Resolvent set of a Banach algebra element (${\rho }_A(x)=\mathbb{C}\setminus {\sigma }_A(x)$)

The algebra resolvent set consists of those scalars for which $\lambda 1-x$ is invertible in the algebra. It is open by the Neumann series, and the inverse-valued map is the resolvent of an element. A closed subalgebra can have a smaller algebra resolvent set because an ambient inverse need not lie in it.