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Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

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Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G… Click to show full abstract

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.

Keywords: operators affiliated; neumann algebra; spaces measurable; ideal spaces; von neumann

Journal Title: Siberian Mathematical Journal
Year Published: 2018

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