Hilbertian Frobenius algebras

Abstract

Commutative Hilbertian Frobenius algebras are those commutative semigroup 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 “comultiplication” is a bimodule map. In this note we show that the Frobenius relation forces the multiplication operators to be normal. We then prove that these algebras have a strong Wedderburn decomposition where the (ortho)complement of the Jacobson radical or equivalently of the annihilator, is the closure of the linear span of elements which essentially are the non-trivial characters. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In particular every commutative special Hilbertian Frobenius algebra, that is, with a coisometric multiplication, is semisimple. Moreover we characterize from a setting a priori free of an involution, Ambrose’s commutative -algebras as the underlying algebras of Hilbertian Frobenius algebras. 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-to-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.