ON THE τ-COMPACTNESS OF PRODUCTS OF τ-MEASURABLE OPERATORS ADJOINT TO SEMI-FINITE VON NEUMANN ALGEBRAS

摘要

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, Lit-tlewood, and Polya) 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.