bio | website | |
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location | University of California, San Diego | |
age | 24 | |
visits | member for | 1 year |
seen | 3 mins ago | |
stats | profile views | 146 |
I am a graduate student at University of California, San Diego under the advice of Jim Agler.
Jan 11 |
awarded | Teacher |
Jul 26 |
accepted | Structure theorem for finite dimensional $C^*$-algebras and their representations |
Jul 26 |
comment |
Structure theorem for finite dimensional $C^*$-algebras and their representations
This proof is interesting. Luckily, I know enough about the theory vN algebra for the context of this proof to make sense. I understand the feeling that this is somehow the decomposition into factors combined with some manipulatorics. However, I'm certain there are a lot of different ways to approach the problem. I was certainly more interested in the location of known solutions so that I could use this for some simplification of some proof. |
Jul 26 |
awarded | Commentator |
Jul 26 |
comment |
Structure theorem for finite dimensional $C^*$-algebras and their representations
This is a nice succinct explanation of the fact. I needed to use this to simplify some argument I was doing, and if someone asks why, this kind of logic/perspective will be useful. |
Jul 24 |
comment |
Structure theorem for finite dimensional $C^*$-algebras and their representations
Thanks, Mike! That's exactly what I was looking for! |
Jul 24 |
asked | Structure theorem for finite dimensional $C^*$-algebras and their representations |
Jun 25 |
awarded | Citizen Patrol |
May 1 |
comment |
Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields
What exactly do you mean by uniform norm? Do you mean you need to use a different norm on $M_n$ than the two norm/maximum singular value? |
Apr 29 |
revised |
Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields
edited title |
Apr 29 |
asked | Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields |
Mar 26 |
revised |
Injectivity bounds for complex analytic functions
added 4 characters in body |
Mar 26 |
awarded | Editor |
Mar 26 |
revised |
Injectivity bounds for complex analytic functions
added 4 characters in body |
Mar 26 |
comment |
Injectivity bounds for complex analytic functions
I posted an answer. It should clear up your multivariable question. Somehow I think it should encode a basic proof, but I don't know what it is. |
Mar 26 |
answered | Injectivity bounds for complex analytic functions |
Mar 24 |
awarded | Supporter |
Mar 24 |
comment |
Injectivity bounds for complex analytic functions
The theorem is an algebraic criterion for the invertibility of functions. It talks about a precise sense in which having nonsingular derivative on a domain implies that function is injective. (However, there are some provisos here! I'll link to it later when I can upload it! (It's not complicated, I just don't want to do all the TeXing involved again just for MO)) |
Mar 24 |
comment |
Injectivity bounds for complex analytic functions
Somehow you need the series to have the superlinear terms have negative coefficients, That is, take $$F(z) = id(z) - \sum_{|I|\geq 2} a_Iz^I$$ where $a_I\geq 0.$ Further suppose $F$ is defined on some polydisk at 0. If the derivative of $F$ is nonsingular throughout the intersection of polydisk and the positive reals, then $F$ is injective. |
Mar 24 |
comment |
Injectivity bounds for complex analytic functions
It is some analogue of an algebraic theorem I proved, I wanted to know if it had a simple proof in complex variables. You proof exhibits this fact. In fact it should be that for holomorphic maps $F:\mathbb{C}^n\rightarrow \mathbb{C}^n$ of the form $$F(z)= \text{id}(z) + O(\|z\|^2)$$ that the above holds (for appropriate statements of what the domain could be, some kind of polydisks.) Thank you for your time. |