H-compactification is defined as such compactification that all autohomeomorphisms on the original space can be continuously extended to the compactification. (Sometimes H-compactifications are called "topological compactifications" too).
For example $\mathbb{R}^n$ has for $n \geq 2$ only two topological compactifications: $\alpha \mathbb{R}^n$ and $\beta \mathbb{R}^n$.
My guess is that $l^2$space would have only $\beta l^2$ as its topological compactification, as $\alpha l^2$ is impossible (since $l^2$ is not locally compact).
What do you think about this and do you know how to prove that?
Disclaimer: I am assuming Hausdorff compactifications (so the spaces are Tychonoff).
Notation: $\alpha$... ALexandroff one-point compactficiation, $\beta$... Stone-Čech compactification.