We study geodesics in noncommutative geometry by means of bimodule connections and completely positive maps using the Kasparov, Stinespring, Gel'fand, Naimark & Segal (KSGNS) construction. This is motivated from classical geometry, and we also consider examples on the algebras M_2(C) and C(Z_n), though restricting to classical real time. On the way we have to consider the reality of a noncommutative vector field, and for this we propose a definition depending on a state on the algebra.

中文翻译：

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非交换测地线和 KSGNS 构造

我们使用 Kasparov、Stinespring、Gel'fand、Naimark & Segal (KSGNS) 构造通过双模连接和完全正映射研究非对易几何中的测地线。这源于经典几何，我们还考虑了代数 M_2(C) 和 C(Z_n) 上的例子，但仅限于经典实时。在此过程中，我们必须考虑非交换向量场的现实，为此我们提出了一个依赖于代数状态的定义。