Hilbertian Frobenius algebras

Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which adjoint multiplication satisfies compatibility relation, that is, this “comultiplication” is a bimodule map. In note we show relation forces operators to be normal. We then prove these have strong Wedderburn decomposition where (ortho)complement Jacobson radical or equivalently annihilator, closure linear span elements essentially non-trivial characters. As consequence such an algebra semisimple if, and only its has dense range. particular every special algebra, with coisometric multiplication, semisimple. Moreover characterize from setting priori free involution, Ambrose’s H*-algebras as underlying algebras. Extending known result finite-dimensional situation, structures on given space one-to-one correspondence bounded above orthogonal sets. show, moreover, dually equivalent pointed