*** Understanding Better the Frontier of Simple Rings ***
If R is a left noetherian simple ring, but not of the form M_n(D) for any n or skew field D, then we know some things about R:
1. R has no minimal left or right ideal 2. R is not artinian on any side, nor it satisfies
the descending chain condition on left principal ideals generated by idempotents. Which can be rewritten as an infinite system of descending idempotents satisfying plenty of hierarchy equations...
But still...
Jun 12, 2023 • 19 tweets • 7 min read
***RADICALS IN NONCOMMUTATIVE ALGEBRA***
(from the smallest to the biggest)
While we are used from commutative algebra to only ever work with 2 very well-behaved radicals, those are, the intersection of all maximal ideals Jac(R), and the intersection of all prime ideals Nil(R)
Where Nil(R) can also be understood as the biggest nil ideal, or just the union of all the nilpotent elements of R
(and indeed, nilpotent, as soon as your ring is noetherian, which is not an expensive assumption)
((even worse, if your ring is artinian then Jac(R)=Nil(R) ))
Jun 12, 2023 • 23 tweets • 6 min read
_-----~~~** The Daily Dualities (1/3) **~~~-----_
- F_0 := Hom_{cts}( . , S^1) = continuous group homomorphisms with S^1 as a Lie group.
If you could generalize Hahn-Banach's theorem to a slightly greater class of "subspaces", but now only for a Hilbert space, it seems to me that by taking duals (exact functor) one could approach better the invariant subspace problem for complex 2nd countable Hilbert spaces
idk though, I just wanted to transform f \in B(H) for H=l_2(N), assuming f is an injection with not dense image, to a surjection by taking duals but need Hahn-Banach for that functor to be exact...
If it were, direct decomposition after dualizing and taking another dual seems 👍
I have just seen a podcast where you appeared addressing @EricRWeinstein's GU theory and the paper you wrote about that.
That was quite enlightening, and I would also expect for Eric to properly address your well put together paper
As a long time follower of Eric and someone that appreciates some of his ideas and points of view as very original and well spoken out: he is very smart and has a lot of knowledge and experience, I don't doubt that
I couldn't help feeling identified in several parts of the video
Jun 10, 2023 • 7 tweets • 2 min read
*Question (research)
within loc compact T2 groups,
+ what makes Lie groups different to prof. groups is that they have a small neighbourhood from which every closed subgroup escapes
+ and (locally) prof. groups don't have such a neighbourhood for open subgroups
(1/x)
It's sad we have to require small subgroups but not for closed subgroups but for something even much stronger
open subgroups
but that also means that there is a structure in between? what is it like?
