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On local representations of von Neumann algebras

Author(s): Florin Pop
Journal: Proc. Amer. Math. Soc. 132 (2004), 3569-3576.
MSC (2000): Primary 46L10, 46K10
Posted: July 22, 2004
MathSciNet review: 2084078
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Abstract | References | Similar articles | Additional information

Abstract: We prove that every bounded, linear, 2-local Hilbert space representation of a von Neumann algebra is a representation. In contrast, 1-local representations may fail to be multiplicative, even at the 2 by 2 matrix algebra level.

References:

1.: M. Bresar and P. Semrl, Mappings which preserve idempotents,local automorphisms, and local derivations, Canad. J. Math. 45 (1993), 483-496. MR 1222512 (94k:47054)
2.: M. Bresar and P. Semrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (1995), 101-108. MR 1318418 (96i:47058)
3.: R. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996), 76-92. MR 1367625 (96m:46128)
4.: R. Crist, Local automorphisms, Proc. Amer. Math. Soc. 128 (2000), 1409-1415. MR 1657786 (2001a:47078)
5.: L. Ge and S. Popa, On some decomposition properties of factors of type $\rm {II}_1$ , Duke Math. J. 94 (1998), 79-101. MR 1635904 (99j:46070)
6.: D. Han and S. Wei, Local derivations of nest algebras, Proc. Amer. Math. Soc. 123 (1995), 3095-3100. MR 1246521 (95m:47077)
7.: R. V. Kadison, Derivations of operator algebras, Ann. of Math. 83 (1966), 280-293. MR 0193527 (33:1747)
8.: R. V. Kadison, Local derivations, J. Algebra 130 (1990), 494-509. MR 1051316 (91f:46092)
9.: R. V. Kadison and J. R Ringrose, Fundamentals of the theory of operator algebras, vol. II, Academic Press, 1986. MR 0859186 (88d:46106)
10.: E. Kirchberg, The derivation and the similarity problem are equivalent, J. Operator Theory. 36 (1996), 59-62. MR 1417186 (97f:46108)
11.: D. R. Larson and A. R. Sourour, Local derivations and local automorphisms of $\mathcal{B}$ (X), Proc. Sympos. Pure Math. 51, Part 2, Providence, Rhode Island 1990, pp. 187-194. MR 1077437 (91k:47106)
12.: L. Molnar, Local automorphisms of operator algebras on Banach spaces, Proc. Amer. Math. Soc. 131 (2003), 1867-1874. MR 1955275 (2003j:47050)
13.: L. Molnar and P. Semrl, Local automorphisms of the unitary group and the general linear group on a Hilbert space, Expo. Math. 18 (2000), 231-238. MR 1763889 (2001a:47083)
14.: C. Pearcy, -algebras with a single generator, Proc. Amer. Math. Soc. 13 (1962), 831-832. MR 0152904 (27:2875)
15.: S. Popa, Notes on Cartan subalgebras in type $\rm {II}_1$ factors, Math. Scand. 57 (1985), 171-188. MR 0815434 (87f:46114)
16.: S. Sakai, Derivations of -algebras, Ann. of Math. 83 (1966), 273-279. MR 0193528 (33:1748)
17.: P. Semrl, Local automorphisms and derivations on $\mathcal{B}$ (H), Proc. Amer. Math. Soc. 125 (1997), 2677-2680. MR 1415338 (98e:46082)
18.: W. Wogen, On generators for von Neumann algebras, Bull. Amer. Math. Soc. 75 (1969), 95-99. MR 0236725 (38:5020)

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