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
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 2 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 4 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 5 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 6 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 7 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 8 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 9 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 10 / 77
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
Kadison duality, in relation to C ∗-algebras
Heisenbergpredicate transformers
︷ ︸︸ ︷Schrodinger
state transformers︷ ︸︸ ︷
BEModop ' BOUSop ' CCHsep
(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
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
Relevant structures/categories
norm
BEMod ' BOUS ' (CCHsep)op
vvmmmmmmm((QQQQQQQ
Bana
OUS
rrfffffffffffffffffff
NVect
((RRRRRRR
AA
OVect
vvlllllll
Vect
order
Jacobs 20-22 May 2014 Lectures on Kadison Duality 12 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 14 / 77
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
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 ∗∗
Jacobs 20-22 May 2014 Lectures on Kadison Duality 15 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 16 / 77
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
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).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 17 / 77
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
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 ′ ?
Jacobs 20-22 May 2014 Lectures on Kadison Duality 18 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 19 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 20 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 21 / 77
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
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‖.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 22 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 23 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 24 / 77
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
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##.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 25 / 77
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
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‖.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 26 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 27 / 77
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
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
@@
Jacobs 20-22 May 2014 Lectures on Kadison Duality 28 / 77
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
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
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
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)
Jacobs 20-22 May 2014 Lectures on Kadison Duality 31 / 77
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
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)
Jacobs 20-22 May 2014 Lectures on Kadison Duality 32 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 33 / 77
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
`∞ : 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 ).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 34 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 35 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 36 / 77
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
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
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
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’)
Jacobs 20-22 May 2014 Lectures on Kadison Duality 38 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 39 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 40 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 41 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 42 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 43 / 77
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
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
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 45 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 46 / 77
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
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
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
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).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 48 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 50 / 77
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
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‖.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 51 / 77
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
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 )
Jacobs 20-22 May 2014 Lectures on Kadison Duality 52 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 53 / 77
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
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#′
Jacobs 20-22 May 2014 Lectures on Kadison Duality 54 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 55 / 77
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
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).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 56 / 77
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
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).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 57 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 58 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 59 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 60 / 77
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
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)
Jacobs 20-22 May 2014 Lectures on Kadison Duality 62 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 63 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 64 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 65 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 66 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 67 / 77
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
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).
Jacobs 20-22 May 2014 Lectures on Kadison Duality 68 / 77
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
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
Jacobs 20-22 May 2014 Lectures on Kadison Duality 69 / 77
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
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 .
Jacobs 20-22 May 2014 Lectures on Kadison Duality 70 / 77
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
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 .
Jacobs 20-22 May 2014 Lectures on Kadison Duality 71 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 72 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 73 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 74 / 77
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
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:
BEModop ' BOUSop ' CCHsep
(CstarUP)opeffects=Hom(−,1+1)
cc
states=Hom(1,−)
;;
self-adjoints
OO
Jacobs 20-22 May 2014 Lectures on Kadison Duality 75 / 77
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
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.
Jacobs 20-22 May 2014 Lectures on Kadison Duality 76 / 77
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
Thanks for your attention. Questions/remarks?
Jacobs 20-22 May 2014 Lectures on Kadison Duality 77 / 77