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I've just started reading representation theory books, and I was wondering about representation intertwiners in a somewhat more general setting. I also think I might have asked this question in the past but it has been phrased badly so got no responses.

Suppose $\mathbb{F}$ is a field and $V,\,W$ vector spaces over $\mathbb{F}$. Let $\phi:End_\mathbb{F}(V)\rightarrow End_\mathbb{F}(W)$ be an algebra isomorphism. Is it necessarily true that there is some $T:V\rightarrow W$ linear map such that $\phi(X)=TXT^{-1}$ for all $X\in End_\mathbb{F}(V)$?

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    $\begingroup$ For finite-dimensional vector spaces, one could try to use the Skolem-Noether theorem. I doubt that it holds for general infinite-dimensional vector spaces though. $\endgroup$
    – abenthy
    Commented Jul 23, 2019 at 16:26

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