Differentiability and Analyticity

Abstract

Let A be a finite-dimensional commutative algebra over R and let CA (U), CO (U, A) and (OA (U) be the ring of A-differentiable functions of class cr, 0 < r < oo, 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'. We prove two basic results concerning A-differentiability and A-analyticity: Ist) (9A(U) = CA (U)q nC-(U, A), 2nd) (9A(U) = CA (U) if and only if A is defined over C. 1. PRELIMINARIES AND STATEMENT OF THE MAIN RESULTS Let A be a finite-dimensional commutative lR-algebra. Let us denote by G(A) the group of units of A. Clearly, G(A) is an open dense subset of A and hence it is endowed with a canonical structure of Lie group. Let U be an open subset of A. A function f: U A is said to be A-differentiable if for every a E U there exists the limit f'(a) = limx,a(f (x) -f (a))/(x a), x a E G(A). We say that f is A-differentiable of class Cr and set f E CA (U) if for every a E U, f'(a), f(a), ..., f(r)(a) exist and are continuous. For further properties on Adifferentiability we refer to [2, 3, 4]. As is well known, if V is a finite-dimensional lR-vector space, for each open subset U C V we have a canonical isomorphism (,: V T (U) given by (,(v)(f) = limt,o 1(f (x + tv) -f (x)) (directional derivative). In the particular case V = A we can define a family of endomorphisms on each tangent space ha: T, (U) TX(U)) a E A, ha(X) = (x(a 1(x-(X)). Then, it is proved that the definition of the analyst coincides with that of the differential geometer (this is not the case for non-commutative algebras, see [2]); i.e., a function f: U A of class C' is A-differentiable if and only if for every a E A, x E U, the mapping f*: TX(U) TX(U) commutes with ha. The product in A induces an lR-bilinear mapping ,u: A x A A. Let eo = 1, el, ...,I em1 be a basis of A as an JR-vector space. Then, ,(e,, ej) =Em1 k_ekj O < i, j < mr-1. Each function f: U A can be written as f (x) Zm-1 fi(x)ei and f is A-differentiable if and only if Of /x or more explicitly, f j = 1,...,m1 (Cauchy-Riemann equations). It follows that CA(U) is a closed A-subalgebra of Cr(U, A) with respect to its natural topology of Frechet algebra. Received by the editors March 1, 1994 and, in revised form, September 16, 1994. 1991 Mathematics Subject Classification. Primary 30G35; Secondary 26E05, 26E10, 16P10. Supported by DGICYT (Spain) grant no. PB89-0004. ( 1996 American Mathematical Society