Operators preserving ideals in C*-algebras

V. Shul'Man

Studia Mathematica (1994)

  • Volume: 109, Issue: 1, page 67-72
  • ISSN: 0039-3223

Abstract

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The aim of this paper is to prove that derivations of a C*-algebra A can be characterized in the space of all linear continuous operators T : A → A by the conditions T(1) = 0, T(L∩R) ⊂ L + R for any closed left ideal L and right ideal R. As a corollary we get an extension of the result of Kadison [5] on local derivations in W*-algebras. Stronger results of this kind are proved under some additional conditions on the cohomologies of A.

How to cite

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Shul'Man, V.. "Operators preserving ideals in C*-algebras." Studia Mathematica 109.1 (1994): 67-72. <http://eudml.org/doc/216061>.

@article{ShulMan1994,
abstract = {The aim of this paper is to prove that derivations of a C*-algebra A can be characterized in the space of all linear continuous operators T : A → A by the conditions T(1) = 0, T(L∩R) ⊂ L + R for any closed left ideal L and right ideal R. As a corollary we get an extension of the result of Kadison [5] on local derivations in W*-algebras. Stronger results of this kind are proved under some additional conditions on the cohomologies of A.},
author = {Shul'Man, V.},
journal = {Studia Mathematica},
keywords = {C*-algebra; derivation; reflexivity; derivations of a -algebra; derivations in -algebras},
language = {eng},
number = {1},
pages = {67-72},
title = {Operators preserving ideals in C*-algebras},
url = {http://eudml.org/doc/216061},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Shul'Man, V.
TI - Operators preserving ideals in C*-algebras
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 1
SP - 67
EP - 72
AB - The aim of this paper is to prove that derivations of a C*-algebra A can be characterized in the space of all linear continuous operators T : A → A by the conditions T(1) = 0, T(L∩R) ⊂ L + R for any closed left ideal L and right ideal R. As a corollary we get an extension of the result of Kadison [5] on local derivations in W*-algebras. Stronger results of this kind are proved under some additional conditions on the cohomologies of A.
LA - eng
KW - C*-algebra; derivation; reflexivity; derivations of a -algebra; derivations in -algebras
UR - http://eudml.org/doc/216061
ER -

References

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  1. [1] C. A. Akemann, The general Stone-Weierstrass problem, J. Funct. Anal. 4 (1969), 277-294. Zbl0177.17603
  2. [2] W. Arveson, Interpolation problems in nest algebras, ibid. 20 (1975), 208-233. Zbl0309.46053
  3. [3] J. W. Bunce and W. L. Paschke, Derivations on a C*-algebra and its dual, ibid. 37 (1980), 235-247. 
  4. [4] U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. Zbl0529.46041
  5. [5] R. Kadison, Local derivations, J. Algebra 130 (1990), 494-509. Zbl0751.46041
  6. [6] J. Kraus and D. R. Larson, Reflexivity and distance formulae, Proc. London Math. Soc. 53 (1986), 340-356. Zbl0623.47046
  7. [7] A. J. Loginov and V. S. Shul'man, Hereditary and intermediate reflexivity of W*-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 1260-1273 (in Russian). Zbl0327.46073
  8. [8] V. S. Shul'man, On the geometry of some pairs of subspaces in C*-algebras, in: Spectral Theory of Operators and its Applications, No. 6, Elm, Baku, 1985, 196-216 (in Russian). 

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