DERIVATIONS ON CONVOLUTION ALGEBRAS

Title & Authors
DERIVATIONS ON CONVOLUTION ALGEBRAS

Abstract
In this paper, we investigate derivations on the noncommutative Banach algebra $\small{L^{\infty}_0({\omega})^*}$ equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra $\small{L^{\infty}_0({\omega})^*}$. We then show that a derivation on $\small{L^{\infty}_0({\omega})^*}$ is continuous if and only if its restriction to rad($\small{L^{\infty}_0({\omega})^*}$) is continuous. We also prove that there is no nonzero centralizing derivation on $\small{L^{\infty}_0({\omega})^*}$. Finally, we prove that the space of all inner derivations of $\small{L^{\infty}_0({\omega})^*}$ is continuously homomorphic to the space $\small{L^{\infty}_0({\omega})^*/L^1({\omega})}$.
Keywords
derivation;inner derivation;centralizing;automatic continuity;
Language
English
Cited by
References
1.
M. Bresar and M. Mathieu, Derivations mapping into the radical III, J. Funct. Anal. 133 (1995), no. 1, 21-29.

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.

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.

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.

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.

11.
M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. 57 (1991), no. 5, 469-474.

12.
M. Mathieu and V. Runde, Derivations mapping into the radical II, Bull. London Math. Soc. 24 (1992), no. 5, 485-487.

13.
A. R. Medghalchi, The second dual algebra of a hypergroup, Math. Z. 210 (1992), no. 4, 615-624.

14.
S. Ouzomgi, Factorization and bounded approximate identities for a class of convolution Banach algebras, Glasg. Math. J. 28 (1986), no. 2, 211-214.

15.
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.

16.
A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), no. 1, 166-170.

17.
I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264.

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.

19.
M. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), no. 3, 435-460.

20.
J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequationes Math. 75 (2008), no. 3, 260-266.