what is mean by complex conjugation of density matrix?

Good day.

Please help to fix my problem..

I got confuse for these expression:

ρ∼AB=(σy⊗σy)ρ∗(σy⊗σy)\rho^\sim_{AB}=(\sigma_y\otimes\sigma_y)\rho^\ast(\sigma_y\otimes\sigma_y)

and ρ∗\rho^\ast is complex conjugation in the standard basis and σy=(0−ii0)\sigma_y=\begin{pmatrix} 0 & -i
\\i & 0 \\
\end{pmatrix}

but may density matrix is real and why it must multiply with σy\sigma_y.

Please help me.

=================

=================

=================