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

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Let   be a topological Hausdorff space, a (continuous) Banach bundle over   is a tuple  , where   is a topological Hausdorff space, and   is a continuous, open surjection, such that each fiber   is a Banach space. Which satisfies the following conditions:

  1. The map   is continuous for all  
  2. The operation   is continuous
  3. For every  , the map   is continuous
  4. If  , and   is a net in  , such that   and  , then  , where   denotes the zero of the fiber  .[1]

If the map   is only upper semi-continuous,   is called upper semi-continuous bundle.

Examples

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Trivial bundle

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Let A be a Banach space, X be a topological Hausdorff space. Define   and   by  . Then   is a Banach bundle, called the trivial bundle

See also

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References

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