$C^{\ast }$-fermi systems and detailed balance

Abstract

A systematic theory of product and diagonal states is developed for tensor products of ${\mathbb{Z}}_2$-graded $\ast$-algebras, as well as ${\mathbb{Z}}_2$-graded $C^{\ast }$-algebras. As a preliminary step to achieve this goal, we provide the construction of a {\it fermionic $C^{\ast }$-tensor product} of ${\mathbb{Z}}_2$-graded $C^{\ast }$-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 $C^{\ast }$-systems with gradation of type ${\mathbb{Z}}_2$, by viewing such a system as part of a compound system and making use of a diagonal state.

44 pages