View source for Banach bundle (non-commutative geometry)

In [[mathematics]], a '''Banach bundle''' is a [[fiber bundle]] over a [[Topological space|topological]] [[Hausdorff space]], such that each fiber has the structure of a [[Banach space]].

== Definition ==
Let <math> X </math> be a topological Hausdorff space, a ('''continuous''') '''Banach bundle''' over <math>X</math> is a tuple <math>\mathfrak{B} = (B, \pi)</math>, where <math>B</math> is a topological Hausdorff space, and <math>\pi\colon B\to X</math> is a [[continuous function|continuous]], [[open function|open]] [[surjective function|surjection]], such that each '''fiber''' <math>B_x := \pi^{-1}(x)</math> is a Banach space. Which satisfies the following conditions:
# The map <math>b\mapsto\|b\|</math> is continuous for all <math>b\in B</math>
# The operation <math>+\colon\{(b_1,b_2)\in B\times B:\pi(b_1)=\pi(b_2)\}\to B</math> is continuous
# For every <math>\lambda\in\mathbb{C}</math>, the map <math>b\mapsto\lambda\cdot b</math> is continuous
# If <math>x\in X</math>, and <math>\{b_i\}</math> is a [[net_(mathematics)|net]] in <math>B</math>, such that <math>\|b_i\|\to 0</math> and <math>\pi(b_i)\to x</math>, then <math>b_i\to 0_x\in B</math>, where <math>0_x</math> denotes the [[zero]] of the fiber <math>B_x</math>.<ref>Fell, M.G., Doran, R.S.: "Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1"</ref>
If the map <math>b\mapsto \|b\|</math> is only [[semi-continuity|upper semi-continuous]], <math>\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> B := A\times X</math> and <math>\pi\colon B\to X</math> by <math>\pi(a,x) := x</math>. Then <math>(B,\pi)</math> is a Banach bundle, called the '''trivial bundle'''

== See also ==
* [[Banach bundle]]s in differential geometry

== References ==
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{{Functional analysis}}

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[[Category:Noncommutative geometry]]
[[Category:Banach spaces]]

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