If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation*} \|ABA\|_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation*}

But can we say there is a constant $C_n > 0$ depending only on $n$ where $\|ABA\|_\text{tr} \geq C_n \| A^2 B\|_\text{tr}$?

Note that it's easy to get the reverse:

$\|ABA\|_\text{tr} \leq \|A^2B\| _\text{tr} $

so it's the above inequality I really need.

More generally though, I'm guessing $\|AB\|_\text{tr}$

and $\|BA\|_\text{tr}$ are not equivalent (modulo a constant depending only on $n$)

trace normseems to be wrong; afaik, the trace-norm is just the sum of the singular values the very first equality in your question seems to be incorrect. $\endgroup$4more comments