|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'506'133
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Inverse and stability theorems for approximate representations of finite groups | | W. T. Gowers
; O. Hatami
; | | Date: |
14 Oct 2015 | | Abstract: | We prove a stability result for approximate representations of finite groups
(though our results can be straightforwardly generalized to compact groups),
which states that for any unitary-matrix-valued function f such that f(xy) is
always approximately equal to f(x)f(y) there exists a unitary representation
rho of G such that rho(x) approximately equals f(x) for every x. Results of
this kind have been proved already. In particular, when the approximations are
in the operator norm, then this is a result of Grove, Karcher and Ruh,
rediscovered and generalized by Kazhdan. However, our approximations are in the
Hilbert-Schmidt norm (also referred to as the Frobenius norm), or more
generally any trace-class norm with p at most 2. These norms are insensitive to
low-rank perturbations, and our result reflects this in the sense that the
approximating representation need not have the same dimension as f, though it
must have approximately the same dimension. We deduce the result from an
inverse theorem for a suitable analogue of the U^2 norm for matrix-valued
functions, which states that if a uniformly bounded function f (in the operator
norm) has a U^2 norm that is bounded below, then there must be a representation
of dimension within a constant of the dimension of f that correlates well with
f. | | Source: | arXiv, 1510.4085 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER