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 organizer: 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. Any cross product 퐴 = 퐶 ( 푋 ) ⋊ 훼 ℤ is 1.1. Example (cross products) : 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 푋 1 , . . . , 푋 푙 ∙ closed subspaces 푋 (0) ⊂ 푋 푖 푖 ∙ numbers 푛 1 , . . . , 푛 푙 ∈ ℕ ∙ unital ∗ -homomorphisms 휙 푘 : 퐴 푘 − 1 → 푀 푛 푘 ( 퐶 ( 푋 (0) 푘 )) (attaching maps) such that 퐴 1 = 푀 푛 1 ( 퐶 ( 푋 1 )), and the following is a pullback (for 푘 = 2 , . . . , 푙 ):

� � � The Cuntz semigroup and its relation to classification 2 퐴 푘 퐴 푘 − 1 휙 푘 ∂ 푘 � 푀 푛 푘 ( 퐶 ( 푋 (0) 푀 푛 푘 ( 퐶 ( 푋 푘 )) 푘 )) Here ∂ 푘 is induced by the inclusion 푋 (0) → 푋 푘 . Such a pullback is often 푘 written as 퐴 푘 = 퐴 푘 − 1 ⊕ 푀 푛푘 ( 퐶 ( 푋 (0) )) 푀 푛 푘 ( 퐶 ( 푋 푘 )), and the standard way to 푘 define that pullback algebra is as follows: 퐴 푘 = { ( 푎, 푏 ) : 푎 ∈ 퐴 푘 − 1 , 푏 ∈ 푀 푛 푘 ( 퐶 ( 푋 푘 )) , 휑 푘 ( 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 ( 퐾 0 -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 0 [ 푝 ] + [ 푞 ] = . In this way 푉 ( 퐴 ) becomes an abelian semigroup. 0 푞 Use the Grothendieck completion process Γ to define an abelian group 퐾 0 ( 퐴 ) := Gr( 푉 ( 퐴 )). This comes with a natural map Γ : 푉 ( 퐴 ) → 퐾 0 ( 퐴 ) and we denote its image as 퐾 0 ( 퐴 ) + := Γ( 푉 ( 퐴 )). This is also called the positive part (or positive cone) in 퐾 0 ( 퐴 ). Then ( 퐾 0 ( 퐴 ) , 퐾 0 ( 퐴 ) + , [1 퐴 ]) 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 푀 푛 ( 퐴 ) are finite (for all 푛 ). In that case 퐾 0 is ordered. 1.5 ( 퐾 1 -group) : Let 풰 ( 퐴 ) denote the set of unitaries in 퐴 , and 풰 0 ( 퐴 ) ⊂ ( 푢 0 ) 풰 ( 퐴 ) its connected component containing 1 퐴 . The map 푢 �→ induces 0 1 퐴

The Cuntz semigroup and its relation to classification 3 a homomorphism 휑 푛 : 풰 ( 푀 푛 퐴 ) / 풰 0 ( 푀 푛 퐴 ) → 풰 ( 푀 푛 +1 퐴 ) / 풰 0 ( 푀 푛 +1 퐴 ). We → 푛 풰 ( 푀 푛 퐴 ) / 풰 0 ( 푀 푛 퐴 ). This is an abelian group with addi- set 퐾 1 ( 퐴 ) := lim − 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 (1 퐴 ) = 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 퐾 0 ( 퐴 ) as follows: first extend 휏 to a trace 휏 ⊗ tr on 푀 푛 ( 퐴 ) using the canonical trace tr : 푀 푛 → ℂ , then for a projection 푝 ∈ 푀 푛 ( 퐴 ) set 휏 ([ 푝 ]) := ( 휏 ⊗ tr)( 푝 ). We get a map 휌 퐴 : 푇 ( 퐴 ) → St( 퐾 0 ( 퐴 ) , 퐾 0 ( 퐴 ) , [1 퐴 ]). For a unital C*-algebra 퐴 the Elliott invariant is: Ell( 퐴 ) := ( 퐾 0 ( 퐴 ) , 퐾 0 ( 퐴 ) , [1 퐴 ] , 퐾 1 ( 퐴 ) , 푇 ( 퐴 ) , 휌 퐴 ) In good cases ( 퐾 0 ( 퐴 ) , 퐾 0 ( 퐴 ) , [1 퐴 ] , 푇 ( 퐴 ) , 휌 퐴 ) is equivalent to the Cuntz semi- group Cu( 퐴 ), and then Ell( 퐴 ) ∼ = (Cu( 퐴 ) , 퐾 1 ( 퐴 )), which amounts to a de- composition in a positive and unitary part. Let 퐴 be unital. For 푎, 푏 ∈ ( 퐴 ⊗ 핂 ) + we 1.7 (The Cuntz semigroup) : say 푎 is Cuntz-dominated by 푏 (denoted 푎 ≾ 푏 ) if there exists a sequence ( 푟 푛 ) ⊂ 퐴 ⊗ 핂 s.t. 푟 푛 푏푟 푛 ∗ → 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: 푀 푛 Let 퐴 = 푀 푛 . Then 푎 ≾ 푏 iff rank( 푎 ) ≤ rank( 푏 ). 1.9. Example: 푀 푛 ( 퐶 [0 , 1]) Let 퐴 = 푀 푛 ( 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: 푀 푛 ( 퐶 ( 푋 )) Let 퐴 = 푀 푛 ( 퐶 ( 푋 )) with 푋 a CW-complex of dim( 푋 ) ≥ 3 and 푛 ≥ 2. Then there exist 푎, 푏 ∈ 푀 푛 ( 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 푆 2 in 푋 . We can find projections 푝, 푞 in 푀 2 ( 퐶 ( 푆 2 )) 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 푆 2 ֒ → 푋 , and then to positive elements 푎, 푏 ∈ 푀 2 ( 퐶 ( 푋 )) ⊂ 푀 푛 ( 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 ⟩ . 〈( 푎 0 )〉 As before we define an addition ⟨ 푎 ⟩ + ⟨ 푏 ⟩ := . If we define ⟨ 푎 ⟩ ≤ ⟨ 푏 ⟩ 0 푏 iff 푎 ≾ 푏 , then we get an ordered abelian semigroup. 1.13. Example: 푀 푛 Let 퐴 = 푀 푛 . Then Cu( 퐴 ) = ℕ ∪ {∞} with 푥 + ∞ = ∞ , ∞ + ∞ = ∞ and ⟨ 1 퐴 ⟩ = 푛 ∈ ℕ . 1.14. Example: 푀 푛 ( 퐶 [0 , 1]) Let 퐴 = 푀 푛 ( 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 푑 휏 ( 푎 ) = lim 푛 →∞ 휏 ( 푎 1 /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 ( ⟨ 1 퐴 ⟩ ) = 1. (this gives exactly the lower semicontinuous dimension functions). For 푎 ∈ ( 푀 푛 ) + we get 푑 휏 ( 푎 ) = rank( 푎 ) /푛 . 1.15. Example: For ⟨ 푎 ⟩ ∈ Cu( 퐴 ) we define 휄 ( ⟨ 푎 ⟩ ) : 푇 ( 퐴 ) → [0 , ∞ ] by 휄 ( ⟨ 푎 ⟩ )( 휏 ) := 푑 휏 ( 푎 ). Then: ∙ 휄 ( ⟨ 푎 ⟩ ) is in 퐿 ( 푇 ( 퐴 )) since 휏 �→ 휏 ( 푎 1 /푛 ) is continuous and 휏 ( 푎 1 /푛 ) ≤ 휏 ( 푎 1 /푛 +1 ) (if ∥ 푎 ∥ ≤ 1, so rescale 푎 ) ∙ if 푎 ≥ 0, 푓 ∈ 퐶 ∗ ( 푎 ), 푓 ≥ 0, then 푑 휏 ( 푓 ( 푎 )) = 휇 휏 (supp( 푓 ) ∩ 휎 ( 푎 )) where 휇 휏 is the spectral measure induced by 휏 ∙ 푎 ≾ 푏 iff ∀ 휀 > ∃ 훿 > 0 such that ( 푎 − 휀 ) + ≾ ( 푏 − 훿 ) + .