TY - GEN

T1 - Erratum: On the Functor ℓ2

AU - Heunen, Chris

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-642-38164-5_26

DO - 10.1007/978-3-642-38164-5_26

M3 - Conference contribution

SN - 978-3-642-38163-8

T3 - Lecture Notes in Computer Science

SP - E1-E1

BT - Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

A2 - Coecke, Bob

A2 - Ong, Luke

A2 - Panangaden, Prakash

PB - Springer Berlin Heidelberg

CY - Berlin, Heidelberg

ER -