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Naimark's dilation theorem in papers and textbooks is usually stated as:

Let $E$ be a regular, positive, $B(\mathcal H)$-valued measure on $X$. Then there exists a Hilbert space $\mathcal K$, a bounded linear operator $V: \mathcal H \rightarrow \mathcal K$, and a regular, self-adjoint, spectral $B(\mathcal K)$-valued measure $F$ on $X$, such that $E(B) = V^*F(B)V$ (from Paulsen's book Theorem 4.6).

What was the original formulation of Naimark's dilation theorem? It seems conceivable that it changed over the 70+ years.

Did he assume regularity, or is this assumption coming from the later version of this theorem proved by Stinespring. Were his operator-valued measures weakly countably additive?

My trouble is that I cannot find the original paper:

Neumark, M. A., On a representation of additive operator set functions, C. R. (Doklady) Acad. Sci. URSS (N.S.), 41, (1943), 359--361

Does anyone know if this paper is legitimately online anywhere, in Russian or an English translation? As Willie Wong mentions in the comments my fall back will be interlibrary loan.

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  • $\begingroup$ Zbmath seems to think it is in English (so does ISI, apparently); also the Doklady from 1935 - 1947 have immediate translations into German French and English published, according to Wikipedia, so you shouldn't be so pessimistic in your final sentence. My university doesn't have a copy, but Stanford does, so Interlibrary loan seems a possibility. $\endgroup$
    – Willie Wong
    Commented Nov 16, 2017 at 20:12
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    $\begingroup$ @WillieWong Thanks for the info. I was hoping for something more immediate than ILL (I think they run it up by dogsled to Winnipeg in the winter) but it's good to know that it probably is a possibility. I really wish that more of these old journals would be digitized. $\endgroup$
    – Chris Ramsey
    Commented Nov 17, 2017 at 1:27
  • $\begingroup$ Amazingly some of them are. I was poking around on elibrary.ru and a few of the articles from the 1940s in Doklady are actually scanned and available. I am not sure how they decided which ones to scan though. $\endgroup$
    – Willie Wong
    Commented Nov 17, 2017 at 16:18
  • $\begingroup$ The original paper is in Russian: M.A. Naimark, On a Representation of Additive Operator Set Functions (in Russian), Dokl. Akad. Nauk SSSR. 41 (1943), 373--375. Your reference is the English translation. $\endgroup$
    – Arnold Neumaier
    Commented Oct 16, 2019 at 15:00

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I found it. It's a (as far as I can make out) legitimate, non-paywall source and in English! Thumbs up for careful googling. It's in an online library called the "Scientific Heritage of Russia" which seems to be an archive of scientific papers from the years of world war 2.

I was correct as well: Naimark's original assumptions are much weaker than Stinespring's and everyone later. His theorem is for positive, operator-valued measures that are additive.

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    $\begingroup$ The link no longer leads to the paper in question.... $\endgroup$
    – Arnold Neumaier
    Commented Oct 16, 2019 at 14:36
  • $\begingroup$ I do not know why the link has disappeared. Seemingly everything beyond 1942 was removed from public access. This paper is still just inside of having copyright. However, if anyone needs a digital copy I might just "know" someone who has it. [I can remove this if any of the moderators have a problem with the above statement]. $\endgroup$
    – Chris Ramsey
    Commented Mar 15, 2022 at 1:36

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