On the τ-Compactness of Products of τ -Measurable Operators Adjoint to Semi-Finite Von Neumann Algebras

Let $M$ be the von Neumann algebra of operators in a Hilbert space $H$ and τ be an exact normal semi-finite trace on $M$. We obtain inequalities for permutations of products of τ-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of τ -measurable operators and a sufficient condition of orthogonality of certain nonnegative τ-measurable operators. We state sufficient conditions of the τ –compactness of products of self-adjoint τ -measurable operators and obtain a criterion of the τ -compactness of the product of a nonnegative τ-measurable operator and an arbitrary τ -measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from $M$ . All results are new for the *-algebra $B$($H$) of all bounded linear operators in $H$ endowed with the canonical trace τ = tr.