The computable functional calculus

Abstract

We show that the continuous functional calculus is computable. As consequences we obtain the computable compactness of the spectrum of any computable normal element of a computably presented $\mathrm{C}^*$-algebra, the existence of effective approximate units for computably presented $\mathrm{C}^*$-algebras, and an effective version of the Spectral Theorem for compact operators on separable Hilbert spaces.