Banach bundle (non-commutative geometry)

In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space.

Definition

Let $X$ be a topological Hausdorff space, a (continuous) Banach bundle over $X$ is a tuple ${\mathfrak {B}}=(B,\pi )$, where $B$ is a topological Hausdorff space, and $\pi \colon B\to X$ is a continuous, open surjection, such that each fiber $B_{x}:=\pi ^{-1}(x)$ is a Banach space. Which satisfies the following conditions:

1. The map $b\mapsto \|b\|$ is continuous for all $b\in B$
2. The operation $+\colon \{(b_{1},b_{2})\in B\times B:\pi (b_{1})=\pi (b_{2})\}\to B$ is continuous
3. For every $\lambda \in \mathbb {C}$, the map $b\mapsto \lambda \cdot b$ is continuous
4. If $x\in X$, and $\{b_{i}\}$ is a net in $B$, such that $\|b_{i}\|\to 0$ and $\pi (b_{i})\to x$, then $b_{i}\to 0_{x}\in B$. Where $0_{x}$ denotes the zero of the fiber $B_{x}$.

If the map $b\mapsto \|b\|$ is only upper semi-continuous, ${\mathfrak {B}}$ is called upper semi-continuous bundle.

Examples

Trivial bundle

Let A be a Banach space, X be a topological Hausdorff space. Define $B:=A\times X$ and $\pi \colon B\to X$ by $\pi (a,x):=x$. Then $(B,\pi )$ is a Banach bundle, called the trivial bundle

See also

• Banach bundles in differential geometry