an extended limit at $\infty$. By an abuse of notation, we write $\lim_{t\to\omega} f(t):=\omega(f)$ for an extended limit $\omega$ and $f\in L^\infty(0,\infty)$. If $\omega=\omega\circ P_\alpha$ for all $\alpha\geq 1$, we say that $\omega$ is an exponentiation invariant extended limit. Associated with an exponentiation invariant extended limit $\omega$ there is a Dixmier trace $\mathrm{Tr}_{\omega,\psi}:\mathcal{M}_\psi\to {{\mathbb C}}$ defined by $$\mathrm{Tr}_{\omega,\psi}(T):=\lim_{t\to \omega} \frac{1}{\psi(t)}\int_0^t \mu_T(t)\mathrm{d}t,$$ for positive $T\in \mathcal{M}_\psi$ and extending to $\mathcal{M}_\psi$ by linearity (see [@GS Proposition 1.12] for the proof).

The $p$:th convexification $\mathcal{M}_\psi^{(p)}$ is defined as the set of operators $T$ for which $|T|^p\in \mathcal{M}_\psi$; it is normed by $\|T\|_{\mathcal{M}_\psi^{(p)}}:=\||T|^p\|_{\mathcal{M}_\psi}^{1/p}$. The separable part $\mathcal{M}_{\psi,0}^{(p)}$ is defined as the closure in $\mathcal{M}_\psi^{(p)}$ of the finite trace operators in $\mathcal{N}$.

The most studied example of Lorentz ideals comes from the function $\psi(t):=\log(1+t)$. In this case, one often writes $\mathcal{M}_{1,\infty}:=\mathcal{M}_\psi$ and $\mathcal{M}_{p,\infty}:=\mathcal{M}_\psi^{(p)}$. The reader should note that in [@EZ], the Lorentz ideal $\mathcal{M}_{1,\infty}$ associated with $\mathcal{N}=\mathbb{B}(\mathcal{H})$ is denoted by $\mathcal{S}^{Dixm}$.

Technical results on extrapolation and Dixmier traces
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The following result takes its starting point in work of Gayral-Sukochev [@GS]. The first statement is found in [@GS Theorem 3.3] and the second statement in  [@GS Proposition 2.17]. The third statement will be proven below, 