I was trying to study the hermitian part of a C*-algebra defined as the tensor product of two C*-algebras.
Let $\mathcal{A}$ and $\mathcal{B}$ be C*-algebras. Consider $\mathcal{A} \otimes \mathcal{B}$ and the completion $\mathcal{A} \otimes_{\max} \mathcal{B}$ with the projective C*-cross-norm. Is it possible to describe the hermitian elements of $\mathcal{A} \otimes_{\max} \mathcal{B}$ as the tensor product of the hermitian elements of $\mathcal{A}$ and the ones in $\mathcal{B}$? Any references or bibliography about this topic?
Thank you.