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August 30, 2007

Question about von Neumann Algebras

Posted by Urs Schreiber

Jim Stasheff asks me to share this quote:

Study is hard work. It is so much easier to find something else to do in its place than to stay at the grind of it. We have excuses aplenty for avoiding the dull, hard, daily attempt to learn. There is always something so much more important to do than reading. There is always some excuse for not stretching our souls with new ideas and insights now or yet or ever.

by Joan Chittister
Quoted in Essential Monastic Wisdom, by Hugh Feiss .

And another email I receive reminds me of the truth of this. Somebody writes

In one of your entries in the nn-category Café blog, you raised a question that is very relevant to what I’m doing. Did you settle the question in the end as to whether all bimodules over von Neumann algebras really do for sure come from homomorphisms? Do you have any suggestions for what I can read to find out?

This reminds me of my feeble attempts to learn von Neumann algebra theory (was it here?), and how I already start forgetting what I did learn. I think the above statement, that all bimodules in fact come from algebra homomorphisms, is at least true for type III factors.

Somebody please help. Me, and, probably more importantly, the person who wrote the above message.

Posted at August 30, 2007 6:38 PM UTC

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Re: Question about von Neumann Algebras

Study is hard work, but how hard? I’ve been idly wondering for a while now how much energy it takes to think. Say we compare people watching for an hour at a corner cafe to thinking really really hard for an hour. Sometimes after this kind of thinking I’m really exhausted. So what’s the difference in calorie use?

Posted by: James on August 30, 2007 11:04 PM | Permalink | Reply to this

Re: Question about von Neumann Algebras

Hmm… I seem to recall that (in natural units) it costs on the order of Tlog(2)T \log(2) in energy to forget one bit of classical information at temperature TT (for the sense of “bit” apropriate to the chosen value of 2). Sounds like “entropy is lost information” … And I suppose one can think of thinking as forgetting the irrelevant data, although I’m sure something more subtle is going on, and it surely isn’t fair to construe the brain as a classical computer. But if we entertain this absurd reduction the question seems to become, in an hour of hard thinking how many bits do you (or your brain, rather) consider and then discard, and what is the natural scale of the temperature 37 degrees C? Maybe I’ll pull out my slide-rule and tables of digits…
Posted by: some guy on the street on August 31, 2007 4:15 AM | Permalink | Reply to this

10,000 hours of study; Re: Question about von Neumann Algebras

I’m not sure on which blog thread this has been debated, but I’ve commented before that there seems to be a rule of thumb that it takes an adult human being on the order of 10,000 hours of study to reach professional competence in almost any subject, from Music to Mathematics.

It is no coincidence that this is how long a full-time student spends in a given university.

Counterexamples have been made to my comments on the likes of Mozart (who appears to have been composing while still in utero), Gauss, Ramanujan, Terry Tao, Feynman, and Newton. Counter-counter comments were made. I’m not sure what conclusion, if any, was reached, but I still hold (perhaps more loosely) the 10,000 hour study rule.

For children, the line between Play and Study is fuzzy at best. My strength and weakness as a scholar is that I still feel that I’m playing with Math, Physics, Computing, Biology, and Literature…

Posted by: Jonathan Vos Post on September 1, 2007 5:52 PM | Permalink | Reply to this

Music: science and 10^4 hours; Re: 10,000 hours of study; Re: Question about von Neumann Algebras

From The Sunday Times
September 2, 2007
The science of music
Why does music affect us like no other art? An American scientist thinks he can explain these ‘glorious illusions’

“… In fact, we can quantify how much work is required. To become a master musician requires 10,000 hours of work, irrespective of any preexisting gift. But what about Mozart, the child prodigy? Well, says Levitin, if he’d started working at his music for 32 hours a week from the age of two – not inconceivable, with a father such as his – then he’d have done 10,000 hours by the age of eight. The idle, effortless genius appears to be a myth…”

Posted by: Jonathan Vos Post on September 10, 2007 6:00 PM | Permalink | Reply to this

Re: Question about von Neumann Algebras

From page 431+- of Adkins and Weintraub (Algebra, an approach via module theory). The answer seems to be “yes” but only up to ismorphism, consequentially, if you replace the bimodules in a double category by homomorphisms, you are making strict a composition that would otherwise be weak.

Posted by: anonymous on September 3, 2007 7:37 AM | Permalink | Reply to this

Re: Question about von Neumann Algebras

You may try Alain Connes’ “Noncommutative Geometry”, p. 542.

Posted by: Pasquale Zito on September 3, 2007 12:07 PM | Permalink | Reply to this

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