Goursat's lemma in the context of Banach algebras (Q887476)

The author introduces a way of integrating along a line segment in a Banach space \(X\). A function \(f\) from an open subset \(V\) of \(X\) to the space \(L(X)\) of bounded linear operators on \(X\) is said to be in the Goursat class if the integral of \(f\) around every triangle, whose convex hull is contained in \(V\), is equal to zero. When \(V\) is convex it is shown that the set of functions in the Goursat class coincides with those functions that are the anti-derivative of a function \(g\) from \(V\) into \(X\). In the case when \(f\) is differentiable, using \(f'_x\) to denote the Fréchet derivative of \(f\) at \(x\), it is shown that a necessary and sufficient condition for \(f\) to be in the Goursat class is that \(((f'_x)a)b=((f'_x)b)a\) for all \(x\) in \(V\) and all \(a,b\) in \(X\). When \(X\) is a real Banach algebra without unit and \(f: V\to X\) is differentiable, it is shown that the left multiplication operator, \(a\mapsto f(x).a\), is in the Goursat class if and only if \(f'_x.[a,b]=0\) where \([a,b]\) denotes the commutator of \(a\) and \(b\).