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

1. The map [math]\displaystyle{ b\mapsto\|b\| }[/math] is continuous for all [math]\displaystyle{ b\in B }[/math]
2. The operation [math]\displaystyle{ +\colon\{(b_1,b_2)\in B\times B:\pi(b_1)=\pi(b_2)\}\to B }[/math] is continuous
3. For every [math]\displaystyle{ \lambda\in\mathbb{C} }[/math], the map [math]\displaystyle{ b\mapsto\lambda\cdot b }[/math] is continuous
4. If [math]\displaystyle{ x\in X }[/math], and [math]\displaystyle{ \{b_i\} }[/math] is a net in [math]\displaystyle{ B }[/math], such that [math]\displaystyle{ \|b_i\|\to 0 }[/math] and [math]\displaystyle{ \pi(b_i)\to x }[/math], then [math]\displaystyle{ b_i\to 0_x\in B }[/math], where [math]\displaystyle{ 0_x }[/math] denotes the zero of the fiber [math]\displaystyle{ B_x }[/math].

If the map [math]\displaystyle{ b\mapsto \|b\| }[/math] is only upper semi-continuous, [math]\displaystyle{ \mathfrak{B} }[/math] is called upper semi-continuous bundle.

Examples

Trivial bundle

Let A be a Banach space, X be a topological Hausdorff space. Define [math]\displaystyle{ B := A\times X }[/math] and [math]\displaystyle{ \pi\colon B\to X }[/math] by [math]\displaystyle{ \pi(a,x) := x }[/math]. Then [math]\displaystyle{ (B,\pi) }[/math] is a Banach bundle, called the trivial bundle

See also

• Banach bundles in differential geometry