Erratum: On the Functor ℓ2
The main purpose of this erratum is to correct a claim made in “On the functor ℓ2” (Computation, Logic, Games, and Quantum Foundations, Lecture Notes in Computer Science Volume 7860, 2013, pp 107–121) in Lemma 5.9. Namely, positive operators on Hilbert space are not necessarily isomorphisms, but merely bimorphisms, i.e. both monic and epic; this is precisely the issue in 2.8. Here is the corrected version.
Lemma 5.9. Positive operators on Hilbert spaces are bimorphisms.
Proof. Let p : H → H be a positive operator in Hilb. If p(x) = 0 then certainly 〈p(x) | x 〉 = 0 which contradicts positivity. Hence \(\ker(p)=0\), and so p is monic.
To see that p is epic, suppose that p ∘ f = p ∘ g for parallel morphisms f,g. Then 〈p ∘ (f − g)(x) | x 〉 = 0 for all x. By positivity, For each x there is p x > 0 such that p ∘ (f − g)x = p x ·(f − g)(x). Hence 〈(f − g)(x) | x 〉 = 0 for all x, that is, f = g and p is epic.
Definition 5.10 then needs to be adapted accordingly: a functor F : C → D is essentially full when for each morphism g in D and bimorphisms u,v in C such that g = v ∘ Ff ∘ u.