When there is exists some non-trivial morphisms between A-bimodules M and M^*?

Question

Let $M$ is finite-dimensional vector space over $\mathbb{C}$ and $A$ is finite-dimensional $\mathbb{C}$-algebra with unit, also $M$ is $A$-bimodule. Can it be such situation when $Hom_{A-A-bimod}(M,M^*) = 0$? There is no canonical morphism from $M$ to $M^*$ but maybe it is always exists non-canonical morphism?