The algebra of thin measurable operators is directly finite

Let M be a semifinite von Neumann algebra on a Hilbert space H equipped with a faithful normal semifinite trace τ, S(M,τ) be the ^*-algebra of all τ-measurable operators. Let S₀(M,τ) be the ^*-algebra of all τ-compact operators and T(M,τ)=S₀(M,τ)+CI be the ^*-algebra of all operators X=A+λ I with A∈ S₀(M,τ) and λ ∈ C. It is proved that every operator of T(M,τ) that is left-invertible in T(M,τ) is in fact invertible in T(M,τ). It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in B (H). For the singular value function μ(t; Q) of Q=Q²∈ S(M,τ), the inclusion μ(t; Q)∈ \0\ [1, +∞) holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.