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 $\Phi =({\mathcal{J}}_1,\dots ,{\mathcal{J}}_n)$ of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus $k_{\Phi }(\alpha )$ and proved that $\alpha$ is diagonal modulo $({\mathcal{J}}_1,\dots ,{\mathcal{J}}_n)$ 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 ${\mathcal{J}}_1,\dots ,{\mathcal{J}}_n$ are replaced with symmetric spaces $E_1(\mathcal{M}),\dots ,E_n(\mathcal{M})$ associated with $\mathcal{M}.$