Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some projection $p$ in a matrix algebra $M\_n(C(M))$. I am wondering if it possible to approximate this projection by a smooth idempotent, call it $e$, and whether it is possible to preserve certain properties of $p$.


More precisely:


**Question 1:** Take any $\varepsilon>0$. Does there exist an idempotent $e\in M\_n(C^\infty(M))$ such that $\|p(x)-e(x)\|<\varepsilon$ for all $x\in M$?


**Question 2:** Suppose $p$ is Lipschitz in the sense that $\|p(x)-p(y)\|\leq d(x,y)$ for all $x,y\in M$, where $\|\cdot\|$ is the operator norm on $M\_n(\mathbb{C})$ and $d$ is the distance determined by a Riemannian metric on $M$. Take any $\varepsilon>0$. Does there exist an idempotent $e\in M\_n(C^\infty(M))$ that is $(1+\varepsilon)$-Lipschitz and such that $\|p(x)-e(x)\|<\varepsilon$ for all $x\in M$?


I'd be grateful for any references that discuss this.



Approximation of continuous projections on a manifold by smooth idempotents
Well, I learned this was Rieffel's question and was already solved by Hanfeng Li. <https://mathscinet.ams.org/mathscinet-getitem?mr=2994680>


**Obsolete:**
The following is not an answer but maybe useful (see the comments above).


**Lemma.** Let $a,b$ be self-adjoint operators and $0<\epsilon<0.01$. Assume that $\|b-a\|<\epsilon$ and that the spectra of $a$ and $b$ are contained in the $\epsilon$ neighborhood of $\{0,1\}$. Then the spectral projections satisfy $$\|1\_{(1-\epsilon,1+\epsilon)}(b)-1\_{(1-\epsilon,1+\epsilon)}(a)\|\le(1+999\epsilon)\|b-a\|.$$


**Proof.** Put $b(t):=a+t(b-a)$ and $p(t):=1\_{(1-2\epsilon,1+2\epsilon)}(b(t))$.
By Cauchy's integral formula, one has
$$p(t)=\frac{1}{2\pi i}\oint\_\Gamma \frac{dz}{z-b(t)},$$
where $\Gamma:=\{|z-1|=\frac{1}{2}\}$. Take the derivative
$$p'(t)= \frac{1}{2\pi i}\oint\_\Gamma (z-b(t))^{-1}b'(t)(z-b(t))^{-1}\,dz.$$
For $p:=p(0)$, one has $\|(z-b(t))^{-1}-(z-p)^{-1}\|<99\epsilon$ for $z\in\Gamma$,
because $\|(z-b(t))^{-1}\|<3$ and $\|b(t)-p\|\le\|b(t)-a\|+\|a-p\|<2\epsilon$.
Hence
\begin{align\*}
p'(t) &\approx\_{999\epsilon\|b-a\|}\frac{1}{2\pi i}\oint\_\Gamma (z-p)^{-1}(b-a)(z-p)^{-1}\,dz\\
 &=p(b-a)p^\perp + p^\perp(b-a)p.
\end{align\*}
Note that $\|p(b-a)p^\perp + p^\perp(b-a)p\|=\|p(b-a)p^\perp\|\le\|b-a\|$.
It follows that
$$\|p(1)-p(0)\|\le(1+999\epsilon)\|b-a\|.$$


