The algebra of thin measurable operators is directly finite

Authors : Airat BİKCHENTAEV

Pages : 1-5

Doi:10.33205/cma.1181495

View : 8 | Download : 6

Publication Date : 2023-03-15

Article Type : Research Paper

Abstract :Let $\\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\\mathcal{H}$ equipped with a faithful normal semifinite trace $\\tau$, $Sinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);$ be the ${}^*$-algebra of all $\\tau$-measurable operators. Let $S_0insert ignore into journalissuearticles values(\\mathcal{M},\\tau);$ be the ${}^*$-algebra of all $\\tau$-compact operators and $Tinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);=S_0insert ignore into journalissuearticles values(\\mathcal{M},\\tau);+\\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\\lambda I$ with $A\\in S_0insert ignore into journalissuearticles values(\\mathcal{M},\\tau);$ and $\\lambda \\in \\mathbb{C}$. It is proved that every operator of $Tinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);$ that is left-invertible in $Tinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);$ is in fact invertible in $Tinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);$. It is a generalization of Sterling Berberian theorem insert ignore into journalissuearticles values(1982); on the subalgebra of thin operators in $\\mathcal{B} insert ignore into journalissuearticles values(\\mathcal{H});$. For the singular value function $\\muinsert ignore into journalissuearticles values(t; Q);$ of $Q=Q^2\\in Sinsert ignore into journalissuearticles values(\\mathcal{M},\\tau);$, the inclusion $\\muinsert ignore into journalissuearticles values(t; Q);\\in \\{0\\}\\bigcup [1, +\\infty);$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.
Keywords : Hilbert space, von Neumann algebra, semifinite trace, \\tau measurable operator, \\tau compact operator, singular value function, idempotent