Preliminaries on double duals of vector spaces Order unit ... · Alfsen and Shultz, State spaces of...

Introduction & overviewPreliminaries on double duals of vector spaces

Order unit spaces, and the easy half of Kadison dualityKadison duality, the difficult half

Appendix on C∗-algebras and effect modulesRadboud University Nijmegen

Lectures on Kadison Duality

Bart Jacobs

Institute for Computing and Information Sciences – Digital SecurityRadboud University Nijmegen

Oxford University, May 20-22, 2014

Jacobs 20-22 May 2014 Lectures on Kadison Duality 1 / 77




Outline

Introduction & overview

Preliminaries on double duals of vector spacesVector spaces with algebraic dualNormed vector spaces with norm dual

Order unit spaces, and the easy half of Kadison duality

Kadison duality, the difficult half

Appendix on C ∗-algebras and effect modulesEffect algebras & modules





Richard V. Kadison (1925)

• Student of Marshal Stone, PhD Univ. Chicago 1956

• Famous ao. for the Kadison-Ringrose books Fundamentals ofthe Theory of Operators (I and II), from 1983 and 1986





Kadison duality

(complete order

unit spaces

)op

'(

convex compactHausdorff spaces

)

• Essentials published in 1951 as Memoirs AMS — one half:representation of order unit spaces

• At the heart of the duality between states andobservables/effects/predicates in quantum foundations

• Crying out for a modern, categorical description





Own contribution in these lectures

• No new mathematical results!

• Systematic presentation of results and proofs — in categoricalterminology — but without historical details

• Proofs of auxiliary results are scattered around in theliterature, and collected here

• Possibly there is some novelty in the different adjointrelationships and resulting monads between:

1 order unit spaces2 sets, and convex sets, and compact Hausdorff spaces, and

convex compact Hausdorff spaces





Relevant / used literature

• Original article• R. Kadison, A representation theory for commutative

topological algebra, Memoirs of the AMS, 7, 1951

• Additional books/papers• Ellis, The duality of partially ordered normed vector spaces, J.

London Math. Soc., 1964.• Alfsen, Compact Convex Sets and Boundary Integrals, 1971• Nagel, Order unit and base norm spaces, in LNP 29, 1974• Asimow and Ellis, Convexity Theory and its Applications in

Functional Analysis, 1980• Kadison and Ringrose, Fundamentals of the Theory of

Operator Algebras. Volume I. Elementary Theory, 1983• Alfsen and Shultz, State spaces of operator algebras: basic

theory, orientations and C∗-products, 2001• Paulsen and Tomforde, Vector Spaces with an order unit,

Indiana Univ. Math. Journ., 2009





Prerequisites

Basic knowledge of:

1 linear algebra

2 topology

3 category theory• categories, functors, adjunctions• mainly used for organising mathematical structures and the

transformations between them





Relevant categories — details follow

• Sets — sets and functions

• Conv — convex sets and affine functions

• CH — compact Hausdorff spaces and continuous maps

• CCH — convex compact Hausdorff spaces and affinecontinuous maps• subcategory CCHsep → CCH where points can be separated

by maps to [0, 1]• equivalently, convex compact subspaces of locally convex space

• EMod — effect modules• subcategory BEMod → EMod of Banach/complete modules

• OUS — order unit spaces• subcategory BOUS → OUS of Banach/complete spaces





Basic states-and-effects adjunction (BJ, Ifip TCS’10)

EModop

EMod(−,[0,1]),,> Conv

Conv(−,[0,1])

ll

With Mandemaker it was shown [QPL’11] that this adjunctionrestricts to an equivalence

BEModop ' CCHsep

between Banach effect modules and convex compact Hausdorffspaces

The proof used Kadison duality:

BEModop ' BOUSop ' CCHsep

proven

II

Kadison,assumed

RR





Kadison duality, in relation to C ∗-algebras

Heisenbergpredicate transformers

︷︸︸︷Schrodinger

state transformers︷︸︸︷


(CstarUP)opeffects

cc

states

;;

self-adjoints

OO

where CstarUP is the category of C ∗-algebras with positive unital(UP) maps.

The effects of a C ∗-algebra A can equivalently be described as:

• [0, 1]A = a ∈ A | 0 ≤ a ≤ 1• UP-maps C2 → A, so maps A→ 1 + 1 in (CstarUP)op

The states are the maps A→ CJacobs 20-22 May 2014 Lectures on Kadison Duality 11 / 77




Relevant structures/categories

norm

BEMod ' BOUS ' (CCHsep)op

vvmmmmmmm((QQQQQQQ

Bana

OUS

rrfffffffffffffffffff

NVect

((RRRRRRR

AA

OVect

vvlllllll

Vect

order





Dualities, via dualising objects

Frequently dualities arise by “homming into” a dualising object D,as in:

Aop

Hom(−,D)**' B

Conv(−,D)

kk

• Set-theoretically there is a “double dual” map:

X // dd // D(DX ) given by dd(x)(f ) = f (x)

• In order to make this an isomorphism one restricts the maps

X → D and also DX → D

This gives a representation of X inside the function spaces.





Examples

• In Stone duality, one proves for a Boolean algebra B,

B∼= // Cont

(BA(B, 2), 2

)

∼= Clopens(

Ultrafilters(B))

• Here the plan is to prove for an order unit space V ,

V // // AffineCont(

PositiveUnital(V ,R), R)

This injection is an isomorphism iff V is complete (Banach).

It is obtained via restriction of the double dual map V → V ∗∗





Overview over part I

For a vector space V , over K = R or K = C, there is the algebraicdual:

V ∗ = ω : V → K | ω is linear.It comes with the double dual map dd : V → V ∗∗, withdd(v)(ω) = ω(v). It is linear & natural & iso if dimV <∞.

If W is a normed vector space there is the norm dual:

W# = ω : W → K | ω is linear and bounded.

Both V ∗ and W# →W ∗ carry a “weak-* topology”

Main results in part I: the double dual map restricts to

Vsurj // // V ∗′ W

∼= // W#′

where (−)′ describes the continuous dual





Algebraic duals and linear combinations

Proposition Let V ∈ VectK . ω ∈ V ∗ is a linear combinationω =

∑i kiωi of ω1, . . . , ωn ∈ V ∗ if and only if

⋂i ker(ωi ) ⊆ ker(ω).

Proof (only-if) is easy; for (if) use induction on n, with base case:

Lemma If ker(ω) ⊆ ker(ρ) for ω, ρ ∈ V ∗, then ρ = k · ω, for somescalar k ∈ K .

Proof If ω 6= 0, use V / ker(ω) K , so quotient V / ker(ω) isone-dimensional. Choose base vector b ∈ V such thatV / ker(ω) = K · (b + ker(ω)). Take k = ρ(b)

ω(b) . Using

ker(ω) ⊆ ker(ρ) we have a well-defined map ρ : V / ker(ω)→ K ,namely ρ(v + ker(ω)) = ρ(v). Pick v and writev + ker(ω) = s · b + ker(ω), to show ρ(v) = k · ω(v).





Weak topology on a vector space

Definition For Ω ⊆ V ∗ the weak topology σ(V ,Ω) on V is theweakest topology that makes each (ω : V → K ) ∈ Ω continuous.

Thus, subbasic opens of this topology are of the form

ω−1(S) = v ∈ V | ω(v) ∈ S

for ω ∈ Ω and S ⊆ K open.

We write the continuous dual as:

V ′ = f : V → K | f is linear and continuous

Question: can we characterise maps in V ′ ?





Weakly open and weakly continuous

Lemma Consider the weak topology σ(V ,Ω), for Ω ⊆ V ∗

1 basic opens around 0 ∈ V are of the form:

x ∈ V | ∀i ≤ n. |ωi (x)| < r

for ω1, . . . , ωn ∈ Ω and r > 0.

2 If f : V → K is weakly continuous, then:

|f (v)| ≤ b ·∨

i|ωi (v)|

for certain ω1, . . . , ωn ∈ Ω and b ≥ 0.





The weak-* topology

Recall the double dual map dd : V → V ∗∗ with dd(v)(ω) = ω(v).

Definition The image dd(V ) ⊆ V ∗∗ yields the weak-* topology onV ∗.

Often it is written as σ(V ∗,V ) instead of σ(V ∗, dd(V )).

It’s subbasic opens are:

dd(v)−1(S) = ω ∈ V ∗ | dd(v)(ω) = ω(v) ∈ S

for v ∈ V and S ⊆ K open.





Double dual surjectivity theorem

Theorem V V ∗′

That is: for each weakly continuous ρ : V ∗ → K there is a v ∈ Vwith ρ = dd(v).

Proof There are b ≥ 0 and v1, . . . , vn ∈ V with, for all ω ∈ V ∗,

|ρ(ω)| ≤ b ·∨

i|dd(vi )(ω)| = b ·

∨i|ω(vi )|.

Then ker(dd(v1)) ∩ · · · ∩ ker(dd(vn)) ⊆ ker(ρ). Thus we can writeρ as linear combination ρ = k1dd(v1) + · · ·+ kndd(vn). But thenρ = dd(k1v1 + · · ·+ knvn) by linearity of dd.





Normed vector spaces

A normed vector space V has a norm ‖ − ‖ : V → R≥0 satisfying:

• ‖v‖ = 0 iff v = 0;

• ‖k · v‖ = |k | · ‖v‖;• ‖v + w‖ ≤ ‖v‖+ ‖w‖

A linear map f : V →W between normed V ,W is bounded ifthere is an s ∈ R≥0 with ‖f (v)‖ ≤ s · ‖v‖, for all v ∈ V .

We write NVectK for the resulting category

The operator norm is defined for such a bounded f as:

‖f ‖op =∧s ∈ R≥0 | ∀v ∈ V . ‖f (v)‖ ≤ s · ‖v‖.

Each homset Hom(V ,W ) is then itself a normed vector space,with ‖f (v)‖ ≤ ‖f ‖op · ‖v‖.





Notation

For a normed space V write the norm dual:

V# = ω : V → K | ω is linear and bounded.

For r ≥ 0 write the r -ball as:

V≤r = v ∈ V | ‖v‖ ≤ r

These two can be combined in:(V#)≤1

= ω ∈ V# | ‖ω‖op ≤ 1.





Hahn-Banach extension & separation results

Proposition Let V be a normed vector space over the reals

1 Let S ⊆ V be a linear subspace with a linear map ω : S → Rsatisfying |ω(x)| ≤ ‖x‖, for all x ∈ S .Then there is an ω′ ∈ (V#)≤1 with ω′|S = ω.

2 For each v ∈ V there is an ω ∈ (V#)≤1 with ω(v) = ‖v‖.3 For v 6= v ′ in V there is an ω ∈ (V#)≤1 with ω(v) 6= ω(v ′).

4 For each closed linear subspace S ⊆ V and each v 6∈ S thereis an ω ∈ V# with ω|S = 0 and ω(v) = 1.





The double dual, in the normed case

Lemma The double dual map dd : V → V ∗∗ restricts todd : V → V##

Proof We have ‖dd(v)‖op ≤ ‖v‖ since:

|dd(v)(ω)| = |ω(v)| ≤ ‖ω‖op · ‖v‖ = ‖v‖ · ‖ω‖op.

Definition The weak-* topology on V# is induced bydd(V ) ⊆ V##.





Main results about the weak-* topology

Theorem For a normed vector space V over R,

1 Each ball (V#)≤r ⊆ V# is compact Hausdorff in the weak-*topology — aka. Banach-Alaoglu

2 dd : V → V## is an isometry: ‖dd(v)‖op = ‖v‖.

Proof sketch (V#)≤r is a closed subset of the compact spaceD =

∏v∈V [−r · ‖v‖, r · ‖v‖ ].

We saw ‖dd(v)‖op ≤ ‖v‖. Then (≥) follows because there is anω ∈ (V#)≤1 with ω(v) = ‖v‖.





Double dual isomorphism theorem

Theorem V∼=−→ V#′

That is: for each weakly continuous ρ : V# → K there is a uniquev ∈ V with ρ = dd(v).

Proof Existence we already know. Uniqueness follows because ddis an isometry, and thus injective.





Complete normed spaces

Definition A Banach space is a complete normed vector space:every Cauchy sequence has a limit — using distanced(v ,w) = ‖v − w‖.We write BanK → NVectK for the full subcategory.

Proposition The standard metric completion construction yields aleft adjoint:

BanK

a NVectK

@@





Recall: we have done the left leg, so far

norm

BEMod ' BOUS ' (CCHsep)op

vvmmmmmmm((QQQQQQQ

Bana

OUS

rrfffffffffffffffffff

NVect

((RRRRRRR

AA

OVect

vvlllllll

Vect

order

With main result, for a normed vector space V ,

Vdd∼=

// V#′ = ρ : V# → R | ρ is weakly continuous

where:• V# = ω : V → R | ω is linear and bounded• the weak-* topology on V# is the least one that makes all

point-evaluation maps dd(v) : V# → R continuousJacobs 20-22 May 2014 Lectures on Kadison Duality 30 / 77




Order on a vector space

Definition A vector space V is ordered if it carries a partial order≤ satisfying:

1 v ≤ v ′ implies v + w ≤ v ′ + w , for all w ∈ V ;

2 v ≤ v ′ implies r · v ≤ r · v ′ for all r ∈ R≥0.

We write OVectK → VectK for the subcategory of ordered vectorspaces with monotone linear maps between them.

(Aside: in a Riesz space there is additionally a join ∨)

For a linear map f : V →W it is equivalent to say:

• f is monotone: v ≤ v ′ implies f (v) ≤ f (v ′)

• f is positive: 0 ≤ v implies 0 ≤ f (v)





Unit in an ordered space

Definition A strong unit in an ordered vector space V is a positiveelement 1 ∈ V with:

• for each v ∈ V there is an n ∈ N with −n · 1 ≤ v ≤ n · 1

An Archimedean unit is a strong unit with:

•[∀r > 0. v ≤ r · 1

]implies v ≤ 0

An order unit space is a ordered vector space over R with anArchimedean unit.

The category OUS → OVectR has as maps that are

• linear, positive/monotone, and unital (preserve 1)

• we call them simple Positive-Unital (UP)





How order unit spaces may arise

Sets `∞

((QQQQQQQ CHC

vvmmmmmmm

OUSop

Conv A∞

66mmmmmmmCCHAC

hhQQQQQQQ

We shall see that all these functors have (right) adjoints, given by“states”

Recall that Conv is the category of convex sets and affine maps

• in a convex set X , finite convex sums∑

i rixi exist, ofelements xi ∈ X and ri ∈ [0, 1] with

∑i ri = 1

• affine functions preserve such convex sums





`∞ : Sets→ OUSop

• For a set X , write `∞(X ) for the bounded maps X → R:

`∞(X ) = φ : X → R | ∃b.∀x . |φ(x)| ≤ b

• Vector space operations on `∞(X ) are inherited pointwisefrom R

• The order is also pointwise: φ ≤ φ′ iff φ(x) ≤ φ′(x) for all x

• The unit 1 : X → R in `∞(X ) is the “constant 1” map• This unit is strong because maps in `∞(X ) are bounded• It’s Archimedean since R is Archimedean

• For a function f : X → Y we get (−) f : `∞(Y )→ `∞(X ).





A∞ : Conv→ OUSop

• For an affine set X ∈ Conv we take

A∞(X ) = φ : X → R | φ is bounded and affine

• OUS-structure is like in `∞(X )• affine maps are closed under (pointwise) vector operations





C : CH→ OUSop and AC : CCH→ OUSop

• For a compact Hausdorff space X ∈ CH take

C(X ) = φ : X → R | φ is continuous

Such maps φ are automatically bounded.• OUS-structure is obtained pointwise, like before

• Similarly, for a convex compact Hausdorff space X ,

AC(X ) = φ : X → R | φ is affine continuous

Aside: the `∞ and C constructions yield C ∗-algebras, but A∞ andAC do not, in the convex case.

• affine maps are not closed under multiplication





States and effects in OUS

• A state in an OUS V is a UP-map ω : V → R• we write Stat(V ) = Hom(V ,R)• R is initial in OUS, so a state is a point 1→ V in OUSop

• Example: Stat(Rn) = D(n) = probability distributions on n

• An effect is an element v ∈ V with 0 ≤ v ≤ 1• we write [0, 1]V = v ∈ V | 0 ≤ v ≤ 1• an effect corresponds to a map R2 → V• and thus to a map V → 1 + 1 in OUSop

• example: [0, 1]Rn = [0, 1]n

• The Born rule gives the validity probability ω |= v• obtained by function application (ω |= v) = ω(v) ∈ [0, 1]• more abstractly,

by composition in OUSop 1state //

probability ((QQQQQQQQ Veffect

1 + 1Jacobs 20-22 May 2014 Lectures on Kadison Duality 37 / 77




Aside about OUS-effect module relationship

• The mapping V 7→ [0, 1]V yields an equivalence of categories:

OUS ' AEMod

where AEMod → EMod is the full subcategory ofArchimedean effect modules

• By imposing (suitable) completeness requirements it restrictsto:

BOUS ' BEMod

as discussed in the introduction (with ‘B’ for ‘Banach’)





Convexity for OUS-maps

Lemma For V ,W ∈ OUS the homset Hom(V ,W ) is convex.Pre- and post-composition (−) g and h (−) are affine.

Proof The key point is that f =∑

i ri fi : V →W is unital again:

f (1) =∑

i ri · fi (1) =∑

i ri · 1 = (∑

i ri ) · 1 = 1 · 1 = 1.

Corollary Taking states yields a functor:

OUSop Stat // Conv





Adjunctions Sets OUSop and Conv OUSop

Proposition The states functor is right adjoint in:

OUSop

Stat

OUSop

Stata a

Sets

`∞DD

Conv

A∞DD

Proof By swapping arguments we get bijective correspondences:

Vf // `∞(X ) in OUS

=============X g

// Stat(V ) in Sets

Vf // A∞(Y ) in OUS

==============Y g

// Stat(V ) in Conv





Relating `∞ and A∞

These adjunctions allow some abstract categorical reasoning.

Proposition A∞ D ∼= `∞, where D is the (discrete probability)distribution monad on Sets

Proof We have Conv = EM(D) and so the result follows bycomposition and uniqueness of adjoints in:

OUSop

Stat

Stat

uu

aConv

forget

A∞DD

aSets

DDD`∞

66





Order unit spaces are normed

Definition Each V ∈ OUS carries a norm defined by:

‖v‖ou =∧r ∈ R≥0 | − r · 1 ≤ v ≤ r · 1.

Example The induced norm on `∞(X ) ∈ OUS is the supremum(aka. uniform) norm:

‖φ‖∞ =∨|φ(x)| | x ∈ X

It exists since φ is bounded. Similarly for A∞(X ), C(X ), AC(X )(The spaces are actually complete)

Lemma Each map f in OUS satisfies ‖f ‖op = 1. We get afunctor:

OUS // NVectR

In particular, Stat(V ) ⊆ (V#)≤1





Weak-* topology on states

Stat(V ) ⊆ V# inherits the weak-* topology, with subbasic opensdd(v)−1(S) = ω ∈ Stat(V ) | ω(v) ∈ S, for v ∈ V , S ⊆ R open.

Proposition

1 This weak-* topology on Stat(V ) is compact Hausdorff

2 Alternative basic opens are, for v ∈ V and s ∈ Q,

s(v) = ω ∈ Stat(V ) | ω(v) > s,

3 For each f : V →W in OUS, the map(−) f : Stat(W )→ Stat(V ) is continuous

Proof Stat(V ) ⊆ (V#)≤1 is a closed subset of a compactHausdorff space. The rest is easy





Adjunctions CH OUSop and CCH OUSop

Proposition The states functor is also right adjoint in:

OUSop

Stat

OUSop

Stata a

CH

C

DD

CCH

AC

DD

As a result, C U ∼= `∞ and AC E ∼= `∞ in:

OUSop

Stat

Stat

uu

OUSop

Stat

Stat

uu

a aCH

forget

C

DD

CCH

forget

AC

DD

a aSets

UDD`∞

66

Sets

EDD`∞

66

Here U = ultrafilter, and E = expectation.Jacobs 20-22 May 2014 Lectures on Kadison Duality 44 / 77




Summarising adjunction diagram, with induced monads

SetsFE@GE AB @@ `∞

!!

CHFECDRAB^^C

OUSop

Stat

bb

Statuu

Stat

==

Stat ))ConvAB@G

unusedFE

A∞66

CCHABCD see belowFE

AC

hh

Here R is the Radon monad, with (Furber & Jacobs, Calco’13):

K`(R) ' (CCstarUP)op

We continue with the lower-right adjunction OUSop CCH.





The easy half of Kadison duality

Theorem The adjunction OUSop CCH restricts toOUSop CCHsep, with separation of points

The adjunction’s unit, for X ∈ CCHsep, is the evaluation map

Xev // Stat(AC(X )) given by ev(x)(φ) = φ(x)

This unit ev is an isomorphism.

Proof It is easy to see that ev is continuous and affine. It isinjective by the separation property: ifev(x)(φ) = φ(x) = φ(y) = ev(y)(φ) for all φ ∈ Stat(X ), thenx = y .For surjectivity we prove that the image ev(X ) ⊆ Stat(AC(X )) isweak-* dense.





Proof, continued, with auxiliary result

Lemma For each φ ∈ AC(X ) and for each state ω ∈ Stat(AC(X ))there is an x ∈ X with ω(φ) = φ(x).

Proof The image φ(X ) ⊆ R is compact, and thus closed andbounded, and also convex. Hence it is a closed interval:φ(X ) = [s1, s2] ⊆ R. Take the middle m = 1

2 (s1 + s2) ∈ [s1, s2] andradius r = 1

2 (s2 − s1), so that φ(X ) = t ∈ R | |m − t| ≤ r.The function φ−m · 1 ∈ AC(X ) has r as upperbound:

‖φ−m · 1‖∞ =∨

x∈X

∣∣(φ−m · 1)(x)∣∣ =

∨

x∈X

∣∣φ(x)−m∣∣ ≤ r

because φ(x) ∈ φ(X ). For ω ∈ Stat(AC(X )), since ‖ω‖op = 1,∣∣ω(φ)−m

∣∣ =∣∣ω(φ−m · 1)

∣∣ ≤ ‖φ−m · 1‖∞ ≤ r

But then ω(φ) ∈ φ(X ), and thus ω(φ) = φ(x), for some x ∈ X . Jacobs 20-22 May 2014 Lectures on Kadison Duality 47 / 77




Proof, continued, denseness

Recall that we still need to prove ev(X ) ⊆ Stat(AC(X )) is dense

Assume a non-empty subbasis open of Stat(AC(X )), namely:

s(φ) = ω | ω(φ) > s

for some φ ∈ AC(X ) and s ∈ Q. We must shows(φ) ∩ ev(X ) 6= ∅.So let ω(φ) > s, using that s(φ) 6= ∅. By the Lemma, there is anx ∈ X with ω(φ) = φ(x) = ev(x)(φ). But then ev(x) ∈ s(φ).





Where do we stand?

• Our aim to prove an equivalence BOUSop ' CCHsep

• We have adjoint functors in two directions:

OUSop

Stat=Hom(−,R)--> CCHsep

AC

ll

• The unit of this adjunction is an isomorphism: forX ∈ CCHsep,

Xev∼=

// Stat(AC(X ))

• The counit is the (restricted) double dual map, for V ∈ OUS,

Vdd // AC(Stat(V ))

• Next, we will prove two things:• this dd is injective — in fact an isometry• it is also surjective — via the Krein-Smulian Theorem





Basic results about order unit spaces

Lemma Let V be an order unit space with, norm ‖− ‖ and v ∈ V

1 one can write v = v+ − v−, for positive vectors v+, v− ∈ V ;

2 −‖v‖ · 1 ≤ v ≤ ‖v‖ · 1;

3 if f (1) = 0 for a positive linear map f : V →W , then f = 0;

4 each positive linear functional ρ : V → R is bounded, with‖ρ‖op = ρ(1);

5 for each non-zero positive linear map ρ : V → R there is aunique r > 0 such that rρ is a state, namely r = 1

ρ(1) .

6 if −w ≤ v ≤ w then ‖v‖ ≤ ‖w‖.





Extension & separation results for order unit spaces

Proposition Let V be an order unit space.

1 If S ⊆ V is a linear subspace containing 1, then each stateω : S → R extends to a state ω′ : V → R with ω′|S = ω

2 Each ρ ∈ V# can be written as ρ = ρ+ − ρ− whereρ+, ρ− ∈ V# are positive and ‖ρ+‖op + ‖ρ−‖op = ‖ρ‖op

3 v ≥ 0 iff ω(v) ≥ 0 for all ω ∈ Stat(V )

4 If v 6= v ′, then there is an ω ∈ Stat(V ) with ω(v) 6= ω(v ′)

5 ‖v‖ =∨|ω(v)| | ω ∈ Stat(V )





The counit of OUSop CCHsep is an isometry

The “double dual” counit map

Vdd // AC(Stat(V ))

• is an isometry: ‖dd(v)‖op = ‖v‖ — and thus injective

• preserves and reflects the order

Proof of the second point

dd(v) ≤ dd(v ′) ⇐⇒ dd(v)(ω) ≤ dd(v ′)(ω), for all ωsince the order is pointwise

⇐⇒ ω(v) ≤ ω(v ′), for all ω⇐⇒ v ≤ v ′ by the previous result





The counit is an isomorphism

Theorem (Kadison) The counit dd : V AC(Stat(V )) is anisomorphism iff V is complete (Banach)

Proof The (only if) part is easy, since AC(Stat(V )) is complete.

The (if) part requires much more work.

Strategy: extend φ ∈ AC(Stat(V )) to a weakly continuousφ : V# → R in:

Stat(V )

affine continuous φ((QQQQQQQQQQQQQQ

// V#

weakly continous φR

Then use the isomorphism V∼=−→ V#′





Proof, step 1, assuming φ : Stat(V )→ R

Recall that each ρ ∈ V# can be written as difference ρ = ρ1 − ρ2

of positive maps ρ1, ρ2 ≥ 0.

If ρi 6= 0, then we can write ρi = riωi for ωi ∈ Stat(V ), whereri = ρi (1) and ωi = 1

ρi (1)ρi .

Then define:

φ(ρ) = r1φ(ω1)− r2φ(ω2) = ρ1(1)φ( 1ρ1(1)ρ1)− ρ2(1)φ( 1

ρ2(1)ρ2)

One now proves that this φ : V# → R• is well-defined, ie. independent of choice of ρ1, ρ2

• is linear — and also bounded

• extends φ : Stat(V )→ RThese verifications use that φ is affine





Auxiliary result

We rely on:

Theorem (Krein-Smulian) If V is a complete order unit space,then f : V# → R is weakly continuous if its restriction to(V#)≤r → R is weakly continuous, for each r ≥ 0.

• The general result is formulated for Banach spaces (not V#)

• The proof is non-trivial and uninsightful, see for instance(Dunford & Schwartz 1957), (Conway 1990), (Pedersen1989), (Asimov & Ellis 1980).





Proof, step 2, given φ : V# → R

We need to prove that the restriction φ : (V#)≤r → R is weaklycontinuous, for an arbitrary r > 0.

Assume a weakly converging net να ∈ (V#)≤r withlim ν = ρ ∈ (V#)≤r . We need to write all maps as difference ofpositive parts

να = ρ+α − ρ−α ρ = ρ+ − ρ−

Next we need to scale them to states. These scaling factors andstates form a new net, in a compact space:

µα ∈ [−r , r ]× Stat(V )× [−r , r ]× Stat(V )

Thus there is a converging subnet. With this subnet one showsφ(lim ν) = limφ(ν).





Finally, Kadison duality

Theorem There is an equivalence of categories

BOUSop ' CCHsep

Main occurrence: relating observables and states in C ∗-algebras:

BOUSop ' CCHsep

(CstarUP)opself-adjoints

UU

states

II which gives the samepicture as eg. in the

non-deterministic case

(CL∧)op ' CL∨

K`(P)2(−)

WW

P

JJ

Such triangles form a general pattern for predicate and statetransformer semantics.





A cousin-duality: Yosida

Recall: A Riesz space is an ordered vector space with binary joins∨ — and thus also ∧.

Write BARsz → BOUS for the category of “Banach-ArchimedeanRiesz spaces”

Theorem (Yosida/Stone) There is an equivalence of categories:

BARszopStat=Hom(−,R)

,,' CHC

mm





Kadison and Yosida

The following diagram commutes

CCHsep

a forget

' // BOUSop

CH

RBB

BARszop'oo

forget

OO

As a result, there is a left adjoint to the forgetful functor in:

BOUS

BARsz

a forget

OO





Maps between C ∗-algebras

• Traditionally, almost always, C ∗-algebras are used with*-homomorphisms as maps

• However, there are good reasons to consider them with the(weaker) unital positive (UP) maps• probabilistic nature of UP-maps, in the commutative case• relation to order unit spaces and effect algebras

• The goal here is to explain some of the differences — and topromote UP-maps a bit

• (I don’t wish to distinguish positive and completely positivemaps at this stage)





Maps between C ∗-algebras, in detail

A linear map f : A→ B between C ∗-algebras is called:

• unital (U), if f (1) = 1(variation: subunital means 0 ≤ f (1) ≤ 1)

• positive (P), if a ≥ 0⇒ f (a) ≥ 0(where a ≥ 0 means a = x∗x , for some x)

• multiplicative (M), if f (a · a′) = f (a) · f (a′)

• involutive (I), if f (a∗) = f (a)∗

FACTS UP ⇒ I and MIU ⇒ UP

We use categories CstarMIU and CstarUP → CstarMIU

MIU-maps are usally called ∗-homomorphisms; they are the“standard” maps in C ∗-algebra theory

• eg. Gelfand duality: CH '(CCstarMIU

)op





Some characteristic differences between MIU and UP

• Functors from Hilbert spaces to C ∗-algebras:(

Hilbert spaceswith unitary maps

)// (CstarMIU)op

(Hilbert spaceswith isometries

)// (CstarUP)op

• Projections and effects arise via different maps:

MIU-maps C2 → A==============

projections in A

UP-maps C2 → A=============

effects in A





MIU-maps example

For n,m ∈ N, there is a bijective correspondence:

MIU-maps Cn // Cm

===================functions m // n

Essentially, this is the finite-dimensional version of Gelfand duality:

FinSets '(FdCCstarMIU

)op





MIU-maps example, continued

Proof of the correspondence.

• Each f : m→ n obviously gives (−) f : Cn → Cm. Itpreserves the (pointwise) structure.

• Assume ϕ : Cn → Cm is a MIU map. Write the standard basevectors as | i 〉 = (0, . . . , 0, 1, 0, . . . , 0) ∈ Cn.

Since | i 〉 · | i 〉 = | i 〉, we get ϕ(| i 〉) · ϕ(| i 〉) = ϕ(| i 〉), so thatϕ(| i 〉) = (ri1, . . . , rim) ∈ Cm consists of rij ∈ 0, 1.Since

∑i | i 〉 = 1 ∈ Cn, we get

∑i ϕ(| i 〉) = ϕ(1) = 1 ∈ Cm,

so that∑

i rij = 1, for each j ≤ m.

But then: for each j ≤ m there is precisely one i ≤ n with arij = 1. This yields a function m→ n.





UP-maps example

For n,m ∈ N, there is a bijective correspondence:

UP-maps Cn // Cm

===================functions m // D(n)

where D is the distribution monad.

This gives “probabilistic” Gelfand duality, in the finite case:

K`N(D) '(FdCCstarUP

)op

where K`N(D) → K`(D) is the full subcategory with numbersn ∈ N as objects.

Thus, FdCCstarUP is the Lawvere theory of the distribution monad





UP-maps example, continued

Proof of the correspondence.

• Each f : m→ D(n) gives a map Cn → Cm by:

v 7−→ λj ≤ m.∑

i≤nf (j)(i) · v(i)

• Assume ϕ : Cn → Cm is a UP map. The base vector | i 〉 ∈ Cn

is positive, and so ϕ(| i 〉) = (ri1, . . . , rim) ∈ Cm consists ofpositive (real) numbers rij

As before,∑

i ϕ(| i 〉) = ϕ(1) = 1 ∈ Cm, so for each j ≤ m wehave

∑i rij = 1.

Thus we get the required map m→ D(n).





Beyond the finite-dimensional case

In (Furber & Jacobs, CALCO’13) it is shown that:

• There is a Radon monad R : CH→ CH on the category CHof compact Hausdorff spaces

• This monad R is given by states of the C ∗-algebra C (X ):

R(X ) = HomUP

(C (X ), C

)

• We then get::K`(R) '

(CCstarUP

)op





Effect algebras, definition

Effect algebras axiomatise the unit interval [0, 1] with its (partial!)addition + and “negation” x 7→ 1− x .

A Partial Commutative Monoid (PCM) consists of a set M withzero 0 ∈ M and partial operation > : M ×M → M, which issuitably commutative and associative.

One writes x ⊥ y if x > y is defined.

An effect algebra is a PCM in which each element x has a unique‘orthosuplement’ x⊥ with x > x⊥ = 1 ( = 0⊥ )Additionally, x ⊥ 1⇒ x = 0 must hold.

There is then a partial order, via x ≤ y iff y = x > z , for some z .





Effect algebras, main examples

1 Projections / closed subspaces on a Hilbert space form aneffect algebra; P⊥ is orthocomplement:

〈x | y〉 = 0 for all x ∈ P, y ∈ P⊥

2 Orthomodular lattices are effect algebras, with > as join x ∨ yonly for elements with x ⊥ y , i.e. x ≤ y⊥

3 Each Boolean algebra is an effect algebra: it is a distributiveorthomodular lattice, in which x ⊥ y iff x ∧ y = 0.

In particular, the Boolean algebra of measurable subsets of ameasure space forms an effect algebra, where U > V isdefined if U ∩ V = ∅, and is then equal to U ∪ V .





Homomorphisms of effect algebras

Definition

A homomorphism of effect algebras f : X → Y satisfies:

• f (1) = 1

• if x ⊥ x ′ then both f (x) ⊥ f (x ′) and f (x > x ′) = f (x) > f (x ′).

This yields a category EA of effect algebras.

Examples:

• There is a functor Measop → EA, via (A,ΣX ) 7→ ΣX

(This forms an adjunction, via homming into 2 = 0, 1)

• A probability measure yields a map ΣX → [0, 1] in EA.





Effect modules

Effect modules are effect algebras with a scalar multiplication, withscalars not from R or C, but from [0, 1].(Or more generally from an “effect monoid”, ie. effect algebra with multiplication)

Definition

An effect module M is a effect algebra with an action[0, 1]×M → M that is a “bihomomorphism”

A map of effect modules is a map of effect algebras that commuteswith scalar multiplication.

We get a category EMod → EA.





Effect modules, main examples

Probabilistic examples

• Fuzzy predicates [0, 1]X on a set X , with scalar multiplication

r · p def= λx ∈ X . r · p(x).

• Measurable predicates Hom(X , [0, 1]), for a measurable spaceX , with the same scalar multiplication.

Quantum examples

• Effects E(H) on a Hilbert space: operators A : H → Hsatisfying 0 ≤ A ≤ I , with scalar multiplication (r ,A) 7→ rA.

• Effects in a C ∗-algebra A: positive elements below the unit:

[0, 1]A = a ∈ A | 0 ≤ a ≤ 1.

This one covers the previous three illustrations.





Full & faithful functors

Since positive unital maps are determined by what they do onpositive elements we get full & faitful functors

CstarUPeffects // EMod CstarUP

self-adjoints// OUS

Together they form the state-effect diagram:


(CstarUP)opeffects=Hom(−,1+1)

cc

states=Hom(1,−)

;;

self-adjoints

OO





Observation about predicates, and sneak-preview

There is a general construction that yields effect algebra/modulestructure on predicates X → 1 + 1 in a category.

The examples below discuss maps X → 1 + 1 as predicates.

• Obviously in Sets

• In K`(D) one gets fuzzy predicates [0, 1]X

• in (CstarUP)op on gets effects

• in rings Rngop one gets idempotents e2 = e, forming aBoolean algebra — used in the sheaf theory of rings

• in distributive lattices DLatop one gets elements withcomplements, ie. x with x ′ satisfying x ∧ x ′ = 0, x ∨ x ′ = 1.





Thanks for your attention. Questions/remarks?


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