nLab Banach bundle

Idea

A Banach bundle is a bundle in which every fibre is a Banach space. Certain other conditions apply.

Definitions

A Banach bundle is an open (necessarily surjective) continuous map of Hausdorff topological spaces $p\colon Y\to B$, each of whose fibers carries a structure of a complex Banach space, this structure being continuous in the base point (in other words, the global operations $\mathbb{C} \times Y \to Y$ of multiplication by a scalar, $Y \times_B Y \to Y$ of addition and $Y \to \mathbb{R}$ of taking the norm are continuous) and such that for every net $\{y_\alpha\}_{\alpha\in A}$, if ${\|y_{\alpha}\|} \to 0$ and $p(y_\alpha) \to b$, then $y_\alpha \to 0 = 0_b \in p^{-1}(b)$.

We distinguish a different concept of Banach algebraic bundle, where the base space $B$ is also a Banach algebra and the multiplication is defined as a map $\cdot\colon Y\times Y \to Y$ (not only $Y \times_B Y \to Y$), that is we can multiply the points in different fibers, and $p(a \cdot b) = p(a) \cdot p(b)$.

A Banach bundle is a Hilbert bundle if each fiber is a separable Hilbert space. As usual, the inner product can be obtained by the polarization formula $(x,y) \coloneqq \frac{1}{4}({\|x+y\|^2} - {\|x-y\|^2})$ from the norm of a Banach space if the norm satisfies the parallelogram identity. From this, we infer that for Hilbert bundles, the inner product is continuous as a map $Y \times_B Y \to \mathbb{C}$. Hilbert bundles are important in the study of induced representations of locally compact groups, and Mackey theory? in particular; more recently their study is connected to the study of Hilbert modules.

A morphism of Banach bundles $(p\colon Y \to B)\to (p'\colon Y' \to B)$ over the same base is a morphism of total spaces commuting with the projections, $\mathbb{C}$-linear in each fiber, and preserving the norm. A Banach bundle is sometimes said to be Hilbertizable if it is isomorphic to the underlying Banach bundle of a Hilbert bundle; structurally, there is no difference between a Hilbert bundle and a Hilbertizable Banach bundle (again using the polarisation formula to prove that being a Hilbert space is a property-like structure).

One also considers Banach $*$-algebraic bundles, where an antilinear involution $*$ preserving the norm is involved, is continuous as a global map $Y \to Y$ and is an antihomomorphism of algebras satisfying $p(y^\ast) = p(y^{-1})$.

References

• Wikipedia: Banach bundle

For Banach bundles see ch. 13 in vol. 1 (from page 125; def. 13.4 on p. 127) and for Banach algebraic bundles see from 783 on in vol. 2 of

• J. M. G. Fell, R. S. Doran, Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles, Vol. 1. Basic representation theory of groups and algebras. Pure and Applied Mathematics, 125, Academic Press 1988. xviii+746 pp. MR90c:46001 Vol. 2, Banach $*$-algebraic bundles, induced representations, and the generalized Mackey analysis. Pure and Applied Mathematics 126, Acad. Press 1988. pp. i–viii and 747–1486, MR90c:46002