Weak amenability of Segal algebras

Authors:
H. G. Dales and S. S. Pandey

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1419-1425

MSC (1991):
Primary 46J10

Published electronically:
October 6, 1999

MathSciNet review:
1641681

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact abelian group, and let . We show that the Segal algebra is always weakly amenable, but that it is amenable only if is discrete.

**[1]**W. G. Bade, P. C. Curtis Jr., and H. G. Dales,*Amenability and weak amenability for Beurling and Lipschitz algebras*, Proc. London Math. Soc. (3)**55**(1987), no. 2, 359–377. MR**896225**, 10.1093/plms/s3-55_2.359**[2]**J. T. Burnham,*Closed ideals in subalgebras of Banach algebras. I*, Proc. Amer. Math. Soc.**32**(1972), 551–555. MR**0295078**, 10.1090/S0002-9939-1972-0295078-5**[3]**H. G. Dales,*Banach algebras and automatic continuity*, Clarendon Press, Oxford, to appear.**[4]**M. Despić and F. Ghahramani,*Weak amenability of group algebras of locally compact groups*, Canad. Math. Bull.**37**(1994), no. 2, 165–167. MR**1275699**, 10.4153/CMB-1994-024-4**[5]**Jośe E. Galé,*Weak amenability of Banach algebras generated by some analytic semigroups*, Proc. Amer. Math. Soc.**104**(1988), no. 2, 546–550. MR**962826**, 10.1090/S0002-9939-1988-0962826-X**[6]**Niels Groenbaek,*A characterization of weakly amenable Banach algebras*, Studia Math.**94**(1989), no. 2, 149–162. MR**1025743****[7]**A. Ya. Helemskii,*The homology of Banach and topological algebras*, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West. MR**1093462****[8]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations*, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0156915****[9]**Barry Edward Johnson,*Cohomology in Banach algebras*, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR**0374934****[10]**Ron Larsen, Teng-sun Liu, and Ju-kwei Wang,*On functions with Fourier transforms in 𝐿_{𝑝}*, Michigan Math. J.**11**(1964), 369–378. MR**0170173****[11]**John C. Martin and Leonard Y. H. Yap,*The algebra of functions with Fourier transforms in 𝐿^{𝑝}*, Proc. Amer. Math. Soc.**24**(1970), 217–219. MR**0247378**, 10.1090/S0002-9939-1970-0247378-0**[12]**S. Poornima,*Multipliers of Sobolev spaces*, J. Funct. Anal.**45**(1982), no. 1, 1–28. MR**645643**, 10.1016/0022-1236(82)90002-7**[13]**Hans Reiter,*Classical harmonic analysis and locally compact groups*, Clarendon Press, Oxford, 1968. MR**0306811****[14]**Hans Reiter,*𝐿¹-algebras and Segal algebras*, Lecture Notes in Mathematics, Vol. 231, Springer-Verlag, Berlin-New York, 1971. MR**0440280****[15]**Walter Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR**0152834****[16]**Allan M. Sinclair,*Continuous semigroups in Banach algebras*, London Mathematical Society Lecture Note Series, vol. 63, Cambridge University Press, Cambridge-New York, 1982. MR**664431****[17]**M. C. White,*Strong Wedderburn decompositions of Banach algebras containing analytic semigroups*, J. London Math. Soc. (2)**49**(1994), no. 2, 331–342. MR**1260116**, 10.1112/jlms/49.2.331

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Additional Information

**H. G. Dales**

Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England

Email:
pmt6hgd@leeds.ac.uk

**S. S. Pandey**

Affiliation:
Department of Mathematics, R. D. University, Jabalpur, India

Email:
ssp@rdunijb.ren.nic.in

DOI:
https://doi.org/10.1090/S0002-9939-99-05139-4

Received by editor(s):
March 10, 1998

Received by editor(s) in revised form:
July 3, 1998

Published electronically:
October 6, 1999

Additional Notes:
The second author acknowledges with thanks the support of the Royal Society-INSA exchange program which enabled him to visit the University of Leeds to work with the first author. He is also thankful to the Department of Pure Mathematics at Leeds for hospitality.

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 2000
American Mathematical Society