The algebra of thin measurable operators is directly finite

2022/05/25 by Bikchentaev, Airat M.
#16E50 #46L51 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2205.12525

Abstract

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+λI with A∈ S₀(M,τ) and λ∈ ℂ. We prove 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) we have μ(t; Q)∈ \0\\bigcup [1, +∞) for all t>0. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

Related

• Inequalities for trace on τ-measurable operators
• Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra
• Structure of derivations on various algebras of measurable operators for type I von Neumann algebras
• Perturbations of self-adjoint operators in semifinite von Neumann algebras: Kato-Rosenblum theorem
• Description of Derivations on Measurable Operator Algebras of Type I