Algebras of power series of elements of a Lie algebra, and Slodkowski spectra

Topological algebras of (convergent) power series of elements of a Lie algebra are introduced and the existence of continuous homomorphisms of these algebras into an operator algebra is studied. For Slodkowski spectra, the spectral mapping theorem $\sigma_{\delta, k}(f(a))=f(\sigma_{\delta,k}(a))$, $\sigma_{\pi,k}(f(a))=f(\sigma_{\pi,k}(a))$ is proved for generators $a$ of a finite-dimensional nilpotent Lie algebra of bounded linear operators whenever the family $f$ of elements of a power series algebra is finite-dimensional.