The case of equality in H\"older's inequality for matrices and operators

Let $p>1$ and $1\mathrm{/}p+1\mathrm{/}q=1$. Consider H"older’s inequality $$\mathrm{\parallel }a{b}^{\ast }{\mathrm{\parallel }}_1\le \mathrm{\parallel }a{\mathrm{\parallel }}_p\mathrm{\parallel }b{\mathrm{\parallel }}_q$$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^{\ast }$-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 $\mathrm{\mid }a{\mathrm{\mid }}^p=\lambda \mathrm{\mid }b{\mathrm{\mid }}^q$ for some $\lambda \ge 0$.