Classical adjoint commuting and determinant preserving linear maps on Kronecker products of Hermitian matrices

Let psi :circle times(d)(i=1) H-ni -> circle times(d)(i=1) H-ni be a linear map on the Kronecker product of spaces of Hermitian matrices H-ni of size n(i) >= 3. (If d= 1, we identify circle times(d)(i=1) H-ni with H-ni.) We establish a condition under which psi (adj (circle times(d )(i=1)A(i))) = adj (psi(circle times(d )(i=1)A(i))) if and only if det (psi(circle times(d )(i=1)A(i))) = det (circle times(d )(i=1)A(i)) for all circle times(d )(i=1)A(i) is an element of circle times(d)(i=1) H-ni. Then for d is an element of {1,2}, we apply this fact to characterize maps psi : circle times(d)(i=1) H-ni -> circle times(d)(i=1) H-ni such that psi (adj (circle times(d )(i=1)A(i))) = adj (psi(circle times(d )(i=1)A(i))) with some mild conditions.