An interpretation for filters of subspaces in Banach spaces

Question

Let $X$ be a separable infinite-dimensional (real or complex) Banach space.

Call a collection $\mathcal{F}$ of closed subspaces of $X$ a filter if it is nonempty, does not contain $\{0\}$, is closed upwards under containment, and closed under intersection.

Alternatively, if $X$ has a (Schauder) basis $(e_n)_n$, and $\Delta=(\delta_n)_n$ is a sequence of non-negative numbers (typically positive and $\to0$), call a collection $\mathcal{F}$ of block subspaces (w.r.t. $(e_n)_n$) a $\Delta$-filter if if it is nonempty, does not contain $\{0\}$, is closed upwards under containment, and whenever $Y,Z\in\mathcal{F}$, there is a $V\in\mathcal{F}$ such that $V\in[Y]_\Delta\cap[Z]_\Delta$.

(Here $[Y]_\Delta$ is the set of all block subspaces $V$ of $X$ with block basis $(v_n)_n$, such that for some block basic sequence $(y_n)_n$ contained in $Y$, we have $\|y_n-v_n\|\leq\delta_n$ for all $n$.)

Is there a natural/interesting interpretation of such objects?

Answer 1

When $X$ is a Hilbert space, Ilijas Farah and I called these things quantum filters and we proved a few things about them, plus the natural generalization to projections in an arbitrary C*-algebra. See his paper with Eric Wofsey Set theory and operator algebras, from Definition 6.38 on. A bit later Tristan Bice got some mileage out of switching from projections to positive elements. The resulting theory is quite nice; see his paper Filters in C*-algebras. Not exactly what you're asking about, but it seems pretty close.