Abstract
This entry formalises complete normed algebras equipped with an involution, so-called C*-algebras. We provide both a class definition, and a locale for C*-algebras on carrier sets in the spirit of existing developments of linear algebra and smooth manifolds. Bounded operators on a complex Hilbert space, with the operator norm and adjoints, form such an algebra. The main theorem of this entry is a result in the converse direction: the Gelfand–Naimark–Segal (GNS) construction, which starts with a single suitable functional on a C*-algebra in order to obtain both a Hilbert space and a representation of the algebra in terms of bounded operators on that space. This is implemented as a type construction in Isabelle/HOL, taking advantage of existing mechanisms for quotient types, and integrating with existing type classes for Hilbert spaces and Cauchy completions.