Algebra von Neumann

Dalam matematik, algebra von Neumann atau algebra-W* ialah algebra-* pengendali yang terikat dengan ruang Hilbert yang tertutup dalam topologi operator yang lemah dan mengandungi identiti pengendali. Ini ialah jenis khas algebra-C*.

Algebra Von Neumann pada awalnya diperkenalkan oleh John von Neumann, didorong oleh kajiannya mengenai pengendali tunggal, perwakilan kumpulan, teori ergodik dan mekanik kuantum. Teorem penggantian berganda menunjukkan bahawa definisi analitik sama dengan definisi algebra yang murni.

Rujukan

• Araki, H.; Woods, E. J. (1968), "A classification of factors", Publ. Res. Inst. Math. Sci. Ser. A, 4 (1): 51–130, doi:10.2977/prims/1195195263
• Blackadar, B. (2005), Operator algebras, Springer, ISBN 3-540-28486-9, corrected manuscript (PDF), 2013
• Connes, A. (1976), "Classification of Injective Factors", Annals of Mathematics, Second Series, 104 (1): 73–115, doi:10.2307/1971057, JSTOR 1971057
• Connes, A. (1994), Non-commutative geometry, Academic Press, ISBN 0-12-185860-X.
• Dixmier, J. (1981), Von Neumann algebras, ISBN 0-444-86308-7 (A translation of Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars, the first book about von Neumann algebras.)
• Jones, V.F.R. (2003), von Neumann algebras (PDF); incomplete notes from a course.
• Kostecki, R.P. (2013), W*-algebras and noncommutative integration, arXiv:1307.4818, Bibcode:2013arXiv1307.4818P.
• McDuff, Dusa (1969), "Uncountably many II₁ factors", Annals of Mathematics, Second Series, 90 (2): 372–377, doi:10.2307/1970730, JSTOR 1970730
• Murray, F. J. (2006), "The rings of operators papers", The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., 50, Providence, RI.: Amer. Math. Soc., m/s. 57–60, ISBN 0-8218-4219-6 A historical account of the discovery of von Neumann algebras.
• Murray, F.J.; von Neumann, J. (1936), "On rings of operators", Annals of Mathematics, Second Series, 37 (1): 116–229, doi:10.2307/1968693, JSTOR 1968693. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I.
• Murray, F.J.; von Neumann, J. (1937), "On rings of operators II", Trans. Amer. Math. Soc., American Mathematical Society, 41 (2): 208–248, doi:10.2307/1989620, JSTOR 1989620. This is a continuation of the previous paper, that studies properties of the trace of a factor.
• Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV", Annals of Mathematics, Second Series, 44 (4): 716–808, doi:10.2307/1969107, JSTOR 1969107. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II₁ are isomorphic.
• Powers, Robert T. (1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings", Annals of Mathematics, Second Series, 86 (1): 138–171, doi:10.2307/1970364, JSTOR 1970364
• Sakai, S. (1971), C*-algebras and W*-algebras, Springer, ISBN 3-540-63633-1
• Schwartz, Jacob (1967), W-* Algebras, ISBN 0-677-00670-5
• Shtern, A.I. (2001), "von Neumann algebra", dalam Hazewinkel, Michiel (penyunting), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Takesaki, M. (1979), Theory of Operator Algebras I, II, III, ISBN 3-540-42248-X
• von Neumann, J. (1930), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren", Math. Ann., 102 (1): 370–427, Bibcode:1930MatAn.102..685E, doi:10.1007/BF01782352. The original paper on von Neumann algebras.
• von Neumann, J. (1936), "On a Certain Topology for Rings of Operators", Annals of Mathematics, Second Series, 37 (1): 111–115, doi:10.2307/1968692, JSTOR 1968692. This defines the ultrastrong topology.
• von Neumann, J. (1938), "On infinite direct products", Compos. Math., 6: 1–77. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.
• von Neumann, J. (1940), "On rings of operators III", Annals of Mathematics, Second Series, 41 (1): 94–161, doi:10.2307/1968823, JSTOR 1968823. This shows the existence of factors of type III.
• von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings", Annals of Mathematics, Second Series, 44 (4): 709–715, doi:10.2307/1969106, JSTOR 1969106. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.
• von Neumann, J. (1949), "On Rings of Operators. Reduction Theory", Annals of Mathematics, Second Series, 50 (2): 401–485, doi:10.2307/1969463, JSTOR 1969463. This discusses how to write a von Neumann algebra as a sum or integral of factors.
• von Neumann, John (1961), Taub, A.H. (penyunting), Collected Works, Volume III: Rings of Operators, NY: Pergamon Press. Reprints von Neumann's papers on von Neumann algebras.
• Wassermann, A. J. (1991), Operators on Hilbert space