Continuity of systems of derivations on -algebras

Author:
R. L. Carpenter

Journal:
Proc. Amer. Math. Soc. **30** (1971), 141-146

MSC:
Primary 46.55; Secondary 32.00

MathSciNet review:
0283574

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *A* be a commutative semisimple *F*-algebra with identity, and let be a system of derivations from *A* into the algebra of all continuous functions on the spectrum of *A*. It is shown that the transformations are necessarily continuous. This result is used to obtain a characterization of derivations on where is an open polynomially convex subset of .

**[1]**Richard Arens,*Dense inverse limit rings*, Michigan Math. J**5**(1958), 169–182. MR**0105034****[2]**R. L. Carpenter,*Uniqueness of topology for commutative semisimple 𝐹-algebras*, Proc. Amer. Math. Soc.**29**(1971), 113–117. MR**0298424**, 10.1090/S0002-9939-1971-0298424-0**[3]**A. G. Dors,*On the spectrum of an F-algebra*, Thesis, University of Utah, Salt Lake City, Utah, 1970.**[4]**Frances Gulick,*Systems of derivations*, Trans. Amer. Math. Soc.**149**(1970), 465–488. MR**0275170**, 10.1090/S0002-9947-1970-0275170-4**[5]**B. E. Johnson,*Continuity of derivations on commutative algebras*, Amer. J. Math.**91**(1969), 1–10. MR**0246127****[6]**Ernest A. Michael,*Locally multiplicatively-convex topological algebras*, Mem. Amer. Math. Soc.,**No. 11**(1952), 79. MR**0051444****[7]**Melvin Rosenfeld,*Commutative 𝐹-algebras*, Pacific J. Math.**16**(1966), 159–166. MR**0190786**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0283574-5

Keywords:
*F*-algebra,
derivation,
systems of derivations,
continuity of derivations

Article copyright:
© Copyright 1971
American Mathematical Society