Timeline for Are all automorphisms of Lin(V) given by similarity transforms?
Current License: CC BY-SA 2.5
12 events
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| Oct 20, 2010 at 15:11 | comment | added | Todd Trimble | I was. If OP meant to consider infinite-dimensional $V$ as well (and it seems there's no indication he doesn't), then I'd have to rethink the problem. The Morita business I had in mind does assume finite-dimensionality. | |
| Oct 20, 2010 at 14:34 | comment | added | t3suji | I am confused: are you assuming that $V$ is finite-dimensional? | |
| Oct 17, 2010 at 16:18 | comment | added | Martin Brandenburg | It surprises me that this shows that every algebra endomorphism of $End(V)$ is actually an automorphism. | |
| Oct 16, 2010 at 3:33 | comment | added | user5810 | Thanks. (I would have accepted it even before you added that, it just took me a while to give up trying to comment first since the site wouldn't let me.) | |
| Oct 16, 2010 at 3:27 | vote | accept | CommunityBot | ||
| Oct 16, 2010 at 3:26 | comment | added | Todd Trimble | I have updated my answer to point you to a slightly different proof. | |
| Oct 16, 2010 at 3:25 | history | edited | Todd Trimble | CC BY-SA 2.5 |
Added a reference
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| Oct 16, 2010 at 3:16 | comment | added | Todd Trimble | That's not an algebra homomorphism -- it doesn't preserve the identity. | |
| Oct 16, 2010 at 3:07 | comment | added | user5810 | Okay, I'm lost in the category theory, but how could we possibly get the zero homomorphism by conjugating the identity with an isomorphism? | |
| Oct 16, 2010 at 2:19 | comment | added | Todd Trimble | I wrote it that way in remembrance of the canonical bijection between linear maps $M \to \hom_F(V, V)$ and linear maps $M \otimes_F V \to V$, and (by specializing) between algebra maps and module structures. I was trying to segue from a question about algebras to a module-theoretic question. | |
| Oct 16, 2010 at 2:12 | comment | added | user5810 | Am I missing a reason for you writing $M \to \hom(V, V)$ instead of $M \to M$? | |
| Oct 16, 2010 at 1:58 | history | answered | Todd Trimble | CC BY-SA 2.5 |