elliott's topological enrichment of the cuntz semigroupkeimel/papers/muenster… · domains....

Elliott’s topological enrichment of theCuntz semigroup

Klaus KeimelFachbereich Mathematik

Technische Universitat Darmstadt

www.mathematik.tu-darmstadt.de/ keimel

7. Mai 2016

Klaus Keimel Elliott’s topological enrichment of the Cuntz semigroup

References

[CEI] K. T. Coward, G. A. Elliott, and C. Ivanescu. The Cuntzsemigroup as an invariant for C∗-algebras. Journal ReineAngew. Math. 623 (2008), pp 161–193.

[ERS] G. A. Elliott, L. Robert, and L. Santiago, The cone oflower semicontinuous traces on a C ∗-algebra. AmericanJournal of Mathematics 133 (2011), 969–1000.

[ABP] R. Antoine, J. Bosa, and F. Perera, Completions ofmonoids with applications to the Cuntz semigroups.International Journal of Mathematics, 22 (6) (2011), 837–861.

[APT] R. Antoine, F. Perera, and H. Thiel, Tensor productsand regularity properties of Cuntz semigroups. arXiv:1410.0483(2015).


References

[AJ] S. Abramsky, A. Jung, Domain Theory. In: Handbook ofLogic in Computer Science, Vol. III, Clarendon Press, 1994,pages 1–168.

[GHKLMS] G. Gierz, K. K. Hofmann, K. Keimel, J. D.Lawson, M. Mislove, D. S. Scott, Continuous Lattices andDomains. Cambridge University Press, 2003.

[P] G. D. Plotkin, A domain-theoretic Banach-Alaoglutheorem. Mathematical Structures in Computer Science 16(2006), pages 299–312.

[K] K. Keimel, Weak topologies and compactness inasymmetric functional analysis. Topology and its Applications185/186 (2015), pages 1–22.


Historical Note 1

Dana S. Scott. Continuous Lattices, LNM 274 (1972), pages97–136. Springer Verlag.

Scott introduced continuous lattices for building models of theuntyped λ-calculus, a logical system introduced by A. Church in the1930ies. Untyped λ-calculus was used in programming languageslike LISP although no model was known before Scott’s discovery.

Scott’s definition: On every partially ordered set P, he defines atopology σ, later called the Scott topology; a subset U is open if,for every directed subset D of P that has a lub supD ∈ U, there isa d ∈ D ∩U. In a complete lattice L he defines x ≺ y if y is in theσ-interior of the upper set ↑ x = {z ∈ L | x ≤ z}. A completelattice is called continuous if every element y is the lub of the setof elements x ≺ y .


Historical Note 2

A. Day. Filter monads, continuous lattices and closure systems,Can. J. Math. 27 (1975), pages 50–59.

In category theory the notion of a monad (also called triple at thetime) was introduced as a category theoretical formulation of ’freeconstructions’ like free groups, etc. The monad of ultrafilters onsets was an example not coming from algebra. The algebras of theultrafilter monad were known to be the compact Hausdorff spaces.

The filters (of subsets) on sets were also known to yield a monad,the filter monad. The question was asked: What are the algebrasof the filter monad. It was shown by A. Day that the algebras ofthe filter monad are the Scott’s continuous lattices. So the freecontinuous lattice over a set is the lattice of all filters (of subsets)ordered by inclusion.


Historical Note 3

Compact (Hausdorff) semigroups were a field of interest ofHofmann, Keimel, Lawson, Mislove in the 1960ies. A particularcase are compact semilattices S . A natural question was whetherthe continuous homomorphisms h : S → [0, 1] separate the pointsof S , where [0, 1] is considered as a compact semilattice withx ∧ y = min(x , y). In

J. D. Lawson. Topological semilattices with small semilattices, J.London Math. Soc. 1 (1969), 719–724,

it was shown that the answer is YES if, and only if, every elementof S had a neighborhood basis sets with a least element.


Historical Note 3

Hofmann and Stralka where looking for a purely order theoreticalcharacterisation of Lawson’s semilattices in 1974:

K H Hofmann, A L Stralka: The algebraic Theory of CompactSemilattices. Dissertationes Mathematicae CXXXVII (1976), 54pages.

On p. 27 one finds the following characterisation of the compactsemilattices characterised by Lawson:


The Definition

”It is now notationally convenient to call an element x in a latticeL relatively compact under y iff it is contained in any ideal I of Lwith sup I ≥ y . This is ostensibly equivalent to saying that for allsubsets D ⊆ L with supD ≥ y there is a finite subset subsetF ⊆ Y with x ≤ supF . Let us call a lattice relatively algebraic if itis complete and every element in it is the l.u.b. of all relativelycompact elements under it.”

It was in the sequel of this work that it was discovered that thisstructure was equivalent to the continuous lattices introduced byD. S. Scott in his 1972 paper.The development of DOMAIN THEORY really then started in1976. The further development was strongly motivated bydenotational semantics. The generalization from lattices topartially ordered sets was forced by this background.


Comparison

For posets and even for directed complete ones, Scott’s relation ≺does not agree with Hofmann and Stralka’s �.

s ≺ y implies x � y always holds. (Indeed, if x ≺ y andy ≤ supi yi for a directed family (yi )i , then yi is in the interior or↑ x for some i , whence x ≤ yi .)x � y implies x ≺ y holds if every y is the supremum of a directedfamily of elements x ≺ y , but not in general.

Counterexample:Begin with a chain x < x1 < x2 < · · · < y = supn xn. Add chainsxi1 < xi2 < · · · < xi = supj xij for every i . Only add comparisonsforced by the ones already given: xij ≤ xk for i ≤ k and xij ≤ y forall i , j . Then x � y , but x 6≺ y , as the Scott interior of ↑ x isempty.


Domain Theory

For a mathematician, the terminology DOMAIN Theory is a bitunfortunate. But the development of the theory was mainly due tothe use of these mathematical structures for building models– semantic domains – in which formal expressions, programs, etc.,can be interpreted.


Predomains

A predomain is a structure (P,�, 0) where P is a set with adistinguished element 0 and a binary relation � which is

(NT) non-trivial: ∀a. 0� a

(T) transitive: a� b � c =⇒ a� c

(Int) interpolative:

ai � c (i = 1, 2) =⇒ ∃b.ai � b � c (i = 1, 2)

A basis of a predomain P is a subset B such that the interpolatingelement b in (Int) can always be chosen from B. Such a basis isagain a predomain with the restriction of � to B. A predomain issecond countable if it has a countable basis. It is first countable iffor every element c there is a countable subset Bc such that theinterpolating element b in (Int) can be chosen from Bc .


Predomains





ai � c (i = 1, 2) =⇒ ∃b.ai � b � c (i = 1, 2)

A basis of a predomain P is a subset B such that the interpolatingelement b in (Int) can always be chosen from B. Such a basis isagain a predomain with the restriction of � to B.

A predomain issecond countable if it has a countable basis. It is first countable iffor every element c there is a countable subset Bc such that theinterpolating element b in (Int) can be chosen from Bc .


Predomains





ai � c (i = 1, 2) =⇒ ∃b.ai � b � c (i = 1, 2)

A basis of a predomain P is a subset B such that the interpolatingelement b in (Int) can always be chosen from B. Such a basis isagain a predomain with the restriction of � to B. A predomain issecond countable if it has a countable basis. It is first countable iffor every element c there is a countable subset Bc such that theinterpolating element b in (Int) can be chosen from Bc .


Very simple examples

Q∗, r � s if r < s or r = s = 0.

R+

R+ = R+ ∪ {∞}


Basic example

Let X be a locally compact Hausdorff space,C0(X ) the C∗-algebra of continuous f : X → C vanishing at infinity,C0(X )+ its positive cone, i.e., the set of all continuous f : X → R+

vanishing at infinity.

Notation: For f ∈ C0(X )+ and ε > 0, let

(f − ε)+ be defined by max(f (x)− ε, 0)

For g , f ∈ C0(X )+, define

g � f ⇐⇒ ∃ε > 0. g ≤ (f − ε)+

Taking ε = 1n we obtain a sequence fn = (f − 1

n )+ � f such thatfor every g � f there is an n such that g ≤ fn; thus,(C0(X )+, 0,�) is a first countable predomain.


Topology on a predomain P

The sets ↑↑a = {c ∈ P | a� c}, a ∈ P,are the basis of a topology σ on P. The topological spaces thatarise in this way are called C-spaces (Erne), also α-spaces(Ershov). The relation � can be recovered from the topology:

x � y if, and only if, the intersection of all the neighborhoods of xis a neighborhood of y .

A predomain is first or second countable if and only if its topologyσ is first or second countable. The topology σ is far from beingHausdorff. It is the topology of (uniform) convergence from below.

Continuity of functions f : P → Q between predomains is meantwith respect to their σ-topologies, equivalently,whenever b � f (a) there is an a′ � a such that b � f (a′).

σ-open sets on R+: half-open intervals ]r ,∞].For function to R+: continuous for σ = lower semicontinuous.


Relative compactness and domains

Let (P,≤) be a directed complete poset, that is, every (upwards)directed subset D has a lub supD. For x , y ∈ P, say thatx is relatively compact in y and write

x � y if y ≤ supD =⇒ ∃d ∈ D. x ≤ d

for every directed subset D ⊆ P .

A domain is a directed complete poset with a least element 0 suchthat, for every element y , there is a directed set of elements x � ythe sup of which is y .

Every domain is a predomain. The predomain topology σ agreeswith the Scott topology. A function between domains isσ-continuous iff it is monotone and preserves suprema of directedsets.


Round ideals and domains

A subset I of a predomain (P,�, 0) is a round ideal if

0 ∈ I

a� b ∈ I =⇒ a ∈ I

ai ∈ I (i = 1, 2) =⇒ ∃b ∈ I .ai � b (i = 1, 2)

Example: For every a ∈ P, the set

↓↓a = {b ∈ P | b � a}

is a round ideal.


Round ideal completion

For a predomain (P,�, 0), the collection I(P) of all round idealsof P ordered by ⊆ is directed complete, and even a domain, calledthe round ideal completion of P.

For round ideals I , J, one has

I � J if I ⊆ ↓↓b for some b ∈ J

The map a 7→ ↓↓a : P → I(P) is σ-continuous and preserves �.

One can look of the image of P in I(P):

P = {↓↓a | a ∈ P}

By the above P is a base of I(P).


Round ideal completion, ctd.

If P is second countable, the same holds for its round idealcompletion.

If P is first countable, the round ideal completion need not be firstcountable. (Counterexample: C0(X )+ for X locally compact, butnot sigma-compact.) In this case, the round ideals ↓↓a arecountably generated. One can restrict to the countably generatedround ideals (= round ω-ideals) and one obtains a first countablepredomain Iω(P), which is an ω-domain in the sense that everycountable directed subset (equivalently, every increasing sequence,)has a least upper bound.


Universal property

The map x 7→ ↓↓x : P → I(P) is the universal completion:

For every continuous map f from P into a domain D there is aunique continuous map f : I(P)→ D such that f (↓↓x) = f (x),

namely f (I ) =∨↑

x∈I f (x):

P↓↓ - I(P)

D

f

?

f-

The extension f preserves � if and only if f preserves �.


The functor I

From the universal property we obtain: For every continuous mapf : P → Q between predomains there is a unique continuous mapI(f ) : I(P)→ I(Q) such that I(f )(↓↓a) = ↓↓f (a). If f preserves�, also I(f ) preserves �.

P↓↓ - I(P)

Q

f

?

↓↓- I(Q)

I(f )

?

Clearly I is a functor.


Basic example

The round ideals of Q+ are the Dedekind cuts [0, r [, r ∈ R+, theround ideal completion is R+.

The round ideal completion of C0(X )+ is (isomorphic to) the setLSC (X )+ of all lower semicontinuous functions f : X → [0,∞].

A subtle point:A predomain (P,�, 0) inherits a natural preorder -nat from theorder ⊆ of its round ideal completion:

a -nat bdef⇐⇒ ↓↓a ⊆ ↓↓b

It often occurs – as in the examples above – that a preorder isalready given on the predomain which is close to the naturalpreorder but need not be the same.


Pre-Cuntz semigroups

(S ,+, 0,�) is a is a pre-Cuntz semigroup if (S ,�, 0) is apredomain and (S ,+, 0) a commutative monoid and (S ,�, 0)predomain such that + is continuous and preserves �.

(Continuity:If c � a + b there are a′ � a, b′ � b such that c � a′ + b′.�-preserving: a′ � a, b′ � b =⇒ a′ + b′ � a + b.)

(S ,+, 0,≤) is a(n abstract) Cuntz semigroup, if (S , 0,≤) is adomain in which + is continuous and preserves �..

Completion

The round ideal completion of a pre-Cuntz semigroup is a Cuntzsemigroup.

If S is first countable pre-Cuntz semigroup, we can form a roundω-ideal completion Iω(S).


Pre-Cuntz semigroups

(S ,+, 0,�) is a is a pre-Cuntz semigroup if (S ,�, 0) is apredomain and (S ,+, 0) a commutative monoid and (S ,�, 0)predomain such that + is continuous and preserves �.(Continuity:If c � a + b there are a′ � a, b′ � b such that c � a′ + b′.�-preserving: a′ � a, b′ � b =⇒ a′ + b′ � a + b.)

(S ,+, 0,≤) is a(n abstract) Cuntz semigroup, if (S , 0,≤) is adomain in which + is continuous and preserves �..

Completion

The round ideal completion of a pre-Cuntz semigroup is a Cuntzsemigroup.

If S is first countable pre-Cuntz semigroup, we can form a roundω-ideal completion Iω(S).


Basic example and proof

Q+ is a pre-Cuntz semigroupR+ a Cuntz semigroup.

C0(X )+ is a pre-Cuntz semigroup andLSC (X )+ a Cuntz semigroup.

Proof: +: S × S → S is continuous and preserves �. ThusI(+): I(S × S)→ I(S) is also continuous and preserves �. Nowuse that the functor I preserves products, i.e.,I(S × S) ∼= I(S)× I(S).One checks that the canonical map a 7→ ↓↓a : S → I(S) is amonoid homomorphism. That the extended addition iscommutative and associative is then straightforward.


Round ideal completion preserves finite products

Claim: I(P × Q) = I(P)× I(Q), that is, every round ideal K ofP × Q is of the form I × J, where I and J are round ideals of Pand Q, respectively.

For a proof we let I = {a ∈ P | (a, 0) ∈ K} andJ = {b ∈ Q | (0, b) ∈ K}. Then I and J are round ideals. Weshow that K = I × J. Indeed, if (a, b) ∈ K , there area′ � a, b′ � b such that (a′, b′) ∈ K . Then (a, 0)� (a′, b′),whence (a, 0) ∈ K and si a ∈ I . Similarly, b ∈ J.Conversely, if a ∈ I and b ∈ J, then there are a� a′ ∈ I andb � b′ ∈ J. Thus (a, 0) ∈ K and (0, b) ∈ K so there is an element(a′, b′) ∈ K such that (a, 0)� (a′, b′) and (0, b)� (a′, b′), thatis, a� a′ and b � b′, whence (a, b)� (a′, b′), that is, (a, b) ∈ K .

Since addition is a continuous function from P × P to P preserving�, the extension to a function from I(P)× I(P) to I(P) is alsocontinuous and preserves �.


Dual S∗ of a pre-Cuntz semigroup S

S∗ is the set of all lower semicontinuous monoid homomorphismsλ : S → R+ with pointwise defined order, addition andmultiplication by real numbers r > 0. Note: S∗ ∼= I(S)∗.

τ weak∗upper topology: weakest topology on S∗ for which theevaluations

λ 7→ λ(x) : S∗ → R+, x ∈ S , are lower semicontinuous,τc the co-compact topology associated to τ ,τp the patch topology (generated by τ and τc).

A net λi in S∗ converges to λfor the topology τ if, and only if,

λ(x) ≤ lim inf λi (x) for all x ∈ S

for the topology τc if, and only if,

lim supλi (y) ≤ λ(x) whenever y � x


Banach-Alaoglu Theorem

Theorem

τ and τc are stably compact topologies, τp is compact Hausdorff.

(S∗, τ) and (S∗, τp) are topological cones in the sense thataddition (λ, µ) 7→ λ+ µ and scalar multiplication(r , λ) 7→ rλ : R>0 × S∗ → S∗ are continuous,where R>0 is endowed with the upper (= Scott) topology in thefirst case, with the usual Hausdorff topology in the second case.

The above Theorem is due to Plotkin for continuous d-cones; ageneralization by myself covers Cuntz semigroups.

Elliott, Robert and Santiago have proved this for the patchtopology in two special cases (see later). They introduce the patchtopology directly through convergence as indicated above.


Bidual S∗∗ of a pre-Cuntz semigroup S

S∗∗ the cone of all lower semicontinuous linear functionalsΛ: S∗ → R+ with pointwise defined order, addition and scalarmultiplication (it does not matter whether S∗ is considered withthe weak∗upper topology τ or the patch topolgy τp.

There is a canonical map δ : S → S∗∗: x is mapped to the pointevaluation δ(x) : λ 7→ λ(x). This canonical map can be extendedto the round ideal completion I(S) of S .

Questions: Is S∗∗ a Cuntz semigroup, can it be constructeddirectly from S?

Very partial answer: C0(X )+∗∗ ∼= LSC (X )

(Special case of a result due to Schroder and Simpson).


Positive cone of a C ∗-algebra

Let A be a C∗-algebra, A+ its cone of positive elements.

With respect to the relationa� b if there is an ε > 0 such that a ≤ (b − ε)+,A+ becomes a (first countable) predomain.

More generally: Suppose given a preorder (= reflexive transitiverelation) - on A+ such that a ≤ b =⇒ a - b. Define

a≺≺ b ⇐⇒ def ∃ε > 0. a - (b − ε)+

Claim

(A+,≺≺, 0) is a predomain.


Proof

Remark first that a≺≺ b implies a - b.

Clearly, 0≺≺ a for every a.

For transitivity, suppose a≺≺ b≺≺ c . Then a - b - (c − ε)+ forsome ε > 0, whence a - (c − ε)+, that is, a≺≺ c .

For i = 1, 2, suppose ai≺≺ b. Then there is an ε > 0 such thatai - (b − ε)+. Note that (b − ε)+≺≺ (b − ε

2)+≺≺ b. So,ai≺≺ (b − ε

2)+≺≺ b.


A+ as a pre-Cuntz semigroup

Claim

(A+,+, 0,≺≺) is a pre-Cuntz semigroup, if addition preserves -and if

∀ε > 0. ∃δ > 0. (a + b − ε)+ - (a− δ)+ + (b − δ)+

∀ε > 0. ∃δ > 0. (a− ε)+ + (b − ε)+ - (a + b − δ)+

Addition preserves≺≺, iff

a′≺≺ a, b′≺≺ b =⇒ a′ + b′≺≺ a + b

iff

∃ε > 0. a′ - (a−ε)+, b′ - (b−ε)+ =⇒ ∃δ > 0. a′+b′ - (a+b−δ)+

A sufficient condition for this is that addition preserves - and that

∀ε > 0. ∃δ > 0. (a− ε)+ + (b − ε)+ - (a + b − δ)+


A+ as a pre-Cuntz semigroup ctd.

Addition is continuous iff

c≺≺ a + b =⇒ ∃a′≺≺ a, b′≺≺ b. c≺≺ a′ + b′

iffc≺≺ a + b =⇒ ∃a′≺≺ a, b′≺≺ b. c - a′ + b′

(Indeed, if c≺≺ a + b interpolate c≺≺ c ′≺≺ a + b; then there area′≺≺ a, b′≺≺ b such that c ′ - a′ + b′, whence c≺≺ a′ + b′.)

iff∃ε > 0. c - (a + b − ε)+=⇒ ∃a′, b′, δ > 0. a′ - (a− δ)+, b

′ - (b − δ)+, c - a′ + b′

A sufficient condition for this is:

∀ε > 0. ∃δ > 0. (a + b − ε)+ - (a− δ)+ + (b − δ)+

Indeed, take a′ = (a− δ)+, b′ = (b − δ)+. then a′≺≺ a andb′≺≺ b and c - (a + b − ε)+ - (a− δ)+ + (b − δ)+ = a′ + b′


A+ as a pre-Cuntz semigroup

Cuntz-Pedersen comparison on A+: a - b if there is a sequence(xn)n of elements in A such that a =

∑n xnx

∗n and

∑n x∗nxn ≤ b.

Definition

a≺≺ b if there is an ε > 0 such that a - (b − ε)+.

Claim

(A+,+, 0,≺≺) is a first countable pre-Cuntz semigroup .

Lower semicontinuous trace on A:Lower semicontinuous monotone linear t : A+ → R+ such thatt(x∗x) = t(xx∗) for all x ∈ A.T (A) denotes the cone of all lower semicontinuous traces t.

Claim

T (A) is the dual of the pre-Cuntz semigroup (A+,+, 0,≺≺).


Proof

Consider a linear map t : A+ → R+ such that t(x∗x) = t(xx∗).

(1) Suppose t is lower semicontinuous w.r.t. the c-space topologyσ associated with≺≺. We show that t is monotone and lowersemicontinuous for the norm topology on A+.For r ∈ R+, let

Ur = {a ∈ A+ | t(a) > r}

By hypotheis, Ur is σ-open. It follows that U is an upper set,which implies that t is monotone. Choose any a ∈ U. There also isan ε > 0 such that (a− ε)+ ∈ U. For every b with ||a− b|| < ε onehas (a− ε)+ - b, whence (a− ε)+ -nat b as Robert has shown.And the latter implies b ∈ U.


Proof, ctd.

(2) Suppose that t is monotone and lower semicontinuous w.r.t.the norm topology on A+. We want to show that t is lowersemicontinuous for the topology σ associated with≺≺.For r ∈ R+, now Ur is open for the norm topology (by lowersemicontinuity) and an upper set (by monotonicity). We want toshow that U is open for the topology σ.Since (a− ε)+ converges to a w.r.t. the norm, when ε tends to 0,we have (a− ε)+ ∈ U for ε small enough. Thus, we have anelement b = (a− ε)+) ∈ U such that b≺≺ a.We now look at any element c ∈ A+ with a≺≺ c . Then there is ana′ such that a ∼ a′ ≤ (c − ε)+ for some ε > 0. By monotonicityand lower semicontinuity, t(a) = t(a′). Thent(a) = t(a′) ≤ t((c − ε)+) ≤ t(c) since t is monotone. Hencer < t(a) ≤ t(c), that is, c ∈ U. Thus, U is open for the c-spacetopology σ.


THE Cuntz semigroup of A

For a, b ∈ M∞(A)+ leta - b if there is a sequence xn in A such that a = limn x

∗nbxn,

a ∼ b if a - b and b - a, and define

a≺≺ bdef⇐⇒ ∃ε > 0. a - (b − ε)+

On M∞(A)+/ ∼ one uses the addition a⊕ b.

Claim

(M∞(A)+/ ∼,⊕, 0,≺≺) is a pre-Cuntz semigroup, first countable.

Its round ω-ideal completion Iω is the Cuntz semigroup Cu(A) ofA in the sense of Coward, Elliott, Ivanescu. Its dual has theproperties given by the Banach-Alaoglu Theorem.


Quasitraces on a C∗-algebra A

The lower semicontinuous monoid homomorphism from(M∞(A)+/ ∼,⊕, 0,≺≺) to R+ have been identified to be the lowersemicontinuous 2-quasi-traces on A.

If I understand it correctly, a 2-quasi-trace on A is a functiont : M∞(A)+ → R+ with the following properties:(1) t is monotone(2) linear on the positive part of every commutative C∗-subalgebra,(3) t(xx∗) = t(x∗x) for all x ∈ M∞(A).

QT2(A) denotes the set of all lower semicontinuous 2-quasi-traceson A with pointwise defined order, addition and multiplication withscalars r > 0.

Claim

QT2(A) is (isomorphic to) the dual of the pre-Cuntz semigroup(M∞(A)+/ ∼,⊕, 0,≺≺).


elliott's topological enrichment of the cuntz semigroupkeimel/papers/muenster… · domains....

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