# Erratum: On the Functor ℓ2

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract / Description of output

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)=0ker⁡(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.
Original language English Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky Essays Dedicated to Samson Abramsky on the Occasion of His 60th Birthday Bob Coecke, Luke Ong, Prakash Panangaden Berlin, Heidelberg Springer Berlin Heidelberg E1-E1 978-3-642-38164-5 978-3-642-38163-8 https://doi.org/10.1007/978-3-642-38164-5_26 Published - 2013

### Publication series

Name Lecture Notes in Computer Science Springer Berlin Heidelberg 7860 0302-9743

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