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Abstract

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,\alpha)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $\alpha:\mathbb{T}^n\rightarrow\operatorname{Aut}(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of trivial principal $\mathbb{T}^n$-bundles based on such dynamical systems, i.e., we call a dynamical system $(A,\mathbb{T}^n,\alpha)$ a trivial noncommutative principal $\mathbb{T}^n$-bundle if each isotypic component contains an invertible element. Each trivial principal bundle $(P,M,\mathbb{T}^n,q,\sigma)$ gives rise to a smooth trivial noncommutative principal $\mathbb{T}^n$-bundle of the form $(C^{\infty}(P),\mathbb{T}^n,\alpha)$. Conversely, if $P$ is a manifold and $(C^{\infty}(P),\mathbb{T}^n,\alpha)$ a smooth trivial noncommutative principal $\mathbb{T}^n$-bundle, then we recover a trivial principal $\mathbb{T}^n$-bundle. While in classical (commutative) differential geometry there exists up to isomorphy only one trivial principal $\mathbb{T}^n$-bundle over a given manifold $M$, we will see that the situation completely changes in the noncommutative world. Moreover, it turns out that each trivial noncommutative principal $\mathbb{T}^n$-bundle possesses an underlying algebraic structure of a $\mathbb{Z}^n$-graded unital associative algebra, which might be thought of an algebraic counterpart of a trivial principal $\mathbb{T}^n$-bundle. In the second part of this paper we provide a complete classification of this underlying algebraic structure, i.e., we classify all possible trivial noncommutative principal $\mathbb{T}^n$-bundles up to completion.