Banach bundle (non-commutative geometry) explained

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 ak{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

1. The operation

+ \colon\{(b _{1,b} _{2)\in} B x B:\pi(b _{1)=\pi(b} _{2)\}\to} B is continuous

1. For every

λ \in C , the map b\mapsto λ ⋅ b is continuous

1. 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, ak{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 x 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

Notes and References

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