Systems of derivations

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, …, $m$, \[ D_k(xy) = \sum _{j = 0}^k \binom {k}{j} (D_j x)(D_{k-j} y).\] 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 $(D_0$, $D_1$, …, $D_m)$ 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 $C(U)(U \subset R \;\mathrm {open})$ 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 $(D_0$, $D_1$, …, $D_m)$ 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 $\mathbf {R}$ 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 $U$.