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

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 $\phi (t)=t^{1/n}$. We prove that for every $\epsilon >0$ and every commuting self-adjoint $n$-tuple $(\alpha (j){)}_{j=1}^n,$ where $\alpha (j)$ is affiliated with $\mathcal{M}$ for each $1\le j\le n,$ there exists a commuting $n$-tuple $(\delta (j){)}_{j=1}^n$ of diagonal operators affiliated with $\mathcal{M}$ such that $\max \{\Vert \alpha (j)-\delta (j){\Vert }_{E(\mathcal{M})},\Vert \alpha (j)-\delta (j){\Vert }_{\infty }\}<\epsilon$ 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.