Inner derivations on ultraprime normed algebras

Cabrera, M.; Martínez, J.

doi:10.1090/S0002-9939-97-03833-1

Inner derivations on ultraprime normed algebras
HTML articles powered by AMS MathViewer

by M. Cabrera and J. Martínez PDF

Proc. Amer. Math. Soc. 125 (1997), 2033-2039 Request permission

Abstract:

We show that, for every ultraprime Banach algebra $A$, there exists a positive number $\gamma$ satisfying $\gamma \|a+Z(A)\|\le \|D_a\|$ for all $a$ in $A$, where $Z(A)$ denotes the centre of $A$ and $D_a$ denotes the inner derivation on $A$ induced by $a$. Moreover, the number $\gamma$ depends only on the “constant of ultraprimeness” of $A$.

References

Pere Ara and Martin Mathieu, On ultraprime Banach algebras with nonzero socle, Proc. Roy. Irish Acad. Sect. A 91 (1991), no. 1, 89–98. MR 1173162
M. Cabrera García and Á. Rodríguez-Palacios, Non-degenerately ultraprime Jordan-Banach algebras: a Zel′manovian treatment, Proc. London Math. Soc. (3) 69 (1994), no. 3, 576–604. MR 1289864, DOI 10.1112/plms/s3-69.3.576
Lawrence A. Fialkow, Structural properties of elementary operators, Elementary operators and applications (Blaubeuren, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 55–113. MR 1183937
B. E. Johnson, Norms of derivations of ${\cal L}({\mathfrak {X}})$, Pacific J. Math. 38 (1971), 465–469. MR 306966
J. Kyle, Norms of derivations, J. London Math. Soc. (2) 16 (1977), no. 2, 297–312. MR 487486, DOI 10.1112/jlms/s2-16.2.297
Martin Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators, Conference on Automatic Continuity and Banach Algebras (Canberra, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 21, Austral. Nat. Univ., Canberra, 1989, pp. 297–317. MR 1022011
Martin Mathieu, Elementary operators on prime $C^*$-algebras. I, Math. Ann. 284 (1989), no. 2, 223–244. MR 1000108, DOI 10.1007/BF01442873
Martin Mathieu, The symmetric algebra of quotients of an ultraprime Banach algebra, J. Austral. Math. Soc. Ser. A 50 (1991), no. 1, 75–87. MR 1094060
Martin Mathieu, The $cb$-norm of a derivation, Algebraic methods in operator theory, Birkhäuser Boston, Boston, MA, 1994, pp. 144–152. MR 1284942
Douglas W. B. Somerset, Inner derivations and primal ideals of $C^*$-algebras, J. London Math. Soc. (2) 50 (1994), no. 3, 568–580. MR 1299458, DOI 10.1112/jlms/50.3.568
Joseph G. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737–747. MR 265952