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 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

Trivial bundle

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

References

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