The necessary and sufficient condition is that $|x|^p$ and $|y|^q$ |y^*|^q$are proportional. This can be deduced from Dixmier's paper (although it is not clearly stated that way there; it is based on Proposition 8). It probably also appears in more modern treatments (Nelson, Terp, Haagerup, Hiai, Kosaki, etc.) but I don't have the sources here to check that. 1 The necessary and sufficient condition is that$|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.