Embeddings of operator ideals into $\mathcal{L}_p-$spaces on finite von Neumann algebras

TL;DR

This paper investigates conditions under which operator ideals can be embedded into $L_p$-spaces on finite von Neumann algebras, revealing new embedding criteria related to their commutative cores and interpolation properties.

Contribution

It establishes new isomorphic embedding results for operator ideals into $L_p$-spaces on hyperfinite II$_1$-factors, linking embeddings to their commutative cores and interpolation spaces.

Findings

Operator ideals embed into $L_p$-spaces if their cores embed into $L_p(0,1)

Orlicz ideals embed iff they are interpolation spaces for $(L_p, L_2)$

Embeddings extend to $L_p$-spaces associated with arbitrary finite von Neumann algebras

Abstract

Let $\mathcal{L}(H)$ be the $*$-algebra of all bounded operators on an infinite dimensional Hilbert space $H$ and let $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ be an ideal in $\mathcal{L}(H)$ equipped with a Banach norm which is distinct from the Schatten-von Neumann ideal $\mathcal{L}_p(\mathcal{H})$, $1\leq p<2$. We prove that $\mathcal{I}$ isomorphically embeds into an $L_p$-space $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ (here, $\mathcal{R}$ is the hyperfinite II$_1$-factor) if its commutative core (that is, Calkin space for $\mathcal{I}$) isomorphically embeds into $L_p(0,1).$ Furthermore, we prove that an Orlicz ideal $\mathcal{L}_M(H)\neq\mathcal{L}_p(H)$ isomorphically embeds into $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ if and only if it is an interpolation space for the Banach couple $(\mathcal{L}_p(H),\mathcal{L}_2(H)).$ Finally, we consider isomorphic embeddings of $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ into $L_p$-spaces associated with arbitrary finite von Neumann algebras.