The noncommutative geometry of frame bundles

Abstract

We apply ourselves to the noncommutative geometry of frame bundles by showing that each C${}^{\ast }$-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${}^{\ast }$-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 $M$.

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