Derivations into iterated duals of Banach algebras

H. Dales; F. Ghahramani; N. Grønbæek

Studia Mathematica (1998)

  • Volume: 128, Issue: 1, page 19-54
  • ISSN: 0039-3223

Abstract

top
We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space A ( n ) is zero; i.e., 1 ( A , A ( n ) ) = 0 . Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C*-algebras, Banach function algebras, and algebras of operators. Our results are summarized and some open questions are raised in the final section.

How to cite

top

Dales, H., Ghahramani, F., and Grønbæek, N.. "Derivations into iterated duals of Banach algebras." Studia Mathematica 128.1 (1998): 19-54. <http://eudml.org/doc/216473>.

@article{Dales1998,
abstract = {We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space $A^\{(n)\}$ is zero; i.e., $ℋ^1(A,A^\{(n)\}) = \{0\}$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C*-algebras, Banach function algebras, and algebras of operators. Our results are summarized and some open questions are raised in the final section.},
author = {Dales, H., Ghahramani, F., Grønbæek, N.},
journal = {Studia Mathematica},
keywords = {derivations; iterated duals of Banach algebras; cohomology group; amenability},
language = {eng},
number = {1},
pages = {19-54},
title = {Derivations into iterated duals of Banach algebras},
url = {http://eudml.org/doc/216473},
volume = {128},
year = {1998},
}

TY - JOUR
AU - Dales, H.
AU - Ghahramani, F.
AU - Grønbæek, N.
TI - Derivations into iterated duals of Banach algebras
JO - Studia Mathematica
PY - 1998
VL - 128
IS - 1
SP - 19
EP - 54
AB - We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space $A^{(n)}$ is zero; i.e., $ℋ^1(A,A^{(n)}) = {0}$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C*-algebras, Banach function algebras, and algebras of operators. Our results are summarized and some open questions are raised in the final section.
LA - eng
KW - derivations; iterated duals of Banach algebras; cohomology group; amenability
UR - http://eudml.org/doc/216473
ER -

References

top
  1. [1] C. A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286-302. Zbl0157.44603
  2. [2] W. G. Bade, P. C. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), 359-377. Zbl0634.46042
  3. [3] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
  4. [4] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. Zbl0119.10903
  5. [5] A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248-252. 
  6. [6] I. G. Craw and N. J. Young, Regularity of multiplication in weighted group and semigroup algebras, Quart. J. Math. Oxford Ser. (2) 25 (1974), 351-358. Zbl0304.46027
  7. [7] M. Despič and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), 165-167. Zbl0813.43001
  8. [8] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977. 
  9. [9] J. Duncan and S. A. Hosseinium, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309-325. Zbl0427.46028
  10. [10] J. F. Feinstein, Weak (F)-amenability of R(X), in: Conference on Automatic Continuity and Banach Algebras, Proc. Centre Math. Anal. Austral. Nat. Univ. 21, Austral. Nat. Univ., Canberra, 1989, 97-125. 
  11. [11] F. Ghahramani, Automorphisms of weighted measure algebras, ibid., 144-154. 
  12. [12] F. Ghahramani and J. P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992), 180-185. Zbl0789.43001
  13. [13] N. Grønbæk, Commutative Banach algebras, module derivations, and semigroups, J. London Math. Soc. (2) 40 (1989), 137-157. Zbl0632.46046
  14. [14] N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), 150-162. 
  15. [15] N. Grønbæk, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (1992), 315-328. Zbl0760.46043
  16. [16] N. Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), 109-140. Zbl0864.46026
  17. [17] N. Grønbæk, B. E. Johnson and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), 289-324. Zbl0806.46058
  18. [18] U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. Zbl0529.46041
  19. [19] A. Ya. Helemskiĭ, The Homology of Banach and Topological Algebras, Kluwer, 1989. 
  20. [20] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
  21. [21] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281-284. Zbl0757.43002
  22. [22] B. E. Johnson, private communication. 
  23. [23] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. Zbl0236.47045
  24. [24] H. Kamowitz and S. Sheinberg, Derivations and automorphisms of L^1(0,1), Trans. Amer. Math. Soc. 135 (1969), 415-427. Zbl0172.41703
  25. [25] D. Lamb, The second dual of certain Beurling algebras, preprint. 
  26. [26] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. I, Algebras and Banach Algebras, Cambridge Univ. Press, 1994. Zbl0809.46052
  27. [27] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986. 
  28. [28] J. Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31-33. Zbl0572.46044
  29. [29] C. J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space, J. London Math. Soc. (2) 40 (1989), 305-326. Zbl0722.46020
  30. [30] S. Sakai, C*-Algebras and W*-Algebras, Springer, New York, 1971. 
  31. [31] Yu. V. Selivanov, Biprojective Banach algebras, Izv. Akad. Nauk SSSR 43 (1979), 1159-1174 (in Russian); English transl.: Math. USSR-Izv. 15 (1980), 381-399. 
  32. [32] M. V. Sheĭnberg, A characterization of the algebra C(Ω) in terms of cohomology groups, Uspekhi Mat. Nauk 32 (5) (1977), 203-204 (in Russian). 
  33. [33] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-212. Zbl0121.10204
  34. [34] S. Watanabe, A Banach algebra which is an ideal in the second dual algebra, Sci. Rep. Niigata Univ. Ser. A 11 (1974), 95-101. Zbl0359.46034
  35. [35] M. Wodzicki, Vanishing of cyclic homology of stable C*-algebras, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 329-334. Zbl0652.46052
  36. [36] N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59-62. Zbl0252.43009

Citations in EuDML Documents

top
  1. A. Jabbari, Mohammad Sal Moslehian, H. R. E. Vishki, Constructions preserving n -weak amenability of Banach algebras
  2. M. Eshaghi-Gordji, A. Ebadian, F. Habibian, B. Hayati, Weak * -continuous derivations in dual Banach algebras
  3. Alireza Medghalchi, Taher Yazdanpanah, Problems concerning n -weak amenability of a Banach algebra
  4. Eshaghi M. Gordji, F. Habibian, B. Hayati, Ideal amenability of module extensions of Banach algebras
  5. Abasalt Bodaghi, Behrouz Shojaee, A generalized notion of n -weak amenability
  6. Fatemeh Anousheh, Davood Ebrahimi Bagha, Abasalt Bodaghi, Weak amenability for the second dual of Banach modules

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.