Let $ V = M_n(C)$ be equipped with the inner-product $(A,B) = tr(B_xA)$, $A,B ∈ V$ and $B_x$ is transpose of $B$. Let $M ∈M_n(C)$. Define $T : V → V$ by $T(A) = MA$. What is $T'(A)$, where $T'$ denotes the adjoint of the mapping $T$ ? we can say by the condition $(T(A), B) = (A, T'(B))$ this implies $tr(B_xMA) = tr((T'(B))_xA)$
from here Can I conclude something?