Closed subalgebras of the Banach algebra of continuously differentiable functions on an interval
For \(f\in D^ 1(I)\), the Banach algebra of \(C^ 1\) complex functions on the closed interval \(I\), \(A_ f\) is the closed subalgebra generated by \(f\). When the range of \(f\) does not separate the plane, then \(g\in A_ f\) if and only if \(f(x)=f(y)\) implies both \(g(x)=g(y)\) and \(g'(x)f'(y)=g'(y)f'(x)\), and \(f'(x)=0\) implies \(g'(x)=0\). For real \(f\in D^ 1(I)\), an exhaustive set of conditions is given for \(A_ f\) to be a natural set defined in terms of point separation and zero sets of the derivatives. The authors establish various conditions for (Shilov) regularity of closed subalgebras (such as being inverse closed on \(I\)). The final section characterizes closed separating subalgebras over which \(D^ 1(I)\) is integral.
scientific article; zbMATH DE number 14136
scientific article