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.



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> 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 $\alpha(e)=f$?
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**Note:** The above question actualy defines an equivalent relation on the space of all $k$ dimensional subbundles of the $n$ dimensional trivial bundle.



A kind of isomorphicity of vector bundles
The antipodal map of $S^{2}$ sends $L$ to $L^{-1}$, where $L$ is the line bundle constructed via clutching along an equatorial $S^1$ with the "identity" map $S^1\rightarrow U(1)$.


So let $\alpha$ be the automorphism of $M\_{n}(C(S^2))$ induced by the antipodal map, let $e$ be the projection corresponding to $L$, and let $f=\alpha(e)$.


