Compressible operators and the continuity of homomorphisms from algebras of operators

G. Willis

Studia Mathematica (1995)

  • Volume: 115, Issue: 3, page 251-259
  • ISSN: 0039-3223

Abstract

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The notion of a compressible operator on a Banach space, E, derives from automatic continuity arguments. It is related to the notion of a cartesian Banach space. The compressible operators on E form an ideal in ℬ(E) and the automatic continuity proofs depend on showing that this ideal is large. In particular, it is shown that each weakly compact operator on the James' space, J, is compressible, whence it follows that all homomorphisms from ℬ(J) are continuous.

How to cite

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Willis, G.. "Compressible operators and the continuity of homomorphisms from algebras of operators." Studia Mathematica 115.3 (1995): 251-259. <http://eudml.org/doc/216211>.

@article{Willis1995,
abstract = {The notion of a compressible operator on a Banach space, E, derives from automatic continuity arguments. It is related to the notion of a cartesian Banach space. The compressible operators on E form an ideal in ℬ(E) and the automatic continuity proofs depend on showing that this ideal is large. In particular, it is shown that each weakly compact operator on the James' space, J, is compressible, whence it follows that all homomorphisms from ℬ(J) are continuous.},
author = {Willis, G.},
journal = {Studia Mathematica},
keywords = {compressible operator on a Banach space; automatic continuity; James' space},
language = {eng},
number = {3},
pages = {251-259},
title = {Compressible operators and the continuity of homomorphisms from algebras of operators},
url = {http://eudml.org/doc/216211},
volume = {115},
year = {1995},
}

TY - JOUR
AU - Willis, G.
TI - Compressible operators and the continuity of homomorphisms from algebras of operators
JO - Studia Mathematica
PY - 1995
VL - 115
IS - 3
SP - 251
EP - 259
AB - The notion of a compressible operator on a Banach space, E, derives from automatic continuity arguments. It is related to the notion of a cartesian Banach space. The compressible operators on E form an ideal in ℬ(E) and the automatic continuity proofs depend on showing that this ideal is large. In particular, it is shown that each weakly compact operator on the James' space, J, is compressible, whence it follows that all homomorphisms from ℬ(J) are continuous.
LA - eng
KW - compressible operator on a Banach space; automatic continuity; James' space
UR - http://eudml.org/doc/216211
ER -

References

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  1. [Da] H. G. Dales, Banach Algebras and Automatic Continuity, Oxford Univ. Press, in preparation. 
  2. [D,L&W] H. G. Dales, R. J. Loy and G. A. Willis, Homomorphisms and derivations from ℬ(E), J. Funct. Anal., to appear. Zbl0815.46063
  3. [Go] W. T. Gowers, A solution to the Schroeder-Bernstein problem for Banach spaces, preprint. 
  4. [GM1] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
  5. [GM2] W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, preprint. Zbl0876.46006
  6. [Ja] R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1950), 174-177. 
  7. [BJo] B. E. Johnson, Continuity of homomorphisms from algebras of operators, J. London Math. Soc. 42 (1967), 537-541. Zbl0152.32805
  8. [WJo1] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. Zbl0236.47045
  9. [WJo2] W. B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, ibid. 13 (1972), 301-310. 
  10. [L&W] R. J. Loy and G. A. Willis, Continuity of derivations on ℬ(E) for certain Banach spaces E, J. London Math. Soc. (2) 40 (1989), 327-346. Zbl0651.47035
  11. [Og] C. Ogden, Homomorphisms from B ( C ω η ) , J. London Math. Soc., to appear. 
  12. [Re] C. J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space, ibid. (2) 40 (1989), 305-326. 

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