Schrödinger Institute in Vienna during the program on “Bivariant $K$-theory in Geometry and Physics".

Lorentz spaces and extrapolation {#prelsection}
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We will in this section provide an overview of Lorentz ideals and Hankel operators. Most results in this section can be found in the literature, and the remainder of the results in this section are variations on well known results.

Operator ideals {#opidelalsos}
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The operators we will consider in this paper will in general belong to some ideal of operators on a Hilbert space. The general theory of operator ideals is well defined starting from a semi-finite von Neumann algebra. While this introduces some additional technicalities, it will allow us to treat ideal of operators on the same footing as $L^p$-spaces on a measure space. We will not go through this theory beyond its salient points. The reader is referred to [@LSZ] for a more thorough presentation.

Let $\mathcal{N}$ denote a semi-finite von Neumann algebra and $\tau$ a normal, faithful, semi-finite tracial weight on $\mathcal{N}$. The two main examples to keep in mind are $\mathcal{N}=\mathbb{B}(\mathcal{H})$ – the bounded operators on a separable Hilbert space – with $\tau$ being the operator trace and $\mathcal{N}=L^\infty(X,\mu)$ with $\tau(a):=\int_Xa\mathrm{d}\mu$ for a $\sigma$-finite measure 