Ideal spaces of measurable operators affiliated to a semifinite von Neumann algebra

Suppose that $\mathscr M$ is a von Neumann algebra of operators on a Hilbert space $\mathscr H$ and $\tau$ is a faithful normal semifinite trace on $\mathscr M$. Let $\mathscr E$, $\mathscr F$ and $\mathscr G$ be ideal spaces on $(\mathscr M,\tau)$. We find when a $\tau$-measurable operator $X$ belongs to $\mathscr E$ in terms of the idempotent $P$ of $\mathscr M$. The sets $\mathscr E+\mathscr F$ and $\mathscr E\cdot\mathscr F$ are also ideal spaces on $(\mathscr M,\tau)$; moreover, $\mathscr E\cdot\mathscr F=\mathscr F\cdot\mathscr E$ and $(\mathscr E+\mathscr F)\cdot\mathscr G=\mathscr E\cdot\mathscr G+\mathscr F\cdot\mathscr G$. The structure of ideal spaces is modular. We establish some new properties of the $L_1(\mathscr M,\tau)$ space of integrable operators affiliated to the algebra $\mathscr M$. The results are new even for the *-algebra $\mathscr M=\mathscr B(\mathscr H)$ of all bounded linear operators on $\mathscr H$ which is endowed with the canonical trace $\tau=\operatorname{tr}$.