Derivations and Embeddings of a Field in Its Power Series Ring

Heerema, Nickolas

doi:10.2307/2032953

Cited by 11 publications

(9 citation statements)

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“…R. J. Loy has shown that the result of B. E. Johnson and A. M. Sinclair [9] giving the automatic continuity of derivations on semi-simple Banach algebras can be extended to higher derivations whose domain algebra is the same as the range algebra and where F o is the identity map. He did this by using results of Heerema [7] to express the higher derivations in terms of derivations. We extend this result (i) by allowing the domain algebra to be any Banach algebra whatsoever, (ii) by allowing the range algebra to include a wider class than just semi-simple algebras and (iii) by weakening the condition that F Q be the identity map.…”

Section: Let B Be a Commutative Banach Algebra With Identity And M A mentioning

confidence: 99%

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Continuity of module and higher derivations

Jewell¹

1977

Pacific J. Math.

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Section: Let B Be a Commutative Banach Algebra With Identity And M A mentioning

confidence: 99%

“…(d) Using the methods of [7] and [10] it is possible to classify all the higher derivations acting on L^O, 1] where F o is the identity map.…”

Section: Let B Be a Commutative Banach Algebra With Identity And M A mentioning

confidence: 99%

Continuity of module and higher derivations

Jewell¹

1977

Pacific J. Math.

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“…The relation between higher derivations and homomorphisms into algebras of power series has been well studied in certain cases [4], [10], [8] Theorem 2 is not applicable to algebras of power series, since the kernel inclusions will not be satisfied; however Theorem 1 has the following consequence.…”

Section: Proofmentioning

confidence: 99%

Continuity of Higher Derivations

Loy¹

1973

Proceedings of the American Mathematical Society

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Abstract.Let A and B be two algebras over the complex field C. A sequence {Fn}0g"gra (resp.{F"}0gnA will denote the (possibly empty) carrier space of all nonzero continuous multiplicative linear functionals on A with the weak topology determined by the algebra Â of Gelfand transforms of elements of A. Since A is ultrabarrelled the Banach-Steinhaus theorem shows that any compact subset of $>A is equicontinuous. A is termed functionally continuous if every multiplicative linear functional on A is continuous. It is an open question whether A is necessarily functionally continuous, even in the locally w-convex case, though it is known that some metrizability condition is necessary.In the case A locally convex with identity having a neighbourhood of invertible elements (so O^ is compact by Tychonoff's theorem), Q)A point separating and B=Â, Johnson [7] has shown that any derivation from A into B is continuous. Indeed, for A a Banach algebra he obtained results for higher derivations (Dk¡k\)k3.0 on A, where D is a derivation on A. In this note we show how the argument of [7] works in the more general situation to give results on the continuity of higher derivations. Our results contain those of the relevant sections of [3] and [12] where regular semisimple £-algebras are considered. After completing an earlier draft of this

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“…A higher derivation on AT is a sequence d={di\0-i< oo} of additive maps of K into K such that dr(ab) = Jt{di(a)di(b)\i+j=r} and d0 is the identity map. The set HX(K) of all higher derivations on K is a group with respect to the composition d° e=f where ft = 2 W»e" \m + n = i} [1,Theorem 1,p. 33].…”

Section: Introductionmentioning

confidence: 99%

Fields of constants of infinite higher derivations

Deveney¹

1973

Proc. Amer. Math. Soc.

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Abstract.Let K be a field of characteristic p^O, and let P be its maximal perfect subfield. Let A be a subfield of K containing P such that K is separable over h. We prove : Every regular subfield of .Äf containing h is the field of constants of a set of higher derivations on K if and only if (1) the transcendence degree of K over h is finite, and (2) K has a separating transcendency basis over h. This result leads to a generalization of the Galois theory developed in [4].

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Derivations and Embeddings of a Field in Its Power Series Ring

Cited by 11 publications

References 0 publications

Continuity of module and higher derivations

Continuity of module and higher derivations

Continuity of Higher Derivations

Fields of constants of infinite higher derivations

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