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We show that an operator $x$ in the unit sphere of $E\\left(\\mathcal{M},\\tau\\right)$ is $k$-extreme, $k\\in\\mathbb N$, whenever its singular value function $\\mu(x)$ is $k$-extreme and one of the following conditions hold (i) $\\mu(\\infty,x)=\\lim_{t\\to\\infty}\\mu(t,x)=0$ or (ii) $n(x)\\mathcal{M} n(x^*)=0$ and $|x|\\geq \\mu(\\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. 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Czerwi\\'nska","submitted_at":"2015-02-13T20:17:55Z","abstract_excerpt":"Let $\\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\\tau$ and $E$ be a strongly symmetric Banach function space on $[0,\\tau(1))$. We show that an operator $x$ in the unit sphere of $E\\left(\\mathcal{M},\\tau\\right)$ is $k$-extreme, $k\\in\\mathbb N$, whenever its singular value function $\\mu(x)$ is $k$-extreme and one of the following conditions hold (i) $\\mu(\\infty,x)=\\lim_{t\\to\\infty}\\mu(t,x)=0$ or (ii) $n(x)\\mathcal{M} n(x^*)=0$ and $|x|\\geq \\mu(\\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. 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