# 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

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topDales, 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] C. A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286-302. Zbl0157.44603
- [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] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
- [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] A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248-252.
- [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] M. Despič and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), 165-167. Zbl0813.43001
- [8] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
- [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] 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] F. Ghahramani, Automorphisms of weighted measure algebras, ibid., 144-154.
- [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] N. Grønbæk, Commutative Banach algebras, module derivations, and semigroups, J. London Math. Soc. (2) 40 (1989), 137-157. Zbl0632.46046
- [14] N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), 150-162.
- [15] N. Grønbæk, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (1992), 315-328. Zbl0760.46043
- [16] N. Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), 109-140. Zbl0864.46026
- [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] U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. Zbl0529.46041
- [19] A. Ya. Helemskiĭ, The Homology of Banach and Topological Algebras, Kluwer, 1989.
- [20] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
- [21] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281-284. Zbl0757.43002
- [22] B. E. Johnson, private communication.
- [23] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. Zbl0236.47045
- [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] D. Lamb, The second dual of certain Beurling algebras, preprint.
- [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] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
- [28] J. Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31-33. Zbl0572.46044
- [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] S. Sakai, C*-Algebras and W*-Algebras, Springer, New York, 1971.
- [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] 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] 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] 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] 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] 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- A. Jabbari, Mohammad Sal Moslehian, H. R. E. Vishki, Constructions preserving $n$-weak amenability of Banach algebras
- Alireza Medghalchi, Taher Yazdanpanah, Problems concerning $n$-weak amenability of a Banach algebra
- M. Eshaghi-Gordji, A. Ebadian, F. Habibian, B. Hayati, Weak${}^{*}$-continuous derivations in dual Banach algebras
- Eshaghi M. Gordji, F. Habibian, B. Hayati, Ideal amenability of module extensions of Banach algebras
- Fatemeh Anousheh, Davood Ebrahimi Bagha, Abasalt Bodaghi, Weak amenability for the second dual of Banach modules
- Abasalt Bodaghi, Behrouz Shojaee, A generalized notion of $n$-weak amenability

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