Consider the following subbundle of $\mathbb{C}P^1\times \mathbb{C}^2$, $$V=\{(x,v)\in \mathbb{C}P^1\times \mathbb{C}^2:\; v\in x\}$$ where we consider $x\in \mathbb{C}P^1$ as a complex "line" passing through the origin. This is called the Bott bundle and can also be seen as a bundle over $S^2$ via the identification of $\mathbb{C}P^1$ as the Riemann sphere. What I'm actually trying to do is to construct a finite Parseval frame of $\Gamma(V)$ (looked as a Hilbert module over $C(\mathbb{C}P^1)$) but in order to even consider this I need to know how to construct a single non-trivial (meaning non zero) section at the very least.

I tried considering $\mathbb{C}P^1=U_1\cup U_2$, where $U_i$ are the canonical open subsets in $\mathbb{C}P^1$, and then trying to extend a couple of functions from $\mathbb{C}\simeq U_1\rightarrow V|_{U_1}$ to functions $\mathbb{C}P^1\rightarrow V$ but these didn't actually formed a Parseval frame (was aiming for a frame with 2 elements). Do sections for this bundle have any particular form I could use to search my desired Parseval frame?

The couple of sections I considered but didn't result in a frame are the following:

Let $U_1=\{[x:1]\in \mathbb{C}P^1:\; x\in \mathbb{C}\}$ and $U_2=\{[1:y]\in \mathbb{C}P^1:\; y\in \mathbb{C}\}$. Now consider \begin{align*} \tilde{\xi_1}: U_1&\rightarrow V|_{U_1}\\ [x:1]&\mapsto \left([x:1],\frac{(x,1)}{\|(x,1)\|_\text{max}^2}\right) \end{align*} We can extend this to a section in $\mathbb{C}P^1$ by considering $\xi_1:\mathbb{C}P^1\rightarrow V$ as $\xi_1(x)=\tilde{\xi_1}(x)$ if $x\in U_1$ and $0$ outisde $U_1$. Similarly we can define $\xi_2$ for $U_2$. The fact that we have a norm squared I feel is required to extend $\tilde{\xi_i}$ but is also in the way for $\{\xi_i\}$ to be a Parseval frame.

Note: A Parseval frame for a Hilbert module $M$ over a C*-algebra $A$ is a family $\{\xi_\lambda\}_{\lambda\in\Lambda}$ such that for any $\xi\in M$ we have $$\xi=\sum_{\lambda\in\Lambda}\xi_\lambda\langle\xi_\lambda ,\xi\rangle$$ If $\Lambda$ is finite we say that the Parseval frame is finite.


I finally found a frame for it, consider \begin{align*} \xi_i:\mathbb{C}P^1 \rightarrow V\\ [x_1:x_2]&\mapsto \left([x_1:x_2],\frac{\overline{x_i}}{|x_1|^2+|x_2|^2}(x_1,x_2) \right) \end{align*} It's not hard to verify that $\{\xi_1,\xi_2\}$ is a Parseval frame for $\Gamma(V)$ if we consider the inner product inherited from $\Gamma(\mathbb{C}P^1\times \mathbb{C}^2)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.