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 $\pi :B\to X$ 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 $b\mapsto \Vert b\Vert$ is continuous for all $b\in B$
2. The operation $+:\{(b_1,b_2)\in B\times B:\pi (b_1)=\pi (b_2)\}\to B$ 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 $\Vert b_i\Vert \to 0$ 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 $b\mapsto \Vert b\Vert$ 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 $\pi :B\to X$ by $\pi (a,x):=x$. Then $(B,\pi )$ is a Banach bundle, called the trivial bundle

See also

• Banach bundles in differential geometry