Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky pp E1-E1 | Cite as

# Erratum: On the Functor ℓ^{2}

## Abstract

The main purpose of this erratum is to correct a claim made in “On the functor ℓ^{2}” (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*.