For any linear operator $A$, the support of $A$ is the orthogonal complement of its kernel. Hence when we say, $supp(A)\subset supp(B)$, we have that for any vector $v$ in the kernel of $B$ i.e. $Bv = 0$, it also must be that $Av = 0$.
My question is if it is true (and if yes, how to show) that support containments for quantum states are preserved under partial trace. That is, if
is it true that