Relations of structures related to conjugate idempotents

Question

If $e$ and $f$ are conjugate idempotents in some algebra $A$, I guess the modules $Ae$ and $Af$ should be isomorphic, as well as the algebras $eAe$ and $fAf$ . Are the maps canonically given by just mapping $ae$ to $af$ resp. $eae$ to $faf$, or can something there go wrong? Thank you very much.

Answer 1

No, this doesn't work: why would the map sending $ae$ to $af$ be well-defined, for instance? This would mean that if $ae=0$ then $af=0$, which need not be true. For instance, if $e$ and $f$ are projections of the same rank in a matrix algebra, this would be saying that if $a$ vanishes on the image of $e$ then it must also vanish on the image of $f$, which certainly isn't true in general.

To define the correct maps, you need to use the fact that $e$ and $f$ are conjugate. This means there is some unit $u$ such that $u^{-1}eu=f$. The isomorphism $Ae\to Af$ is then just the map $x\mapsto xu$ (notice that $aeu=auf$ so this does map $Ae$ to $Af$), and the isomorphism $eAe\to fAf$ is $x\mapsto u^{-1}xu$. (Note in particular that these isomorphisms may not be unique or canonical, and depend on a choice of $u$.)