$C^*$-fermi systems and detailed balance

TL;DR

This paper develops a systematic theory of fermionic $C^*$-systems, introducing a fermionic tensor product, and applies these concepts to define detailed balance in graded $C^*$-systems, advancing the mathematical framework of quantum statistical mechanics.

Contribution

It introduces a fermionic $C^*$-tensor product and extends the concept of detailed balance to graded $C^*$-systems, providing new tools for quantum statistical mechanics.

Findings

Constructed a fermionic $C^*$-tensor product for graded $C^*$-algebras.

Solved a positivity problem on the infinite Fermi lattice.

Defined fermionic detailed balance in general $C^*$-systems with $bZ_2$-gradation.

Abstract

A systematic theory of product and diagonal states is developed for tensor products of $\mathbb Z_2$-graded $*$-algebras, as well as $\mathbb Z_2$-graded $C^*$-algebras. As a preliminary step to achieve this goal, we provide the construction of a {\it fermionic $C^*$-tensor product} of $\mathbb Z_2$-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 $\mathbb Z_2$, by viewing such a system as part of a compound system and making use of a diagonal state.