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Selected works on functional analysis. In two volumes. (Избранные труды по функциональному анализу. В двух томах.) (Russian) Zbl 0654.46003
Klassiki Nauki. Moskva: Izdatel’stvo “Nauka”. Vol. I: 376 p. R. 3.20; Vol. II: 372 p. R. 3.40 (1987).
The papers of John von Neumann were published mostly in German or English, and in various journals. It is the first time that an important part of his papers appear in Russian. They are arranged according to their subjects and are accompanied by valuable comments by first rate experts, namely, by A. M. Vershik, A. V. Marchenko, P. M. Bleher, and L. A. Bunimovich.
Vol. I. begins with papers on operator methods in classical mechanics ([Ann. Math. (2) 33, 587–642 (1932; Zbl 0005.12203, JFM 58.1270.04)] and [Ann. Math. (2) 43, 332–350 (1942; Zbl 0063.01888)]), the second one written together with P. R. Halmos). There follow the papers from 1933/34 on analytic parameters in topological groups and on almost periodic functions on groups (partly with S. Bochner) [Ann. Math. (2) 34, 170–190 (1933; Zbl 0006.30003, JFM 59.0433.01), Trans. Am. Math. Soc. 36, 445–492 (1934; Zbl 0009.34902, JFM 60.0357.01), ibid. 37, 21–50 (1935; Zbl 0011.16004, JFM 61.0470.06)]: first important applications of the powerful tool of Haar measure [Compos. Math. 1, 106–114 (1934; Zbl 0008.24602, JFM 60.0356.11)]. Next follow the paper [Fundam. Math. 13, 73–116 (1929; JFM 55.0151.01), ibid. 13, 333 (1929; JFM 55.0151.02)] on the general theory of measure (relating in particular to the Hausdorff-Banach-Tarski paradox), the paper [Compos. Math. 6, 1–77 (1938; Zbl 0019.31103, JFM 64.0377.01)] on infinite tensor products, and the paper [Port. Math. 3, 1–62 (1942; Zbl 0026.23302, JFM 68.0029.02)] on approximative properties of matrices of high finite order. The last of the papers included [Math. Ann. 104, 570–578 (1931; Zbl 0001.24703, JFM 57.1446.01)] contains the ingenious proof of the uniqueness of the Weyl form of the commutation equation for Schrödinger equations in quantum mechanics.
Vol. II. contains the four papers on rings of operators (called today “von Neumann algebras”), of fundamental importance in the development of mathematics in our century [Ann. Math. (2) 37, 116–229 (1936; Zbl 0014.16101, JFM 62.0449.03), Trans. Am. Math. Soc. 41, 208–248 (1937; Zbl 0017.36001, JFM 63.1008.03), Ann. Math. (2) 41, 94–161 (1940; Zbl 0023.13303, JFM 66.0547.01), Ann. Math. (2) 44, 716–808 (1943; Zbl 0060.26903)], all (except the third paper) written in collaboration with F. J. Murray. These papers fill the first 336 pages. The rest of the volume (13 pages) contains comments on the modern development of Neumann’s ideas in the theory of factors, representations of groups and algebras, and also measure theory and ergodic theory, with a list 143 items, book and papers from the still fast growing literature.
The editor of both volumes is Ya. G. Sinai. He and his collaborators can be congratulated upon this fine and most useful endeavour.
Reviewer: B. Sz.-Nagy

MSC:
46-03 History of functional analysis
46L10 General theory of von Neumann algebras
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
43-03 History of abstract harmonic analysis
46L30 States of selfadjoint operator algebras
01A60 History of mathematics in the 20th century
01A75 Collected or selected works; reprintings or translations of classics