The many classical faces of quantum structures Chris Heunen - - PowerPoint PPT Presentation

The many classical faces of quantum structures

Chris Heunen University of Oxford November 30, 2013

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Relationship between classical and quantum

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Relationship between classical and quantum

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Relationship between classical and quantum

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Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

3 / 31

Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

• rthomodular lattice

1 a a⊥ b b⊥

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Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

• rthomodular lattice not distributive
• r
• and

=

• and
• r
• and
• 4 / 31

Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

• rthomodular lattice not distributive

tea coffee biscuit nothing

5 / 31

Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

• rthomodular lattice not distributive

tea coffee biscuit nothing More problems: no good →, ∀, ∃, ⊗

5 / 31

Quantum logic

✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤

Subsets of a set Subspaces of a Hilbert space

• rthomodular lattice not distributive

tea coffee biscuit nothing More problems: no good →, ∀, ∃, ⊗ However: fine when within Boolean block / orthogonal basis!

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Part I Algebras of observables

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Algebras of observables

Observables are primitive, states are derived

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Algebras of observables

Observables are primitive, states are derived C*-algebras ∗-algebra of operators that is closed AW*-algebras abstract/algebraic version of W*-algebra von Neumann algebras / W*-algebras ∗-algebra of operators that is weakly closed

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Algebras of observables

Observables are primitive, states are derived C*-algebras ∗-algebra of operators that is closed AW*-algebras abstract/algebraic version of W*-algebra von Neumann algebras / W*-algebras ∗-algebra of operators that is weakly closed Jordan algebras JC/JW-algebras: real version of above

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Classical mechanics

◮ If X is a state space,

then C(X) = {f : X → C} is an operator algebra.

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Classical mechanics

◮ If X is a state space,

then C(X) = {f : X → C} is an operator algebra.

◮

Theorem: Every commutative operator algebra is of this form.

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Classical mechanics

◮ If X is a state space,

then C(X) = {f : X → C} is an operator algebra.

◮

Theorem: Every commutative operator algebra is of this form.

◮ Can recover states (as maps C(X) → C): “spectrum”

Constructions on states transfer to observables: X + Y → C(X) ⊗ C(Y ) X × Y → C(X) ⊕ C(Y Equivalence of categories: states determine everything

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Quantum mechanics

◮ If H is a Hilbert space,

then B(H) = {f : H → H} is an operator algebra.

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Quantum mechanics

◮ If H is a Hilbert space,

then B(H) = {f : H → H} is an operator algebra.

◮

Theorem: Every operator algebra embeds into one of this form.

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Quantum mechanics

◮ If H is a Hilbert space,

then B(H) = {f : H → H} is an operator algebra.

◮

Theorem: Every operator algebra embeds into one of this form.

◮ Recover states?

Do states determine everything? “Noncommutative spectrum”?

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Quantum state spaces?

certain convex sets (states) sheaves over locales (prime ideals) quantales (maximal ideals)

• rthomodular lattices (projections)

q-spaces (projections of enveloping W*-algebra)

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Quantum state spaces?

commutative

• perator algebras

spectrum

• ≃
• state spaces
• ✤

✤ ✤ ✤

• perator algebras
• ❴

❴ ❴ ❴ ❴

quantum state spaces

11 / 31

Quantum state spaces? No!

commutative

• perator algebras

spectrum

• ≃
• state spaces
• G

✤ ✤ ✤ ✤

• perator algebras

F

• ❴

❴ ❴ ❴ ❴

quantum state spaces

◮

Theorem: If G is continuous, then F degenerates.

11 / 31

Quantum state spaces? No!

commutative

• perator algebras

spectrum

• ≃
• state spaces
• G

✤ ✤ ✤ ✤

• perator algebras

F

• ❴

❴ ❴ ❴ ❴

quantum state spaces

◮

Theorem: If G is continuous, then F degenerates.

◮

That’s right. (F(Mn) = ∅ for n ≥ 3.)

11 / 31

Quantum state spaces? No!?

commutative

• perator algebras

spectrum

• ≃
• state spaces
• G

✤ ✤ ✤ ✤

• perator algebras

F

• ❴

❴ ❴ ❴ ❴

quantum state spaces

◮

Theorem: If G is continuous, then F degenerates.

◮

That’s right. (F(Mn) = ∅ for n ≥ 3.)

◮ So G better not be continuous

So quantum state spaces must be radically different ...

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Part II Classical viewpoints

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Doctrine of classical concepts

“However far the phenomena transcend the scope of classical physical explanation, the ac- count of all evidence must be expressed in classi- cal terms.... The argument is simply that by the word experiment we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account

• f the experimental arrangements and of the re-

sults of the observations must be expressed in unambiguous language with suitable application

• f the terminology of classical physics.”

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Classical viewpoints

◮ Invariant that circumvents the obstruction:

Given an operator algebra A, consider C(A) = {C ⊆ A commutative subalgebra}, the collection of classical viewpoints.

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Classical viewpoints

◮ Invariant that circumvents the obstruction:

Given an operator algebra A, consider C(A) = {C ⊆ A commutative subalgebra}, the collection of classical viewpoints.

◮

Theorem: Can reconstruct A as a piecewise algebra. (A ∼ = colim C(A))

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Piecewise structures

◮

A piecewise widget is a widget that forgot

• perations between noncommuting elements.

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Piecewise structures

◮

A piecewise widget is a widget that forgot

• perations between noncommuting elements.

◮ A piecewise complex *-algebra is a set A with:

◮ a reflexive binary relation ⊙ ⊆ A2; ◮ (partial) binary operations +, ·: ⊙ → A; ◮ (total) functions ∗: A → A and ·: C × A → A;

such that every S ⊆ A with S2 ⊆ ⊙ is contained in a T ⊆ A with T 2 ⊆ ⊙ where (T, +, ·, ∗) is a commutative ∗-algebra.

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Piecewise structures

◮

A piecewise widget is a widget that forgot

• perations between noncommuting elements.

◮ A piecewise Boolean algebra is a set B with:

◮ a reflexive binary relation ⊙ ⊆ B2; ◮ (partial) binary operations ∨, ∧: ⊙ → B; ◮ a (total) function ¬: B → B;

such that every S ⊆ B with S2 ⊆ ⊙ is contained in a T ⊆ B with T 2 ⊆ ⊙ where (T, ∧, ∨, ¬) is a Boolean algebra.

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Piecewise structures

◮

A piecewise widget is a widget that forgot

• perations between noncommuting elements.

◮ Every projection lattice gives a piecewise Boolean algebra:

• ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣

❥❥❥❥❥❥❥❥❥❥❥❥❥ ✈✈✈✈✈✈ ❍ ❍ ❍ ❍ ❍ ❍ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲

• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈

❍ ❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈

❍ ❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❲

❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲

• ❚

❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚

• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❥❥❥❥❥❥❥❥❥❥❥❥❥
• ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
• 15 / 31

Piecewise structures

◮

A piecewise widget is a widget that forgot

• perations between noncommuting elements.

◮ Every projection lattice gives a piecewise Boolean algebra:

• ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣

❥❥❥❥❥❥❥❥❥❥❥❥❥ ✈✈✈✈✈✈ ❍ ❍ ❍ ❍ ❍ ❍ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲

• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈

❍ ❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈

❍ ❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❲

❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲

• ❚

❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚

• ❍

❍ ❍ ❍ ❍ ❍

• ✈✈✈✈✈✈
• ❥❥❥❥❥❥❥❥❥❥❥❥❥
• ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
• ◮

Theorem: There is no piecewise morphism Proj(C3) → {0, 1}

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Piecewise structures

◮

Theorem: Can reconstruct A as a piecewise algebra. (A ∼ = colim C(A))

◮ What we can say about a quantum system

= what we can say about it from classical viewpoints = what we can say about A using just C(A) = what we can say about A as a piecewise algebra

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Piecewise structures

◮

Theorem: Can reconstruct A as a piecewise algebra. (A ∼ = colim C(A))

◮ What we can say about a quantum system

= what we can say about it from classical viewpoints = what we can say about A using just C(A) = what we can say about A as a piecewise algebra

◮ How much is this? Quite a bit:

◮ Quantum foundations: Bohrification ◮ Quantum logic: Bohrification ◮ Quantum information theory: entropy 16 / 31

Contextual entropy

Define: contextual entropy of state ρ of A function Eρ : C(A) → R, C → Shannon entropy H(tr(ρ −))

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Contextual entropy

Define: contextual entropy of state ρ of A function Eρ : C(A) → R, C → Shannon entropy H(tr(ρ −)) Theorem: contextual entropy generalises von Neumann entropy S(ρ) = min{Eρ(C) | C ∈ C(A)}

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Contextual entropy

Define: contextual entropy of state ρ of A function Eρ : C(A) → R, C → Shannon entropy H(tr(ρ −)) Theorem: contextual entropy generalises von Neumann entropy S(ρ) = min{Eρ(C) | C ∈ C(A)} Theorem: Eρ determines ρ! (in dim ≥ 3)

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Bohrification: history

reformulate with classical viewpoints general topos approach to physics Bohrification attempts at dynamics

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Bohrification: idea

◮ Consider “contextual sets”

assignment of set S(C) to each classical viewpoint C ∈ C(A) such that C ⊆ D implies S(C) ⊆ S(D)

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Bohrification: idea

◮ Consider “contextual sets”

assignment of set S(C) to each classical viewpoint C ∈ C(A) such that C ⊆ D implies S(C) ⊆ S(D)

◮ They form a topos T (A)!

category whose objects behave a lot like sets in particular, it has a logic of its own!

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Bohrification: idea

◮ Consider “contextual sets”

assignment of set S(C) to each classical viewpoint C ∈ C(A) such that C ⊆ D implies S(C) ⊆ S(D)

◮ They form a topos T (A)!

category whose objects behave a lot like sets in particular, it has a logic of its own!

◮ There is one canonical contextual set A

A(C) = C

19 / 31

Bohrification: idea

◮ Consider “contextual sets”

assignment of set S(C) to each classical viewpoint C ∈ C(A) such that C ⊆ D implies S(C) ⊆ S(D)

◮ They form a topos T (A)!

category whose objects behave a lot like sets in particular, it has a logic of its own!

◮ There is one canonical contextual set A

A(C) = C

◮

Theorem: T (A) believes that A is a commutative operator algebra!

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Bohrification: caveats

Change rules to make quantum system classical. Price:

◮ No proof by contradiction. (P ∨ ¬P) ◮ No choice. (Si = ∅ =

⇒

i Si = ∅) ◮ No real numbers. (completions of Q differ)

20 / 31

Bohrification: caveats

Change rules to make quantum system classical. Price:

◮ No proof by contradiction. (P ∨ ¬P) ◮ No choice. (Si = ∅ =

⇒

i Si = ∅) ◮ No real numbers. (completions of Q differ)

No matter! Theorem: A determined by state space (within T (A))

20 / 31

Bohrification: quantum state space?

Change rules to make quantum system classical. Price:

◮ No proof by contradiction. (P ∨ ¬P) ◮ No choice. (Si = ∅ =

⇒

i Si = ∅) ◮ No real numbers. (completions of Q differ)

No matter! Theorem: A determined by state space (within T (A)) Circumvents obstruction ...

20 / 31

Piecewise structures: how far can we get?

Theorem: If C(A) ∼ = C(B), then A ∼ = B as Jordan algebras (for W*-algebras without I2 term) Theorem: If C(A) ∼ = C(B), then A ∼ = B as piecewise Jordan algebras (for all C*-algebras except C2 and M2)

21 / 31

Piecewise structures: how far can we get?

Theorem: If C(A) ∼ = C(B), then A ∼ = B as Jordan algebras (for W*-algebras without I2 term) Theorem: If C(A) ∼ = C(B), then A ∼ = B as piecewise Jordan algebras (for all C*-algebras except C2 and M2)

◮ So need to add more information to C(A) ...

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Part III Interaction between classical viewpoints

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Five stages of grief

Established psychology:

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Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”

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Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”
• 2. Anger: “Why are you destroying my groups? I hate you!”

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Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”
• 2. Anger: “Why are you destroying my groups? I hate you!”
• 3. Bargaining: “At least think in terms of commutative groups?”

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Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”
• 2. Anger: “Why are you destroying my groups? I hate you!”
• 3. Bargaining: “At least think in terms of commutative groups?”
• 4. Depression: “I wasted my life on the wrong groups!”

23 / 31

Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”
• 2. Anger: “Why are you destroying my groups? I hate you!”
• 3. Bargaining: “At least think in terms of commutative groups?”
• 4. Depression: “I wasted my life on the wrong groups!”
• 5. Acceptance: “Noncommutative groups are cool!”

23 / 31

Five stages of grief

Established psychology:

• 1. Denial: “These are not groups!”
• 2. Anger: “Why are you destroying my groups? I hate you!”
• 3. Bargaining: “At least think in terms of commutative groups?”
• 4. Depression: “I wasted my life on the wrong groups!”
• 5. Acceptance: “Noncommutative groups are cool!”
• 6. Stockholm syndrome: “Commutative groups? Don’t care!”

23 / 31

Active lattices: idea

• perator

algebra

qqqqqqqqqqqqqqq

classical viewpoints

24 / 31

Active lattices: idea

• perator

algebra

projections

qqqqqqqqqqqqqqq

lattice

◮ Replace classical viewpoints C(A)

by projection lattice {p ∈ A | p∗ = p = p2}

24 / 31

Active lattices: idea

• perator

algebra

projections

qqqqqqqqqqqqqqq

unitaries

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

lattice group

◮ Replace classical viewpoints C(A)

by projection lattice {p ∈ A | p∗ = p = p2}

◮ Any ∗-algebra has unitary group {u ∈ A | uu∗ = 1 = u∗u}

24 / 31

Active lattices: idea

• perator

algebra

projections

qqqqqqqqqqqqqqq

unitaries

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

lattice group

◮ Replace classical viewpoints C(A)

by projection lattice {p ∈ A | p∗ = p = p2}

◮ Any ∗-algebra has unitary group {u ∈ A | uu∗ = 1 = u∗u} ◮ Unitaries act on projections (u · p = upu∗)

Projections inject into unitaries (p → 1 − 2p)

24 / 31

Active lattices: idea

• perator

algebra

projections

qqqqqqqqqqqqqqq

unitaries

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ✤ ✤ ✤

lattice active lattice

• group

◮ Replace classical viewpoints C(A)

by projection lattice {p ∈ A | p∗ = p = p2}

◮ Any ∗-algebra has unitary group {u ∈ A | uu∗ = 1 = u∗u} ◮ Unitaries act on projections (u · p = upu∗)

Projections inject into unitaries (p → 1 − 2p) So projections act on themselves!

24 / 31

Symmetries

◮

− → p − → 1 − 2p projections ∼ = self-adjoint unitaries =“symmetries”

25 / 31

Symmetries

◮

− → p − → 1 − 2p projections ∼ = self-adjoint unitaries =“symmetries”

◮ Sym(A) is subgroup of unitaries generated by symmetries

25 / 31

Symmetries

◮

− → p − → 1 − 2p projections ∼ = self-adjoint unitaries =“symmetries”

◮ Sym(A) is subgroup of unitaries generated by symmetries ◮ if A type I1, then

Sym(A) = { all symmetries }

◮ if A type I2/I3/..., then Sym(A) = { u | det(u)2 = 1 } ◮ if A type I∞/II/III, then Sym(A) = { all unitaries }

25 / 31

Active lattices

◮ An action of a (piecewise) group G on a (piecewise) lattice P

is a homomorphism G → Aut(P)

26 / 31

Active lattices

◮ An action of a (piecewise) group G on a (piecewise) lattice P

is a homomorphism G → Aut(P)

◮ An active lattice is:

◮ a piecewise AW*-algebra A ◮ a lattice structure P on the projections ◮ a group structure G on the symmetries ◮ an action of G on P 26 / 31

Active lattices

◮ An action of a (piecewise) group G on a (piecewise) lattice P

is a homomorphism G → Aut(P)

◮ An active lattice is:

◮ a complete orthomodular lattice P ◮ a group G generated by P ◮ an action of G on P 26 / 31

Active lattices

◮ An action of a (piecewise) group G on a (piecewise) lattice P

is a homomorphism G → Aut(P)

◮ An active lattice is:

every AW*-algebra A has one:

◮ a complete orthomodular lattice P

Proj(A)

◮ a group G generated by P

Sym(A)

◮ an action of G on P

u · p = upu∗

26 / 31

Active lattices

◮ An action of a (piecewise) group G on a (piecewise) lattice P

is a homomorphism G → Aut(P)

◮ An active lattice is:

every AW*-algebra A has one:

◮ a complete orthomodular lattice P

Proj(A)

◮ a group G generated by P

Sym(A)

◮ an action of G on P

u · p = upu∗

◮

Theorem: Its active lattice determines A (full and faithful functor)

26 / 31

Matrix algebras

◮ If A is an operator algebra, then so is Mn(A)

27 / 31

Matrix algebras

◮ If A is an operator algebra, then so is Mn(A) ◮ “All AW*-algebras are matrix algebras”

If A type In, then A ∼ = Mn(C) If A type I∞/II∞/III, then A ∼ = Mn(A)

27 / 31

Matrix algebras

◮ If A is an operator algebra, then so is Mn(A) ◮ “All AW*-algebras are matrix algebras”

If A type In, then A ∼ = Mn(C) If A type I∞/II∞/III, then A ∼ = Mn(A)

◮

Theorem: Classical viewpoints in Mn(A) are diagonal. (∀C ∈ C(Mn(A)) ∃u ∈ U(Mn(A)): uCu∗ diagonal)

27 / 31

Matrix algebras: projections

◮

Even if A has few projections, Mn(A) has lots!

28 / 31

Matrix algebras: projections

◮

Even if A has few projections, Mn(A) has lots! pij(a) = (1 + aa∗)−1 (1 + aa∗)−1a a∗(1 + aa∗)−1 a∗(1 + aa∗)−1a

• a

1 p12(a)

28 / 31

Matrix algebras: projections

◮

Even if A has few projections, Mn(A) has lots! pij(a) = (1 + aa∗)−1 (1 + aa∗)−1a a∗(1 + aa∗)−1 a∗(1 + aa∗)−1a

• a

1 p12(a)

◮ These vector projections encode algebraic structure of A!

pij(a + b) = polynomial in pij(a), pik(b), pjk(c), . . . pij(ab) = polynomial in pik(a), pkj(b), . . . pij(a∗) = polynomial in pji(a), . . .

28 / 31

Active lattices determine operator algebras

◮

Theorem: Its active lattice determines A (full and faithful functor)

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Active lattices determine operator algebras

◮ Lemma: If f : Proj(Mn(A)) → Proj(Mn(B)) equivariant,

then f (pij(a)) = pij(ϕ(a)) for some ϕ: A → B.

◮

Theorem: Its active lattice determines A (full and faithful functor)

29 / 31

Active lattices determine operator algebras

◮ Lemma: If f : Proj(Mn(A)) → Proj(Mn(B)) equivariant,

then f (pij(a)) = pij(ϕ(a)) for some ϕ: A → B.

◮ Lemma: The vector projections generate Proj(Mn(A)). ◮

Theorem: Its active lattice determines A (full and faithful functor)

29 / 31

Active lattices determine operator algebras

◮ Lemma: If f : Proj(Mn(A)) → Proj(Mn(B)) equivariant,

then f (pij(a)) = pij(ϕ(a)) for some ϕ: A → B.

◮ Lemma: The vector projections generate Proj(Mn(A)). ◮ Recall: “All AW*-algebras are matrix algebras” ◮

Theorem: Its active lattice determines A (full and faithful functor)

29 / 31

Active lattices determine operator algebras

◮ Lemma: If f : Proj(Mn(A)) → Proj(Mn(B)) equivariant,

then f (pij(a)) = pij(ϕ(a)) for some ϕ: A → B.

◮ Lemma: The vector projections generate Proj(Mn(A)). ◮ Recall: “All AW*-algebras are matrix algebras” ◮

Theorem: Its active lattice determines A (full and faithful functor)

◮ What is the logic of such things ... ??

29 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

30 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

◮ Physics = dynamics and kinematics in one

30 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

◮ Physics = dynamics and kinematics in one ◮ Quantum logic = modal / dynamic

30 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

◮ Physics = dynamics and kinematics in one ◮ Quantum logic = modal / dynamic ◮ Logic of contextuality

30 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

◮ Physics = dynamics and kinematics in one ◮ Quantum logic = modal / dynamic ◮ Logic of contextuality ◮ Protocol specification language

30 / 31

Whither quantum structures?

“Knowing a quantum system = all classical viewpoints + switching between them”

◮ Physics = dynamics and kinematics in one ◮ Quantum logic = modal / dynamic ◮ Logic of contextuality ◮ Protocol specification language ◮ Noncommutative topology, game theory, database theory ...

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References

“Extending obstructions to noncommutative functorial spectra”

Journal of Pure and Applied Algebra, 2013

“Noncommutativity as a colimit”

Applied Categorical Structures 20(4):393–414, 2012

“A topos for algebraic quantum theory”

Communications in Mathematical Physics 291:63–110, 2009

“Diagonalizing matrices over AW*-algebras”

Journal of Functional Analysis 264(8):1873–1898, 2013

“Active lattices determine AW*-algebras”

Advances in Mathematics, 2013 31 / 31