Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
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1$\begingroup$ Also, could the moderators or/and high-reputation users kindly add here a tag "fractals" or something like that? $\endgroup$– Igor KorepanovCommented Jan 4, 2010 at 12:45
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2$\begingroup$ How are they "self-similar" or "fractal"? What Hilbert space are these operators acting on? $\endgroup$– john mangualCommented Jan 4, 2010 at 12:47
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1$\begingroup$ Well, John, give me some time to think how to explain this... Or, in the case if you can read some Russian, here are two short texts with examples of such matrices: csc.ac.ru/ej/file/4381 and csc.ac.ru/ej/file/4641 . And thanks to Dmitri for creating the fractals tag! $\endgroup$– Igor KorepanovCommented Jan 4, 2010 at 13:29
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8$\begingroup$ This question should be closed, I think. If you can turn it into something more concrete with maybe a motivation and a small explanation, then I would be quite interested in reading it! As it stands, the answer is probably yes, as apparently someone knows something about that kind of matrices. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jan 4, 2010 at 19:22
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3$\begingroup$ "nobody could say anything" perhaps because the question was too vague? Maybe they even had some information you're looking for, but they had no way to know it! $\endgroup$– Reid BartonCommented Jan 5, 2010 at 22:41
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1 Answer
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You should learn about the hyperfinite II_1 factor, which is the limit of the inclusions
M_1 --> M_2 --> M_4 --> M_8 --> ....
(here M_k is the k by k matrices over $\mathbb{C}$) where each inclusion is given by tensoring with the identity matrix in M_2. Every 'finite' element is "self-similar" in a sense.
answered Jan 5, 2010 at 22:20
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4$\begingroup$ For the record, btw, I'm in the camp that thinks this is a poor question: too vague, and it's unclear what you mean by "self-similar". Moreover, not knowing whether this answer is basic stuff you already know, or somewthing new, I have little motivation to expand on it. $\endgroup$ Commented Jan 5, 2010 at 22:28
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2$\begingroup$ Scott, thanks for you answer. I am just studying what this mathoverflow is, and your answer is helpful for this. Perhaps I will explain in the next version of my question that just tensor products are not "fractal enough" - or, otherwise, I may decide that this was a wrong site to ask serious questions. But thank you anyhow! $\endgroup$ Commented Jan 5, 2010 at 22:37
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6$\begingroup$ Igor, this is "the wrong site" to ask questions that are so vague that it's not clear what an answer would even look like! In a comment above you equate creativity with vagueness, but I don't see how creativity prevents you from defining the terms that you use in the question. $\endgroup$– HJRWCommented Jan 5, 2010 at 23:06
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7$\begingroup$ Igor, "anything vaguely resembling self-similarity would work" Except that Scott's answer vaguely resembles self-similarity, and apparently it doesn't work! You could have saved Scott the trouble of giving an answer you didn't need if you'd given more details in the question. $\endgroup$– HJRWCommented Jan 5, 2010 at 23:26
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3$\begingroup$ Actually, mostly no --- I saw the high noise/signal discussion going on on meta, and gave an answer in order to more clearly highlight the fact that your question was too vague to receive a useful answer. :-) $\endgroup$ Commented Jan 6, 2010 at 9:09