator $X \in S(\mathscr{M},\tau)$ is defined by setting
$$\mu(s;X) = \mathrm{inf}\{\|X P\| : P \in \mathscr{P (M)} \text{ and } \tau(I -P) \le s\}.$$
Theorem 2. If $Q \in S(\mathscr{M},\tau)$ is such that $Q^2 = Q$, then $\mu(t;Q) \in {0}\cup [1,+\infty)$ for all $t > 0$.
Note that for $Q \in \mathscr{M}$ such that $Q^2 = Q$ the relation $\mu(t;Q) \in {0}\cup[1,\|Q\|]$ for all $t > 0$ was obtained by another way in [2, Lemma 3.8, item 1)]. Theorem 2 gives the positive answer to the problem suggested by D. Mushtari in 2010.
Acknowledgements. The work is performed under the development program of the Volga Region Mathematical Center (agreement no. 075-02-2022-882).