Banach bundle (non-commutative geometry)

Last updated November 03, 2020

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:

The map $b\mapsto \|b\|$ is continuous for all $b\in B$
The operation $+\colon \{(b_{1},b_{2})\in B\times B:\pi (b_{1})=\pi (b_{2})\}\to B$ is continuous
For every $\lambda \in \mathbb {C}$ , the map $b\mapsto \lambda \cdot b$ is continuous
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 $B_{x}$ .^[1]

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

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References

↑ Fell, M.G., Doran, R.S.: "Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1"

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[1] Fell, M.G., Doran, R.S.: "Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1"

v t e Functional analysis (topics – glossary)
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