When trying to derive the probability density from the Pauli equation, I face a problem.
Starting from the Pauli equation
$$ i\hbar \frac{\partial \Psi}{\partial t}=\hat H\_0 \Psi +\mu\_B \ \hat \sigma \cdot \mathbf{B} \Psi, $$
I need to adjoint it:
$$ -i\hbar \frac{\partial \Psi^+ }{\partial t}=\hat H\_0^\* \Psi^+ +\mu\_B \ \left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+. $$
So, I'm trying to calculate the multiplication in brackets, using the properties of conjugation $(AB)^+=B^+A^+$:
$$
\left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+ \equiv 
\bigg( \left( \hat \sigma \cdot \mathbf{B} \right) \Psi \bigg)^+= 
\Psi^+ \left( \hat \sigma \cdot \mathbf{B} \right)^+ =
\Psi^+ \mathbf{B}^+ \hat \sigma^+= \Psi^+ \mathbf{B}^T \hat \sigma.
$$
Here I've used the facts that the magnet field is real $(\mathbf{B}^+=\mathbf{B}^T)$ and pauli matrices are hermitian $(\hat \sigma^+ =\hat \sigma)$.

However, in the book (Greiner, Quantum Mechanics: an introduction) there is another answer:
$$ \left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+ =
\Psi^+ \hat \sigma \cdot \mathbf{B}.
$$
Where is the mistake? Thanks in advance.

P.S. I understand that $\hat \sigma$ is an operator, and so it must act some function, but... I still don't see my mistake.