space $(X,\mu)$. By definition, a von Neumann algebra is a weak operator closed $*$-subalgebra of $\mathbb{B}(\mathcal{H})$ for a Hilbert space $\mathcal{H}$. We will tacitly assume that $\mathcal{N}$ has a separable pre-dual, which is equivalent to $\mathcal{H}$ being separable. For any closed densely defined positive operator $T$ affiliated with $\mathcal{N}$, we define its singular value function $$\begin{aligned}
\mu_T(t):&=\inf\{\|PT\|_{\mathcal{N}}: P\in \mathcal{N} \,\mbox{a projection with} \;\tau(1-P)\leq t\}=\\
&=\inf\{s\geq 0: \tau(\chi_{[s,\infty)}(T))\leq t\}\end{aligned}$$ There is a rich theory of so called symmetrically normed operator ideals, see more in [@LSZ Chapter 3], which carries over to the theory of ideals in $L^\infty(0,\infty)$ by means of the singular value function. We are mainly interested in the following two classes.\
[**$L^p$-spaces.**]{} The noncommutative $L^p$-space $\mathcal{L}^p$ is defined as the set of operators affiliated with $\mathcal{N}$ such that $\mu_T\in L^p(0,\infty)$. The space $\mathcal{L}^p$ is a symmetrically normed operator ideal, in particular a Banach space, in the norm $$\|T\|_{\mathcal{L}^p}:=\|\mu_T\|_{L^p(0,\infty)}.$$

In the case that $\mathcal{N}=\mathbb{B}(\mathcal{H})$, we write $\mathcal{L}^p(\mathcal{H})$ for the associated noncommutative $L^p$-space. The space $\mathcal{L}^p(\mathcal{H})$ coincides with the $p$:th Schatten ideal with the same norm.

In the case that $\mathcal{N}=L^\infty(X,\mu)$ with $\tau(a):=\int_Xa\mathrm{d}\mu$ for a $\sigma$-finite measure space $(X,\mu)$, it holds that $\mathcal{L}^p=L^p(X,\mu)$ with the same norm.\
[**Lorentz ideals.**]{} Let $\psi:[0,\infty)\to 