Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras

TL;DR

This paper extends Kuroda's theorem to $n$-tuples of commuting self-adjoint operators in semifinite von Neumann algebras, showing they can be approximated by diagonal operators within certain non-commutative symmetric spaces.

Contribution

It generalizes classical results to a broader non-commutative setting involving symmetric function spaces and $n$-tuples, providing new approximation techniques.

Findings

Approximation of commuting self-adjoint $n$-tuples by diagonal operators within $E()$-norms.

Extension of classical Kuroda and Bercovici-Voiculescu theorems to semifinite von Neumann algebras.

Conditions on symmetric spaces ensuring the approximation property.

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra and let $E$ be a symmetric function space on $(0,\infty)$. Denote by $E(\mathcal{M})$ the non-commutative symmetric space of measurable operators affiliated with $\mathcal{M}$ and associated with $E.$ Suppose $n\in \mathbb{N}$ and $E\cap L_{\infty}\not\subset L_{n,1}$, where $L_{n,1}$ is the Lorentz function space with the fundamental function $\varphi(t)=t^{1/n}$. We prove that for every $\varepsilon>0$ and every commuting self-adjoint $n$-tuple $(\alpha(j))_{j=1}^n,$ where $\alpha(j)$ is affiliated with $\mathcal{M}$ for each $1\leq j\leq n,$ there exists a commuting $n$-tuple $(\delta(j))_{j=1}^n$ of diagonal operators affiliated with $\mathcal{M}$ such that $\max\{\|\alpha(j)-\delta(j)\|_{E(\mathcal{M})},\|\alpha(j)-\delta(j)\|_{\infty}\}<\varepsilon$ for each $1\le j\le n$. In the special case when $\mathcal{M}=B(H)$, our results yield the classical Kuroda and Bercovici-Voiculescu theorems.