Let ${\mathcal K}$ be the space of Fredholm operators on a Hilbert space. It is well known that ${\mathcal K}$ represents $K$-theory. Let ${\mathcal K}_0$ be the path component of ${\mathcal K}$ of operators of index zero. Then ${\mathcal K}_0$ is a model for the space that algebraic topologists usually call $BU$ - the classifying space of the infinite unitary group.

My question is: is it possible to realise the filtration of $BU$ by the subspaces $BU(n)$ in terms of Fredholm operators? 

An obvious idea is to consider the subspace of ${\mathcal K}_0$ consisting of operators whose kernel/cokernel has rank at most $n$. Is this space weakly homotopy equivalent to $BU(n)$?

Hopefully this question is interesting enough on its own (I am prepared to find out that it is stupid). My motivation is to understand better <a href="https://mathoverflow.net/questions/248302/is-it-possible-to-construct-an-action-of-an-e-infty-operad-on-bu-that-respe"> Jesse McKeown's answer </a> to my previous question. This is my attempt to understand the statement that the space of subspaces of a Hilbert space of corank at most $n$ is a model for $BU(n)$. If there are other ways to make it precise, I would be very interested in learning about it.

**EDIT:** I think that <a href="https://mathoverflow.net/questions/248302/is-it-possible-to-construct-an-action-of-an-e-infty-operad-on-bu-that-respe/248518#248518"> Tyler Lawson's negative answer </a> to the previous question makes it very likely that the answer to this question is negative as well.