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A certain class of Frobenius algebras has been used to characterizeorthonormal bases and observables on finite-dimensional Hilbert spaces. Thepresence of units in these algebras means that they can only be realizedfinite-dimensionally. We seek a suitable generalization, which will allowarbitrary bases and observables to be described within categoricalaxiomatizations of quantum mechanics. We develop a definition of H*-algebrathat can be interpreted in any symmetric monoidal dagger category, reduces tothe classical notion from functional analysis in the category of (possiblyinfinite-dimensional) Hilbert spaces, and hence provides a categorical way tospeak about orthonormal bases and quantum observables in arbitrary dimension.Moreover, these algebras reduce to the usual notion of Frobenius algebra incompact categories. We then investigate the relations between nonunitalFrobenius algebras and H*-algebras. We give a number of equivalent conditionsto characterize when they coincide in the category of Hilbert spaces. We alsoshow that they always coincide in categories of generalized relations andpositive matrices.

H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics