Let $A$ be a noncommutative (unital and associative) $\ast$-algebra over $\mathbb{C}$ and let $M$ be a left $A$-module. There is a left $A$-module structure on $$M'=\{f:M\longrightarrow A\mid f \mbox{ is a left }A\mbox{-module morphism} \}$$ given by $$(a,f)\longmapsto fa^{\ast}\;...\; (*)$$ where $$fa^{\ast}:M\longrightarrow A,\;\;\;\; m\longmapsto f(m)a^{\ast}$$ Also if $M$ is an $A$-bimodule, we can define another $A$-bimodule structure on $M$ by means of \begin{equation*} (a,m) \longmapsto m\, a^{\ast}, \qquad (m,a) \longmapsto a^{\ast} m \end{equation*}
We will denote by $M^*$ this new $A$-bimodule. My questions are
1) If $M$ is finitely generated and projective left $A$-module, is $M'$ (with (*)) finitely generated and projective left $A$-module too?
2) If $M$ is finitely generated and projective $A$-bimodule, is $M^*$ finitely generated and projective $A$-bimodule as well?.
In the second question, the part "finitely generated" is true but I am not so sure about the part "projective".