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November 16, 2009

A Rose by Any Other Name

Posted by David Corfield

Pity Ola Bratteli who, when checking out how cited he is, has to take into account the “common misspellings Bratelli, Brattelli and the less common Brateli and Blatteli” of his last name. For someone who has given his name to the rather important Bratteli diagram, this is unfortunate.

Bratteli diagrams are ways of depicting approximately finite C *C^*-algebras.

Posted at November 16, 2009 1:50 PM UTC

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Re: A Rose by Any Other Name

Do you realise that in your google link you’ve used a “common misspelling”? I guess this illustrates your point.

Posted by: anon on November 16, 2009 4:35 PM | Permalink | Reply to this

Re: A Rose by Any Other Name

No that was deliberate to show people how frequent the misspelling is. Even Vaughan Jones only scores 60% accuracy in the second Google entry.

Posted by: David Corfield on November 16, 2009 4:43 PM | Permalink | Reply to this

Trygve

Ola is apparently the son of former prime minister Trygve Bratteli.

Posted by: Thomas Larsson on November 16, 2009 4:49 PM | Permalink | Reply to this

Re: Trygve

Nothing “apparent” about it: He really is the son of Trygve Bratteli. I am certain of this having known him for about thirty years. But also, this information is in wikipedia. BTW, the name “Bratteli” roughly means “steep hillside”, possibly referring to the less than ideal location of the family farm from which the name is derived.

Posted by: Harald Hanche-Olsen on November 22, 2009 4:55 PM | Permalink | Reply to this

Re: A Rose by Any Other Name

I’m sure readers have lots of interesting things to tell us about the diagrams. It appears, for instance, that the Bratteli diagram for the algebra of the Penrose tiling (fig. 12) involves the Fibonacci numbers.

The reason I got to looking them up was following a few trails to the answers Greg Kuperberg gave me to a question at Math Overflow (variant 1 and 2).

I think I’m right in saying that this Bratteli diagram governs John’s construction of the matrix algebra which gets completed to the hyperfinite II 1II_1 factor.

Posted by: David Corfield on November 16, 2009 5:50 PM | Permalink | Reply to this

Re: A Rose by Any Other Name

I’d be interested in “John’s construction of the matrix algebra”, was that a topic of “this weeks finds”? Can I read about it online?
I’m also interested in a textbook or an expository paper where I could learn more about Bratteli diagrams.

Posted by: Tim vB on November 17, 2009 9:01 AM | Permalink | Reply to this

Re: A Rose by Any Other Name

Week 175.

Posted by: David Corfield on November 17, 2009 9:32 AM | Permalink | Reply to this

Re: A Rose by Any Other Name

Thanks, meanwhile I found an explanation of Bratteli diagrams that I actually could make some sense of.

You can find it in “Classification of nuclear C*-algebras” by Mikael Rørdam in the “Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry, Volume 126”.

According to this source the algebra constructed in Week 175 is also known as the CAR (canonical anticommutaion relations) algebra, which would explain the remark that this “is the right algebra of observables for a free quantum field theory with only fermions.”

The explanation of Bratteli diagrams contains a description how you can determine if two diagrams are equivalent = define isomorphic AF algebras (p.15):
Take a Bratteli diagram and remove a finite or infinite number of rows from it such that infinitely many rows are left afterwards. The edges of the new Bratteli diagram are obtained by connecting the nodes that were connected (directly or via intermediate nodes) in the old diagram. Two diagrams are equivalent if one can be obtained from the other by finitley many steps of this sort.

(Unless I got something horribly wrong this explains why the Bratteli diagram of the construction of week 175 is equivalent to the one that David refers to in his post).

Posted by: Tim vB on November 17, 2009 12:23 PM | Permalink | Reply to this

Re: A Rose by Any Other Name

The version I learned at school was “That which we call a rose / By any other word would smell as sweet”. Anyone know if this one of those “multiple edition/Folio/Quarto” issues?

Posted by: Yemon Choi on November 16, 2009 8:42 PM | Permalink | Reply to this

Re: A Rose by Any Other Name

In a word, yes. The First Quarto (1597) reads,

Whats Mountague? It is nor hand nor foote, Nor arme, nor face, nor any other part. Whats in a name? That which we call a Rose, By any other name would smell as sweet.

The s in smell and sweet and Rose should, of course, look more like an f. The analogous passage in the Second Quarto (1599) reads,

Whats Mountague? it is nor hand nor foote, Nor arme nor face, ô be some other name Belonging to a man. Whats in a name that which we call a rose, by any other word would smell as sweete,

And the First Folio, published after Shakespeare’s death in 1623, has it this way:

What’s Mountague? it is nor hand nor foote, Nor arme, nor face, O be some other name Belonging to a man. What? in a names that which we call a Rose, By any other word would smell as sweete,

None of the early texts read quite the same as the version standardized upon in the eighteenth century, which takes the “O be some other name!” part from the Folio and uses the First Quarto’s “by any other name”.

Posted by: Blake Stacey on November 17, 2009 5:42 AM | Permalink | Reply to this

Re: A Rose by Any Other Name

Thanks, Blake!

Posted by: Yemon Choi on November 17, 2009 9:00 AM | Permalink | Reply to this

Re: A Rose by Any Other Name

Hmmm. My line breaks seem to have gotten mangled. Oh, well. That’ll learn me to hit “Post” too quickly.

Posted by: Blake Stacey on November 17, 2009 5:45 AM | Permalink | Reply to this

Re: A Rose by Any Other Name

There’s an interview with Bratteli in norwegian at: http://www.apollon.uio.no/vis/art/2008_2/Artikler/bratteli

He sounds like a nice guy and makes an interesting observation I thought I’d translate:

It’s odd that I have never been invited anywere, to talk about the work I’m most famous for. I guess it’s because I was so young when I did it [the Bratteli diagrams was his master thesis] and that it took some years before the value of it was discovered.
On the other hand I could have wished that some of my later works had gotten just as much attention. Myself, I’m quite satisfied with what I’ve written on quasi product actions. It probably dosn’t appeal so much to the younger generation, being so hard to read and understand.

Deep complex works that give a full exposition often get the unlucky fate that they’r not read because they close the doors on a field.
Mathematicians like to solve new problems, they’r not as interested anymore when someone has found an explanation. I feel it myself too, so I can surely sympathise with it. Before I turned 40, I always kept up to date with what others published. Now I’m more of an old stubborn geezer and think best on my own.

Posted by: M. on November 17, 2009 5:03 PM | Permalink | Reply to this

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