On local automorphisms and mappings that preserve idempotents

Matej Brešar; Peter Šemrl

Studia Mathematica (1995)

  • Volume: 113, Issue: 2, page 101-108
  • ISSN: 0039-3223

Abstract

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Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Automorphisms and antiautomorphisms are the only bijective linear mappings θ of B(H) with the property that θ(P) is an idempotent whenever P ∈ B(H) is. In case H is separable and infinite-dimensional, every local automorphism of B(H) is an automorphism.

How to cite

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Brešar, Matej, and Šemrl, Peter. "On local automorphisms and mappings that preserve idempotents." Studia Mathematica 113.2 (1995): 101-108. <http://eudml.org/doc/216163>.

@article{Brešar1995,
abstract = {Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Automorphisms and antiautomorphisms are the only bijective linear mappings θ of B(H) with the property that θ(P) is an idempotent whenever P ∈ B(H) is. In case H is separable and infinite-dimensional, every local automorphism of B(H) is an automorphism.},
author = {Brešar, Matej, Šemrl, Peter},
journal = {Studia Mathematica},
keywords = {algebra of all bounded linear operators on a Hilbert space; antiautomorphisms; idempotent; local automorphism},
language = {eng},
number = {2},
pages = {101-108},
title = {On local automorphisms and mappings that preserve idempotents},
url = {http://eudml.org/doc/216163},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Brešar, Matej
AU - Šemrl, Peter
TI - On local automorphisms and mappings that preserve idempotents
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 2
SP - 101
EP - 108
AB - Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Automorphisms and antiautomorphisms are the only bijective linear mappings θ of B(H) with the property that θ(P) is an idempotent whenever P ∈ B(H) is. In case H is separable and infinite-dimensional, every local automorphism of B(H) is an automorphism.
LA - eng
KW - algebra of all bounded linear operators on a Hilbert space; antiautomorphisms; idempotent; local automorphism
UR - http://eudml.org/doc/216163
ER -

References

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  1. [1] L. B. Beasley and N. J. Pullman, Linear operators preserving idempotent matrices over fields, Linear Algebra Appl. 146 (1991), 7-20. Zbl0718.15004
  2. [2] M. Brešar, Characterizations of derivations on some normed algebras with involution, J. Algebra 152 (1992), 454-462. Zbl0769.16015
  3. [3] M. Brešar and P. Šemrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (1993), 483-496. Zbl0796.15001
  4. [4] G. H. Chan, M. H. Lim and K. K. Tan, Linear preservers on matrices, Linear Algebra Appl. 93 (1987), 67-80. Zbl0619.15003
  5. [5] P. R. Chernoff, Representations, automorphisms and derivations of some operator algebras, J. Funct. Anal. 12 (1973), 275-289. Zbl0252.46086
  6. [6] M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math. 9 (1940), 97-105. Zbl0061.25301
  7. [7] I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, 1969. Zbl0232.16001
  8. [8] N. Jacobson and C. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479-502. Zbl0039.26402
  9. [9] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255-261. Zbl0589.47003
  10. [10] R. V. Kadison, Local derivations, J. Algebra 130 (1990), 494-509. Zbl0751.46041
  11. [11] 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. Zbl0713.47045
  12. [12] C. Pearcy and D. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453-465. 

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