$$C^*$$-fermi systems and detailed balance

On $$C^*$$-norms on $${{\mathbb {Z}}}_2$$-graded tensor products

“…It turns out that a ℤ 2 -graded * -algebra is a pair ( , ) , where is a * -algebra, and an involutive * -automorphism on . Following [3], we say that is a ℤ 2 -grading of . Moreover, we denote the * -subalgebra + ∶= 1 the even part, and the subspace − ∶= −1 the odd part of , respectively.…”

“…Another example of paramount importance for its applications to Physics is of course the CAR algebra, see e.g. [2,3].…”

“…) and ( 2 , 2 ) are ℤ 2 -graded * -algebras, and consider the linear space 1 ⊙ 2 . In what follows, we recall the definition of the involutive ℤ 2 -graded tensor product, which will be henceforth denoted by 1 F 2 , as in [3]. For homogeneous elements a 1 ∈ 1 , a 2 ∈ 2 and i, j ∈ ℤ 2 , we set Given x, y ∈ 1 ⊙ 2 with the involution, which by a minor abuse of notation we continue to denote by * , and the multiplication on 1 F 2 are defined as (see also e.g.…”

“…[8]. In particular, graded tensor products of ℤ 2 -graded C * -algebras have recently been given a good deal of attention in [3], where a definition of quantum detailed balance for product systems endowed with a ℤ 2 -grading has been proposed. Among other things, the aforementioned paper provides an in-depth analysis of the maximal norm on the graded tensor products of ℤ 2 -graded C * -algebras, as well as showing that a product state on the algebraic Fermi product of two ℤ 2 -graded C * -algebras is well defined whenever just one of the two states is even, namely it is invariant with respect to the grading.…”

“…In Sect. 2 we set the notation and recall the basic notions on graded C * -algebras and their graded tensor products, which we also refer to as Fermi tensor products, as is done in [3]. We then focus on even states on a given graded C * -algebra.…”