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 $p:Y\to B$, 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 $ℂ\times Y\to Y$ of multiplication by a scalar, $Y{\times }_{B}Y\to Y$ of addition and $Y\to ℝ$ of taking the norm are continuous) and such that for every net $\{y_{\alpha }{\}}_{\alpha \in A}$, if $\parallel y_{\alpha }\parallel \to 0$ 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 $\cdot :Y\times Y\to Y$ (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 $(x,y)≔\frac{1}{4}(\parallel x+y{\parallel }^2-\parallel x-y{\parallel }^2)$ 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 $Y{\times }_{B}Y\to ℂ$. 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 $(p:Y\to B)\to (p\prime :Y\prime \to B)$ over the same base is a morphism of total spaces commuting with the projections, $ℂ$-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 $p(y^{*})=p(y^{-1})$.

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