LaurentPoinsot HilbertianFrobeniusalgebras

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Section 1: Basic definitions Section 2: The main result Section 3: Some direct consequences Section 4: Category-theoretic observations

Hilbertian Frobenius algebras

Laurent Poinsot

University Sorbonne Paris North and

French Air School Academy

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Main result (B. Coecke et al.,arXived in 2008, published in 2013) Every commutative†-Frobenius monoid inFdHilb determines an orthogonal basis (consisting of its group-like elements) and every orthogonal basis determines a commutative†-Frobenius monoid in FdHilb. These correspondences are inverse to each other.

What about the infinite-dimensional situation? In a joint paper (arXived 2010, published 2012), S. Abramsky and C. Heunen explore the notion of commutative†-Frobenius semigroupsin Hilb to “provide a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension”.

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Main result (B. Coecke et al.,arXived in 2008, published in 2013) Every commutative†-Frobenius monoid inFdHilb determines an orthogonal basis (consisting of its group-like elements) and every orthogonal basis determines a commutative†-Frobenius monoid in FdHilb. These correspondences are inverse to each other.

What about the infinite-dimensional situation? In a joint paper (arXived 2010, published 2012), S. Abramsky and C. Heunen explore the notion of commutative†-Frobenius semigroupsin Hilb to “provide a categorical way to speak about orthonormal bases

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Objective of the talk

To provide asuitableextension of Coecke et al.’s main result for arbitrary dimensions.

⇐Conditions under which a commutative †-Frobenius semigroup inHilb is semisimple.

⇐Structure Theorem for Hilbertian Frobenius algebras.

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Objective of the talk

⇐Conditions under which a commutative†-Frobenius semigroup inHilb is semisimple.

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Objective of the talk

⇐Conditions under which a commutative†-Frobenius semigroup inHilb is semisimple.

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Semisimple commutative Banach algebras

A (not necessarily unital) commutative algebraAis 1. semisimplewhen J(A) = (0),

2. radicalwhen A=J(A).

Jacobson radicalJ(A)

= intersection of all maximal modular ideals ofA,

= intersection of the kernels of the characters ontoC (ABanach),

= kernel of the Gelfand representation.

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Hilbertian algebras

= (commutative) semigroups inHilb

Hilbertian algebras

are (commutative) semigroups inHilb,

“are” Banach algebras (forgetful functor), aren’t Hilbert algebras (no involutions).

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Hilbertian algebras

Group-like elements

By Riesz Representation Theorem, Aˆ'G(A)

whereG(A) = set of non-zero group-like elementsofA= (H, µ), that is, those 06=x∈H s.t. µ^†(x) =x⊗x.

Consequently,

J(A) =G(A)^⊥ and

H =J(A)⊕₂J(A)^⊥=G(A)^⊥⊕₂hG(A)i (qua Hilbert spaces).

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Hilbertian algebras

Group-like elements

By Riesz Representation Theorem, Aˆ'G(A)

whereG(A) = set of non-zero group-like elementsofA= (H, µ), that is, those 06=x∈H s.t. µ^†(x) =x⊗x.

Consequently,

J(A) =G(A)^⊥ and

H =J(A)⊕₂J(A)^⊥ =G(A)^⊥⊕₂hG(A)i (qua Hilbert spaces).

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Hilbertian algebras

Semisimplicity

A= (H, µ) is

1. semisimplewhen H =hG(A)i=J(A)^⊥, 2. radicalwhen G(A) =∅.

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Hilbertian Frobenius algebras

a.k.a. commutative†-Frobenius algebras

A commutative semigroupH⊗2H−→^µ Hin Hilbs.t.

H⊗2H

µ

''

µ^†⊗₂id

id⊗₂µ^†//_H⊗₂_(H⊗₂H)

(H⊗2H)⊗2H H

µ^† ''

(H⊗2H)⊗2H

µ⊗₂id

H⊗2(H⊗2H)

id⊗₂µ //H⊗2H

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Call itspecialwhen furthermoreµis a coisometry, that is,µ◦µ^†=id.

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Structure Theorem for Hilbertian Frobenius Algebras

The main result

Theorem

LetA= (H, µ) be a Hilbertian Frobenius algebra.

Then,H=J(A)⊕₂J(A)^⊥, whereJ(A) andJ(A)^⊥ are both ideals (J(A) is the annihilator of A) and subcoalgebras. J(A) andJ(A)^⊥ are Frobenius, radical and semisimple respectively.

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hG(A)i=J(A)^⊥ is a closed subalgebra and subcoalgebra ofA

for a Hilbertian Frobenius algebraA= (H, µ)

Proof:

IshG(A)ia subcoalgebra?

By general properties of the Hilbert adjoint, for a Hilbertian algebraA= (H, µ),

I closed ideal ofA⇔ I^⊥ is a subcoalgebra of (H, µ^†).

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Proof:

IshG(A)ia subcoalgebra?

By general properties of the Hilbert adjoint, for a Hilbertian algebraA= (H, µ),

I closed ideal ofA⇔ I^⊥ is a subcoalgebra of (H, µ^†).

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IshG(A)ia subalgebra?

Letx,y ∈G(A),x 6=y. It is essentially due to the Frobenius conditions, thatxy ∈J(A).

But then _kxk^x2

y

kyk² is an idempotent element that belongs toJ(A) so it is equal to 0, andxy = 0 too.

It remains to check thatx² ∈J(A)^⊥. Letu ∈J(A). Then, 0 =hx⊗x²,u⊗xi=hµ^†(x²),u⊗xi=hx²⊗x,u⊗xi= kxk²hx²,ui ⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

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But then _kxk^x2

y

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But then _kxk^x2

y

It remains to check thatx² ∈J(A)^⊥.

Let u ∈J(A). Then, 0 =hx⊗x²,u⊗xi=hµ^†(x²),u⊗xi=hx²⊗x,u⊗xi= kxk²hx²,ui ⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

It remains to check thatx² ∈J(A)^⊥. Letu ∈J(A). Then, 0 =hx⊗x²,u⊗xi

=hµ^†(x²),u⊗xi=hx²⊗x,u⊗xi= kxk²hx²,ui ⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

It remains to check thatx² ∈J(A)^⊥. Letu ∈J(A). Then, 0 =hx⊗x²,u⊗xi=hµ^†(x²),u⊗xi

=hx²⊗x,u⊗xi= kxk²hx²,ui ⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

It remains to check thatx² ∈J(A)^⊥. Letu ∈J(A). Then, 0 =hx⊗x²,u⊗xi=hµ^†(x²),u⊗xi=hx²⊗x,u⊗xi

= kxk²hx²,ui ⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

It remains to check thatx² ∈J(A)^⊥. Letu ∈J(A). Then, 0 =hx⊗x²,u⊗xi=hµ^†(x²),u⊗xi=hx²⊗x,u⊗xi=

⇒x²∈J(A)^⊥.

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But then _kxk^x2

y

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Quasi-nilpotent elements

LetAbe a (not necessarily commutative) Banach algebra and let u∈A.

u is quasi-nilpotent when its spectral radius is equal to zero, that is,kuⁿk¹ⁿ →0.

WhenAis commutative, J(A) coincides with the set of all quasi-nilpotent elements ofA.

For a Hilbert spaceH, call quasi-nilpotent operator a bounded

f

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Quasi-nilpotent elements

For a Hilbert spaceH, call quasi-nilpotent operator a bounded linear mapH −→^f H quasi-nilpotent as a member of B(H).

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Quasi-nilpotent elements

For a Hilbert spaceH, call quasi-nilpotent operator a bounded

f

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Quasi-nilpotent elements

For a Hilbert spaceH, callquasi-nilpotent operatora bounded linear mapH −→^f H quasi-nilpotent as a member of B(H).

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Frobenius ⇒ multiplication operators are normal

The statement

LetA= (H, µ) be a Hilbertian algebra. For u ∈H, define M_u:H →H,M_u(v) :=uv.

Proposition

Let us assume thatA= (H, µ) is Frobenius. Then, for each u∈H,Mu is normal, that is, Mu◦Mu^†=Mu^†◦Mu.

Consequently,J(A) is equal to the annihilator of A.

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Frobenius ⇒ multiplication operators are normal

The proof

In any Hilbertian algebraA= (H, µ), u∈J(A)⇒M_u is a quasi-nilpotent operator.

If furthermoreMu is normal, then u belongs to the annihilator of A, that is, M_u(v) = 0 for all v, because the spectral radius of a normal operator coincides with its operator norm.

As the annihilator is contained intoJ(A), the second assertion of the proposition is proved.

The first assertion is obtained by direct inspection (using the Frobenius conditions).

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Frobenius ⇒ multiplication operators are normal

The proof

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Frobenius ⇒ multiplication operators are normal

The proof

The first assertion is obtained by direct inspection (using the Frobenius conditions).

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Frobenius ⇒ multiplication operators are normal

The proof

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Structure Theorem for Hilbertian Frobenius Algebras

Main result

Then,H=J(A)⊕₂J(A)^⊥, whereJ(A) andJ(A)^⊥ are both ideals (J(A) is the annihilator ofA) and subcoalgebras and subcoalgebras.

J(A) andJ(A)^⊥ are Frobenius, radical and semisimple respectively.

Proof:

ThatJ(A)^⊥ is an ideal follows from the facts that J(A)^⊥ is a subalgebra andJ(A) is the annihilator of A. Consequently, J(A)^⊥⊥=J(A) is a subcoalgebra.

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Structure Theorem for Hilbertian Frobenius Algebras

Main result

Then,H=J(A)⊕₂J(A)^⊥, whereJ(A) andJ(A)^⊥ are both ideals (J(A) is the annihilator ofA) and subcoalgebras and subcoalgebras.

J(A) andJ(A)^⊥ are Frobenius, radical and semisimple respectively.

Proof:

ThatJ(A)^⊥ is an ideal follows from the facts that J(A)^⊥ is a subalgebra andJ(A) is the annihilator of A. Consequently,

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The conditions for semisimplicity

In arbitrary dimension

Corollary

LetA= (H, µ) be a Hilbertian Frobenius algebra. The following assertions are equivalent.

1 A is semisimple.

2 µ has a dense range.

3 µ^† is one-to-one.

4 µ◦µ^† is one-to-one.

In particular, ifA is special, thenA is semisimple.

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Remark: Commutativity is necessary

LetX be a non void set andx₀ ∈X.

Then, (`²(X), µ_x₀) is a non commutative(whenever |X|>1) special Hilbertian Frobenius algebra, withµx0(u⊗v) :=v(x0)u. However it isnot (Jacobson) semisimple. E.g.,{δ_x₀}^⊥ consists entirely of nilpotent elements! Its Jacobson radical is precisely {δx0}^⊥, while its annihilator is (0). (The left annihilator is (0) and the right annihilator is{δ_x₀}^⊥.)

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Remark: Commutativity is necessary

Then, (`²(X), µx0) is a non commutative(whenever |X|>1) special Hilbertian Frobenius algebra, withµx0(u⊗v) :=v(x0)u.

However it isnot (Jacobson) semisimple. E.g.,{δ_x₀}^⊥ consists entirely of nilpotent elements! Its Jacobson radical is precisely {δx0}^⊥, while its annihilator is (0). (The left annihilator is (0) and the right annihilator is{δ_x₀}^⊥.)

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Remark: Commutativity is necessary

Then, (`²(X), µx0) is a non commutative(whenever |X|>1) special Hilbertian Frobenius algebra, withµx0(u⊗v) :=v(x0)u.

However it isnot (Jacobson) semisimple. E.g., {δ_x₀}^⊥ consists entirely of nilpotent elements! Its Jacobson radical is precisely {δx0}^⊥, while its annihilator is (0). (The left annihilator is (0) and the right annihilator is{δ_x₀}^⊥.)

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The conditions for semisimplicity

In finite dimension

Corollary

LetA= (H, µ) be a finite-dimensional Hilbertian Frobenius algebra. The following assertions are equivalent.

1 A has a unit.

2 µ is onto.

3 µ^† is one-to-one.

4 A is semisimple.

In particular, ifA is special, thenA is unital.

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In the finite-dimensional situation, structures of (resp. special) Frobenius algebras onH correspondone-one to orthogonal (resp.

orthonormal) bases ofH.

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This is almost the same when is dropped the finite-dimensional requirement.

Callbounded above (resp. bounded below) a setX of a Hilbert space such that there existsC >0 with kxk ≤C (resp. C ≤ kxk) for eachx∈X.

Bounded below or bounded above: a matter of taste!

x7→_kxk^x2 transforms bijectively a bounded above orthogonal set into a bounded below orthogonal set (and vice versa). Moreover it is a bijection between the sets of all bounded below orthogonal sets and of all bounded above orthogonal sets.

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Relations with Hilbertian bases

Theorem

LetH be a Hilbert space. There are (among others) one-one correspondences between

1 Bounded above orthogonal bases ofH and bounded linear mapsµ:H⊗₂H→H with a dense range such that (H, µ) is a Frobenius algebra.

2 Orthonormal bases of H and bounded linear coisometries µ:H⊗ H→H such that (H, µ) is a Frobenius algebra.

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Relations with Hilbertian bases

Elements of the proof

LetX be a bounded above orthogonal basis ofH. Define µX(u⊗v) :=P

x∈X 1

kxkhu,xihv,xi_kxk^x . Then, (H, µX) is a semisimple Hilbertian algebra.

Let (H, µ) be a semisimple Frobenius algebra. Then, G(H, µ) is a bounded above orthogonal basis ofH (because for each

x∈G(H, µ),kxk ≤ kµk_op).

The operations are invertible and inverse one from the other.

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Relations with Hilbertian bases

x∈X 1

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Relations with Hilbertian bases

x∈X 1

The operations are invertible and inverse one from the other.

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The main theorem as an equivalence of categories

Frob (resp. _semisimpleFrob) = full subcategory of cSem(Hilb) spanned by the (resp. semisimple) Frobenius algebras.

Proposition

The categoriesFrob and_semisimpleFrob×Hilb are equivalent.

Proof: By the main theorem, each radical Frobenius algebra is trivial, that is, it has the zero multiplication. Consequently it is essentially a Hilbert space.

There is a functorFrob×Frob→Frob,

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The main theorem as an equivalence of categories

Frob (resp. _semisimpleFrob) = full subcategory of cSem(Hilb) spanned by the (resp. semisimple) Frobenius algebras.

Proposition

The categoriesFrob and_semisimpleFrob×Hilb are equivalent.

((H, µ),(K, γ))7→(H⊕₂K, ρ), whereρ acts like µ onH⊗₂H, likeγ on K⊗₂K, and is zero everywhere else (here is used additivity of⊗₂). Moreover J(H⊕₂K, ρ) =J(H, µ)⊕₂J(K, γ).

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The main theorem as an equivalence of categories

Frob (resp. _semisimpleFrob) = full subcategory of _cSem(Hilb) spanned by the (resp. semisimple) Frobenius algebras.

Proposition

The categoriesFrob andsemisimpleFrob×Hilb are equivalent.

((H, µ),(K, γ))7→(H⊕₂K, ρ), whereρ acts like µ onH⊗₂H,

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The main theorem as an equivalence of categories

Frob (resp. _semisimpleFrob) = full subcategory of _cSem(Hilb) spanned by the (resp. semisimple) Frobenius algebras.

Proposition

The categoriesFrob andsemisimpleFrob×Hilb are equivalent.

((H, µ),(K, γ))7→(H⊕₂K, ρ), whereρ acts like µ onH⊗₂H, likeγ on K⊗₂K, and is zero everywhere else (here is used additivity of⊗₂). Moreover J(H⊕₂K, ρ) =J(H, µ)⊕₂J(K, γ).

This functors restricts to the desired equivalence of categories.

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Weighted pointed sets

Aweighted pointed setis a triple (X,x0, α), where (X,x0) is a pointed set andα:X\ {x0} →[C,+∞[ is a map, whereC >0.

A morphism (X,x0, α)−→^f (Y,y0, β) is a base-point preserving map (X,x₀)−→^f (Y,y₀) such that

1 for eachy 6=y₀,|f⁻¹({y})|<+∞,

2 there exists a real numberM_f ≥0 such that for ally 6=y₀, P

x∈f⁻¹({y})α(x)≤M_fβ(y).

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The `

²_•

-functor

Let (X,x₀, α) be a weighted pointed set.

Define the following subspace of`²(X)

`²_•(X,x0, α) :={f ∈C^X:f(x0) = 0, P

x∈X\{x₀}α(x)|f(x)|² <+∞ }.

Under pointwise product of maps, it is a semisimple Frobenius algebra.

This construction extends to a functor`²_•:WSet•^op →Frob.

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The set of minimal ideals functor

LetA= (H, µ) be a Frobenius algebra. Each minimal ideal of Ais a one-dimensional space generated by a unique group-like element.

LetMin(A) :={Cx:x∈G(A)} andw_A:Min(A)→[C,+∞[, w_A(I) := _kg¹

Ik², whereg_I is the group-like generator ofI. (Min•(A),0,wA), with Min•(A) =Min(A)∪ {0}, is a weighted pointed set, and this construction extends to a functor

Min•:Frob→WSet•^op.

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The set of minimal ideals functor

LetMin(A) :={Cx:x∈G(A)}and wA:Min(A)→[C,+∞[, w_A(I) := _kg¹

Ik², where g_I is the group-like generator ofI.

(Min•(A),0,wA), with Min•(A) =Min(A)∪ {0}, is a weighted pointed set, and this construction extends to a functor

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The set of minimal ideals functor

LetMin(A) :={Cx:x∈G(A)}and wA:Min(A)→[C,+∞[, w_A(I) := _kg¹

Ik², where g_I is the group-like generator ofI. (Min•(A),0,wA), with Min•(A) =Min(A)∪ {0}, is a weighted pointed set, and this construction extends to a functor

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An adjunction and its restricted equivalence

One has an adjunctionMin• a`²_•:Frob→WSet•^op

which restricts to an equivalence of categories

semisimpleFrob'WSet_•^op. Consequently,Frob'WSet•^op×Hilb (equivalence).

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Some derived equivalences

FromFrob'WSet•^op×Hilb, by restriction:

FdFrob'FinSet_•^op×FdHilb

1FdFrob'FdFrobproper 'FinSet^op×FdHilb.

A morphism (H, µ)−→^f (K, γ) of Hilbertian algebras is calledproperwhen ran(f) is included in no maximal modular ideals of (K, γ). Alternatively, for each group-like elementy of (K, γ), there existsu∈Hsuch thathf(u),yi 6= 0.

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Some derived equivalences

FromFrob'WSet•^op×Hilb, by restriction:

FdFrob'FinSet_•^op×FdHilb and

1FdFrob'FdFrobproper 'FinSet^op×FdHilb.

A morphism (H, µ)−→^f (K, γ) of Hilbertian algebras is calledproperwhen ran(f) is included in no maximal modular ideals of (K, γ). Alternatively, for each group-like elementy of (K, γ), there existsu∈Hsuch thathf(u),yi 6= 0.

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