Locally analytic vectors of a quotient space

Question

My question here is in connection with one of my previous question "A definition of a (amalgamated) direct sum" Following the notations there, my question is:

Why the locally analytic vectors of $B(V)$ is not isomorphic to $A(\alpha)/L^{loc}(\alpha)$ where $L^{loc}(\alpha)$ is the locally analytic vectors of $L(\alpha)$?

In general, is it true that if $W_1/W_2$ is a $\mathbb{Q}_p$-Banach representation of $GL_2(\mathbb{Q}_p)$, then $$(W_1/W_2)^{loc}\cong W_1^{loc}/W_2^{loc},$$,

where $(-)^{loc}$ denotes the locally analytic vectors of $(-)$?

Answer 1

It holds if you work in the category of admissible representations (Schneider-Teitelbaum). Otherwise it can fail.

For a natural counterexample, see for instance my paper with Colmez "Théorie de Sen et vecteurs localement analytiques", section 3.3