Editing John von Neumann (section)

== Mathematics ==

=== Set theory ===
{{See also|Von Neumann–Bernays–Gödel set theory}}
[[File:NBG Evolution svg.svg|thumb|upright=1.5|History of approaches that led to NBG set theory]]
At the beginning of the 20th century, efforts to base mathematics on [[naive set theory]] suffered a setback due to [[Russell's paradox]] (on the set of all sets that do not belong to themselves).{{sfn|Macrae|1992|pp=104–105}} The problem of an adequate axiomatization of [[set theory]] was resolved implicitly about twenty years later by [[Ernst Zermelo]] and [[Abraham Fraenkel]]. [[Zermelo–Fraenkel set theory]] provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—the ''[[axiom of regularity|axiom of foundation]]'' and the notion of ''[[Class (set theory)|class]].''<ref name=vanheijenoort>{{cite book |last=Van Heijenoort |first=Jean |author-link=Jean van Heijenoort |year=1967 |title=From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931 |url=https://archive.org/details/fromfregetogodel0000vanh |url-access=registration |location=Cambridge, Massachusetts |publisher=Harvard University Press |isbn=978-0-674-32450-3|oclc=523838 }}</ref>

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced the ''method of [[inner model]]s'', which became an essential demonstration instrument in set theory.{{r|vanheijenoort}}

The second approach to the problem of sets belonging to themselves took as its base the notion of [[Class (set theory)|class]], and defines a set as a class that belongs to other classes, while a ''proper class'' is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a ''proper class'', not a set.{{r|vanheijenoort}}

Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the [[Ordinal number|ordinal]] and [[cardinal number]]s as well as the first strict formulation of principles of definitions by the [[transfinite induction]]".{{sfn|Murawski|2010|p=196}}

====Von Neumann paradox====
{{main|Von Neumann paradox}}
Building on the [[Hausdorff paradox]] of [[Felix Hausdorff]] (1914), [[Stefan Banach]] and [[Alfred Tarski]] in 1924 showed how to subdivide a three-dimensional [[ball (mathematics)|ball]] into [[disjoint sets]], then translate and rotate these sets to form two identical copies of the same ball; this is the [[Banach–Tarski paradox]]. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929,<ref>{{citation | first=J. | last=von Neumann | author-link=John von Neumann | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm13/fm1316.pdf | title=Zur allgemeinen Theorie des Masses |trans-title=On the general theory of mass |language=de | journal=[[Fundamenta Mathematicae]] | volume=13 | pages=73–116 | year=1929 | doi=10.4064/fm-13-1-73-116 | doi-access=free }}</ref> von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving [[affine transformation]]s instead of translations and rotations. The result depended on finding [[free group]]s of affine transformations, an important technique extended later by von Neumann in [[#Measure theory|his work on measure theory]].{{sfn|Ulam|1958|pages=14–15}}

=== Proof theory ===
{{See also|Hilbert's program}}

With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its [[consistency]]. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger [[Axiom#Mathematical logic|axioms]] that could be used to prove a broader class of theorems.<ref>{{cite encyclopedia |last=Von Plato |first=Jan |title=The Development of Proof Theory |encyclopedia=The Stanford Encyclopedia of Philosophy |year=2018 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/proof-theory-development/ |publisher=Stanford University |edition=Winter 2018 |access-date=2023-09-25 }}</ref>

By 1927, von Neumann was involving himself in discussions in Göttingen on whether [[elementary arithmetic]] followed from [[Peano axioms]].<ref>{{cite journal |last1=van der Waerden |first1=B. L. |author-link1=Bartel Leendert van der Waerden |title=On the sources of my book Moderne algebra |journal=Historia Mathematica |date=1975 |volume=2 |issue=1 |pages=31–40 |doi=10.1016/0315-0860(75)90034-8 |doi-access=free }}</ref> Building on the work of [[Wilhelm Ackermann|Ackermann]], he began attempting to prove (using the [[Finitism|finistic]] methods of [[Hilbert's program|Hilbert's school]]) the consistency of [[Peano axioms#Peano arithmetic as first-order theory|first-order arithmetic]]. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on [[Mathematical induction|induction]]).<ref>{{cite journal |last1=Neumann |first1=J. v. |title=Zur Hilbertschen Beweistheorie |journal=Mathematische Zeitschrift |date=1927 |volume=24 |pages=1–46 |language=German |doi=10.1007/BF01475439 |s2cid=122617390 |url=https://eudml.org/doc/167910}}</ref> He continued looking for a more general proof of the consistency of classical mathematics using methods from [[proof theory]].{{sfn|Murawski|2010|pp=204-206}}

A strongly negative answer to whether it was definitive arrived in September 1930 at the [[Second Conference on the Epistemology of the Exact Sciences]], in which [[Kurt Gödel]] announced his [[Gödel's incompleteness theorems|first theorem of incompleteness]]: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.{{sfn|Rédei|2005|p=123}} At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers.{{sfn|von Plato|2018|p=4080}}

Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency.{{sfn|Rédei|2005|p=123}} Gödel replied that he had already discovered this consequence, now known as his [[second incompleteness theorem]], and that he would send a preprint of his article containing both results, which never appeared.<ref>{{cite book |last=Dawson |first=John W. Jr. |author-link=John W. Dawson, Jr. |year=1997 |title=Logical Dilemmas: The Life and Work of Kurt Gödel |location=Wellesley, Massachusetts |publisher=A. K. Peters |isbn=978-1-56881-256-4 |page=70}}</ref>{{sfn|von Plato|2018|pp=4083-4088}}{{sfn|von Plato|2020|pp=24-28}} Von Neumann acknowledged Gödel's priority in his next letter.{{sfn|Rédei|2005|p=124}} However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did.{{sfn|von Plato|2020|p=22}}<ref>{{cite book |last1=Sieg |first1=Wilfried |title=Hilbert's Programs and Beyond |date=2013 |publisher=Oxford University Press |isbn=978-0195372229 |url=https://books.google.com/books?id=4lDrwqo-8TkC&pg=PA149 |page=149}}</ref> With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the [[foundations of mathematics]] and [[metamathematics]] and instead spent time on problems connected with applications.{{sfn|Murawski|2010|p=209}}

=== Ergodic theory ===
In a series of papers published in 1932, von Neumann made foundational contributions to [[ergodic theory]], a branch of mathematics that involves the states of [[dynamical systems]] with an [[invariant measure]].<ref>{{cite journal|author-link=Eberhard Hopf|first=Eberhard|last=Hopf|title=Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung|year=1939|journal=Leipzig Ber. Verhandl. Sächs. Akad. Wiss.|volume=91|pages=261–304|language=German}} {{pb}}
Two of the papers are: {{pb}}
{{cite journal|first=John|last=von Neumann|title=Proof of the Quasi-ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=70–82|doi=10.1073/pnas.18.1.70|pmid=16577432|issue=1|pmc=1076162|bibcode=1932PNAS...18...70N |doi-access=free |bibcode-access=free }} {{pb}}
{{cite journal|first=John|last=von Neumann|title=Physical Applications of the Ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=263–266|doi=10.1073/pnas.18.3.263|pmid=16587674|issue=3|pmc=1076204|jstor=86260|bibcode=1932PNAS...18..263N|doi-access=free}}.</ref> Of the 1932 papers on ergodic theory, [[Paul Halmos]] wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality".{{sfn|Halmos|1958|p=93}} By then von Neumann had already written his articles on [[operator theory]], and the application of this work was instrumental in his [[Ergodic theory#Mean ergodic theorem|mean ergodic theorem]].{{sfn|Halmos|1958|p=91}}

The theorem is about arbitrary [[One-parameter group|one-parameter]] [[unitary group]]s <math>\mathit{t} \to \mathit{V_t}</math> and states that for every vector <math>\phi</math> in the [[Hilbert space]], <math display=inline>\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} V_t(\phi) \, dt</math> exists in the sense of the metric defined by the Hilbert norm and is a vector <math>\psi</math> which is such that <math>V_t(\psi) = \psi</math> for all <math>t</math>. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to [[Ludwig Boltzmann|Boltzmann's]] [[ergodic hypothesis]]. He also pointed out that [[ergodicity]] had not yet been achieved and isolated this for future work.<ref name=mackey1990>{{harvc |last1=Mackey |first1=George W. |author-link=George Mackey |year=1990 |chapter=Von Neumann and the Early Days of Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=27–30}}</ref>

Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic [[Measure-preserving dynamical system|measure preserving actions]] of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with [[Paul Halmos]] have significant applications in other areas of mathematics.{{r|mackey1990}}<ref>{{harvc |last1=Ornstein |first1=Donald S. |author-link=Donald Samuel Ornstein |year=1990 |chapter=Von Neumann and Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |page=39}}</ref>

=== Measure theory ===
{{See also|Lifting theory}}

In [[Measure (mathematics)|measure theory]], the "problem of measure" for an {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of {{math|'''R'''<sup>''n''</sup>}}?"{{sfn|Halmos|1958|p=86}} The work of [[Felix Hausdorff]] and [[Stefan Banach]] had implied that the problem of measure has a positive solution if {{math|1=''n'' = 1}} or {{math|1=''n'' = 2}} and a negative solution (because of the [[Banach–Tarski paradox]]) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the [[transformation group]] of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the [[Euclidean group]] is a [[solvable group]] for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."{{sfn|Halmos|1958|p=87}} Around 1942 he told [[Dorothy Maharam]] how to prove that every [[Complete measure|complete]] [[σ-finite measure|σ-finite]] [[measure space]] has a multiplicative lifting; he did not publish this proof and she later came up with a new one.{{sfn|Pietsch|2007|p=168}}

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions.{{sfn|Halmos|1958|p=88}} A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of [[Alfréd Haar|Haar]] regarding whether there existed an [[Algebra over a field|algebra]] of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions".{{sfn|Dieudonné|2008}} He proved this in the positive, and in later papers with [[Marshall Harvey Stone|Stone]] discussed various generalizations and algebraic aspects of this problem.<ref>{{cite book |last1=Ionescu-Tulcea |first1=Alexandra |author1-link=Alexandra Bellow |last2=Ionescu-Tulcea |first2=Cassius |author2-link=Cassius Ionescu-Tulcea |title=Topics in the Theory of Lifting |date=1969 |publisher=Springer-Verlag Berlin Heidelberg |isbn=978-3-642-88509-9 |page=V |url=https://www.springer.com/gp/book/9783642885099}}</ref> He also proved by new methods the existence of [[Disintegration theorem|disintegrations]] for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for [[compact group]]s.{{sfn|Dieudonné|2008}} He had to create entirely new techniques to apply this to [[locally compact group]]s.{{sfn|Halmos|1958|p=89}} He also gave a new, ingenious proof for the [[Radon–Nikodym theorem]].<ref>{{cite journal |last1=Neumann |first1=J. v. |title=On Rings of Operators. III. |journal=Annals of Mathematics |date=1940 |volume=41 |issue=1 |pages=94–161 |doi=10.2307/1968823 |jstor=1968823 |url=https://www.jstor.org/stable/1968823}}</ref> His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.{{sfn|Halmos|1958|p=90}}<ref>{{cite book |last1=Neumann |first1=John von |title=Functional Operators, Volume 1: Measures and Integrals |date=January 21, 1950 |orig-date=1950 |publisher=Princeton University Press |isbn=((9780691079660))<!-- This appears to be a valid isbn for this edition and publisher despite the date conflict -->}}</ref><ref>{{cite book |last1=von Neumann |first1=John |title=Invariant Measures |date=1999 |publisher=American Mathematical Society |isbn=978-0-8218-0912-9 |url=https://bookstore.ams.org/inmeas}}</ref>

=== Topological groups ===
Using his previous work on measure theory, von Neumann made several contributions to the theory of [[topological group]]s, beginning with a paper on almost periodic functions on groups, where von Neumann extended [[Harald Bohr|Bohr's]] theory of [[almost periodic function]]s to arbitrary [[Group (mathematics)|groups]].<ref>{{cite journal |last1=von Neumann |first1=John |title=Almost Periodic Functions in a Group. I. |journal=Transactions of the American Mathematical Society |date=1934 |volume=36 |issue=3 |pages=445–492 |doi=10.2307/1989792 |jstor=1989792 |url=https://www.jstor.org/stable/1989792}}</ref> He continued this work with another paper in conjunction with [[Salomon Bochner|Bochner]] that improved the theory of almost [[Periodic function|periodicity]] to include [[Function (mathematics)|functions]] that took on elements of [[Vector space|linear spaces]] as values rather than numbers.<ref>{{cite journal |last1=von Neumann |first1=John |last2=Bochner |first2=Salomon |title=Almost Periodic Functions in Groups, II. |journal=Transactions of the American Mathematical Society |date=1935 |volume=37 |issue=1 |pages=21–50 |doi=10.2307/1989694 |jstor=1989694 |url=https://www.jstor.org/stable/1989694}}</ref> In 1938, he was awarded the [[Bôcher Memorial Prize]] for his work in [[Mathematical analysis|analysis]] in relation to these papers.<ref>{{cite web |url=https://www.ams.org/profession/prizes-awards/pabrowse?purl=bocher-prize#year=1938 |title=AMS Bôcher Prize |publisher=AMS |date=January 5, 2016 |access-date=2018-01-12}}</ref>{{sfn|Bochner|1958|p=440}}

In a 1933 paper, he used the newly discovered [[Haar measure]] in the solution of [[Hilbert's fifth problem]] for the case of [[compact group]]s.<ref>{{cite journal|first=J.|last=von Neumann|title=Die Einfuhrung Analytischer Parameter in Topologischen Gruppen|journal=[[Annals of Mathematics]]|volume=34|issue=1|series=2|year=1933|pages=170–190|doi=10.2307/1968347|jstor=1968347|language=German}}</ref> The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of [[Linear map|linear transformations]] and found that closed [[subgroup]]s of a general [[linear group]] are [[Lie group]]s.<ref>{{cite journal |last1=v. Neumann |first1=J. |title=Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen |journal=Mathematische Zeitschrift |date=1929 |volume=30 |issue=1 |pages=3–42 |doi=10.1007/BF01187749 |s2cid=122565679 |language=German}}</ref> This was later extended by [[Élie Cartan|Cartan]] to arbitrary Lie groups in the form of the [[closed-subgroup theorem]].{{sfn|Bochner|1958|p=441}}{{sfn|Dieudonné|2008}}

=== Functional analysis ===
{{Main|Operator theory}}{{See also|Spectral theorem}}
Von Neumann was the first to axiomatically define an abstract [[Hilbert space]]. He defined it as a [[Vector space|complex vector space]] with a [[Inner product space|Hermitian scalar product]], with the corresponding [[Norm (mathematics)|norm]] being both separable and complete. In the same papers he also proved the general form of the [[Cauchy–Schwarz inequality]] that had previously been known only in specific examples.{{sfn|Pietsch|2007|p=11}} He continued with the development of the [[spectral theory]] of operators in Hilbert space in three seminal papers between 1929 and 1932.{{sfn|Dieudonné|1981|p=172}} This work cumulated in his ''[[Mathematical Foundations of Quantum Mechanics]]'' which alongside two other books by [[Marshall Harvey Stone|Stone]] and [[Stefan Banach|Banach]] in the same year were the first monographs on Hilbert space theory.{{sfn|Pietsch|2007|p=14}} Previous work by others showed that a theory of [[Weak topology|weak topologies]] could not be obtained by using [[Weak convergence (Hilbert space)|sequences]]. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining [[Locally convex topological vector space|locally convex spaces]] and [[topological vector spaces]] for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from [[Felix Hausdorff|Hausdorff]] from Euclidean to Hilbert spaces){{sfn|Dieudonné|1981|pp=211,218}} such as [[Bounded set (topological vector space)|boundness]] and [[Totally bounded space|total boundness]] are still used today.{{sfn|Pietsch|2007|pp=58,65-66}} For twenty years von Neumann was considered the 'undisputed master' of this area.{{sfn|Dieudonné|2008}} These developments were primarily prompted by needs in [[quantum mechanics]] where von Neumann realized the need to extend [[Self-adjoint operator#Spectral theorem|the spectral theory of Hermitian operators]] from the bounded to the [[Unbounded operator|unbounded]] case.<ref name=steen>{{cite journal |last1=Steen |first1=L. A. |author-link1=Lynn Steen |title=Highlights in the History of Spectral Theory |journal=The American Mathematical Monthly |date=April 1973 |volume=80 |issue=4 |pages=359–381, esp. 370–373 |doi=10.1080/00029890.1973.11993292 |jstor=2319079 |url=https://doi.org/10.2307/2319079}}</ref> Other major achievements in these papers include a complete elucidation of spectral theory for [[normal operator]]s, the first abstract presentation of the [[Trace (linear algebra)|trace]] of a [[Positive operator (Hilbert space)|positive operator]],<ref>{{cite journal |last1=Pietsch |first1=Albrecht |author-link=:de:Albrecht Pietsch |title=Traces of operators and their history |journal=Acta et Commentationes Universitatis Tartuensis de Mathematica |date=2014 |volume=18 |issue=1 |pages=51–64 |doi=10.12697/ACUTM.2014.18.06 |url=https://acutm.math.ut.ee/index.php/acutm/article/download/ACUTM.2014.18.06/22|doi-access=free }}</ref>{{sfn|Lord|Sukochev|Zanin|2012|p=1}} a generalisation of [[Frigyes Riesz|Riesz]]'s presentation of [[David Hilbert|Hilbert]]'s spectral theorems at the time, and the discovery of [[Self-adjoint operator#Definitions|Hermitian operators]] in a Hilbert space, as distinct from [[self-adjoint operator]]s, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of [[Matrix (mathematics)#Infinite matrices|infinite matrices]], common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of [[operator algebra]]s.{{sfn|Dieudonné|1981|pp=175–176, 178–179, 181, 183}}

His later work on rings of operators lead to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces.{{r|steen}} Like in his work on measure theory he proved several theorems that he did not find time to publish. He told [[Nachman Aronszajn]] and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the [[invariant subspace problem]].{{sfn|Pietsch|2007|p=202}}

With [[Isaac Jacob Schoenberg|I. J. Schoenberg]] he wrote several items investigating [[Translational symmetry|translation invariant]] Hilbertian [[Metric (mathematics)|metrics]] on the [[Number line|real number line]] which resulted in their complete classification. Their motivation lie in various questions related to embedding [[metric space]]s into Hilbert spaces.<ref>{{cite arXiv |last1=Kar |first1=Purushottam |last2=Karnick |first2=Harish |title=On Translation Invariant Kernels and Screw Functions |date=2013 |page=2 |class=math.FA |eprint=1302.4343}}</ref><ref>{{cite journal |last1=Alpay |first1=Daniel |last2=Levanony |first2=David |title=On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions |journal=Potential Analysis |date=2008 |volume=28 |issue=2 |pages=163–184 |doi=10.1007/s11118-007-9070-4 |arxiv=0705.2863 |s2cid=15895847 }}</ref>

With [[Pascual Jordan]] he wrote a short paper giving the first derivation of a given norm from an [[inner product space|inner product]] by means of the [[Parallelogram law#The parallelogram law in inner product spaces|parallelogram identity]].{{sfn|Horn|Johnson|2013|p=320}} His [[Trace inequality#Von Neumann's trace inequality and related results|trace inequality]] is a key result of matrix theory used in matrix approximation problems.{{sfn|Horn|Johnson|2013|p=458}} He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as  symmetric absolute norms).<ref>{{cite book |last1=Horn |first1=Roger A. |author1-link=Roger Horn |last2=Johnson |first2=Charles R. |author2-link=Charles Royal Johnson |title=Topics in Matrix Analysis |date=1991 |publisher=Cambridge University Press |isbn=0-521-30587-X |url=https://www.cambridge.org/core/books/topics-in-matrix-analysis/B988495A235F1C3406EA484A2C477B03 |page=139}}</ref>{{sfn|Horn|Johnson|2013|p=335}}<ref>{{cite book |last1=Bhatia |first1=Rajendra |title=Matrix Analysis |series=Graduate Texts in Mathematics |date=1997 |volume=169 |publisher=Springer |location=New York |isbn=978-1-4612-0653-8 |page=109 |doi=10.1007/978-1-4612-0653-8 |url=https://link.springer.com/book/10.1007/978-1-4612-0653-8}}</ref> This paper leads naturally to the study of symmetric [[operator ideal]]s and is the beginning point for modern studies of symmetric [[operator space]]s.{{sfn|Lord|Sukochev|Zanin|2021|p=73}}

Later with [[Robert Schatten]] he initiated the study of [[nuclear operator]]s on Hilbert spaces,<ref>{{cite journal |last1=Prochnoa |first1=Joscha |last2=Strzelecki |first2=Michał |title=Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings |journal=Journal of Approximation Theory |date=2022 |volume=277 |page=105736 |doi=10.1016/j.jat.2022.105736 |arxiv=2103.13050 |s2cid=232335769 }}</ref><ref>{{cite web |url=https://encyclopediaofmath.org/wiki/Nuclear_operator |archive-url=https://web.archive.org/web/20210623153701/https://encyclopediaofmath.org/wiki/Nuclear_operator |archive-date=2021-06-23 |title=Nuclear operator |access-date=August 7, 2022 |publisher=Encyclopedia of Mathematics}}</ref> [[Topological tensor product#Cross norms and tensor products of Banach spaces|tensor products of Banach spaces]],{{sfn|Pietsch|2007|p=372}} introduced and studied [[trace class]] operators,{{sfn|Pietsch|2014|p=54}} their [[Ideal (ring theory)|ideals]], and their [[Duality (mathematics)|duality]] with [[compact operator]]s, and [[predual]]ity with [[bounded operator]]s.{{sfn|Lord|Sukochev|Zanin|2012|p=73}} The generalization of this topic to the study of [[Nuclear operators between Banach spaces|nuclear operators on Banach spaces]] was among the first achievements of [[Alexander Grothendieck]].{{sfn|Lord|Sukochev|Zanin|2021|p=26}}{{sfn|Pietsch|2007|p=272}} Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on <math>\textit{l}\,_2^n\otimes\textit{l}\,_2^n</math> and proving several other results on what are now known as Schatten–von Neumann ideals.{{sfn|Pietsch|2007|pp=272,338}}

=== Operator algebras ===
{{Main|Von Neumann algebra}}{{See also|Direct integral}}
Von Neumann founded the study of rings of operators, through the [[von Neumann algebra]]s (originally called W*-algebras). While his original ideas for [[Noncommutative ring|rings]] of [[Linear map|operators]] existed already in 1930, he did not begin studying them in depth until he met [[Francis Joseph Murray|F. J. Murray]] several years later.{{sfn|Pietsch|2007|p=140}}<ref>{{harvc |last1=Murray |first1=Francis J. |author-link=Francis Joseph Murray |year=1990 |chapter=The Rings of Operators Papers |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=57–59}}</ref> A von Neumann algebra is a [[*-algebra]] of bounded operators on a [[Hilbert space]] that is closed in the [[weak operator topology]] and contains the [[Identity function|identity operator]].<ref>{{harvc |last1=Petz |first1=D. |author-link1=Dénes Petz |last2=Rédei |first2=M. R. |contribution=John von Neumann And The Theory Of Operator Algebras |in1=Bródy |in2=Vámos |year=1995 |pages=163–181}}</ref> The [[von Neumann bicommutant theorem]] shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the [[bicommutant]].<ref>{{cite web |url=https://www.princeton.edu/~hhalvors/restricted/jones.pdf |title=Von Neumann Algebras |access-date=January 6, 2016 |publisher=Princeton University }}</ref> After elucidating the study of the [[Commutative ring|commutative algebra]] case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the [[Noncommutative ring|noncommutative]] case, the general study of [[von Neumann algebra|factors]] classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century";{{sfn|Dieudonné|2008|p=90}} they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of [[Von Neumann algebra#Factors|factors]].{{sfn|Pietsch|2007|pp=151}} In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949.{{sfn|Pietsch|2007|p=146}}<ref>{{cite web |url=https://www.math.ucla.edu/~brh6/DirectIntegral.pdf |archive-url=https://web.archive.org/web/20150702001911/http://www.math.ucla.edu/~brh6/DirectIntegral.pdf |archive-date=2015-07-02 |title=Direct Integrals of Hilbert Spaces and von Neumann Algebras |access-date=January 6, 2016 |publisher=University of California at Los Angeles }}</ref> Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out.{{sfn|Segal|1965}}<ref>{{harvc |last1=Kadison |first1=Richard V. |author-link=Richard Kadison |year=1990 |chapter=Operator Algebras - An Overview |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=65,71,74}}</ref> Another important result on [[polar decomposition]] was published in 1932.{{sfn|Pietsch|2007|p=148}}

=== Lattice theory ===
{{Main|Continuous geometry}}{{See also|Complemented lattice#Orthomodular lattices}}

Between 1935 and 1937, von Neumann worked on [[Lattice (order)|lattice theory]], the theory of [[partially ordered set]]s in which every two elements have a greatest lower bound and a least upper bound. As [[Garrett Birkhoff]] wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor".{{sfn|Birkhoff|1958|p=50}} Von Neumann combined traditional projective geometry with modern algebra ([[linear algebra]], [[Ring (mathematics)|ring theory]], lattice theory). Many previously geometric results could then be interpreted in the case of general [[Module (mathematics)|modules]] over rings. His work laid the foundations for some of the modern work in projective geometry.<ref name=lashkhi1995>{{cite journal |last=Lashkhi |first=A. A. |title=General geometric lattices and projective geometry of modules |journal=Journal of Mathematical Sciences |date=1995 |volume=74 |issue=3 |pages=1044–1077 |doi=10.1007/BF02362832 |s2cid=120897087 |doi-access=free }}</ref>

His biggest contribution was founding the field of [[continuous geometry]].<ref>{{cite journal | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Examples of continuous geometries | jstor=86391 | doi=10.1073/pnas.22.2.101 | jfm=62.0648.03 | year=1936 | journal=Proc. Natl. Acad. Sci. USA | volume=22 | issue=2 | pages=101–108 | pmid=16588050 | pmc=1076713| bibcode=1936PNAS...22..101N | doi-access=free}} {{pb}}
{{cite journal | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Continuous geometry | journal=Proceedings of the National Academy of Sciences of the United States of America| orig-year=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=[[Princeton University Press]] | series=Princeton Landmarks in Mathematics | isbn=978-0-691-05893-1 | mr=0120174 | year=1998| volume=22| issue=2| pages=92–100| doi=10.1073/pnas.22.2.92| pmid=16588062| pmc=1076712| doi-access=free}} {{pb}}
{{cite book | last1=von Neumann | first1=John | author1-link=John von Neumann | editor1-last=Taub | editor1-first=A. H. | title=Collected works. Vol. IV: Continuous geometry and other topics | url=https://books.google.com/books?id=HOTXAAAAMAAJ | publisher=Pergamon Press | location=Oxford | mr=0157874 | year=1962}} {{pb}}
{{cite journal | last1=von Neumann | first1=John| author1-link=John von Neumann | editor1-last=Halperin | editor1-first=Israel | title=Continuous geometries with a transition probability | orig-year=1937 | url=https://books.google.com/books?id=ZPkVGr8NXugC | mr=634656 | year=1981 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=34 | issue=252 | isbn=978-0-8218-2252-4 | doi=10.1090/memo/0252}}</ref> It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex [[projective geometry]], where instead of the [[Dimension (vector space)|dimension]] of a [[Linear subspace|subspace]] being in a discrete set <math>0, 1, ..., \mathit{n}</math> it can be an element of the [[unit interval]] <math>[0,1]</math>. Earlier, [[Karl Menger|Menger]] and Birkhoff had axiomatized [[Complex projective space|complex projective geometry]] in terms of the properties of its [[Linear subspace#Lattice of subspaces|lattice of linear subspaces]]. Von Neumann, following his work on rings of operators, weakened those [[axiom]]s to describe a broader class of lattices, the continuous geometries.

While the dimensions of the subspaces of projective geometries are a discrete set (the [[Natural number|non-negative integers]]), the dimensions of the elements of a continuous geometry can range continuously across the unit interval <math>[0,1]</math>. Von Neumann was motivated by his discovery of [[von Neumann algebra]]s with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the [[Von Neumann algebra#Projections|projections]] of the [[hyperfinite type II factor]].{{sfn|Macrae|1992|p=140}}<ref>{{cite journal|first=John |last=von Neumann|doi=10.1007/BF01782352|title= Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren|language=de |journal= [[Mathematische Annalen]]|volume=102 |issue=1 |year=1930|pages= 370–427|bibcode=1930MatAn.102..685E|s2cid=121141866}}. The original paper on von Neumann algebras.</ref>

In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of <math>\mathit{CG(F)}</math> (continuous-dimensional projective geometry over an arbitrary [[division ring]] <math>\mathit{F}\,</math>) in abstract language of lattice theory.{{sfn|Birkhoff|1958|pp=50-51}} Von Neumann provided an abstract exploration of dimension in completed [[Complemented lattice|complemented]] [[Modular lattice|modular]] topological lattices (properties that arise in the [[Linear subspace#Lattice of subspaces|lattices of subspaces]] of [[inner product space]]s): <blockquote>Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity.</blockquote>

For any integer <math>n > 3</math> every <math>\mathit{n}</math>-dimensional abstract projective geometry is [[Isomorphism|isomorphic]] to the subspace-lattice of an <math>\mathit{n}</math>-dimensional [[vector space]] <math>V_n(F)</math> over a (unique) corresponding  division ring <math>F</math>. This is known as the [[Veblen–Young theorem]]. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case.{{sfn|Birkhoff|1958|p=51}} This [[Continuous geometry#Coordinatization theorem|coordinatization theorem]] stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques.{{r|lashkhi1995}}<ref>{{cite journal |last1=Wehrung |first1=Friedrich |title=Von Neumann coordinatization is not first-order |journal=Journal of Mathematical Logic |date=2006 |volume=6 |issue=1 |pages=1–24 |doi=10.1142/S0219061306000499 |arxiv=math/0409250 |s2cid=39438451 }}</ref> Birkhoff described this theorem as follows: <blockquote>Any complemented modular lattice {{mvar|L}} having a "basis" of {{math|''n'' ≥ 4}} pairwise perspective elements, is isomorphic with the lattice {{math|ℛ(''R'')}} of all principal [[Ideal (ring theory)|right-ideals]] of a suitable [[Von Neumann regular ring|regular ring]] {{mvar|R}}. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe.{{sfn|Birkhoff|1958|p=52}}</blockquote>

This work required the creation of [[Von Neumann regular ring|regular rings]].<ref>{{cite book |last1=Goodearl |first1=Ken R. |title=Von Neumann Regular Rings |date=1979 |publisher=Pitman Publishing |isbn=0-273-08400-3 |page=ix}}</ref> A von Neumann regular ring is a [[Ring (mathematics)|ring]] where for every <math>a</math>, an element <math>x</math> exists such that <math>axa = a</math>.{{sfn|Birkhoff|1958|p=52}} These rings came from and have connections to his work on von Neumann algebras, as well as [[AW*-algebra]]s and various kinds of [[C*-algebra]]s.<ref>{{cite journal |last1=Goodearl |first1=Ken R. |title=Von Neumann regular rings: connections with functional analysis |journal=Bulletin of the American Mathematical Society |date=1981 |volume=4 |issue=2 |pages=125–134 |doi=10.1090/S0273-0979-1981-14865-5 |doi-access=free }}</ref>

Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding [[Distributive property|distributivity]] (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of [[metric lattice]]s.{{sfn|Birkhoff|1958|pp=52-53}}

Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication.{{sfn|Birkhoff|1958|pp=55-56}}

=== Mathematical statistics ===
Von Neumann made fundamental contributions to [[mathematical statistics]]. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically [[Normal distribution|normally]] distributed variables.<ref>{{cite journal|last=von Neumann |first=John|year=1941|title=Distribution of the ratio of the mean square successive difference to the variance|journal=[[Annals of Mathematical Statistics]]|volume=12|pages=367–395|jstor=2235951|doi=10.1214/aoms/1177731677|issue=4|doi-access=free}}</ref> This ratio was applied to the residuals from regression models and is commonly known as the [[Durbin–Watson statistic]]<ref name="jstor.org">{{cite journal |last1=Durbin |first1=J. |last2=Watson |first2=G. S. |year=1950 |title=Testing for Serial Correlation in Least Squares Regression, I |journal=[[Biometrika]] |volume=37 |pages=409–428 |pmid=14801065 |issue=3–4 |doi=10.2307/2332391 |jstor=2332391}}</ref> for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order [[Autoregressive model|autoregression]].<ref name="jstor.org"/>

Subsequently, [[Denis Sargan]] and [[Alok Bhargava]] extended the results for testing whether the errors on a regression model follow a Gaussian [[random walk]] (''i.e.'', possess a [[unit root]]) against the alternative that they are a stationary first order autoregression.<ref>{{cite journal |last1=Sargan |first1=J.D. |last2=Bhargava |first2=Alok |year=1983 |jstor=1912252 |title=Testing residuals from least squares regression for being generated by the Gaussian random walk|journal=[[Econometrica]]|volume=51|issue=1 |pages=153–174|doi=10.2307/1912252}}</ref>

=== Other work ===
In his early years, von Neumann published several papers related to set-theoretical real analysis and number theory.<ref>{{cite journal |last1=Rédei |first1=László |author-link1=László Rédei |title=Neumann János munkássága az algebrában és számelméletben |journal=Matematikai Lapok |date=1959 |volume=10 |pages=226–230 |language=Hungarian |url=http://real-j.mtak.hu/id/eprint/9389}}</ref> In a paper from 1925, he proved that for any dense sequence of points in <math>[0,1]</math>, there existed a rearrangement of those points that is [[Equidistributed sequence|uniformly distributed]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Egyenletesen sürü szämsorozatok (Gleichmässig dichte Zahlenfolgen) |journal=Mat. Fiz. Lapok |date=1925 |volume=32 |pages=32–40 |url=http://real-j.mtak.hu/7301/}}</ref><ref>{{cite journal |last1=Carbone |first1=Ingrid |last2=Volcic |first2=Aljosa |title=A von Neumann theorem for uniformly distributed sequences of partitions |journal=Rend. Circ. Mat. Palermo |date=2011 |volume=60 |issue=1–2 |pages=83–88 |doi=10.1007/s12215-011-0030-x|arxiv=0901.2531 |s2cid=7270857 }}</ref><ref>{{cite journal |last1=Niederreiter |first1=Harald |author1-link=Harald Niederreiter |title=Rearrangement theorems for sequences |journal=Astérisque |date=1975 |volume=24-25 |pages=243–261 |url=http://www.numdam.org/item/?id=AST_1975__24-25__243_0}}</ref> In 1926 his sole publication was on [[Heinz Prüfer|Prüfer's]] theory of [[Ideal number|ideal algebraic numbers]] where he found a new way of constructing them, thus extending Prüfer's theory to the [[Field (mathematics)|field]] of all [[algebraic number]]s, and clarified their relation to [[p-adic number]]s.<ref>{{cite journal |last1=von Neumann |first1=J. |title=Zur Prüferschen Theorie der idealen Zahlen |journal=Acta Szeged |date=1926 |volume=2 |pages=193–227 |url=http://acta.bibl.u-szeged.hu/13323/ |jfm=52.0151.02}}</ref>{{sfn|Ulam|1958|pp=9-10}}<ref>{{cite book |last1=Narkiewicz |first1=Wladyslaw |title=Elementary and Analytic Theory of Algebraic Numbers |series=Springer Monographs in Mathematics |date=2004 |publisher=Springer |isbn=978-3-662-07001-7  |doi=10.1007/978-3-662-07001-7 |edition=3rd |page=120}} {{pb}} {{cite book |last1=Narkiewicz |first1=Władysław |title=The Story of Algebraic Numbers in the First Half of the 20th Century: From Hilbert to Tate |series=Springer Monographs in Mathematics |date=2018 |publisher=Springer |doi=10.1007/978-3-030-03754-3 |isbn=978-3-030-03754-3 |page=144}}</ref><ref>{{cite journal |last=van Dantzig |first=D. |author-link=David van Dantzig |title=Nombres universels ou p-adiques avec une introduction sur l'algèbre topologique |journal=Annales scientifiques de l'École Normale Supérieure |date=1936 |volume=53 |pages=282–283 |language=fr |doi=10.24033/asens.858 |url=https://eudml.org/doc/81525|doi-access=free }}</ref><ref>{{cite book |last1=Warner |first1=Seth |title=Topological Rings |date=1993 |publisher=North-Hollywood |isbn=9780080872896 |url=https://www.elsevier.com/books/topological-rings/warner/978-0-444-89446-5 |page=428}}</ref> 
In 1928 he published two additional papers continuing with these themes. The first dealt with [[Partition of a set|partitioning]] an [[Interval (mathematics)|interval]] into [[Countable set|countably]] many [[Congruence relation|congruent]] [[subset]]s. It solved a problem of [[Hugo Steinhaus]] asking whether an interval is <math>\aleph_0</math>-divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are <math>\aleph_0</math>-divisible by translations (i.e. that these intervals can be decomposed into <math>\aleph_0</math> subsets that are congruent by translation).<ref>{{cite journal |last1=von Neumann |first1=J. |title=Die Zerlegung eines Intervalles in abzählbar viele kongruente Teilmengen |journal=Fundamenta Mathematicae |date=1928 |volume=11 |issue=1 |pages=230–238 |doi=10.4064/fm-11-1-230-238 |url=https://eudml.org/doc/211437 |jfm=54.0096.03|doi-access=free }}</ref>{{sfn|Wagon|Tomkowicz|2016|p=73}}{{sfn|Dyson|2013|p=156}}<ref>{{cite journal |last1=Harzheim |first1=Egbert |title=A Construction of Subsets of the Reals which have a Similarity Decomposition |journal=Order |date=2008 |volume=25 |issue=2 |pages=79–83 |doi=10.1007/s11083-008-9079-3|s2cid=45005704 }}</ref> His next paper dealt with giving a [[constructive proof]] without the [[axiom of choice]] that <math>2^{\aleph_0}</math> [[Algebraic independence|algebraically independent]] [[Real number|reals]] exist. He proved that <math>A_r = \textstyle\sum_{n=0}^{\infty} 2^{2^{[nr]}}\! \big/ \, 2^{2^{n^2}}</math> are algebraically independent for <math>r > 0</math>. Consequently, there exists a perfect algebraically independent set of reals the size of the [[Continuum (set theory)|continuum]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Ein System algebraisch unabhängiger Zahlen |journal=Mathematische Annalen |date=1928 |volume=99 |pages=134–141 |doi=10.1007/BF01459089 |url=https://eudml.org/doc/159249 |jfm=54.0096.02|s2cid=119788605 }}</ref><ref>{{cite journal |last1=Kuiper |first1=F. |last2=Popken |first2=Jan |title=On the So-Called von Neumann-Numbers |journal=Indagationes Mathematicae (Proceedings) |date=1962 |volume=65 |pages=385–390 |doi=10.1016/S1385-7258(62)50037-1|doi-access=free }}</ref><ref>{{cite journal |last1=Mycielski |first1=Jan |author1-link=Jan Mycielski |title=Independent sets in topological algebras |journal=Fundamenta Mathematicae |date=1964 |volume=55 |issue=2 |pages=139–147 |doi=10.4064/fm-55-2-139-147 |url=https://eudml.org/doc/213780 |doi-access=free }}</ref>{{sfn|Wagon|Tomkowicz|2016|p=114}} Other minor results from his early career include a proof of a [[maximum principle]] for the gradient of a minimizing function in the field of [[calculus of variations]],<!-- specifically proving the following theorem: Let <math>u: \mathbb{R}^n \rightarrow \mathbb{R}</math> be a [[Lipschitz continuity|Lipschitz function]] with constant <math>K</math>, and <math>\Omega</math> an open and bounded set in <math>\mathbb{R}^n</math>. If <math>u</math> is a minimum for <math>F</math> in <math>Lip_K(\Omega)</math>, then <math>\sup_{x \in \Omega, y \in \delta\Omega} \frac{|u(x) - u(y)|}{|x - y|} = \sup_{x \neq y \in \Omega} \frac{|u(x) - u(y)|}{|x - y|}</math> (unnecessary detail for a minor result) --><ref>{{cite journal |last1=von Neumann |first1=J. |title=Über einen Hilfssatz der Variationsrechnung |journal=Abhandlungen Hamburg |date=1930 |volume=8 |pages=28–31 |url=https://abhandlungen.math.uni-hamburg.de/en/archiv.php?vol=8 |jfm=56.0440.04}}</ref><ref>{{cite journal |last=Miranda |first=Mario |title=Maximum principles and minimal surfaces |journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |date=1997 |volume=4, 25 |issue=3–4 |pages=667–681 |url=http://www.numdam.org/item/ASNSP_1997_4_25_3-4_667_0/}}</ref><ref>{{cite book |last1=Gilbarg |first1=David |last2=Trudinger |first2=Neil S. |author1-link=David Gilbarg |author2-link=Neil Trudinger |title=Elliptic Partial Differential Equations of Second Order |date=2001 |publisher=Springer |doi=10.1007/978-3-642-61798-0 |isbn=978-3-642-61798-0 |edition=2 |url=https://link.springer.com/book/10.1007/978-3-642-61798-0 |page=316}}</ref><ref>{{cite book |last1=Ladyzhenskaya |first1=Olga A. |author1-link=Olga Ladyzhenskaya |last2=Ural'tseva |first2=Nina N. |author2-link=Nina Uraltseva |title=Linear and Quasilinear Elliptic Equations |date=1968 |publisher=Academic Press |isbn=978-1483253329 |pages= 14, 243}}</ref> and a small simplification of [[Hermann Minkowski]]'s theorem for linear forms in [[Geometry of numbers|geometric number theory]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Zum Beweise des Minkowskischen Stazes über Linearformen |journal=Mathematische Zeitschrift |date=1929 |volume=30 |pages=1–2 |doi=10.1007/BF01187748 |url=https://eudml.org/doc/168103 |jfm=55.0065.04|s2cid=123066944 }}</ref><ref>{{cite book |last=Koksma |first=J. F. |author-link=Jurjen Ferdinand Koksma |title=Diophantische Approximationen |date=1974|orig-date=1936 |publisher=Springer |doi=10.1007/978-3-642-65618-7 |isbn=978-3-642-65618-7 |language=German |url=https://link.springer.com/book/10.1007/978-3-642-65618-7 |page=15}}</ref>{{sfn|Ulam|1958|pp=10,23}}
Later in his career together with [[Pascual Jordan]] and [[Eugene Wigner]] he wrote a foundational paper classifying all [[Dimension (vector space)|finite-dimensional]] [[Jordan algebra#Formally real Jordan algebras|formally real Jordan algebras]] and discovering the [[Albert algebra]]s while attempting to look for a better [[Mathematical formulation of quantum mechanics|mathematical formalism for quantum theory]].<ref>{{cite web |last1=Baez |first1=John |author-link1=John C. Baez |title=State-Observable Duality (Part 2) |url=https://golem.ph.utexas.edu/category/2010/11/stateobservable_duality_part_2.html |website=The n-Category Café |access-date=20 August 2022}}</ref><ref>{{cite book |last1=McCrimmon |first1=Kevin |author-link1=Kevin McCrimmon |title=A Taste of Jordan Algebras |series=Universitext |date=2004 |publisher=Springer |location=New York |doi=10.1007/b97489 |isbn=978-0-387-21796-3 |url=https://link.springer.com/book/10.1007/b97489 |page=68}}</ref> In 1936 he attempted to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras<ref>{{cite journal |last=Rédei |first=Miklós |title=Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead) |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |date=1996 |volume=27 |issue=4 |pages=493–510 |doi=10.1016/S1355-2198(96)00017-2 |bibcode=1996SHPMP..27..493R |url=https://www.sciencedirect.com/science/article/abs/pii/S1355219896000172 }}</ref> in a paper investigating the infinite-dimensional case; he planned to write at least one further paper on the topic but never did.<ref>{{cite journal |last1=Wang |first1=Shuzhou |last2=Wang |first2=Zhenhua |title=Operator means in JB-algebras |journal=Reports on Mathematical Physics |date=2021 |volume=88 |issue=3 |page=383 |doi=10.1016/S0034-4877(21)00087-2 |arxiv=2012.13127 |bibcode=2021RpMP...88..383W |s2cid=229371549 }}</ref> Nevertheless, these axioms formed the basis for further investigations of algebraic quantum mechanics started by [[Irving Segal]].<ref>{{cite book |last=Landsman |first=Nicolaas P. |year=2009 |chapter=Algebraic Quantum Mechanics |editor-last1=Greenberger |editor-first1=Daniel |editor-link1=Daniel Greenberger |editor-last2=Hentschel |editor-first2=Klaus |editor-link2=Klaus Hentschel |editor-last3=Weinert |editor-first3=Friedel |title=Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy |publisher=Springer |isbn=978-3-540-70626-7 |doi=10.1007/978-3-540-70626-7 |pages=6–7}}</ref><ref>{{cite encyclopedia |last1=Kronz |first1=Fred |last2=Lupher |first2=Tracy |year=2021 |title=Quantum Theory and Mathematical Rigor |url=https://plato.stanford.edu/entries/qt-nvd/ |edition=Winter 2021 |editor-last=Zalta |editor-first=Edward N. |encyclopedia=Stanford Encyclopedia of Philosophy |publisher=Stanford University |access-date=2022-12-21}}</ref>