On the functional equation defined by Lie's product formula
Let E be a real normed space and A a complex Banach algebra with unit. We characterize the continuous solutions f : E → A of the functional equation f(x + y) = limn→∞(f(x/n)f(y/n)) n. Let A be a complex Banach algebra with unit 1. In this setting the famous Lie product formula reads (1) exp(a + b) = lim n→∞ (exp(a/n) exp(b/n)) (a, b ∈ A); see [4, Theorem VIII.29] for matrices and a proof which also holds for Banach algebras, and Trotter’s version for semigroups [6]. Let p ∈ A be a projection (that is, p2 = p), and consider the complex Banach algebra pAp which has unit p. The exponential function in pAp will be denoted by expp. Now, let E be a real normed space, let A : E → pAp be a continuous and linear mapping (here pAp is considered as a real vector space), and set (2) f(x) := expp(A(x)) = p exp(A(x))p (x ∈ E). As an immediate consequence of (1) the function f is a continuous solution of the functional equation (3) f(x + y) = lim n→∞ (f(x/n)f(y/n)) (x, y ∈ E). In this paper we prove conversely that all continuous solutions of (3) are of type (2). More precisely we have Theorem 1. Let f : E → A be a continuous function which satisfies (3). Then p = f(0) is a projection, and there exists a unique continuous linear mapping A : E → pAp such that f(x) = expp(A(x)) (x ∈ E). 2000 Mathematics Subject Classification: 39B52, 46H99.