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

Abstract

We systematically investigate $$C^*$$ C ∗ -norms on the algebraic graded product of $${{\mathbb {Z}}}_2$$ Z 2 -graded $$C^*$$ C ∗ -algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $$C^*$$ C ∗ -norms. To this end, we first show that commutative $${{\mathbb {Z}}}_2$$ Z 2 -graded $$C^*$$ C ∗ -algebras enjoy a nuclearity property in the category of graded $$C^*$$ C ∗ -algebras. In addition, we provide a characterization of the extreme even states of a given graded $$C^*$$ C ∗ -algebra in terms of their restriction to its even part.

Symmetric states for $C^*$-Fermi systems II: Klein transformation and their structure

“…In that paper, it was also firstly defined a norm on the C * -Fermi tensor product by using an "universal" representation associated to all even product states, by suggesting that such a norm would have been the analogous of the minimal C * -cross norm of the usual tensor product. Later on, in [8] it was shown that such a norm is indeed minimal among all C * -cross norms on the (algebraic) Fermi product of two Z 2 -graded C * -algebras.…”

“…Although the detailed knowledge of such a GNS construction plays no role in the analysis in [12], it is relevant here to prove that the Klein transformation we are building in the foregoing section is realising a * -isomorphism between the Fermi tensor product and the usual tensor product. Such a detailed analysis is carried out in [8], Section 3, which we report here for the convenience of the reader. 3 Indeed, for Z 2 -graded C * -algebras (A, α) and (B, β) and even states ω and ϕ on A and B as above, let (H ω , π ω , V ω , ξ ω ) and (H ϕ , π ϕ , V ϕ , ξ ϕ ) be the corresponding covariant (w.r.t.…”

“…4 With H ⊗ K, we denotes the usual Hilbertian tensor product. This differs from the notation H F K in [8]. Since the, perhaps crucial, (2.1) is not frequently used in the forthcoming section, we prefer to keep the original notation not to weigh down too much the notations.…”

“…That σ is not a homomorphism is natural, and emerges already in the simplest case of the CAR algebra, see e.g [23],. Exercise XIV.1.6 Since π ϕ (u) and V ψ are both even, their Fermi tensor product π ϕ (u) F V ψ , defined according the rule in[8] and reported in Section 1, coincides with the usual tensor product π ϕ (u) X V ψ ≡ π ϕ (u) ⊗ V ψ .…”

Quantum diffusion on almost commutative spectral triples and spinor bundles

“…If the tensor product was ungraded, and either A i is a nuclear C * -algebra, that is, all crossnorms on the tensor product agree, then there's no ambiguity on the norm with respect to which to complete. But in Z 2 graded tensor product category, only the commutative C * -algebra are characterized as nuclear [22]. While noncommutative geometry does not require that A i be closed, for example, the requirement [D i , a] is bounded is only needed for a ∈ A i ; however, the questions about quantum dynamical semigroups often presume norm closure.…”