Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a topological algebra).


Does the algebra $A$ remember $V$?


Namely, is it true that every representation of $A$ on a locally convex topological vector space $W$ for which the associated map $A\to \mathrm{End}(W)$ is an isomorphism is isomorphic to $V$ as an $A$-representation, and the isomorphism $V\cong W$ is a homeomorphism?



Does $End(V)$ remember $V$, where $V$ is a locally convex space?
If $V$ is a Mackey space (for example, a Banach space), $\text{End} (V)$ coincides with $\text{End} (V\_\sigma)$, where $V\_\sigma$ denotes $V$ with the weak topology.


