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I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed.

Other authors, like Dautray and Lions, or Birman and Solomjak define measurable Hilbert bundles and do not seem to insist on isomorphic fibers.

My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.

Some other questions:

  1. Is the "constant rank" condition necessary in order to put a differentiable (or continuous) structure on the bundle?

  2. In contrast, is it true that the "constant rank" condition is not imposed on measurable Hilbert bundles because measurability is a weaker condition that allows for variation of fibers?

  3. Are "constant rank" fibrations more "natural" in some way in mathematics?

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    $\begingroup$ Only slightly related but: in algebraic/holomorphic geometry you have coherent sheaves that are pretty much like vector bundles in which the dimension of fibers can jump (in an upper semi-continuous way, in this case), and they are as "natural" objects as bundles are. $\endgroup$
    – Qfwfq
    Apr 26 '16 at 22:18
  • $\begingroup$ @Qfwfq On the other hand coherent sheaves are to vector bundles as finitely presented modules are to finitely generated projective modules (in a sense, exactly so). From this point of view one might argue that correct analogs of vector bundles should somehow carry some properties related to projectivity. $\endgroup$ Apr 26 '16 at 22:26
  • $\begingroup$ @Qfwfq Are there any nice references exploring this aspect, viz. upper-semicontinuity of dimension? Is there a parallel notion of sections in this context? $\endgroup$ Apr 26 '16 at 22:27
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    $\begingroup$ For example, there is a well known characterization of finitely generated projectives: a $k$-module $P$ is such iff the canonical map $P^*\otimes_kM\to\operatorname{Hom}_k(P,M)$ is an isomorphism for all modules $M$. I wonder if one gets a sensible notion of a Hilbert bundle by using appropriate duals and tensor products with this in the infinite-dimensional situation. $\endgroup$ Apr 26 '16 at 22:30
  • $\begingroup$ @TanujD., Yes, coherent sheaves (and their possibly "infinite dimensional" cousins, quasicoherent sheaves) have sections, which form vector spaces. But I don't think looking directly into the algebraic geometry literature would be the most sensible path... Maybe there is something analogous to QC sheaves in the noncommutative geometry literature? $\endgroup$
    – Qfwfq
    Apr 26 '16 at 23:36
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"Why not simply use Bochner spaces like $C(X;L^2(\Omega))$?" --- do you mean that this would be the space of continuous sections of the bundle with fiber $L^2(\Omega)$? Yes, that is correct if the bundle is trivial, i.e., a bundle of the form $X \times H$ where $H$ is the fiber Hilbert space. But of course not all bundles have this form. A natural class of examples where the bundle is usually not trivial arises when $X$ is a Riemannian manifold and the fiber at a point is the tangent space there. Each tangent space carries an inner product, so this is a Hilbert bundle.

"Is the constant rank condition necessary in order to put a continuous structure on the bundle?" --- yes. It's hard to see how to reasonably topologize a bundle whose fibers could vary in dimension. One does sometimes consider "dimension drop" conditions where you only consider sections which, say, vanish at some point, or lie in a proper subspace of the fiber at some point.

"Is it true that the constant rank condition is not imposed on measurable Hilbert bundles because measurability is a weaker condition that allows for variation of fibers?" --- again, I basically agree. This setting is very different from the topological setting; measurable sections will be totally insensitive to any global topological features of a bundle. So one can just take a measurable Hilbert bundle over a measure space $X$ to be something of the form $\bigcup (X_n\times H_n)$, with $n$ ranging over $\mathbb{N} \cup \{\infty\}$, $(X_n)$ a measurable partition of $X$, and $H_n$ an $n$-dimensional Hilbert space. (The nonseparable setting introduces some bad pathology, so I prefer to stick to the separable case.)

While I'm on the subject, measurable Hilbert bundles provide a setting for spectral theory that nicely accomodates multiplicity:

$\bullet$ If $A$ is a (bounded or unbounded) self-adjoint operator on a Hilbert space $H$, then there is a measurable Hilbert bundle over the spectrum of $A$, and an isometric isomorphism between $H$ and the $L^2$ sections of this bundle which turns $A$ into multiplication by $x$.

$\bullet$ If $\mathcal{A} \subset B(H)$ is a separable abelian C*-algebra then there is a metrizable locally compact Hausdorff space $X$, a Borel measurable Hilbert bundle over $X$, and an isometric isomorphism between $H$ and the $L^2$ sections of the bundle which turns $\mathcal{A}$ into the set of multiplication operators by functions in $C_0(X)$.

$\bullet$ If $\mathcal{M} \subseteq B(H)$ is an abelian von Neumann algebra, then there is a metrizable locally compact Hausdorff space $X$, a Borel measurable Hilbert bundle over $X$, and an isometric isomorphism between $H$ and the $L^2$ sections of the bundle which turns $\mathcal{M}$ into the set of multiplication operators by functions in $L^\infty(X)$.

I would say that these statements cleanly exhibit the way the abstract C${}^*$- and von Neumann algebras are situated within $B(H)$.

(Details can be found in my book Measure Theory and Functional Analysis.)

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  • $\begingroup$ Thank you for the reference. I was wondering if there was a bridge between the two notions. Clearly, topological Hilbert Bundles would necessarily be measurable. But, something to transfer notions like Chern classes, etc. in the reverse direction. I have read in one place, how topological bundles could be generalized in the setting of distributions and currents. Could measurable bundles be also accomodated there? $\endgroup$ Apr 27 '16 at 4:01
  • $\begingroup$ I've never heard of this and I'm not completely sure what you have in mind, but it could be interesting to look into. $\endgroup$
    – Nik Weaver
    Apr 27 '16 at 4:16

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