Nonvanishing section of infinite-dimensional tautological bundle

Question

Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have infinite dimension.

Fix one half-dimensional subspace $W_0$. The Grassmannian of $H$ is $$\mathrm{Gr}(H, W_0) = \{W \subset H ~|~ W \text{ is half-dimensional}, P_W - P_{W_0} \text{ is Hilbert-Schmidt}\}. $$ Here for $W \subset H$ a subspace, $P_W$ denotes the orthogonal projection onto $W$. $\mathrm{Gr}(H, W_0)$ can be given the structure of a Hilbert manifold in a natural way (see e.g. the book "Loop Groups" of Pressley and Segal).

The space $\mathrm{Gr}(H, W_0)$ has a tautological vector bundle $\tau$ over it, where the fiber is given by $\tau(L) = L$.

Question: Does $\tau$ have a nowhere vanishing section?

I believe that in the case that $H$ is finite-dimensional (say of dimension $2n$), the answer is no, as one can show that the Euler class of $\tau$ is non-zero. But how would one proceed in the infinite-dimensional case?

Answer 1

Let $X$ be any paracompact space. Then Hilbert vector bundles over $X$ are classified by homotopy classes of maps $[X, BU(\mathcal H)]$. But when $\mathcal H$ is infinite-dimensional, the group $U(\mathcal H)$ is contractible (this is Kuiper's theorem), and hence every infinite-dimensional Hilbert bundle over a paracompact space is trivializable.

Hilbert manifolds modelled on a separable Hilbert space are metrizable. Separable metric spaces are paracompact. As long as your $H$ is a separable Hilbert space, the above argument implies that your bundle is trivializable. There is probably a straightforward extension of this argument in the non-separable case but I didn't think about it.

In particular, your bundle has infinitely many linearly independent sections.

Answer 2

As Mike notes Hilbert bundles are trivial over most spaces.

The paper

MR2481802 (2010e:46083) Reviewed Abbondandolo, Alberto (I-PISA); Majer, Pietro (I-PISA) Infinite dimensional Grassmannians. (English summary) J. Operator Theory 61 (2009), no. 1, 19–62. 46T05 (47A53 58B15)

https://arxiv.org/pdf/math/0307192.pdf

Studies the homotopy type of the relative Grasmannian when the projector is compact. It has been too long since I've thought about Hilbert Schmidt operators to see if this is the same answer.