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

Suppose that <italic>M</italic> is a von Neumann algebra of operators on a Hilbert space <italic>H</italic> and <italic>τ</italic> is a faithful normal semifinite trace on <italic>M</italic>. Let <italic>E</italic>, <italic>F</italic> and <italic>G</italic> be ideal spaces on (<italic>M</italic>, <italic>τ</italic>). We find when a <italic>τ</italic>-measurable operator <italic>X</italic> belongs to <italic>E</italic> in terms of the idempotent <italic>P</italic> of <italic>M</italic>. The sets <italic>E</italic>+<italic>F</italic> and <italic>E</italic>·<italic>F</italic> are also ideal spaces on (<italic>M</italic>, <italic>τ</italic>); moreover, <italic>E</italic>·<italic>F</italic> = <italic>F</italic>·<italic>E</italic> and (<italic>E</italic>+<italic>F</italic>)·<italic>G</italic> = <italic>E</italic>·<italic>G</italic>+<italic>F</italic>·<italic>G</italic>. The structure of ideal spaces is modular. We establish some new properties of the <italic>L</italic>1(<italic>M</italic>, <italic>τ</italic>) space of integrable operators affiliated to the algebra <italic>M</italic>. The results are new even for the *-algebra <italic>M</italic> = <italic>B</italic>(<italic>H</italic>) of all bounded linear operators on <italic>H</italic> which is endowed with the canonical trace <italic>τ</italic> = tr. [ABSTRACT FROM AUTHOR]