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
.[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"