recalls some background on monoidal dagger categories and Frobenius algebras, and poses the problem. Section \[sec:hstar\] introduces the key notion of [H\*]{}-algebra, in the general setting of symmetric monoidal dagger categories. In Section \[sec:hstarinhilb\], we prove our results relating to ${{\ensuremath{\mathbf{Hilb}}}}$, the category of Hilbert spaces (of unrestricted dimension). We show how [H\*]{}-algebras provide exactly the right algebraic notion to characterize orthonormal bases in arbitrary dimension. We give several equivalent characterizations of when [H\*]{}-algebras and nonunital Frobenius algebras coincide in the category of Hilbert spaces. Section \[sec:hstarinrel\] studies [H\*]{}-algebras in categories of generalized relations and positive matrices. We show that in these settings, where no phenomena of ‘destructive interference’ arise, [H\*]{}-algebras and nonunital Frobenius algebras always coincide. Finally, Section \[sec:outlook\] provides an outlook for future work.

Background {#sec:background}
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The basic setting is that of *dagger symmetric monoidal categories*. We briefly recall the definitions, referring to [@abramskycoecke:categoricalquantummechanics] for further details and motivation.

A *dagger category* is a category ${\ensuremath{\mathbf{D}}}$ equipped with an identity-on-objects, contravariant, strictly involutive functor. Concretely, for each arrow $f \colon A \to B$, there is an arrow $f^\dag \colon B \to A$, and this assignment satisfies $${\ensuremath{\mathrm{id}_{}}}^\dag = {\ensuremath{\mathrm{id}_{}}}, \qquad 
  (g \circ f)^\dag = f^\dag \circ g^\dag, 