Multinormed 𝑊*-algebras and unbounded operators

Dosi, Anar

doi:10.1090/S0002-9939-2012-11358-9

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

Multinormed $W^{\ast}$ -algebras and unbounded operators

Author: Anar Dosi
Journal: Proc. Amer. Math. Soc. 140 (2012), 4187-4202
MSC (2010): Primary 46K10; Secondary 47L25, 47L60
Posted: April 3, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate multinormed $W^{\ast }$ -algebras in terms of the central topologies of $W^{\ast }$ -algebras. The main result asserts that each multinormed $W^{\ast }$ -algebra can be realized as a local von Neumann algebra on a certain domain in a Hilbert space. Moreover, it admits the predual (unique up to an isometry), which is the $\ell _{1}$ -normed space. In the normed case the assertion is reduced to the known Sakai theorem.

1. Jean-Pierre Antoine, Atsushi Inoue, and Camillo Trapani, Partial *-algebras and their operator realizations, Mathematics and its Applications, vol. 553, Kluwer Academic Publishers, Dordrecht, 2002. MR 1947892 (2005d:47116)
2. Fabio Bagarello, Atsushi Inoue, and Camillo Trapani, Bicommutants of reduced unbounded operator algebras, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3709–3716. MR 2529878 (2010h:47148), http://dx.doi.org/10.1090/S0002-9939-09-10023-0
3. David P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), no. 1, 15–30. MR 1145913 (93d:47083)
4. John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
5. Anar Dosiev, Local operator spaces, unbounded operators and multinormed 𝐶*-algebras, J. Funct. Anal. 255 (2008), no. 7, 1724–1760. MR 2442081 (2010a:46133), http://dx.doi.org/10.1016/j.jfa.2008.07.006
6. Anar Dosiev, Quantized moment problem, C. R. Math. Acad. Sci. Paris 344 (2007), no. 10, 627–630 (English, with English and French summaries). MR 2334073 (2008f:46086), http://dx.doi.org/10.1016/j.crma.2007.03.022
7. Anar Dosi, Local operator algebras, fractional positivity and the quantum moment problem, Trans. Amer. Math. Soc. 363 (2011), no. 2, 801–856. MR 2728586 (2011j:47229), http://dx.doi.org/10.1090/S0002-9947-2010-05145-1
8. Anar Dosi, Noncommutative Mackey theorem, Internat. J. Math. 22 (2011), no. 4, 535–544. MR 2794460 (2012i:47094), http://dx.doi.org/10.1142/S0129167X11006891
9. Anar Dosi, Quotients of quantum bornological spaces, Taiwanese J. Math. 15 (2011), no. 3, 1287–1303. MR 2829912 (2012h:47150)
10. Anar Dosi, Quantum duality, unbounded operators, and inductive limits, J. Math. Phys. 51 (2010), no. 6, 063511, 43. MR 2676488 (2011k:46081), http://dx.doi.org/10.1063/1.3419771
11. Edward G. Effros and Corran Webster, Operator analogues of locally convex spaces, Operator algebras and applications (Samos, 1996) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 495, Kluwer Acad. Publ., Dordrecht, 1997, pp. 163–207. MR 1462680 (99b:46081)
12. Edward G. Effros and Soren Winkler, Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems, J. Funct. Anal. 144 (1997), no. 1, 117–152. MR 1430718 (98e:46066), http://dx.doi.org/10.1006/jfan.1996.2958
13. Maria Fragoulopoulou, On locally 𝑊*-algebras, Yokohama Math. J. 34 (1986), no. 1-2, 35–51. MR 886056 (88g:46084)
14. Maria Fragoulopoulou, Topological algebras with involution, North-Holland Mathematics Studies, vol. 200, Elsevier Science B.V., Amsterdam, 2005. MR 2172581 (2006m:46067)
15. Rachid El Harti and Gábor Lukács, Bounded and unitary elements in pro-𝐶*-algebras, Appl. Categ. Structures 14 (2006), no. 2, 151–164. MR 2247450 (2007b:46134), http://dx.doi.org/10.1007/s10485-006-9012-0
16. A. Ya. Khelemskiĭ, Banakhovy i polinormirovannye algebry: obshchaya teoriya, predstavleniya, gomologii, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1031991 (91h:46001)
17. Henri Hogbe-Nlend, Bornologies and functional analysis, North-Holland Publishing Co., Amsterdam, 1977. Introductory course on the theory of duality topology-bornology and its use in functional analysis; Translated from the French by V. B. Moscatelli; North-Holland Mathematics Studies, Vol. 26; Notas de Matemática, No. 62. [Notes on Mathematics, No. 62]. MR 0500064 (58 #17774)
18. Atsushi Inoue, Locally 𝐶*-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A 25 (1971), 197–235. MR 0305089 (46 #4219)
19. Atsushi Inoue, Tomita-Takesaki theory in algebras of unbounded operators, Lecture Notes in Mathematics, vol. 1699, Springer-Verlag, Berlin, 1998. MR 1725388 (2000j:47129)
20. Maria Joiţa, Locally von Neumann algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(90) (1999), no. 1, 51–64. MR 1880655 (2002k:46157)
21. Maria Joiţa, Locally von Neumann algebras. II, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 1, 83–94. MR 1905709 (2003e:46107), http://dx.doi.org/10.1007/BF02871453
22. S. S. Kutateladze, Fundamentals of functional analysis, Kluwer Texts in the Mathematical Sciences, vol. 12, Kluwer Academic Publishers Group, Dordrecht, 1996. Translated from the second (1995) edition. MR 1384068 (97a:46001)
23. Gerard J. Murphy, 𝐶*-algebras and operator theory, Academic Press Inc., Boston, MA, 1990. MR 1074574 (91m:46084)
24. N. Christopher Phillips, Inverse limits of 𝐶*-algebras, J. Operator Theory 19 (1988), no. 1, 159–195. MR 950831 (90c:46090)
25. Shôichirô Sakai, 𝐶*-algebras and 𝑊*-algebras, Springer-Verlag, New York, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. MR 0442701 (56 #1082)
26. Helmut H. Schaefer, Topological vector spaces, Springer-Verlag, New York, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR 0342978 (49 #7722)
27. Konrad Schmüdgen, Über LMC-Algebren, Math. Nachr. 68 (1975), 167–182 (German). MR 0394227 (52 #15030)
28. Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR 1056697 (91f:47062)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|