The necessary and sufficient condition is that $|x|^p$ and $|y|^q$ |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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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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