Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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DERIVATIONS ON CONVOLUTION ALGEBRAS

  • Received : 2014.04.08
  • Published : 2015.07.31
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Abstract

In this paper, we investigate derivations on the noncommutative Banach algebra $L^{\infty}_0({\omega})^*$ equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra $L^{\infty}_0({\omega})^*$. We then show that a derivation on $L^{\infty}_0({\omega})^*$ is continuous if and only if its restriction to rad($L^{\infty}_0({\omega})^*$) is continuous. We also prove that there is no nonzero centralizing derivation on $L^{\infty}_0({\omega})^*$. Finally, we prove that the space of all inner derivations of $L^{\infty}_0({\omega})^*$ is continuously homomorphic to the space $L^{\infty}_0({\omega})^*/L^1({\omega})$.

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References

  1. M. Bresar and M. Mathieu, Derivations mapping into the radical III, J. Funct. Anal. 133 (1995), no. 1, 21-29. https://doi.org/10.1006/jfan.1995.1116
  2. J. W. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985.
  3. H. G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, New York, 2000.
  4. N. Isik, J. Pym, and A. Ulger, The second dual of the group algebra of a compact group, J. London Math. Soc. 35 (1987), no. 1, 135-148.
  5. B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1-10. https://doi.org/10.2307/2373262
  6. K. W. Jun and H. M. Kim, Derivations on prime rings and Banach algebras, Bull. Korean Math. Soc. 38 (2001), no. 4, 709-718.
  7. K. W. Jun and H. M. Kim, Approximate derivations mapping into the radicals of Banach algebras, Taiwanese J. Math. 11 (2007), no. 1, 277-288. https://doi.org/10.11650/twjm/1500404652
  8. A. T. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. London Math. Soc. 41 (1990), no. 3, 445-460.
  9. S. Maghsoudi, M. J. Mehdipour, and R. Nasr-Isfahani, Compact right multipliers on a Banach algebra related to locally compact semigroups, Semigroup Forum 83 (2011), no. 2, 205-213. https://doi.org/10.1007/s00233-011-9312-z
  10. S. Maghsoudi, R. Nasr-Isfahani, and A. Rejali, Arens multiplication on Banach algebras related to locally compact semigroups, Math. Nachr. 281 (2008), no. 10, 1495-1510. https://doi.org/10.1002/mana.200710691
  11. M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. 57 (1991), no. 5, 469-474. https://doi.org/10.1007/BF01246745
  12. M. Mathieu and V. Runde, Derivations mapping into the radical II, Bull. London Math. Soc. 24 (1992), no. 5, 485-487. https://doi.org/10.1112/blms/24.5.485
  13. A. R. Medghalchi, The second dual algebra of a hypergroup, Math. Z. 210 (1992), no. 4, 615-624. https://doi.org/10.1007/BF02571818
  14. S. Ouzomgi, Factorization and bounded approximate identities for a class of convolution Banach algebras, Glasg. Math. J. 28 (1986), no. 2, 211-214. https://doi.org/10.1017/S0017089500006522
  15. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
  16. A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), no. 1, 166-170. https://doi.org/10.1090/S0002-9939-1969-0233207-X
  17. I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. https://doi.org/10.1007/BF01362370
  18. A. I. Singh, $L^{\infty}_0(G)^*$ as the second dual of the group algebra $L^1$(G) with a locally convex toplogy, Michigan Math. J. 46 (1999), no. 1, 143-150. https://doi.org/10.1307/mmj/1030132365
  19. M. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), no. 3, 435-460. https://doi.org/10.2307/1971432
  20. J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequationes Math. 75 (2008), no. 3, 260-266. https://doi.org/10.1007/s00010-007-2872-z

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