A note (to big to fit into a comment) about Francois Ziegler's investigation. I must say that the Plesner-Rokhlin (1946) article is exceptionally bad at citing references. I have only noted 2 references in the first 70+ pages of the text. Though, that may be understandable since it essentially consists of the pedagogical notes for a course that had been given by Plesner. Here's a translation of a footnote to the title, explaining the origin of these notes:
The first part was published in issue IX of Uspekhi (old series). [Uspekhi Mat. Nauk, 1941:9, 3–125]
The current, second part records, with insignificant changes, the content of the lectures that I read in the spring semester of 1939 at the Faculty of Mechanics and Mathematics, MSU [Moscow State University]. The typesetting of this part of my course was mostly finished by V. A. Rokhlin in the summer of 1941, when the war interrupted him in this task. Thereafter, I myself added only §22 (Hellinger types) part 3, §26 and parts of the Appendix (parts 3 and 5), though I also omitted parts of the treatment of real operators. However, an addition was made (§28) of the theory of generalized functions of a hermitian operator, which I developed in 1942. In relation to this, some rearrangements were made in the preceding chapters and §29 suffered some revisions. --- A. Plesner (A short exposition of the basics necessary for the understanding of part II of the article by A. I. Plesner and V. A. Rokhlin can be found in A. I. Plesner's article Fundamental concepts of the spectral theory of Hermitian operators within the current issue of Uspekhi.
For reference, the part with the theorem cited by Francois Ziegler is in §23.1. The two works that Plesner-Rokhlin article does cite by that point are Stone (Ann. Math. 33, 1932) and von Neumann (Ann. Math. 31, 1931). It seems that the content of Plesner's lectures was fixed a few years before the work of Gelfand-Naimark (1943), so Dieudonné's remark does not seem consistent.