C∗ -fermi systems and detailed balance

A systematic theory of product and diagonal states is developed for tensor products of Z2-graded ∗ -algebras, as well as Z2-graded C∗-algebras. As a preliminary step to achieve this goal, we provide the construction of a fermionicC∗-tensor product of Z2-graded C∗-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general C∗-systems with gradation of type Z2, by viewing such a system as part of a compound system and making use of a diagonal state.