Consider $B$ a C*-algebra and $A$ a C*-subalgebra such that $1_A=1_B$. If $E:B\rightarrow A$ is a faithful conditional expectation (that is a projection of norm 1, by Tomiyama's theorem) then is it true that $E(x^*)=E(x)^*$ for any $x\in B$?
This question popped up because I was reading a book by Yasuo Watatani (Index for C*-subalgebras) and found that in page 39 when trying to prove that the bounded operator $e_A:\mathcal{E}\rightarrow \mathcal{E}$, defined by $e_A(\eta(x))=\eta(E(x))$, is adjointable he used that for any $x,y\in B$ we have \begin{align}\tag{1} \left\langle \eta(E(x)),\eta(y) \right\rangle &= E(E(x^*)E(y))=E(x^*)E(y) \end{align} In this case $\mathcal{E}$ is the Hilbert $A$-module obtained by completing $B$ with respect to the norm induced by the $A$-inner product $\langle x,y\rangle= E(x^*y)$ and $\eta:B\rightarrow \mathcal{E}$ is just the inclusion function. Taking this into account, by taking $y\in A$ in $(1)$ we have that $$\left\langle \eta(E(x)),\eta(y) \right\rangle=E(E(x)^*y)=E(x)^*E(y)=E(x)^*y$$ but also $$\left\langle \eta(E(x)),\eta(y) \right\rangle=E(x^*)E(y)=E(x^*)y$$ and this would mean that $E(x^*)=E(x)^*$.
Conditional expectations are in particular positive maps. From that you can check that they preserve adjoints.
Indeed, let $A,B$ be $C^*$-algebras and $\phi \colon A \to B$ a positive linear map. We show that $\phi$ preserves adjoints.
First, assume $c \in A$ is self-adjoint. Then $\phi(c)^* = \phi(c_+ -c_-)^* = \phi(c_+)^*-\phi(c_-)^* = \phi(c_+)-\phi(c_-) = \phi(c)$, where $c_+ = \max(c,0)$ and $c_- = -\min(c,0)$ are positive. Now, let $x = a + ib \in A$, with $a,b$ self-adjoint. Then, $$ \phi(x^*) = \phi(a-ib) = \phi(a)-i\phi(b) = (\phi(a)+i\phi(b))^* = \phi(x)^*. $$