Diagonality modulo symmetric spaces in semifinite von Neumann algebras

Abstract

In the study on the diagonality of an $n$-tuple $\alpha=(\alpha(j))_{j=1}^n$ of commuting self-adjoint operators modulo a given $n$-tuple $\Phi=(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus $k_{\Phi}(\alpha)$ and proved that $\alpha$ is diagonal modulo $(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ if and only if $k_{\Phi}(\alpha)=0.$ We prove that the same assertion holds true when $B(H)$ is replaced with a $\sigma$-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 $\mathcal{M}.$