I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_\infty (C(X))$ and the Vector bundles on $X$ by $p$ goes to $\xi_p=(E_p,\pi,X)$ where $E_p =\{(x,v) \in X\times \mathbb{C}^n: v\in p(x)(\mathbb{C}^n)\}$ and $\pi$ is the natural projection. I wonder why this is well defined in the equivalence classes of $P_\infty (C(X))$. i.e why $p\sim q$ $\iff$ there exists an isomorphism of vector bundles between $\xi_p$ and $\xi_q$.


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