The Cuntz semigroup and its relation to classification Andrew Toms - - PDF document

The Cuntz semigroup and its relation to classification

Andrew Toms

Notes taken by Hannes Thiel at the Master Class on the Classification of C*-algebras at the University of Copenhagen November 16-27, 2009

• rganizer: Mikael Rørdam
• 1. Part 1 - Lecture from 16.November 2009

We consider C*-algebras 퐴 which are: ∙ separable ∙ unital ∙ nuclear (which is equivalent to being amenable) ∙ usually simple 퐴 is nuclear if for any other C*-algebra 퐵 there is only one way to complete the algebraic tensor product 퐴 ⊙ 퐵 to get a C*-algebra. 1.1. Example (cross products): Any cross product 퐴 = 퐶(푋) ⋊훼 ℤ is nuclear, where 푋 is a compact Hausdorff space, 훼 : 푋 → 푋 is a homeomor-

• phism. Recall that 퐶(푋) ⋊훼 ℤ = 퐶∗(퐶(푋), 푢) where 푢 is a unitary which

implements 훼, i.e. 푢푓푢∗ = 푓 ∘ 훼−1 for any 푓 ∈ 퐶(푋) ⊂ 퐶(푋) ⋊훼 ℤ. 1.2. Example (recursive subhomogeneous algebras): Any recursive sub- homogeneous algebras (RSH-algebra) 퐴 is nuclear. Recall that these are defined as iterated pullbacks using the following data: ∙ compact metric spaces 푋1, . . . , 푋푙 ∙ closed subspaces 푋(0)

푖

⊂ 푋푖 ∙ numbers 푛1, . . . , 푛푙 ∈ ℕ ∙ unital ∗-homomorphisms 휙푘 : 퐴푘−1 → 푀푛푘(퐶(푋(0)

푘 )) (attaching

maps) such that 퐴1 = 푀푛1(퐶(푋1)), and the following is a pullback (for 푘 = 2, . . . , 푙):

The Cuntz semigroup and its relation to classification 2 퐴푘

• 퐴푘−1

휙푘

• 푀푛푘(퐶(푋푘))

∂푘 푀푛푘(퐶(푋(0) 푘 ))

Here ∂푘 is induced by the inclusion 푋(0)

푘

→ 푋푘. Such a pullback is often written as 퐴푘 = 퐴푘−1 ⊕푀푛푘(퐶(푋(0)

푘

)) 푀푛푘(퐶(푋푘)), and the standard way to

define that pullback algebra is as follows: 퐴푘 = {(푎, 푏) : 푎 ∈ 퐴푘−1, 푏 ∈ 푀푛푘(퐶(푋푘)), 휑푘(푎) = ∂푘(푏) = 푏∣푋(0)

푘 }

These algebras are interesting because one can try to extend results form homogeneous to RSH-algebras. Possibly all stably finite C*-algebras are di- rect limits of RSH-algebras. Note also that all RSH-algebras are of type 퐼. What kind of theorem do we want? 1.3. Theorem: Let 퐴, 퐵 be simple, unital, separable, nuclear C*-algebras in some class ℭ. There exists a functor 퐹 : ℭ → ℭ′ such that if 휑 : 퐹(퐴)

∼ =

− → 퐹(퐵) is an isomorphism, then there exists a ∗-isomorphism Φ : 퐴 → 퐵 s.t. 퐹(Φ) = 휑. What is 퐹 typically? It is K-theory and traces. (we do not need quasitraces, since we only consider nuclear C*-algebras, where every quasitrace is auto- matically a trace) 1.4 (퐾0-group): For simplicity let us only consider the unital case. For projections 푝, 푞 ∈ 퐴 ⊗ 핂 say 푝 ∼ 푞 :⇔ there exists some 푣 ∈ 퐴 ⊗ 핂 s.t. 푝 = 푣∗푣, 푣푣∗ = 푞 Set 푉 (퐴) := { the projections in 퐴 ⊗ 핂}/∼. For a projection 푝 ∈ 퐴 ⊗ 핂 we denote its equivalence class in 푉 (퐴) by [푝]. Define an addition on 푉 (퐴) by [푝] + [푞] = [(

푝 0 0 푞

)] . In this way 푉 (퐴) becomes an abelian semigroup. Use the Grothendieck completion process Γ to define an abelian group 퐾0(퐴) := Gr(푉 (퐴)). This comes with a natural map Γ : 푉 (퐴) → 퐾0(퐴) and we denote its image as 퐾0(퐴)+ := Γ(푉 (퐴)). This is also called the positive part (or positive cone) in 퐾0(퐴). Then (퐾0(퐴), 퐾0(퐴)+, [1퐴]) is a pre-ordered, pointed abelian group. A projection 푝 is called infinite if it is equivalent to a proper subprojec- tion, otherwise it is called finite. We call 퐴 stably finite, if all projections in 푀푛(퐴) are finite (for all 푛). In that case 퐾0 is ordered. 1.5 (퐾1-group): Let 풰(퐴) denote the set of unitaries in 퐴, and 풰0(퐴) ⊂ 풰(퐴) its connected component containing 1퐴. The map 푢 → ( 푢 0

0 1퐴

) induces

The Cuntz semigroup and its relation to classification 3 a homomorphism 휑푛 : 풰(푀푛퐴)/풰0(푀푛퐴) → 풰(푀푛+1퐴)/풰0(푀푛+1퐴). We set 퐾1(퐴) := lim − →푛 풰(푀푛퐴)/풰0(푀푛퐴). This is an abelian group with addi- tion defined via [푢] + [푣] = [푢푣]. 1.6 (Traces): A tracial stat on 퐴 is a positive linear functional 푡 : 퐴 → ℂ such that 휏(1퐴) = 1, and 휏(푥푦) = 휏(푦푥) for all 푥, 푦 ∈ 퐴. The set 푇(퐴) of all traces on 퐴 is a metrizable Choquet simplex. A trace defines a state on 퐾0(퐴) as follows: first extend 휏 to a trace 휏⊗tr on 푀푛(퐴) using the canonical trace tr : 푀푛 → ℂ, then for a projection 푝 ∈ 푀푛(퐴) set 휏([푝]) := (휏 ⊗tr)(푝). We get a map 휌퐴 : 푇(퐴) → St(퐾0(퐴), 퐾0(퐴), [1퐴]). For a unital C*-algebra 퐴 the Elliott invariant is: Ell(퐴) := (퐾0(퐴), 퐾0(퐴), [1퐴], 퐾1(퐴), 푇(퐴), 휌퐴) In good cases (퐾0(퐴), 퐾0(퐴), [1퐴], 푇(퐴), 휌퐴) is equivalent to the Cuntz semi- group Cu(퐴), and then Ell(퐴) ∼ = (Cu(퐴), 퐾1(퐴)), which amounts to a de- composition in a positive and unitary part. 1.7 (The Cuntz semigroup): Let 퐴 be unital. For 푎, 푏 ∈ (퐴 ⊗ 핂)+ we say 푎 is Cuntz-dominated by 푏 (denoted 푎 ≾ 푏) if there exists a sequence (푟푛) ⊂ 퐴 ⊗ 핂 s.t. 푟푛푏푟푛∗ → 푎 (in norm). Say 푎 is Cuntz-equivalent to 푏 (denoted 푎 ∼ 푏) if 푎 ≾ 푏 and 푏 ≾ 푎. On projections this agrees with the earlier defined equivalence for stably finite algebras. Note that for any 휆 > 0 and 푎 ∈ (퐴 ⊗ 핂)+ we have 푎 ∼ 휆푎. 1.8. Example: 푀푛 Let 퐴 = 푀푛. Then 푎 ≾ 푏 iff rank(푎) ≤ rank(푏). 1.9. Example: 푀푛(퐶[0, 1]) Let 퐴 = 푀푛(퐶[0, 1]). Then 푎 ≾ 푏 iff rank(푎)(푡) ≤ rank(푏)(푡) for all 푡 ∈ [0, 1]. The reason is that 푎 and 푏 can be approximately unitarily diago- nalized. 1.10. Example: 푀푛(퐶(푋)) Let 퐴 = 푀푛(퐶(푋)) with 푋 a CW-complex of dim(푋) ≥ 3 and 푛 ≥ 2. Then there exist 푎, 푏 ∈ 푀푛(퐶(푋)) s.t. rank(푎)(푡) = rank(푏)(푡) for all 푡 ∈ [0, 1], yet 푎 ≁ 푏. The reason is that dim(푋) ≥ 3 ensures that we can find 푆2 in 푋. We can find projections 푝, 푞 in 푀2(퐶(푆2)) that both have constant rank one, yet 푝 ≁ 푞 (e.g. the trivial line bundle, and the Bott line bundle). Extend this to a small neighborhood of 푆2 ֒ → 푋, and then to positive elements 푎, 푏 ∈ 푀2(퐶(푋)) ⊂ 푀푛(퐶(푋)).

The Cuntz semigroup and its relation to classification 4 1.11. Example: 퐶(푋) Let 퐴 = 퐶(푋) and 푓, 푔 ∈ 퐴+. Then 푓 ≾ 퐺 iff supp(푓) ⊂ supp(푔). 1.12 (The Cuntz semigroup): Define Cu(퐴) := { positive elements in 퐴 ⊗ 핂}/∼. We denote the equivalence class of 푎 ∈ (퐴 ⊗ 핂)+ in Cu(퐴) by ⟨푎⟩. As before we define an addition ⟨푎⟩ + ⟨푏⟩ := 〈( 푎 0

0 푏

)〉 . If we define ⟨푎⟩ ≤ ⟨푏⟩ iff 푎 ≾ 푏, then we get an ordered abelian semigroup. 1.13. Example: 푀푛 Let 퐴 = 푀푛. Then Cu(퐴) = ℕ ∪ {∞} with 푥 + ∞ = ∞, ∞ + ∞ = ∞ and ⟨1퐴⟩ = 푛 ∈ ℕ. 1.14. Example: 푀푛(퐶[0, 1]) Let 퐴 = 푀푛(퐶[0, 1]). Then Cu(퐴) consists of all functions 푓 : [0, 1] → ℕ ∪ {∞} that are the supremum of an increasing sequence of functions 푓(푛) : [0, 1] → {0, . . . , 푛}}. We denote by Aff(푇(퐴)) the continuous affine ℝ-valued functions on 푇(퐴), and by 퐿(푇(퐴)) the functions 푇(퐴) → ℝ ∪ {∞} that are the supremum of an increasing sequence of functions 푓(푛) ∈ Aff(푇(퐴)). Why are we interested in Cu(퐴)? ∙ if Cu(퐴) is nice, you can prove classification theorems for such 퐴 ∙ Cu(퐴) is more sensitive that K-theory and traces Assume 퐴 is unital, exact and 푇(퐴) ∕= ∅. Then every 휏 ∈ 푇(퐴) extends to an unbounded trace on 퐴 ⊗ 핂 as follows: if 푎 ∈ (퐴 ⊗ 핂)+, then define 푑휏(푎) = lim푛→∞ 휏(푎1/푛). This is an example of a dimension function on 퐴, i.e. an additive order- preserving map 휑 : Cu(퐴) → [0, ∞] s.t. 휑(⟨1퐴⟩) = 1. (this gives exactly the lower semicontinuous dimension functions). 1.15. Example: For 푎 ∈ (푀푛)+ we get 푑휏(푎) = rank(푎)/푛. For ⟨푎⟩ ∈ Cu(퐴) we define 휄(⟨푎⟩) : 푇(퐴) → [0, ∞] by 휄(⟨푎⟩)(휏) := 푑휏(푎). Then: ∙ 휄(⟨푎⟩) is in 퐿(푇(퐴)) since 휏 → 휏(푎1/푛) is continuous and 휏(푎1/푛) ≤ 휏(푎1/푛+1) (if ∥푎∥ ≤ 1, so rescale 푎) ∙ if 푎 ≥ 0, 푓 ∈ 퐶∗(푎), 푓 ≥ 0, then 푑휏(푓(푎)) = 휇휏(supp(푓) ∩ 휎(푎)) where 휇휏 is the spectral measure induced by 휏 ∙ 푎 ≾ 푏 iff ∀휀 > ∃훿 > 0 such that (푎 − 휀)+ ≾ (푏 − 훿)+.

The Cuntz semigroup and its relation to classification 5 Question: When is ⟨푎⟩ = ⟨푝⟩ for some projection 푝? 1.16. Lemma: If 퐴 is unital, simple and 푇(퐴) ∕= ∅, then ⟨푎⟩ = ⟨푝⟩ for a projection 푝 iff 0 is not a limit point of 휎(푎). Proof: ⇐: then 푎 ∼ 휒푋(푎) where 휒푋 is the characteristic function on the set (0, ∞) ∩ 휎(푎), and 휒푋(푎) is a projection ⇒: then 푝 ∼ (푝−휀)+ ≾ (푎−훿)+ ≾ 푎 ∼ 푝, whence 푑휏((푎−훿)+) = 푑휏(푝) for all 훿 small enough. But (푎−훿)+ ≤ 푔(푎)+(푎−훿)+ ≤ 푎 for some small function 푔 with supp(푔) ⊂ [0, 훿]. Then 푑휏((푎 − 훿)+) = 푑휏(푔(푎)) + 푑휏((푎 − 훿)+), and therefore 푑휏(푔(푎)) = 0 for all 휏 while 푔(푎) ∕= 0. This is a contradiction. □ Now for 퐴 unital, simple with 푇(퐴) ∕= ∅ we have Cu(퐴) = 푉 (퐴) ⊔ Cu(퐴)+ where Cu(퐴)+ = {⟨푎⟩ : 0 is a limit point of 휎(푎)}. Cu(퐴)+ is absorbing in the sense that 푥 + 푦 ∈ Cu(퐴)+ whenever 푦 ∈ Cu(퐴)+. 1.17. Definition: Let 퐴 be unital. We say 퐴 has strict comparison of pos- itive elements (often abbreviated by just saying ”strict comparison”) if ≾ 푏 whenever 푑휏(푎) < 푑휏(푏) for all 휏 ∈ 푇(퐴) such that 푑휏(푏) < ∞.

The Cuntz semigroup and its relation to classification 6

• 2. Part 2 - Lecture from 17.November 2009

Let 퐴 be simple, unital with 푇(퐴) ∕= ∅. Then Cu(퐴) = 푉 (퐴)⊔Cu(퐴)+. We define a map 휑 : Cu(퐴) → 푉 (퐴) ⊔ 퐿(푇퐴) as 휑(⟨푎⟩) := [푝] whenever 푎 ∼ 푝 for a projection 푝, and for ⟨푎⟩ ∈ Cu(퐴)+ we set 휑(⟨푎⟩) := 휄(⟨푎⟩)(휏) := 푑휏(푎). When is this map injective, when is it surjective? Suppose 퐴 has strict comparison, ⟨푎⟩ ∈ Cu(퐴)+, ⟨푏⟩ ∈ Cu(퐴), and 푑휏(푎) ≤ 푑휏(푏) for all 휏 ∈ 푇(퐴) with 푑휏(푏) < ∞. Since 0 is a limit point of 휎(푎), we have 푑휏((푎−휀)+) < 푑휏(푏) for all 휀 > 0 small enough. From strict comparison

• f 퐴 we get (푎−휀)+ ≾ 푏 for all 휀 > 0 small enough, and therefore also 푎 ≾ 푏.

Thus, if ⟨푎⟩, ⟨푏⟩ ∈ Cu(퐴)+, then ⟨푎⟩ = ⟨푏⟩ iff 푑휏(푎) = 푑휏(푏) for all 휏. Now 휑 is at least injective if 퐴 has strict comparison. When is im(휄) = 퐿푇(퐴)>0? 2.1. Proposition: Let 퐴 be simple, unital with strict comparison and 푇퐴 ∕= ∅. Suppose that for any 푓 ∈ Aff(푇퐴), 휀 > 0 there exists 푎 ∈ (퐴 ⊗ 핂)+, s.t. ∣푑휏(푎) − 푓(휏)∣ < 휀 for all 휏 ∈ 푇퐴. Then for any 푔 ∈ 퐿푇(퐴)>0 there exists 푏 ∈ (퐴 ⊗ 핂)+ s.t. 푑휏(푏) = 푔휏(). Proof: Let 푔 be given. There exists a sequence (푓푛) ⊂ Aff(푇퐴) s.t. 푓푛 > 0, 푓푛 < 푓푛+1 and sup푛 푓푛(휏) = 푔(휏), Find a sequence 휀푛 > 0 s.t. 푓푛 − 휀푛 < 푓푛+1 − 휀푛+1. Then find 푎푛 ∈ (퐴 ⊗ 핂)+ s.t. ∣푑휏(푎푛) − 푓푛(휏)∣ < 휀푛. Then 푑휏(푎푛) < 푑휏(푎푛+1) and sup푛 푑휏(푎푛) = 푔(휏). By strict comparison 푎푛 ≾ 푎푛+1. Suprema of increasing sequences in Cu(퐴) exist, and 푑휏 is sup-preserving. Let ⟨푎⟩ = sup⟨푎푛⟩ ∈ Cu(퐴). Then 푑휏(푎) = 푔(휏). □ So when do we have density (in the sense of the proposition)? 2.2. Definition: We say Cu(퐴) is almost divisible if for any 푥 ∈ Cu(퐴), 푛 ∈ ℕ there exists 푦 ∈ Cu(퐴) s.t. 푛푦 ≤ 푥 ≤ (푛 + 1)푦. 2.3. Proposition: Let 퐴 be simple, unital with 푇(퐴) ∕= ∅ and Cu(퐴) almost

• divisible. It follows that for any 푓 ∈ Aff(푇퐴)>0, 휀 > 0 there exists 푎 ∈

(퐴 ⊗ 핂)+, s.t. ∣푑휏(푎) − 푓(휏)∣ < 휀 for all 휏 ∈ 푇(퐴). Proof:

The Cuntz semigroup and its relation to classification 7 We can assume ∥푓∥ ≤ 1. By a theorem of Lin / Cuntz, Pedersen there exists 푏 ∈ 퐴+ s.t. 휏(푏) = 푓(휏) and ∥푏∥ ≤ 1 + 휀. Then: 푓(휏) = 휏(푏) ≈

푛

∑

푖=1

1/푛휏(휒(푖/푛,∥푏∥](푏)) =

푛

∑

푖=1

1/푛푑휏(푓푖(푏)) for functions 푓푖 with supp(푓푖) = (푖/푛, ∥푏∥] =

푛

∑

푖=1

푑휏(푐푖) Set 푐 = ⊕푛

푖=1 푐푖, then 푑휏(푐) ≈ 푓(휏).

□ 2.4. Theorem: Let 퐴 be simple, unital with strict comparison, 푇(퐴) ∕= ∅ and Cu(퐴) almost divisible. Then Cu(퐴) ∼ = 푉 (퐴) ⊔ 퐿(푇퐴)>0 is an order-

• isomorphism. Here addition on the right hand side is as usual in each of

푉 (퐴) and 퐿(푇퐴)>0, and if 푥 ∈ 푉 (퐴), 푦 ∈ 퐿(푇퐴)>0 then 푥 + 푦 = 휄(푥) + 푦. Also, the order on the right hand side is the usual in each of 푉 (퐴) and 퐿(푇퐴)>0, and if 푥 ∈ 푉 (퐴), 푦 ∈ 퐿(푇퐴)>0 then 푥 ≤ 푦 if 휏(푥) < 푦 in 퐿(푇퐴)>0, and 푦 ≤ 푥 if 푦 ≤ 휄(푥). 2.5. Example: If 퐴 is UHF-algebra with 퐾0(퐴) ∼ = ℚ, then Cu(퐴) ∼ = ℚ+ ⊔ (ℝ+ ∖ {0}) ∪ {∞}. Also Cu(푀푛) = ℕ ∪ {∞}. 2.6. Theorem: (Winter, Lin-Niu) Let 퐴, 퐵 be simple, unital with UCT and locally finite decomposition rank. Also suppose Cu(퐴) = 푉 (퐴) ⊔ 퐿(푇퐴)>0 (similarly for 퐵) and projections separate traces. If there exists an isomor- phism 휑 : 퐾∗(퐴) → 퐾∗(퐵), then there exists a ∗-isomorphism Φ : 퐴 → 퐵 s.t. 퐾(Φ) = 휑. Note that these algebras will have real rank zero after tensoring with an UHF algebra. 2.7. Example: Let 퐴 be simple, unital, exact, finite, 풵-stable. Then 퐴 has strict comparison (the proof uses that strict comparison is equivalent to almost unperforation of Cu(퐴), i.e. if 푥, 푦 ∈ Cu(퐴) with (푛 + 1)푥 ≤ 푦푛 for some 푛, then 푥 ≤ 푦). Also Cu(퐴) is almost divisible. The proof uses: (1) Under the isomorphism 퐴 ⊗ 풵 ∼ = 퐴 we have ⟨푎 ⊗ 1풵⟩ = ⟨푎⟩ (2) There exists an embedding 훾 : 퐶[0, 1] ֒ → 풵 s.t. the image of 휏 ∈ 푇(풵) = {휏} is the Lebesgue measure on [0, 1]. Thus, for any 0 < 휆 < 1 there exists 푎휆 ∈ 퐶[0, 1] s.t. 푑휏(푎휆) = 휆 for all 휏 ∈ 푇(퐴)

The Cuntz semigroup and its relation to classification 8 (3) Compute 푑휏(푎 ⊗ 푎휆) = 휆푑휏(푎) (so Cu(퐴) is a cone) 2.8. Theorem: If 퐴 is a simple, unital ASH-algebra with slow dimension growth, then Cu(퐴) ∼ = 퐶(퐴) ⊔ 퐿(푇퐴)>0 2.9. Definition: 퐴 has slow dimension growth (s.d.g.) if there exist RSH- algebras 퐴푘 and connecting maps 휑푘 : 퐴푘 → 퐴푘+1 s.t. 퐴 ∼ = lim − →푘 퐴푘, and for the underlying spaces 푋푘1, 푋푘2, . . . and matrix sizes 푛푘1, 푛푘2, . . . of the RSH-algebras 퐴푘 we have: lim sup

푘

(max

푖

dim 푋푘푖/푛푘푖) = 0 How to prove strict comparison? Does s.d.g. imply 풵-stability for ASH- algebras? For projections 푝, 푞 ∈ 푀푛(퐶(푋)) with rank(푝) + (dim(푋) − 1)/2 < rank(푞), we have 푝 ≾ 푞. We want to show that a similar result holds for positive elements. Assume 퐴 = lim − → 퐴푘, 퐴푘 = 푀푛푘(퐶(푋푘)). Then s.d.g. means dim(푋푘)/푛푘 →

• 0. Assume (푛 + 1)⟨푎⟩ ≤ 푛⟨푏⟩ for 푎, 푏 ∈ 퐴푘. Does it follow that rank(푎(푥)) ≤

rank(푏(푥)) for all 푥 ∈ 푋푘? 2.10. Theorem: If rank(푎(푥)) + dim(푋)/2 < rank(푏(푥)) for all 푥 ∈ 푋푘, then 푎 ≾ 푏. The proceeding is a sketch why strict comparison holds for simple, unital ASH-algebras with s.d.g. Why is 휄(Cu(퐴)+) ”dense” in Aff(푇퐴)>0 (in the above sense)? Consider 푀푛(퐶(푋)), and 푓 ∈ Aff(푇(푀푛(퐶(푋))))>0 ∼ = 퐶ℝ(푋) (since 푇(...) is a Bauer simplex, with compact boundary 푋). We want 푎 ∈ 푀푛(퐶(푋))+ s.t. ∣푑휏(푎) − 푓(휏)∣ < 1/푛. Can assume 휏 = 훿푥 for some 푥 ∈ 푋, so 푑휏(푎) = rank(푎(푥))/푛. Thus want ∣ rank(푎(푥))/푛 − 푓(푥)∣ < 1/푛. Take 푝 = 푒11 ⊗ id푥, and fix 푓 ∈ 퐶(푋) s.d. supp(푓푖) = 푈푖. Set 푎푖 = 푓푖(푝). Then 푎 = 푎1 ⊕ . . . ⊕ 푎푛 does the trick.

The Cuntz semigroup and its relation to classification 9

• 3. Part 3 - Lecture from 18.November 2009

Are there simple, unital, separable, nuclear C*-algebras with the same K- theory and traces, but which are not isomorphic? Yes, first examples have been given by Rørdam, and there are even ex- amples in the stably finite case. Strategy: Construct 퐴 as inductive limit 퐴 = lim − → 푀푛푘(퐶(푋푘)) with each 푋푘 contractible. Then 퐾0(퐴푘) = ℤ and 퐾1(퐴푘) = 0, so also 퐾1(퐴) = 0. Assume we can achieve that the elements of 퐾0(퐴푘) get divisible in the limit, i.e. for each 푛 and 푘 there is some 푁 > 푘 such that 1 ∈ 퐾0(퐴푘) is divisible by 푛 in 퐴푁. Then 퐾0(퐴) = ℚ, and hence St(퐾0(퐴)) = {휏}, so the pairing between traces and 퐾0 is uninteresting. Let 푄 be the universal UHF-algebra (i.e. 퐾0(푄) = ℚ), then (퐾0(퐴 ⊗ 푄), 퐾1(퐴 ⊗ 푄), 푇(퐴 ⊗ 푄), 휌퐴⊗푄) ∼ = (퐾0(퐴), 퐾1(퐴), 푇(퐴), 휌퐴) For a counterexample we just need 퐴 ≇ 퐴 ⊗ 푄. We will show that AUP (almost unperforation property) fails in Cu(퐴), but Cu(퐴 ⊗ 푄) has AUP. Let us first see how AUP can fail in 푀푛(퐶(푋)) using the fact that AUP is equivalent to: (푛 + 1)푥 ≤ 푛푦 ⇒ 푥 ≤ 푦 How do we show that 푝 푞 for projections 푝, 푞 ∈ 푀푛(퐶(푋))? View 푝, 푞 as VB (vector bundles) over 푋: the fibre of 푝 at 푥 ∈ 푋 is 푝(푥)ℂ푛. Villadsen used Chern classes to get comparability obstructions. 3.1 (Chern classes): The (full) Chern class is a map 푐(⋅) : Vect(푋) → 퐻ev(푋 : ℤ) with the following properties: (i) 푐(휉 ⊕ 휉′) = 푐(휉) ∪ 푐(휉′) (ii) 푐(푒푟) = 1 ∈ 퐻0(푋) where 푒푟 = 푋 × ℂ푟 is the trivial VB (iii) if 푓 : 푋 → 푌 is continuous, then 푐(푓∗(휉)) = 푓∗(푐(휉)) (iv) 푐(휉) = 1 + 푐1(휉) + . . . + 푐dim 휉(휉) with 푐푖(휉) ∈ 퐻2푖(푋) 3.2. Lemma: (Villadsen) Let 훾, 푒푟 be VB over 푋. Assume 푐푗(훾) ∕= 0 for some 푘 > dim(훾) − 푟. Then 푒푟 훾. Proof: If 푒푟 ≾ 훾, then there exists 휔 s.t. 푒푟 ⊕휔 ∼ = 훾. Then 푐(푒푟 ⊕휔) = 푐(푒푟)∪푐(휔) = 푐(휔) = 푐(훾), but dim(휔) < dim(훾) − 푖. □

The Cuntz semigroup and its relation to classification 10 On the other hand, if rank(휔) + (dim(푋) − 1)/2 < rank(훾), then 휔 ≾ 훾. Thus, if rank(휔) < rank(훾), then (푛 + 1)⟨휔⟩ ≤ 푛⟨훾⟩ for large enough 푛. 3.3. Example: Let 휌 be the Bott bundle over 푆2. Then 푐(휌) = 1 + 1 ∈ 퐻0(푆2) ⊕ 퐻2(푆2). 휌 × 휌 is a bundle over 푆2 × 푆2 defined by 휋∗

1(휌) ⊕ 휋∗ 2(휌)

where 휋푖 : 푆2 × 푆2 → 푆2 are the coordinate projections. Then 푐(휋∗

1(휌) ⊕ 휋∗ 2(휌)) = 휋∗ 1(푐(휌))휋∗ 2(푐(휌)) ′′ =′′ (1 + 1)(1 + 1)

in particular 푐2(휌 × 휌) ∕= 0: Thus 푒1 휌 × 휌. Consider 푆2 × 푆2 ⊂ [0, 1]3 × [0, 1]3 = 푋1. Extend 휌 × 휌 to an open neighborhood 푈 of 푆2 × 푆2, choose 푓 : 푋1 → [0, 1] with 푓 = 1 on 푆2 × 푆2 and 푓 = 0 on 푈 푐 (the complement of 푈). Set 푎 = 푓 ⋅ 푒1, 푏 = 푓 ⋅ 휌 × 휌. Then 푎, 푏 ∈ 푀푛(퐶(푋1))+ and (푛 + 1)⟨푎⟩ ≤ 푛⟨푏⟩ for large 푛, but ⟨푎⟩ ≰ ⟨푏⟩ since

• therwise ⟨푎∣푆2×푆2⟩ = ⟨푒1⟩ ≤ ⟨휌 × 휌⟩ = ⟨푏∣푆2×푆2⟩.

Set 푋2 := 푋×푚2

1

. Define 휑1 : 푀푛1(퐶(푋1)) → 푀푛2(퐶(푋2)) as: 휑1(푓) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 푓 ∘ 휋1 . . . 푓 ∘ 휋푚1 푓(푥푖) . . . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Note that we add the evaluations at points 푥푖 to ensure simplicity of the

• limit. (so want these points to be eventually dense). Then:

휑1(푏)∣(푆2×푆2)×푚1 = (휌 × 휌)×푚1 and 푐2푚1((휌 × 휌)×푚1) ∕= 0. Thus ⟨휑1(푎)⟩ ≰ ⟨휑1(푏)⟩. If we proceed this way, a similar result will hold for all forward images. In fact there ex- ists 훿 > 0 such that for all 푖 and 푥 ∈ 퐴푖: ∥푥휑1,푖(푏)푥∗ − 휑1,푖(푎)∥ ≥ 훿, so ⟨휑1,∞(푎)⟩ ≰ ⟨휑1,∞(푏)⟩. Thus AUP fails in 퐴. 3.4. Definition: Let 퐴 be unital, exact. Define the radius of comparison for 퐴 to be: rc(퐴) := inf{푟 > 0 : 푎 ≾ 푏 whenever 푑휏(푎) + 푟 < 푑휏(푏)∀휏} (where 휏 runs over all normalized traces, and 푎, 푏 ∈ (퐴 ⊗ 핂)+). One can show that rc(퐴) = inf{푚/푛 : 푎 ≾ 푏 whenever 푛푎 + 푚⟨1퐴⟩ ≤ 푛푦} 3.5. Proposition: If 푋 is a CW-complex with dim(푋) = 푑 < ∞, then: (푑 − 2)/2 ≤ rc(퐶(푋)) ≤ (푑 − 1)/2

The Cuntz semigroup and its relation to classification 11 Proof: The upper bound was already discussed (and it works for all 푋, not just CW-complexes). To get the lower bound note that one can immerse 푆2푑′ into 푋 (for some large 푑′). □ If 퐴 is simple, then rc(퐴) = 0 if and only if Cu(퐴) is almost unperforated. We also have the following properties: (i) rc(lim − →푘 퐴푘) ≤ lim inf푘 rc(퐴푘) (ii) rc(퐴/퐼) ≤ rc(퐴) (iii) rc(푀푛(퐴)) = 1/푛 rc(퐴) 3.6. Theorem: There exists a family 퐴푟 of simple AH-algebras indexed over 푟 ∈ [0, ∞] s.t.: (1) The Elliott invariant of 퐴푟 (K-theory and traces) is the same for all 푟 (2) rc(퐴푟) = 푟, so the algebras are pairwise not isomorphic The algebras 퐴푟 of the theorem are all shape equivalent, since they are con- structed as AH-algebras over contractible spaces, so all homotopy invariant continuous functors agree on the 퐴푟. Further 퐾0(퐴푟) = ℚ and sr(퐴푟) = 1. This means we have uncountably many different Morita equivalence classes among the 퐴푟. 3.7 (Mean dimension): Let 푋 be compact, metric, 훼 : 푋 → 푋 a homeo- morphism, and 풰 an open cover of 푋. Define

• rd(풰) := sup{(

∑

푈∈풰

휒푈(푥)) − 1 : 푥 ∈ 푋} and write 풱 > 풰 if 풱 refines 풰. Set: 퐷(풰) := min{ord(풱) : 풱 > 풰} We have 퐷(풰 ∪ 풱) ≤ 퐷(풰) + 퐷(풱), since one can show that 퐷(풰) ≤ 푑 if and only if there exists a continuous map 푓 : 푋 → 퐾 with dim(퐾) ≤ 푑 such that 푓 is compatible with 풰. Set 풰푛 := 풰 ∨ 훼−1(풰) ∨ . . . ∨ 훼−(푛−1)(풰) where 풱 ∨ 풲 means the union and also all intersections of set in 풱, 풲. Set mdim(푋, 훼) := sup

풰

lim

푛→∞ 퐷(풰푛)/푛

3.8. Example: Let 푌 be a CW-complex, 푋 = 푌 ℤ, and 훼 : 푋 → 푋 the bilateral shift. Then mdim(푋, 훼) = dim(푌 ). Problem: If dim(푋) < ∞, then mdim(푋, 훼) = 0 for all 훼.

The Cuntz semigroup and its relation to classification 12 3.9. Theorem: (Kerr, Giol) For any 푘 > 0 there exists a minimal system (푋푘, 훼푘) s.t. 푘 ≤ rc(퐶(푋푘) ⋊훼푘 ℤ). Also mdim(푋, 훼)/2 ≈ 푘. If 훼 : 푌 ∞ → 푌 ∞ is the bilateral shift, then let 푌2푛 be the 2푛-periodic points. Then: 퐶(푌2푛) ⋊훼 ℤ

• . . .

퐶(푌2푛) ⋊훼 ℤ 퐶(푌2푛+1) ⋊훼 ℤ . . .

Proposal: Define a dynamical dimension ddim(푋, 퐺) for a countable, dis- crete group 퐺 acting on 푋 via: ddim(푋, 훼) := rc(퐶(푋) ⋊훼 퐺) The reasons are: (1) It looks like one could recover mdim for the bilateral shift (2) If 퐺 = {1}, then ddim(푋, 퐺) ≈ dim(푋)/2 (3) If 퐺 = ℤ acting trivially, then ddim(푋, 퐺) = (dim(푋) + 1)/2 (4) If 푋 = 푌 푚 with 훼 the cyclic shift, then ddim(푋, 훼) ≈ dim(푌 )/2 Outlook: Hopefully for minimal systems (푋, 훼) we have ddim(푋, 훼) ≤ mdim(푋, 훼)/2 and that this is sharp (see results of Kerr and Giol). Why are we hopeful? We have that 퐶∗(퐶(푋), 푢퐶(푋 ∖ {푦})) = 퐴{푦} is ASH, but the RSH- subalgebras have infinite dimension. Idea: fix 푎, 푏 ∈ 퐴{푦}+, 푎 = ∑푁

푖=1 푓푖푢.

Take 풰 a finite open cover, iterate under 훼−1, get covers 풱푛 s.t. ord(풱푛) = 푛 ⋅ ddim, thus 푢 corresponds to the size of the matrices.