# Talk:Banach bundle

WikiProject Mathematics (Rated Start-class, Low-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Low Importance
Field:  Geometry

The definition made here seems rather uncommon.

I think the most commonly used definition (at least in non-commutative geometry) is having two Hausdorff topological Spaces X, B and a continuous open, surjectionn ${\displaystyle \pi \colon B\to X}$, such that each fiber ${\displaystyle B_{x}:=\pi ^{-1}(x)}$ is a Banach space for each ${\displaystyle x\in X}$ fulfilling ${\displaystyle b\mapsto \|b\|}$ is (upper semi-) continuous from B to ${\displaystyle \mathbb {R} }$, + is continuous on ${\displaystyle B\times _{\pi }B}$ to B, ${\displaystyle b\mapsto \lambda b}$ is continuous on B to B for all ${\displaystyle \lambda \in \mathbb {C} }$, and any net ${\displaystyle \{b_{i}\}}$ in B fulfilling ${\displaystyle \|b_{i}\|\to 0}$ and ${\displaystyle \pi (b_{i})\to x}$ converges to ${\displaystyle 0_{x}\in B}$. The most famous reference for this be "Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles." by M.G. Fell and R.S. Doran.

An equivalent definition requires a family of Banach spaces over X, together with a complex linear space ${\displaystyle \Gamma }$. This might be found in some work of Etienne Blanchard, Dixmier or Nielsen, I'd have to check this.

In my opinion, both definitions should be found in this article in the first place. But I'm not sure how, respectively where to include them. Probably splitting into "Banach bundle (Differential Geometry)" and "Banach Bundle (Noncommutative Geometry) might be an option -- Roman3 (talk) 20:46, 3 June 2010 (UTC)

This seems to be a sensible suggestion, except that the article as it presently stands is entirely about the notion in differential geometry rather than the one in non-commutative geometry. I would move this article to Banach bundle (differential geometry) and start a new article Banach bundle (noncommutative geometry), then maybe leave a disambiguation page here to avoid potential confusion in the future. Sławomir Biały (talk) 20:51, 3 June 2010 (UTC)