Banach bundle (non-commutative geometry)
Appearance
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:
- The map is continuous for all
- The operation is continuous
- For every , the map is continuous
- 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
- Banach bundles in differential geometry
References
- ^ Fell, M.G., Doran, R.S.: "Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1"