Finite rank elements in semisimple Banach algebras

Matej Brešar; Peter Šemrl

Studia Mathematica (1998)

  • Volume: 128, Issue: 3, page 287-298
  • ISSN: 0039-3223

Abstract

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Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.

How to cite

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Brešar, Matej, and Šemrl, Peter. "Finite rank elements in semisimple Banach algebras." Studia Mathematica 128.3 (1998): 287-298. <http://eudml.org/doc/216487>.

@article{Brešar1998,
abstract = {Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.},
author = {Brešar, Matej, Šemrl, Peter},
journal = {Studia Mathematica},
keywords = {rank; semisimple Banach algebra; decomposability; indecomposable elements},
language = {eng},
number = {3},
pages = {287-298},
title = {Finite rank elements in semisimple Banach algebras},
url = {http://eudml.org/doc/216487},
volume = {128},
year = {1998},
}

TY - JOUR
AU - Brešar, Matej
AU - Šemrl, Peter
TI - Finite rank elements in semisimple Banach algebras
JO - Studia Mathematica
PY - 1998
VL - 128
IS - 3
SP - 287
EP - 298
AB - Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.
LA - eng
KW - rank; semisimple Banach algebra; decomposability; indecomposable elements
UR - http://eudml.org/doc/216487
ER -

References

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  1. [1] J. C. Alexander, Compact Banach algebras, Proc. London Math. Soc. 18 (1968), 1-18. Zbl0184.16502
  2. [2] B. Aupetit and T. Mouton, Spectrum-preserving linear mappings in Banach algebras, Studia Math. 109 (1994), 91-100. 
  3. [3] B. Aupetit and T. Mouton, Trace and determinant in Banach algebras, ibid. 121 (1996), 115-136. 
  4. [4] M. Brešar and P. Šemrl, Derivations mapping into the socle, Math. Proc. Cambridge Philos. Soc. 120 (1996), 339-346. 
  5. [5] J. M. G. Fell and R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1, Academic Press, Boston, 1988. Zbl0652.46050
  6. [6] A. A. Jafarian and A. R. Sourour, Spectrum preserving linear maps, J. Funct. Anal. 66 (1986), 255-261. Zbl0589.47003
  7. [7] B. V. Limaye, A spectral characterization of operators having rank k, Linear Algebra Appl. 143 (1991), 57-66. Zbl0721.47026
  8. [8] T. Mouton and H. Raubenheimer, On rank one and finite elements of Banach algebras, Studia Math. 104 (1993), 211-219. Zbl0814.46035
  9. [9] P. Nylen and L. Rodman, Approximation numbers and Yamamoto's theorem in Banach algebras, Integral Equations Operator Theory 13 (1990), 728-749. Zbl0731.46028
  10. [10] C. E. Rickart, General Theory of Banach Algebras, The University Series in Higher Mathematics, D. van Nostrand, 1960. Zbl0095.09702

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