The set of automorphisms of B(H) is topologically reflexive in B(B(H))

Lajos Molnár

Studia Mathematica (1997)

  • Volume: 122, Issue: 2, page 183-193
  • ISSN: 0039-3223

Abstract

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The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence ( Φ n ) of automorphisms of B(H) (depending on A) such that Φ ( A ) = l i m n Φ n ( A ) . Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).

How to cite

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Molnár, Lajos. "The set of automorphisms of B(H) is topologically reflexive in B(B(H))." Studia Mathematica 122.2 (1997): 183-193. <http://eudml.org/doc/216369>.

@article{Molnár1997,
abstract = {The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence $(Φ_n)$ of automorphisms of B(H) (depending on A) such that $Φ(A)= lim_n Φ_n(A)$. Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).},
author = {Molnár, Lajos},
journal = {Studia Mathematica},
keywords = {reflexivity; automorphism; Jordan homomorphism; automatic surjectivity; automorphisms; surjective isometries},
language = {eng},
number = {2},
pages = {183-193},
title = {The set of automorphisms of B(H) is topologically reflexive in B(B(H))},
url = {http://eudml.org/doc/216369},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Molnár, Lajos
TI - The set of automorphisms of B(H) is topologically reflexive in B(B(H))
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 2
SP - 183
EP - 193
AB - The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence $(Φ_n)$ of automorphisms of B(H) (depending on A) such that $Φ(A)= lim_n Φ_n(A)$. Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).
LA - eng
KW - reflexivity; automorphism; Jordan homomorphism; automatic surjectivity; automorphisms; surjective isometries
UR - http://eudml.org/doc/216369
ER -

References

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  1. [Bre] M. Brešar, Characterizations of derivations on some normed algebras with involution, J. Algebra 152 (1992), 454-462. Zbl0769.16015
  2. [BS1] M. Brešar and P. Šemrl, Mappings which preserve idempotents, local automorphisms and local derivations, Canad. J. Math. 45 (1993), 483-496. Zbl0796.15001
  3. [BS2] M. Brešar and P. Šemrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (1995), 101-108. Zbl0835.47020
  4. [FMS] C. K. Fong, C. R. Miers and A. R. Sourour, Lie and Jordan ideals of operators on Hilbert space, Proc. Amer. Math. Soc. 84 (1982), 516-520. Zbl0509.47035
  5. [Her] I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331-341. 
  6. [JR] N. Jacobson and C. Rickart, Jordan homomorphisms of rings, ibid. 69 (1950), 479-502. Zbl0039.26402
  7. [Kad1] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. Zbl0045.06201
  8. [Kad2] R. V. Kadison, Local derivations, J. Algebra 130 (1990), 494-509. Zbl0751.46041
  9. [LS] D. R. Larson and A. R. Sourour, Local derivations and local automorphisms of B(X), in: Proc. Sympos. Pure Math. 51, Part 2, Providence, R.I., 1990, 187-194. 
  10. [LoS] A. I. Loginov and V. S. Shul'man, Hereditary and intermediate reflexivity of W*-algebras, Izv. Akad. Nauk SSSR 39 (1975), 1260-1273 (in Russian); English transl.: Math. USSR-Izv. 9 (1975), 1189-1201. 
  11. [Pal] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. I, Encyclopedia Math. Appl. 49, Cambridge University Press, 1994. 
  12. [PS] H. Porta and J. T. Schwartz, Representations of the algebra of all operators in Hilbert space, and related analytic function algebras, Comm. Pure Appl. Math. 20 (1967), 457-492. Zbl0148.37702
  13. [RD] B. Russo and H. A. Dye, A note on unitary operators in C*-algebras, Duke Math. J. 33 (1966), 413-416. Zbl0171.11503
  14. [Shu] V. S. Shul'man, Operators preserving ideals in C*-algebras, Studia Math. 109 (1994), 67-72. 

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