Abstract
The initial development of the theory of von Neumann algebras, proposed by von Neumann [12] and carried out by him in collaboration with F.J. Murray [9,10,11,13] can be viewed as consisting of two parts, an “algebraic theory” and a “spatial theory.” In the algebraic theory, the results refer to the von Neumann algebra R and make no reference to the corn-mutant; in the spatial theory, the results involve the commutant either explicitly or implicitly. Recognizing this mathematical dichotomy, Kaplan-sky [7,8] studied the algebraic structure of von Neumann algebras, without reference to their action on a space, isolating and putting in sharp focus many of the natural techniques that are basic to our subject. Of course, Murray and von Neumann had taken the algebraic theory to an advanced stage in their own way [9,10,11,13].
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References
A. Connes, Classification of injective factors, Cases II1, II∞, IIλ, λ ≠ 1, Ann. of Math. 104 (1976), 73–115.
J. Dixmier, Les Algè bres d’Opérateurs dans l’Espace Hilbertien, Gauthier-Villars, Paris, 1957.
U. Haagerup, A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space, J. Fnal. Anal. 62 (1985), 160–201.
R. Kadison, Isomorphisms of factors of infinite type, Canad. J. Math. 7 (1955), 322–327.
R. Kadison, Centralizers and diagonalizing states, in preparation.
R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, Orlando, Vol. I, 1983, Vol. II, 1986.
I. Kaplansky, Projections in Banach Algebras, Ann. of Math. 53 (1951), 235–249.
I. Kaplansky, Algebras of type I, Ann. of Math. 56 (1952), 460–472.
F. Murray and J. von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116–229.
F. Murray and J. von Neumann, On rings of operators, II, Trans. Amer. Math. Soc. 41 (1937), 208–248.
F. Murray and J. von Neumann, On rings of operators, IV, Ann. of Math. 44 (1943), 716–808.
J. von Neumann, Zur Algebra der Funktional operation en und Theorie der normalen Operatoren, Math. Ann. 102 (1930), 49–131.
J. von Neumann, On rings of operators, III, Ann. of Math. 41 (1940), 94–161.
S. Popa, A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, J. Operator Theory 16 (1986), 261–272.
S. Sakai, A Radon-Nikodym theorem in W*-algebras, Bull. Amer. Math. Soc. 71 (1965), 149–151.
M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and Its Applications, LNM Vol. 128, Springer-Verlag, Heidelberg, 1970.
M. Tomita, Standard forms of von Neumann algebras, Fifth Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, 1967.
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Kadison, R.V. (1991). Reflections Relating a von Neumann Algebra and Its Commutant. In: Araki, H., Kadison, R.V. (eds) Mappings of Operator Algebras. Progress in Mathematics, vol 84. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0453-4_18
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DOI: https://doi.org/10.1007/978-1-4612-0453-4_18
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