The context for this paper comes from the ongoing work on *categorical quantum mechanics* [@abramskycoecke:categoricalsemantics; @abramskycoecke:categoricalquantummechanics]. This work has shown how large parts of quantum mechanics can be axiomatized in terms of monoidal dagger categories and structures definable within them. This axiomatization can be used to perform high-level reasoning and calculations relating to quantum information, using diagrammatic methods [@selinger:graphicallanguages]; and also as a basis for exploring foundational issues in quantum mechanics and quantum computation. In particular, a form of Frobenius algebras has been used to give an algebraic axiomatization of *orthonormal bases* and *observables* [@coeckepavlovic:classicalobjects; @coeckepavlovicvicary:bases].

The structures used so far ([*e.g.* ]{}compact closure, Frobenius algebras) have only finite-dimensional realizations in Hilbert spaces. This raises some interesting questions and challenges:

-   Find a good general notion of Frobenius structure which works in the infinite-dimensional case in ${{\ensuremath{\mathbf{Hilb}}}}$.

-   Use this to characterize general bases and therefore general observables with discrete spectra.

-   Similarly extend the analysis for other categories.

-   Clarify the mathematics, and relate it to the wider literature.

As we shall see, an intriguing problem remains open, but much of this program of work has been accomplished.

The further contents of the paper are as follows. Section \[sec:background\] 