$A$-differentiability and $A$-analyticity

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 $\mathbb {C}$.