Multiplicativity of Connes' calculus

Abstract

We consider the quadruples $(\mathcal{A},\mathbb{V},D,\gamma )$ where $\mathcal{A}$ is a unital, associative $\mathbb{K}$-algebra represented on the $\mathbb{K}$-vector space $\mathbb{V}$, $D\in \mathcal{E}nd(\mathbb{V})$, $\gamma \in \mathcal{E}nd(\mathbb{V})$ is a ${\mathbb{Z}}_2$-grading operator which commutes with $\mathcal{A}$ and anticommutes with $D$. We prove that the collection of such quadruples, denoted by $\widetilde{\mathcal{S}pec}$, is a monoidal category. We consider the monoidal subcategory $\widetilde{\mathcal{S}pe{c}_{sub}}$ of objects of $\widetilde{\mathcal{S}pec}$ for which $\gamma \in \pi (\mathcal{A})$. We show that there is a covariant functor $\mathcal{G}:\widetilde{\mathcal{S}pec}⟶\widetilde{\mathcal{S}pe{c}_{sub}}$. Let ${\Omega }_D^{\bullet }$ be the differential graded algebra defined by Connes ([Con2]) and $DGA$ denotes the category of differential graded algebras over the field $\mathbb{K}$. We show that $\mathcal{F}:\widetilde{\mathcal{S}pe{c}_{sub}}⟶DGA$, given by $(\mathcal{A},\mathbb{V},D,\gamma )⟼{\Omega }_D^{\bullet }(\mathcal{A})$, is a monoidal functor. To show that $\mathcal{F}\circ \mathcal{G}$ is not trivial we explicitly compute it for the cases of compact manifold and the noncommutative torus along with the associated cohomologies.