\indent Categorical quantum mechanics seeks to distill quantum mechanics to minimal assumptions, based on categories with tensor products. We address the question of how to usefully represent observables in this setting. Orthonormal bases in the category of finite-dimensional Hilbert spaces turn out to correspond to Frobenius algebras. We show that for arbitrary dimensions one needs H*-algebras instead, which can be defined in any monoidal category. Finally we compare the notion of H*-algebra to that of nonunital Frobenius algebra in various categories