Banach bundle (non commutative geometry)

Definition

Let X be a topological Hausdorff space, a (continuous) Banach bundle over X is a tuple B = ( B , π ) , where B is a topological Hausdorff space, and π : B → X is a continuous, open surjection, such that each fiber B x := π − 1 ( x ) is a Banach space. Which satisfies the following conditions:

1. The map b ↦ ∥ b ∥ is continuous for all b ∈ B
2. The operation + : { ( b 1 , b 2 ) ∈ B × B : π ( b 1 ) = π ( b 2 ) } → B is continuous
3. For every λ ∈ C , the map b ↦ λ ⋅ b is continuous
4. If x ∈ X , and { b i } is a net in B , such that ∥ b i ∥ → 0 and π ( b i ) → x , then b i → 0 x ∈ B . Where 0 x denotes the zero of the fiber B x .

If the map b ↦ ∥ b ∥ is only upper semi-continuous, B is called upper semi-continuous bundle.

Trivial bundle

Let A be a Banach space, X be a topological Hausdorff space. Define B := A × X and π : B → X by π ( a , x ) := x . Then ( B , π ) is a Banach bundle, called the trivial bundle