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Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S_p$ denote the Schatten space. For any $x\in S_p$ and any $y\in S_q$ we have $$ ||xy||_{S_r} \leq ||x||_{S_p}||y||_{S_q} $$ (noncommutative Hölder's inequality).

Does it exists necessary and sufficient conditions on $x,y$ in order to have an equality in this inequality?

More generally, I ask the same question replacing $S_p$ by the noncommutative $L_p$-space $L_p(M)$ associated with a semifinite von Neumann algebra $M$ equipped with a normal semifinite faithful trace $\tau$.

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The necessary and sufficient condition is that $|x|^p$ and $|y^*|^q$ are proportional. This can be deduced from Dixmier's paper (although it is not clearly stated that way there; it is based on Proposition 8). It probably also appears in more modern treatments (Nelson, Terp, Haagerup, Hiai, Kosaki, etc.) but I don't have the sources here to check that.

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    $\begingroup$ You probably mean that $|x|^p$ and $|y^*|^p$ are proportional. $\endgroup$
    – Mikael de la Salle
    Commented Jun 2, 2011 at 19:52
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    $\begingroup$ Right. Thanks, Mikael. I'll correct it right away. $\endgroup$
    – Martin Argerami
    Commented Jun 2, 2011 at 20:53

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