The case of equality in Hölder's inequality for matrices and operators

Abstract: Let $p>1$ and $1/p+1/q=1$. Consider H\"older's inequality $$ |ab^*|_1\le |a|_p|b|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case $p=2$) of the fact that equality holds iff $|a|^p=\lambda |b|^q$ for some $\lambda\ge 0$.