nLab
Banach bundle

Banach bundles

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:YBp\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 ×YY\mathbb{C} \times Y \to Y of multiplication by a scalar, Y× BYYY \times_B Y \to Y of addition and YY \to \mathbb{R} of taking the norm are continuous) and such that for every net {y α} αA\{y_\alpha\}_{\alpha\in A}, if y α0{\|y_{\alpha}\|} \to 0 and p(y α)bp(y_\alpha) \to b, then y α0=0 bp 1(b)y_\alpha \to 0 = 0_b \in p^{-1}(b).

We distinguish a different concept of Banach algebraic bundle, where the base space BB is also a Banach algebra and the multiplication is defined as a map :Y×YY\cdot\colon Y\times Y \to Y (not only Y× BYYY \times_B Y \to Y), that is we can multiply the points in different fibers, and p(ab)=p(a)p(b)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)14(x+y 2xy 2)(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× BYY \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:YB)(p:YB)(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 YYY \to Y and is an antihomomorphism of algebras satisfying p(y *)=p(y 1)p(y^\ast) = p(y^{-1}).

References

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

Revised on December 9, 2011 16:51:13 by Toby Bartels (75.88.105.185)