If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two idempotents $p$ and $q$ in $M_{\infty}(C^{\infty}(X))$ (the square matrices of arbitrary size over $C^{\infty}(X)$) are equivalent if they are similar, that is, if there exists an invertible $u$ in $M_{\infty}(C^{\infty}(X))$ such that $p=uqu^{-1}$.

My question is very naive: what if we change this equivalence relation by requiring that in addition to $p=uqu^{-1}$ we have $dp=u\cdot dq\cdot u^{-1}$ ? (Here $d$ is the exterior differential.) Is such a definition known? How could this new equivalence relation be described in terms of vector bundles (possibly endowed with some additional structure)? Could it be that what we get is somehow related to differential K-theory?