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

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