VON NEUMANN, John. Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren.
Berlin: Julius Springer, 1929. First edition.

Offprint of one of his most important papers in which he “introduced the concept of a ring of operators that later became known as a von Neumann algebra.” (Bradley: Modern Mathematics, p.89). “In two papers [Math. Ann., 102, 49-131, and 370-427, 1929] von Neumann presented an axiomatic approach to Hilbert space and to operators in Hilbert space” (Morris Kline). “Von Neumann’s most famous work in theoretical physics is his axiomatization of quantum mechanics. When he began work in that field in 1927, the methods used by its founders were hard to formulate in precise mathematical terms; ‘operator’ on ‘functions’ were handled without much consideration of their domain or definition to their topological properties: and it was blithely assumed that such ‘operators,’ when self-adjoint, could always be ‘diagonalized’ (as in the finite dimensional case), at the expense of introducing ‘Dirac functions’ as ‘eigenvectors.’ Von Neumann showed that mathematical rigor could be restored by taking as basic axioms the assumptions that the states of a physical system were points of a Hilbert space and that the measurable quantities were Hermitian (generally unbounded) operators densely defined in that space. This formalism, the practical use of which became available after von Neumann had developed the spectral theory of unbounded Hermitian operators [in the first of the two 1929 papers mentioned by Kline], has survived subsequent developments of quantum mechanics and is still the basis of non relativistic quantum theory; with the introduction of the theory of distributions, it has even become possible to interpret its results in a way similar to Dirac’s original intuition.” (D.S.B. XIV, p.91). The spectral theory of operators in Hilbert space “is by far the dominant theme in Von Neumann’s work. For twenty years he was the undisputed master in this area, which contains what is now considered his most profound and most original creation, the theory of rings of operators [introduced in the offered paper]. … Most of von Neumann’s results on unbounded operators in Hilbert space were independently discovered a little later by M.H.Stone. But von Neumann’s ideas on rings of operators broke entirely new ground. He was well acquainted with the noncommutative algebra beautifully developed by Emmy Noether and E. Artin in the 1920’s and he realized how these concepts simplified and illuminated the theory of matrices. This probably provided the motivation for extending such concepts to algebras consisting of (bounded) operators in a given separable Hilbert space, to which he gave the vague name ‘rings of operators’ and which are now known as ‘von Neumann algebras.’ He introduced their theory in the same year [the offered paper] as his first paper on unbouned operators [the first of the two mentioned by Kline, see above].” (D.S.B. XIV, p.91) For a detailed exposition of von Neumann’s two 1929 papers see Morris Kline’s ‘Mathematical Thought from Ancient to Modern Times’, p.1092-95.

8vo: 233 x 157 mm. Offprint from: Mathematischen Annalen, vol. 102, heft 3, pp. 370-427.Original printed wrappers, some very light foxing to the right margin of the front wrapper, and 5 mm closed tear to the upper right corner, top and bottom of spine strip with a little chipping, in all fine a fine copy. Offprints of von Neumann's papers before 1930, prior to his move to Princeton, are rare.