Hilbertian Frobenius algebras

Abstract

Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint is a bimodule map. In this note we prove that they split as an orthogonal direct sum of two closed ideals, their Jacobson radical which in fact is nothing but their annihilator, and the closure of the linear span of their group-like elements. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In particular every commutative special Hilbertian algebra, that is, with a coisometric multiplication, is semisimple. Extending a known result in the finite-dimensional situation, we prove that the structures of such Frobenius algebras on a given Hilbert space are in one-one correspondence with its bounded above orthogonal sets. We show, moreover, that the category of commutative Hilbertian Frobenius algebras is dually equivalent to a category of pointed sets. Thus, each semigroup morphism between commutative Hilbertian Frobenius semigroups arises from a unique base-point preserving map (of some specific kind), from the set of minimal ideals of its codomain to the set of minimal ideals of its domain, both with zero added. MSC 2010: Primary 46J40, Secondary 16T15.