Continuity of higher derivations
HTML articles powered by AMS MathViewer
- by R. J. Loy PDF
- Proc. Amer. Math. Soc. 37 (1973), 505-510 Request permission
Abstract:
Let A and B be two algebras over the complex field C. A sequence ${\{ {F_n}\} _{0 \leqq n \leqq m}}$ (resp. ${\{ {F_n}\} _{0 \leqq n < \infty }}$) of linear operators from A into B is a higher derivation of rank m (resp. infinite rank) if, for each $n = 0,1, \cdots ,m$ (resp. $n = 0,1, \cdots$) and any $x,y \in A$, \[ {F_n}(xy) = \sum \limits _{i = 0}^n {{F_i}(x){F_{n - i}}(y).} \] We consider the continuity of such $\{ {F_n}\}$ when A and B are commutative topological algebras with complete metrizable topology. Some applications are given to algebras of formal power series.References
- R. L. Carpenter, Continuity of systems of derivations on $F$-algebras, Proc. Amer. Math. Soc. 30 (1971), 141–146. MR 283574, DOI 10.1090/S0002-9939-1971-0283574-5 A. G. Dors, Fréchet algebras without compactly generated spectra (preprint).
- Frances Gulick, Systems of derivations, Trans. Amer. Math. Soc. 149 (1970), 465–488. MR 275170, DOI 10.1090/S0002-9947-1970-0275170-4
- Nickolas Heerema, Derivations and embeddings of a field in its power series ring, Proc. Amer. Math. Soc. 11 (1960), 188–194. MR 123568, DOI 10.1090/S0002-9939-1960-0123568-6
- Nickolas Heerema, Derivations and embeddings of a field in its power series ring. II, Michigan Math. J. 8 (1961), 129–134. MR 136601
- Nickolas Heerema, Derivations and automorphisms of complete regular local rings, Amer. J. Math. 88 (1966), 33–42. MR 197500, DOI 10.2307/2373045
- B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1–10. MR 246127, DOI 10.2307/2373262
- Richard J. Loy, A note on the preceding paper by J. B. Miller, Acta Sci. Math. (Szeged) 28 (1967), 233–236. MR 213398
- R. J. Loy, Uniqueness of the complete norm topology and continuity of derivations on Banach algebras, Tohoku Math. J. (2) 22 (1970), 371–378. MR 276768, DOI 10.2748/tmj/1178242764
- John Boris Miller, Homomorphisms, higher derivations, and derivations on associative algebras, Acta Sci. Math. (Szeged) 28 (1967), 221–231. MR 212052
- John Boris Miller, Higher derivations on Banach algebras, Amer. J. Math. 92 (1970), 301–331. MR 287313, DOI 10.2307/2373325
- Melvin Rosenfeld, Commutative $F$-algebras, Pacific J. Math. 16 (1966), 159–166. MR 190786
- I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260–264. MR 70061, DOI 10.1007/BF01362370
- W. Żelazko, Metric generalizations of Banach algebras, Rozprawy Mat. 47 (1965), 70. MR 193532
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 505-510
- MSC: Primary 46H05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0312265-9
- MathSciNet review: 0312265