{ "id": "ep44cve44f", "slug": "mark-naimark", "title": "Mark Naimark", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists" }, { "id": "physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics" }, { "id": "physical.scientists.mathematics-statistics.analysts-and-pde-researchers", "label": "Analysts and PDE researchers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.analysts-and-pde-researchers" }, { "id": "physical.scientists.mathematics-statistics.analysts-and-pde-researchers.functional-analysis-and-operator-theorists", "label": "Functional analysis and operator theorists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.analysts-and-pde-researchers.functional-analysis-and-operator-theorists" } ], "geo": [ { "id": "geo.eeu.t1946.physical.scientists.mathematics-statistics.analysts-and-pde-researchers", "label": "Eastern Europe · 1946 to 2000: Analysts and PDE researchers", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946.physical.scientists.mathematics-statistics.analysts-and-pde-researchers", "path": [ { "id": "geo.eeu", "label": "Eastern Europe", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu" }, { "id": "geo.eeu.t1946", "label": "Eastern Europe · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946" }, { "id": "geo.eeu.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946.physical" }, { "id": "geo.eeu.t1946.physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946.physical.scientists" }, { "id": "geo.eeu.t1946.physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946.physical.scientists.mathematics-statistics" }, { "id": "geo.eeu.t1946.physical.scientists.mathematics-statistics.analysts-and-pde-researchers", "label": "Analysts and PDE researchers", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.eeu.t1946.physical.scientists.mathematics-statistics.analysts-and-pde-researchers" } ] } ], "excerpt": "Mark Aronovich Naimark (1909–1978) was a Soviet mathematician considered a founder of functional analysis, known for the 1943 Gelfand–Naimark theorem, the Naimark dilation theorem, and the GNS construction.", "snippet": "Mark Aronovich Naimark (1909–1978) was a Soviet mathematician considered a founder of functional analysis, known for the 1943 Gelfand–Naimark theorem, the Naimark dilation theorem, and the GNS construction.", "node": "physical.scientists.mathematics-statistics.analysts-and-pde-researchers.functional-analysis-and-operator-theorists", "markdown": "# Mark Naimark\n\n**Mark Aronovich Naimark** (На́ймарк (Не́ймарк) Марк Аронович; 5 December 1909, Odessa – 30 December 1978, Moscow) was a Soviet mathematician considered one of the founders of functional analysis and the theory of group representations.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) He is best known for the 1943 [Gelfand–Naimark theorem](https://www.edgepedia.org/gelfand-naimark-theorem) characterizing C*-algebras, the Naimark dilation theorem on [positive operator-valued measures](https://www.edgepedia.org/positive-operator-valued-measure), and his half of the construction now called GNS, the [Gelfand–Naimark–Segal construction](https://www.edgepedia.org/gelfand-naimark-segal-construction).[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) • [3](https://arxiv.org/html/0902.3989)\n\n| Key fact | Detail |\n|---|---|\n| Born / died | Odessa, 5 December 1909; Moscow, 30 December 1978[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) |\n| Doctorate | Ph.D. Odesa University 1936, advisor Mark Grigorievich Krein; doctor of physico-mathematical sciences, Steklov Institute, April 1941[4](https://mathgenealogy.org/id.php?id=74288) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) |\n| Signature theorem | Gelfand–Naimark theorem (1943): norm-closed self-adjoint algebras of Hilbert-space operators are described by a few simple axioms[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) |\n| Dilation theorem | A normalized positive operator-valued measure on a Hilbert space H is the compression of a spectral measure on a larger space K containing H[3](https://arxiv.org/html/0902.3989) |\n| Books | 123 research papers and 5 books; *Normed Rings* (1956) in three editions, translated into German, French, and English[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) |\n| Recognition | Never a full academician, obstructed by antisemitic discrimination; omitted from the Great Soviet Encyclopedia[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) |\n\n## Life and career\n\nNaimark was the son of Aron Iakovlevich Naimark, a professional artist, and Zefir Moiseevna Naimark.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) He began his scientific work under [Mark Krein](https://www.edgepedia.org/mark-krein) in Odessa, his first writings being joint papers with Krein on the separation of roots of algebraic equations.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) He graduated from the Odessa Physics and Chemistry Institute in 1933, became a graduate student at Odessa University, worked at Odessa University from 1933 to 1938, and defended his candidate's dissertation, *The theory of normal operators in Hilbert space*, in 1936.[6](https://encyclopedia.yivo.org/article/2264) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/)\n\n**Moscow and the war.** He moved to Moscow in 1938 to study at the Steklov Mathematical Institute and received his doctorate there in April 1941, after which he was appointed to a chair at the Seismological Institute of the USSR Academy of Sciences.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) During World War II he spent eighteen months in Tashkent, evacuated with the Seismological Institute and the Institute of Chemical Physics, on military-related work.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) • [7](https://esu.com.ua/pdf/file/71742.pdf) He became professor at the Moscow Physical-Technical Institute in 1954, and in 1962 became professor in the Department of the Theory of Functions and Functional Analysis at the Steklov Institute, holding that post until his death.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) The Mathematics Genealogy Project records 7 students and 41 descendants, including Alexandr Helemskii and Alexander Shtern, both of whom completed doctorates in 1967.[4](https://mathgenealogy.org/id.php?id=74288)\n\n## Soviet academic politics and discrimination\n\nNaimark's career was shaped by institutional antisemitism. He aspired for years to a position at the Steklov Institute and was thwarted by the antisemitic proclivities of its director, [Ivan Vinogradov](https://www.edgepedia.org/ivan-vinogradov); when he finally joined in 1962 he was the only Jewish mathematician among a staff of roughly 140.[6](https://encyclopedia.yivo.org/article/2264) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) Despite international recognition he never became a full academician, and the [Great Soviet Encyclopedia](https://www.edgepedia.org/great-soviet-encyclopedia) found no place for him.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) In 1967 he undertook a lecture tour of Canada that improved relations between Soviet and Western scientists.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/)\n\n## Mathematical work\n\n**The Gelfand–Naimark theorem.** In 1943, with [Israel Gelfand](https://www.edgepedia.org/israel-gelfand), Naimark published *On the imbedding of normed rings into a ring of operators in Hilbert space* in *Matematicheskii Sbornik* (Rec. Math. N.S. 12 (54), no. 2, pp. 197–217); in the operator-algebra reading, norm-closed self-adjoint algebras of operators in [Hilbert space](https://www.edgepedia.org/hilbert-space) are described by a few simple axioms.[8](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=25509) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) • [1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich)\n\n**Spectral theory.** In 1943 Naimark also generalized von Neumann's spectral theorem to locally compact abelian groups, and his theory of self-adjoint extensions of symmetric operators, with extension of the original Hilbert space, was complementary to von Neumann's.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) His 1949 survey on resolvents of self-adjoint operators in Hilbert space appeared in *Uspekhi Matematicheskikh Nauk* 4:4(32), pp. 83–147.[8](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=25509)\n\n**Group representations.** The 1950 Gelfand–Naimark treatise *Unitary representations of the classical groups* obtained enough irreducible [unitary representations](https://www.edgepedia.org/unitary-representation) of the classical matrix groups for Plancherel's theorem. These results strongly influenced J. M. G. Fell's work on group representations in the 1950s and 1960s and opened the way for Harish-Chandra's definitive work on Plancherel's theorem, done from about 1953 to about 1970.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) In 1948 Naimark also proved that any two irreducible representations of K(H), the [compact operators](https://www.edgepedia.org/compact-operator) on a Hilbert space, are equivalent.[9](https://arxiv.org/pdf/1708.04368)\n\n## The dilation theorem and the Naimark problem\n\n**Naimark's dilation theorem** states that a normalized positive operator-valued measure E on a Hilbert space H can be written as E(S) = \\( P_{H} \\) Q(S) restricted to H, where Q is a spectral measure on a larger Hilbert space K containing H and \\( P_{H} \\) is the orthogonal projection onto H.[3](https://arxiv.org/html/0902.3989) The theorem guarantees existence of a minimal spectral dilation and uniqueness of minimal dilations up to isomorphism over the original Hilbert space.[10](https://projecteuclid.org/journalArticle/Download?urlId=10.1307%2Fmmj%2F1028999543&isResultClick=False)\n\nThe theorem's modern importance comes from quantum measurement theory: positive operator-valued measures, the very objects Naimark dilated, are now central in quantum physics and quantum information theory under the acronym POVM.[3](https://arxiv.org/html/0902.3989) • [11](https://www.mdpi.com/2075-1680/4/3/400)\n\n**Naimark's problem** asks about C*-algebras with a unique irreducible representation. Naimark's 1948 equivalence result for K(H) is connected to the problem, and the problem remains an active research topic: a recent research report constructs a unital [C*-algebra](https://www.edgepedia.org/c-algebra) containing the [CAR algebra](https://www.edgepedia.org/ccr-and-car-algebras) whose normalized trace has a unique extension among all states, a construction in the problem's orbit.[9](https://arxiv.org/pdf/1708.04368) • [12](https://exactmathematics.org/reports/naimark-problem/draft-r1/naimark-brief-report.pdf)\n\n## The GNS construction\n\nIts \"GN\" component is the 1943 Gelfand–Naimark imbedding paper, which appeared in English in *Matematicheskii Sbornik*.[8](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=25509) • [13](https://www.isibang.ac.in/~soumyashant/misc/collected-works-of-kadison/1990s/1994_NotesOnTheGelfand_NeumarkTheorem.pdf) The construction has since been formalized in the Lean proof assistant and merged into Mathlib, and GNS representations have become a useful tool in algebraic quantum field theory.[14](https://lean.nyc/assets/files/formalizing-the-gns-construction-in-lean.pdf)\n\n## Textbooks and influence\n\nNaimark's scientific oeuvre consists of 123 research papers and 5 books.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) MathSciNet's bibliographic count is 135 total publications with 2,763 citations in 2,554 publications, and 2,280 unique citing authors; by subject class his most-cited areas are operator theory (7 publications) and functional analysis (55 publications).[5](https://mathscinet.ams.org/mathscinet/MRAuthorID/129140)\n\n**The books.** *Normirovannye kol'tsa* (*Normed Rings*, 1956) went through three editions and was translated into German, French, and English; the English translation by Leo Boron was published by P. Noordhoff in 1959.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich) • [15](https://ncatlab.org/nlab/show/Mark%20Naimark) The obituary in *Russian Mathematical Surveys* calls it the first comprehensive treatment of its subject.[16](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=3652&what=fullteng) The reviewer E. Hewitt called it a compendium of functional analysis and \"a unique, monumental, and highly valuable work\".[2](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/) *Lineinye differentsial'nye operatory* (*Linear Differential Operators*, 1954) was the first work ever on that topic.[6](https://encyclopedia.yivo.org/article/2264) His last book, *Teoriya predstavlenii grupp* (*Theory of Group Representations*, 1976), was written with A. I. Shtern while Naimark was gravely ill, and also appeared in French and English.[1](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich)\n\n## References\n\n1. [Naimark, Mark Aronovich, Dictionary of Scientific Biography (Encyclopedia.com)](https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/naimark-mark-aronovich)\n2. [Mark Aronovich Naimark (1909–1978), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Naimark/)\n3. [Vern Paulsen, Dilation theory yesterday and today (arXiv)](https://arxiv.org/html/0902.3989)\n4. [Mark Naimark, Mathematics Genealogy Project](https://mathgenealogy.org/id.php?id=74288)\n5. [Naĭmark, Mark Aronovich, MathSciNet author profile](https://mathscinet.ams.org/mathscinet/MRAuthorID/129140)\n6. [Naimark, Mark Aronovich, YIVO Encyclopedia of Jews in Eastern Europe](https://encyclopedia.yivo.org/article/2264)\n7. [Наймарк Марк Аронович, Енциклопедія Сучасної України](https://esu.com.ua/pdf/file/71742.pdf)\n8. [Naimark, Mark Aronovich, Math-Net.Ru person record](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=25509)\n9. [arXiv preprint on Naimark's problem and representations of K(H)](https://arxiv.org/pdf/1708.04368)\n10. [Michigan Mathematical Journal article on spectral dilation](https://projecteuclid.org/journalArticle/Download?urlId=10.1307%2Fmmj%2F1028999543&isResultClick=False)\n11. [POVMs and the Two Theorems of Naimark and Sz.-Nagy, Logics (MDPI)](https://www.mdpi.com/2075-1680/4/3/400)\n12. [Naimark's problem, Draft R1, Exact Mathematics project report](https://exactmathematics.org/reports/naimark-problem/draft-r1/naimark-brief-report.pdf)\n13. [Richard V. Kadison, Notes on the Gelfand-Neumark Theorem](https://www.isibang.ac.in/~soumyashant/misc/collected-works-of-kadison/1990s/1994_NotesOnTheGelfand_NeumarkTheorem.pdf)\n14. [Formalizing the Gelfand-Naimark-Segal Construction in Lean](https://lean.nyc/assets/files/formalizing-the-gns-construction-in-lean.pdf)\n15. [Mark Naimark, nLab](https://ncatlab.org/nlab/show/Mark%20Naimark)\n16. [Obituary of M. A. Naimark, Russian Mathematical Surveys (Math-Net.Ru)](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=3652&what=fullteng)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Analysts and PDE researchers › Functional analysis and operator theorists*\n\n*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/mark-naimark", "markdown_url": "https://www.edgepedia.org/mark-naimark.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Mark Naimark\", Edgepedia (EdgeChat), https://www.edgepedia.org/mark-naimark. Edgepedia Community License 1.0.", "credit_md": "\"[Mark Naimark](https://www.edgepedia.org/mark-naimark)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/mark-naimark](https://www.edgepedia.org/mark-naimark). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Mark Naimark\", Edgepedia (EdgeChat), https://www.edgepedia.org/mark-naimark. Edgepedia Community License 1.0.", "speakable": "Mark Aronovich Naimark was a Soviet mathematician considered a founder of functional analysis, known for the 1943 Gelfand–Naimark theorem, the Naimark dilation theorem, and the GNS construction." }