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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Systems of derivations


Author: Frances Gulick
Journal: Trans. Amer. Math. Soc. 149 (1970), 465-488
MSC: Primary 46.55
MathSciNet review: 0275170
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Abstract: Let A and B be two complex algebras. A system of derivations of order m from A into B is a set of $ m + 1$ linear operators $ {D_k}:A \to B(k = 0,1, \ldots ,m)$ such that for $ x,y \in A$ and $ k = 0,1,2, \ldots ,m$,

$\displaystyle {D_k}(xy) = \sum\limits_{j = 0}^k {\left( {_j^k} \right)} ({D_j}x)({D_{k - j}}y).$

If A is a commutative, regular, semisimple F-algebra with an identity, B the algebra of continuous functions on the closed maximal ideal space of A and $ ({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from A into B with $ {D_0}$ the Gelfand mapping, then each $ {D_k}$ is continuous. The continuity of the operators in a system of derivations from $ {C^n}(U)$ into $ C(U)(U \subset R \;{\text{open}})$ is used to obtain a formula for $ {D_k}f,f \in {C^n}(U)$, in terms of the ordinary derivatives of f and functions in $ C(U)$. Each system of derivations from A into B and each multiplicative seminorm on B determine a multiplicative seminorm on A. Let U be a subset of C and $ ({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from the algebra $ P(x)$ of polynomials on U into $ C(U)$ with $ {D_0}$ the identity operator. Then the system of derivations determines a Hausdorff topology on $ P(x)$. If U is open in R and $ {D_1}x(t) \ne 0$ for $ t \in U(x(t) = t)$, then the completion of $ P(x)$ in this topology is $ {C^m}(U)$. If U is open in C, then the completion of $ P(x)$ in this topology is the algebra of functions analytic on U.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0275170-4
PII: S 0002-9947(1970)0275170-4
Keywords: Systems of derivations, Leibniz rule, systems of order m, systems of point derivations, F-algebra, multiplicative seminorm
Article copyright: © Copyright 1970 American Mathematical Society