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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 18th 2009
    I thought I'd add something about the Gelfand-Neumark theorem

    The category of commutative von Neumann algebras is contravariantly equivalent to the category of localizable measurable spaces.

    to measurable space, but see we don't have anything on localizability.

    I also mentioned the theorem at von Neumann algebra. But what should one then say that a general von Neumann algebra is a noncommutative localizable measurable space? Normally one says noncommutative measure space.
  1. The Gelfand-Neumark theorem is to be contrasted with the–Naimark_theorem?
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 18th 2009
    I guess its a question of transliterating


    which seem more plausibly Naimark. And Google gives more hits that way.

    Hmm, but the usual G-N theorem says

    "An arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space."

    So perhaps what I was quoting (from the second answer here) is better described as a consequence of the commutative G-N theorem?
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 18th 2009
    The Cyrillic didn't come out. You can see it here.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2009
    • (edited Nov 18th 2009)

    I know the spelling "Neumark" only from the German Wikipedia site. Otherwise I always hear "Naimark".

    I have before at times been confused about what precisely of a few closely related statements is called "Gelfand theorem", "Gelfand-Naimark theorem" or "Gelfand-Naimark-Segal" theorem.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeNov 18th 2009

    Ha, ha! I was confused too at first, but since the theorem as David stated it was not the Gelfand–Naimark theorem that I know, I figured that it must be just one more of Gelfand's many theorems with a similar name. Then I wondered if this Neumark guy had ever done anything else!

    I know that ‘eu’ and ‘ai’ (or ‘ei’) sound similar in German (enough that they may be accepted as rhymes in poetry), but surely the Russian original of ‘Naimark’ is unambiguously transliterated with ‘ai’? H'm, but maybe the etymology is the other way around; a lot of Russian surnames come from Germany originals, and ‘Neumark’ would be a good surname for the periphery of the Prussian Empire.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2009
    • (edited Nov 18th 2009)

    They sound somewhat similar in German, but not at all to the extent that they would be taken to be the same sound. In poetry I'd say rhyming "eu" with "ai" would not count as realy successful rhyme.

    Also, I never heard any German person tell me about anyone called "Neumark". One should check who wrote these German Wikipedia entries. Currently my guess is that some author there rather exhibits his personal ideosyncracy than a wide-spread use of this word. Maybe "Neumark" adheres to some old rule-set for Rusisan-German transliteration, but if so, I don't see that people follow that these days.

    In conclusion, I would opt for using "Naimark" here.

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeNov 18th 2009

    The German poetry that I know best is 19th-century Lieder; there, they would rhyme ‘eu’ with ‘ei’, ‘ö’ with ‘e’, and ‘ü’ with ‘i’.

    Example: ‘Deine Zauber binden wieder was die Mode streng geteilt. Alle Menschen werden Brüder wo dein sanfter Flügel weilt.’

    To be fair, the other rhymes are closer (IMO) than ‘eu’/‘ei’ is. But I'm pretty sure that I saw it done too, although I can't think of an example offhand.

    Kind of off-topic. Well, yes, let's use ‘Naimark’.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2009
    • (edited Nov 18th 2009)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Deine Zauber binden wieder was die Mode streng geteilt. Alle Menschen werden Brüder wo dein sanfter Flügel weilt. </blockquote> <p>Yes, nice.</p> <p>We should say such things once in a while to remind us of the power of language. Whenever anyone comes along and says "category theory is just a language."</p> <p>By the way, now off-off-topic: I plan to create an nLab page titled <a href="">category theory is just a language</a>. At that page I want to list some historical examples of math that are "just langage".</p> <p>Such as this: there is an old theoretical physics textbook from the eaarly 20th century that knows no linear algebra. The authors talk about "direction cosines" instead of components of a vector. At one point they derive the gravitational field of a spherical body, showing that it is the same as that of a point source of the same mass, outside the body.</p> <p>They remark at that point that Newton allegedly postponed the publication of his Principia because he wasn't able to show this result using his "Fluxion"-language for 20 years. Because he didn't have their advanced notation for differential calculus.</p> <p>Today, even their notation looks very antiquated. We do this in two lines. But the functional analysis is "just a language", since, after all, it yields nothing that Newton couldn't derive, too. After 20 years.</p> <p>Sorry, went off a tangent here. But it was the power of the language that you quoted which made me do so...</p> </div>
    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeNov 18th 2009

    How about category theory as a language? We can not just answer the critics who believe that it is only that, but also embrace that already that is much of its power.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2009

    Yeah, but I thought it would be nice to have this half-joking effect. A title that is what many people think of as a major critique. The first line would be just


    Then the next line would be

    "Here some similar historical examples:"

    Towards the end the reader would have gone through a classical catharsis and will never use the word "just" in conjunction with the word "language" again.

    That was my idea. But maybe a more sober title is better in the long run, as you suggest.

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeNov 18th 2009

    Well, you could write your ironic article anyway, then see what you want to do with it. (^_^)

    • CommentRowNumber13.
    • CommentAuthorGuest
    • CommentTimeNov 18th 2009
    And here's me thinking the Gelfand-Neumark theorem was a portmanteau of Gelfand-*Naim*ark and von *Neum*ann...

    Completely inappropriate, I know, to invent mathematicians to which to attribute cough Bourbaki cough results :)

    David Roberts
    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeNov 19th 2009
    I think Urs's title category is a language is better for provocation and for justification of his revolutionary-spirited agenda of showing how short sighted this statement is/was (though the best visionary in this area Grothendieck was allegedly favorable of the statement).