Let $X$ be a compact Hausdorff topological space and consider the Hilbert space $\ell^2(\mathbb N)$. As shown here, any $T\in C(X,\ell^2(\mathbb N))$ induces a $C(X)$-Fredholm operator $$ \begin{array}{rccc} \widehat{T}: &\mathscr H_{C(X)}& \rightarrow &\mathscr H_{C(X)} \\ & \xi & \mapsto & \underbrace{\big(x \mapsto T_x\xi(x)\big)}_{T_{(\,\cdot\,)}\xi{(\,\cdot\,)}} \end{array} $$ in the standard Hilbert $C(X)$-module $\mathscr H_{C(X)}$. Following Exel's approach, I'm trying to see a connection between the Atiyah-Jänich theorem and its ``non-commutative'' counterpart, Corollary 3.17. One can think $F(A)$ to the be set of classes of $A$-Fredholm operators, where each class has all operators with the same index.
Can this construction $\widehat{(\,\cdot\,)}: C(X,\ell^2(\mathbb N)) \longrightarrow \mathscr F(\mathscr H_{C(X)})$ induce a map such that the following commutes?
$$ \begin{array}{ccc} {[X,\mathcal F(\ell^2(\mathbb N))]} & \longrightarrow & F(C(X)) \\ \text{ind}\downarrow \,\,\,\,\,\,\,\,\,\,\,\,& & \,\,\,\,\,\,\,\,\,\,\,\,\downarrow \text{ind}\\ K^0(X) & \longrightarrow & K_0(C(X)) \end{array} $$ I'm asking this because it doesn't seems too trivial to just calculate the index with bare hands. I was trying to pair up the index family approach $[\ker P_n\circ T]_0$ with $rank(\ker \widehat T)$, where $\ker P_n\circ T = \bigcup_{x\in X} \ker P_n\circ T_x$ and $\ker \widehat T = \bigcup_{x\in X} \ker T_x ev_x$.
