  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.
