Planes in Schatten-$3$

We prove that any two-dimensional real subspace of Schatten-3 can be linearly isometrically embedded into $L_{3}$. This resolves the case $p = 3$ of Hanner's inequality for Schatten classes, conjectured by Ball, Carlen and Lieb. We conjecture that similar isometric embedding is possible for any $p \geq 1$.

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