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Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras
Operator Algebras · Submitted 24 Sep 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $1\le j\le n$.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathcal{M}$ be a semifinite von Neumann algebra and let $E$ be a symmetric function space on $(0,\infty)$. | background 1.00 | 0.04 | 0.01 | unclear 0.54 |
| s1 | 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 $\varphi(t)=t^{1/n}$. | background 0.96 | 0.09 | 0.03 | asserted 0.66 |
| s2 | 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 $1\le j\le n$. | result 0.93 | 0.96 | 0.96 | asserted 1.00 |
| s3 | In the special case when $\mathcal{M}=B(H)$, our results yield the classical Kuroda and Bercovici-Voiculescu theorems. | interpretation 0.95 | 0.69 | 0.05 | asserted 0.99 |
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Original 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.
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Cites as background 2 papers
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Here, $k_E()(α)$ is a number called the quasicentral modulus, introduced by Voiculescu [ref] ; see Definition [ref].
The condenser quasicentral modulus · Introduction · 0.93
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We now recall a theorem in [ref] concerning the characterization of the diagonality of an $n$-tuple in $$ modulo a given symmetric space, using quasicentral modulus.
Diagonality modulo symmetric spaces in semifinite von Neumann algebras · Quasicentral modulus · 0.90
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