Hilbertian (function) algebras

Abstract

A Hilbertian (co)algebra is defined as a (co)semigroup object in the monoidal category of Hilbert spaces. The carrier Hilbert space of such an algebra splits as an orthogonal direct sum of its Jacobson radical and the closure of the linear span of a special class of elements, the group-like elements of its adjoint coalgebra which by the Riesz representation, correspond to closed maximal modular ideals. When the coproduct is isometric, that is, when semisimplicity is shown to be equivalent to the existence of adjoints in the sense of Ambrose’s H*-algebras. We also prove that the category of semisimple special Hilbertian algebras, that is, semisimple Hilbertian algebras with an isometric coproduct, i.e., essentially the algebras of the form with the pointwise product, are dually equivalent to a subcategory of pointed sets and base-point preserving maps.