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