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arXiv · 2404.00259

Diagonality modulo symmetric spaces in semifinite von Neumann algebras

Aleksey Ber·Fedor Sukochev·Dmitriy Zanin·Hongyin Zhao
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Abstract

In the study on the diagonality of an $n$-tuple $α=(α(j))_{j=1}^n$ of commuting self-adjoint operators modulo a given $n$-tuple $Φ=(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus $k_Φ(α)$ and proved that $α$ is diagonal modulo $(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ if and only if $k_Φ(α)=0.$ We prove that the same assertion holds true when $B(H)$ is replaced with a $σ$-finite semifinite von Neumann algebra $\mathcal{M}$, and $\mathcal{J}_1,\ldots,\mathcal{J}_n$ are replaced with symmetric spaces $E_1(\mathcal{M}),\ldots,E_n(\mathcal{M})$ associated with $\mathcal{M}.$

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Aleksey Ber, Fedor Sukochev, Dmitriy Zanin, Hongyin Zhao. 2024-06-17. Diagonality modulo symmetric spaces in semifinite von Neumann algebras. https://arxiv.org/abs/2404.00259

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