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$A$-differentiability and $A$-analyticity


Authors: P. M. Gadea and J. Muñoz Masqué
Journal: Proc. Amer. Math. Soc. 124 (1996), 1437-1443
MSC (1991): Primary 30G35; Secondary 26E05, 26E10, 16P10
MathSciNet review: 1301495
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Abstract: Let $A$ be a finite-dimensional commutative algebra over $\mathbb{R}$ and let $C_{A}^{r}(U)$, $C^{\omega}(U,A)$ and $\mathcal{ O}_{A}(U)$ be the ring of $A$-differentiable functions of class $C^{r},\,0 \leq r \leq \infty$, the ring of real analytic mappings with values in $A$ and the ring of $A$-analytic functions, respectively, defined on an open subset $U$ of $A^{n}$. We prove two basic results concerning $A$-differentiability and $A$-analyticity: $1^{st}$) $\mathcal{ O}_{A}(U) = C^{\infty}_{A}(U) \bigcap C^{\omega}(U,A)$, $2^{nd}$) $\mathcal{ O}_{A}(U) = C^{\infty}_{A}(U)$ if and only if $A$ is defined over $\mathbb{C}$.


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Additional Information

P. M. Gadea
Affiliation: Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, 28006-Madrid, Spain
Email: pmgadea@gugu.usal.es

J. Muñoz Masqué
Affiliation: Instituto de Electrónica de Comunicaciones Consejo Superior de Investigaciones Científicas Serrano 144, 28006-Madrid, Spain
Email: vctqu01@cc.csic.es

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03070-5
Received by editor(s): March 1, 1994
Received by editor(s) in revised form: September 16, 1994
Additional Notes: Supported by DGICYT (Spain) grant no. PB89-0004
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society