The noncommutative geometry of frame bundles

We apply ourselves to the noncommutative geometry of frame bundles by showing that each C$^*$-algebraic noncommutative principal $\mathrm{SO}(n)$-bundle is, up to isomorphism, uniquely determined by its associated noncommutative vector bundle with respect to the standard representation of $\mathrm{SO}(n)$. For this, we provide a construction procedure, via unitary tensor functors, that for a certain type of correspondence, let's say $M$, attaches a free C$^*$-dynamical system $(\mathcal{A}_M,\mathrm{SO}(n),\alpha_M)$ with the property that its associated noncommutative vector bundle with respect to the standard representation of $\mathrm{SO}(n)$ is isomorphic to $M$.