Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?