Ranks of operators in simple C*-algebras

RANKS OF OPERA TORS IN SIMP L E C ∗ -ALGEBRA S MARIUS DADARLA T AND ANDREW S. TOMS A B S T R A C T . Let A be a unital simp le separable C ∗ -algebra with strict compariso n of pos itive ele- ments. W e prove that the Cuntz semigroup of A is recovered functorially from the Murray-von Neumann semig roup and the tracial state space T( A ) whenever the extreme boundary o f T( A ) is compact and of fin ite covering dimension. Combined with a result of Win ter , we obtain Z ⊗ A ∼ = A whenever A moreover has locally finite de composition rank. As a corollary , we confirm Elliott’s classification conjecture under reasonably general hypotheses which, notably , do not require any inductive li mi t structure. These results all stem from our investigation of a basic ques tio n: what are the possible ranks of operators in a unital simple C ∗ -algebra? 1. I N T R O D U C T I O N A N D S TA T E M E N T O F M A I N R E S U LT S The not ion of rank for operators in C ∗ -algebras is of fundamental importance. Indeed, one may view Murray and von Neumann’s type classification of factors as a complete answer to the question of which ranks occur for projections in a factor . For se parable simple C ∗ -algebras the question of which ranks may o ccur has received rather less attention. This is un fo r t unate given the many interesting areas it bears up on: calcula ting the Cuntz semigroup, the question of whethe r or not a C ∗ -algebra is Z -stable, the classification of nuclear simple C ∗ -algebras by K -theory , and the classification of countably generated Hilbert mod u les, to name a few . In this article we e xplore the rank question for a gene r al tr acial simple C ∗ -algebra, and give app lica- tions in each of these areas. What do we mean by the rank of an operato r x in a unital C ∗ -algebra A ? It is reasonable to assume that whatever the d efinition, x and x ∗ x should have the same rank. W e t h e refor e consider only po sitive elements of A . Let τ : A → C be a tracial st ate. For a po sitive element a of A we d e fine d τ ( a ) = l im n →∞ τ ( a 1 /n ) . This defines the r ank of a with respect t o τ . Our no t ion of rank for a is then the rank fun ction of a , a map ι ( a ) from the tracial s t ate space T( A ) to R + given by the formula ι ( a )( τ ) = d τ ( a ) . (The maps d τ and ι exte nd naturally t o po sitive elements in A ⊗ K , and we always take th is se t of positive elements to be the domain of ι .) Thes e rank functions are lower se micontinuous, af fine, and nonneg ative; if A is simple, t hen they are strictly positive. W e will be working extensively with various subsets of af fine functions o n T( A ) , so let u s fix s ome convenient no tation: Aff ( K ) denote s the se t of real -valued continuous affine functions o n a (typically compact metrizable) Date : November 9, 2018. M.D. was partially suppo r ted by NSF gr ant #DMS-0801173; A.T . was partially suppor ted by NSERC. 1 2 MARIUS DADARLA T AND ANDREW S. TOMS Choquet simplex K , LAff ( K ) denote s the se t o f bounde d, strictly p ositive, lower semicontinu- ous and af fine functions on K , and SAff ( K ) denote s the set of exte n d ed real-val ued functions which can be obtained as th e pointwise supremum of an increasing sequence fr om LAff ( K ) . The range question for ranks of operators in a unital simple tracial C ∗ -algebra A can the n be phrased as follows: When is the range of ι equal to all of SAff (T( A )) ? This brings us to our main r esult. Theorem 1.1. Let A be a unital simple sepa rable infinite-di mensional C ∗ -algebr a with nonempty tracial state spa ce T( A ) . Consider the follo wing thr ee con ditions: (i) The extr eme boun dary of T( A ) is compact and of finite covering dimensio n. (ii) The extr eme boundary of T( A ) is compact, and for each n ∈ N ther e is a positive a ∈ A ⊗ K with the prope rty that nd τ ( a n ) ≤ 1 ≤ ( n + 1) d τ ( a ) , ∀ τ ∈ T( A ) . (iii) For each positive b ∈ A ⊗ K such that d τ ( b ) < ∞ , ∀ τ ∈ T( A ) , and each δ > 0 , ther e is a positive c ∈ A ⊗ K such that | 2 d τ ( c ) − d τ ( b ) | < δ, ∀ τ ∈ T( A ) . If (ii) or (iii) is satisfied, then the uniform closur e of the range of the map ι contains Aff (T( A )) . If any of (i)-(iii ) is satisfied and A moreo ver has strict comparison of posit ive elements, then the range of ι is all of SAff (T( A )) . Some remarks are in or der . • Finite-dimensional approximation properties such as nuclearity or exactness are not re- quir ed by Theorem 1.1. • In the presence of strict comparison, cond itions (i) , (ii), and (iii) ar e equivalent to the range of ι being all of SAff (T( A )) . • Condition (ii) ask s fo r the existe n ce of pos itive operators with ”almost constant rank”. These, we shall prove, ar e connected to the existe nce of unital ∗ -homomorp hisms from so-called dimension drop algebras in to A , and so to t h e important property of Z -stabili ty . These op erators exist, for example, in any crossed product of the form C ( X ) ⋊ G , where X is an infinite, finite dimensional compact s pace and at least one element of G acts by a minimal ho meomorphism ([TW09b]), and in many ot h e r C ∗ -algebras whos e finer structure is presently out of r each. • Condition (iii) is automatically satisfied in Z -stable C ∗ -algebras ([PT07]) and in unital simple ASH algebras with slow dimension gr owth [T o 09b]. • It is p ossible that ι (( A ⊗ K ) + ) = SAff (T( A )) for any unital simple sep arable infinite- dimensional C ∗ -algebra with nonempt y tracial s t ate space. Conditions (ii) and (iii) guar- antee that the r ang e o f ι is de nse in th e sens e that any e lement of S Aff (T( A )) is a supre- mum of an increasing sequen ce from ι (( A ⊗ K ) + ) . In the absence of strict comparison, however , the element s of ( A ⊗ K ) + giving rise to this sequence ar e not thems e lves in- cr easing fo r the Cuntz r elation, and so one cannot readily find a positive a ∈ A ⊗ K for which ι ( a ) is the supremum o f the said seque nce. RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 3 Theorem 1.1 has several conseque n ce s for the algebras it covers. W e d escribe them briefly here, and in detail in Section 6. I f A as in Theorem 1.1 satisfies any of (i)-(iii) and has strict com- parison, the n th e Cunt z semigroup Cu( A ) is recover ed functorially from the Murray-von Neu - mann s emigroup of A and T( A ) , leading to the confirmation of tw o conjectures of Blackadar- Handelman concerning d imension functions ([BPT08], [BT07]). In fact, Cu( A ) ∼ = Cu( A ⊗ Z ) , and so if A moreover has locally finite decomposition rank—a mild condition satisfied by unital separable ASH algebras, for instance—th e n A ∼ = A ⊗ Z by a result of W inter ([NW06], [W i0 9]). This leads to the complet e classification of countably generate d Hilbert modules over A up to isomorphism via K 0 ( A ) and T( A ) in a manner analogous to the classification of W ∗ -modules over a I I 1 factor ([BT07]). Finally , if C is t he cla ss of all such A which, additionally , have projec- tions se parating traces, t h e n C satisfie s Elliott’s classification conjecture: the members of C ar e determined up to ∗ -isomorphism by their graded ordered K -theory ([W i06 ], [W i07 ], [LN 08]). The paper is o rgani zed as follows: Section 2 reviews the Cuntz se migr oup and dimension functions; Section 3 develops a criterion for embedding dimension drop algebras in C ∗ -algebras with st rict comparison; Section 4 develops se veral techniques for constructing positive eleme n t s with prescribed rank function; Section 5 contains the proof of Theorem 1.1; Se ction 6 details our applications. 2. P R E L I M I N A R I E S Let A be a C ∗ -algebra and let K denote the algebra of compact ope rators on a separable infinite-dimensional Hilbert space. Let ( A ⊗ K ) + denote th e set o f pos itive elements in A ⊗ K . Given a, b ∈ ( A ⊗ K ) + , we write a - b if there is a s equence ( v n ) in A ⊗ K such that k v n bv ∗ n − a k → 0 . W e the n s ay that a is Cu ntz subequival ent to b (this relation restricts to usual Murray-von Neu- mann comparison o n projections). W e write a ∼ b if a - b and b - a , and say that a is Cun tz equivalent to b . Set Cu( A ) = ( A ⊗ K ) + / ∼ , and write h a i for the equivalence class of a . W e equ ip Cu( A ) w ith the binary operation h a i + h b i = h a ⊕ b i (using an isomorphism be t ween M 2 ( K ) and K ) and the p artial order h a i ≤ h b i ⇔ a - b. This or dered Abe lian semigroup is the Cuntz semigr oup of A . It was show n in [CEI08] that increasing seque n ce s in Cu( A ) always have a supremum, and we s hall use this fact freely in t h e seq u el. Suppose now t hat A is u nital and τ : A → C is a tracial state. The function d τ introduced in Section 1 is const ant on Cuntz equivalence classes, and d rops to an additive order-pr eserving map o n those cla sses in Cu( A ) coming from po sitive elements in matrices o ver A . Th is map has a u n ique su premum- and order-pr eserving ext ension to all of Cu( A ) , and we also de note this e xtension by d τ (see [BT 07, Lemma 2.3]). Definition 2.1. Let A be a unital C ∗ -algebr a. We say that A has strict compariso n of positi ve elements (or simply strict comparison) if a - b for a, b ∈ ( A ⊗ K ) + whenever d τ ( a ) < d τ ( b ) , ∀ τ ∈ { γ ∈ T( A ) | d γ ( b ) < ∞} . 4 MARIUS DADARLA T AND ANDREW S. TOMS For pos itive elements a, b in a C*-algebra A we w rite that a ≈ b if here is x ∈ A such that x ∗ x = a and xx ∗ = b . The rela tion ≈ is an equivalence relation [Pd98 ], so metimes referred to as Cuntz-Pedersen equivalence . I t is known that a ≈ b implies a ∼ b . If a ∈ A is a positive element and τ ∈ T( A ) we denote by µ τ the measure induce d on the spectrum σ ( a ) of a by τ . T h e n d τ ( a ) = µ τ ((0 , ∞ ) ∩ σ ( a )) and more gene r ally d τ ( f ( a )) = µ τ ( { t ∈ σ ( a ) : f ( t ) > 0 } ) for all nonnegative continuous functions f de fined of σ ( a ) . 3. A C R I T E R I O N F O R E M B E D D I N G Q U O T I E N T S O F D I M E N S I O N D R O P A L G E B R A S The Jiang-Su algebra Z is an important o bject in the st ructur e theory of sep arable nuclear C ∗ -algebras. One wants to k now when a given algebra A has t he property that A ⊗ Z ∼ = A , as one can then frequently o btain detailed information about A through its K -theory and positive tracial functionals. I f A is sep arable, then A ⊗ Z ∼ = A whe n e ver one can find, for any natural number m > 1 , an approximately central sequence ( φ n ) of unital ∗ -homomorphisms from the prime d imension drop algebra I m,m +1 := { f ∈ C([0 , 1]; M m ⊗ M m +1 ) | f (0) ∈ 1 m ⊗ M m +1 , f (1) ∈ M m ⊗ 1 m +1 } into A . It is t herefor e of interest to characterize when A admits a unital ∗ -homomorphism φ : I m,m +1 → A . In this section we obtain such a characterization in the case that A has strict comparison (Theorem 3.6). Our result should be compar ed with [R W09, Proposition 5.1], which uses the assumption of stable rank one instead of strict comparison. So me of the results w e develop here will be employed later to construct pos itive elements with sp ecified rank functions. Lemma 3.1. Let A be a u nital C ∗ -algebr a with T( A ) 6 = ∅ , and let a ∈ M k ( A ) be positive. Suppose that ther e ar e 0 < α < β such that α < d τ ( a ) < β for every τ in a close d subset X of T( A ) . Then the r e exist ǫ > 0 and an open n eigbour ho od U of X with the pr ope rty that for α < d τ (( a − ǫ ) + ) < β , ∀ τ ∈ U. Pro of. S ince d τ (( a − ǫ ) + ) ր d τ ( a ) as ǫ ց 0 for each τ , we can fix ǫ τ > 0 s uch that d τ (( a − ǫ τ ) + ) > α ; since γ 7→ d γ (( a − ǫ τ ) + ) is lower semicontinuous, we can find an op e n neighbourhood V τ of τ such that d γ (( a − ǫ τ ) + ) > α, ∀ γ ∈ V τ . The family { V τ } τ ∈ X is an open cover of X , and s o X ⊂ V τ 1 ∪ · · · ∪ V τ n for some τ 1 , . . . , τ n ∈ X . Set ǫ := min { ǫ τ 1 , . . . , ǫ τ n } and V := V τ 1 ∪ · · · ∪ V τ n , so that for each τ ∈ V , τ ∈ V τ i for some i , and we have d τ (( a − ǫ ) + ) ≥ d τ (( a − ǫ τ i ) + ) > α. Let µ τ be the measur e induced on σ ( a ) by τ . W e also have d τ (( a − ǫ ) + ) = µ τ (( ǫ, ∞ ) ∩ σ ( a )) ≤ µ τ ([ ǫ, ∞ ) ∩ σ ( a )) ≤ µ τ ((0 , ∞ ) ∩ σ ( a )) ≤ d τ ( a ) , ∀ τ ∈ T( A ) . In particular , d τ (( a − ǫ ) + ) ≤ µ τ ([ ǫ, ∞ ) ∩ σ ( a )) < β for all τ ∈ X . By t he Portmanteau theorem, the map γ 7→ µ τ ([ ǫ, ∞ ) ∩ σ ( a )) is uppe r semicontinuous, and so the set W = { γ ∈ T( A ) : µ γ ([ ǫ, ∞ ) ∩ σ ( a )) < β } RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 5 is o p en and contains X . Moreover , for any γ ∈ W , we have d γ (( a − ǫ ) + ) < β . W e conclude the pr oof by setting U = V ∩ W .  For e ach η > 0 define a continuous map f η : R + → [0 , 1] by t he following formula: (1) f η ( t ) =  t/η , 0 < t < η 1 , t ≥ η . Lemma 3.2. Let A be a unital separabl e C ∗ -algebr a w ith nonempty tracial state space, and let X ⊆ T ( A ) be closed. Suppose that a ∈ M k ( A ) is a positive element with the prope rty that β − α < d τ ( a ) ≤ β , ∀ τ ∈ X for some 0 < α < β . Then the r e is η > 0 such that k − β ≤ d τ (1 − f η ( a )) < k − β + 2 α, ∀ τ ∈ X . Pro of. L et µ τ be t he measure induced on σ ( a ) by τ . By Le mma 3.1 t h e re is η > 0 such that d τ (( a − η ) + ) > β − α for all τ ∈ X . Therefore d τ (( a − η ) + ) = µ τ (( η , ∞ ) ∩ σ ( a )) > β − α, ∀ τ ∈ X . It follows that µ τ ((0 , η ]) = d τ ( a ) − µ τ (( η , ∞ ) ∩ σ ( a )) < β − ( β − α ) = α, ∀ τ ∈ X . Then, d τ (1 − f η ( a )) = µ τ ([0 , η ) ∩ σ ( a )) = µ τ ((0 , η ) ∩ σ ( a )) + µ τ ( { 0 } ∩ σ ( a )) and hence d τ (1 − f η ( a )) = µ τ ((0 , η ) ∩ σ ( a )) + k − d τ ( a ) < α + k − ( β − α ) = k − β + 2 α, ∀ τ ∈ X . Moreover , d τ (1 − f η ( a )) ≥ k − d τ ( a ) ≥ k − β .  Lemma 3.3. L et a, b be positive elements in a C*-algebra A . If a ≈ b then f ( a ) ≈ f ( b ) for any con tinu ous function f : [0 , ∞ ) → [0 , ∞ ) with f (0) = 0 . Pro of. B y ass umption t here is x ∈ A such that x ∗ x = a and xx ∗ = b . L e t x = v | x | be the polar decomposition of x where v is a partial isometry in the enveloping von Neumann algebra A ∗∗ . Then as in [KR00, Lemma 2.4] the map d 7→ v dv ∗ defines a isomorphism fr om aAa to bAb which maps a to b and hence f ( a ) t o f ( b ) . Therefor e v f ( a ) v ∗ = f ( b ) . Let us note t hat y := v f ( a ) 1 / 2 is an elemen t of A . Indee d , since f (0) = 0 , v f ( a ) 1 / 2 ∈ v a 1 / 2 A = v | x | A = xA . It follows th at f ( b ) = v f ( a ) v ∗ = y y ∗ ≈ y ∗ y = f ( a ) 1 / 2 v ∗ v f ( a ) 1 / 2 = f ( a ) . as required.  Lemma 3.4. Let A be a un ital separable C ∗ -algebr a with strict compariso n of positi ve elements. Also suppose that for each m ∈ N , ther e is x ∈ Cu( A ) such that md τ ( x ) ≤ 1 ≤ ( m + 1) d τ ( x ) , ∀ τ ∈ T( A ) 6 = ∅ . It follows that for each n ∈ N and for any 0 < ǫ < 1 /n , ther e exist mutually orthog onal positive elements a 1 , . . . , a n ∈ A with the foll owing pr op erties: (i) a i ≈ a j for all i, j ∈ { 1 , . . . , n } ; 6 MARIUS DADARLA T AND ANDREW S. TOMS (ii) 1 /n − ǫ < d τ ( a i ) < 1 /n for each τ ∈ T( A ) and i ∈ { 1 , . . . , n } . Pro of. L et us fir s t sho w that for any ǫ > 0 , there is a in M k ( A ) , k ≥ 1 , with the property that (2) 1 /n − ǫ < d τ ( a ) < 1 /n, ∀ τ ∈ T( A ) . Let r , m be a natural numbers s uch that [ r / ( m + 1) , r /m ] ⊂ (1 /n − ǫ, 1 /n ) . By hy pothesis there is x ∈ C u( A ) such that d τ ( x ) ∈ [1 / ( m + 1) , 1 /m ] for all τ ∈ T ( A ) . It follows that d τ ( r x ) ∈ [ r / ( m + 1) , r /m ] for all τ ∈ T( A ) . Let a ∈ ( A ⊗ K ) + repr esenting r x ∈ C u( A ) . Then a satisfies (2). By [BT07, Lemma 2.2] we may arrange t hat a ∈ M k ( A ) fo r some k ≥ 1 . W e w ill no w prove the Lemma by induction. Let n and ǫ > 0 be given. Find a ∈ M k ( A ) satisfying (2). By L emma 3.1 ther e is η > 0 such that 1 /n − ǫ < d τ (( a − η ) + ) < 1 /n, ∀ τ ∈ T( A ) . Since d τ ( a ) < 1 /n ≤ 1 for each τ ∈ T( A ) , we have a - 1 A by strict comparison. Ther e is v in M k ( A ) s uch that v 1 A v ∗ = ( a − η ) + , w hence A ∋ a 1 := 1 A v ∗ v 1 A ≈ ( a − η ) + also s atisfie s 1 /n − ǫ < d τ ( a 1 ) < 1 /n, ∀ τ ∈ T( A ) . This pr oves the L emma in the case n = 1 ; for larger n we proceed inductively . Suppose that for so me k < n and 0 < ǫ < 1 /n w e have found mutually orthog o nal positive elements a 1 , . . . , a k ∈ A with the following properties: (i) a i ≈ a j for all i, j ∈ { 1 , . . . , k } ; (ii) 1 /n − ǫ < d τ ( a i ) < 1 /n for each i ∈ { 1 , . . . , k } and τ ∈ T( A ) . W e will explain how to use thes e a i to construct mutually o r t hogonal positive elements ˜ a 1 , . . . , ˜ a k +1 in A which satisfy (i) and (ii) above with k replac ed by k + 1 . R e peated applica tion of this con- struction yields the Lemma in full. By L e mma 3.1 t here is η > 0 such th at 1 /n − ǫ < d τ (( a i − η ) + ) < 1 /n, ∀ τ ∈ T( A ) , ∀ i ∈ { 1 , . . . , k } . Set a := P k i =1 a i and c := 1 − f η ( a ) , where f η is given by (1). No te that c is orthogon al to ( a − η ) + and he nce to each ( a i − η ) + . For each τ ∈ T( A ) , d τ ( a ) = k X i =1 d τ ( a i ) ∈ ( k /n − k ǫ, k /n ) . By L e mma 3.2 it follows that for all τ ∈ T( A ) d τ ( c ) = d τ (1 − f η ( a )) ≥ 1 − k /n ≥ 1 /n. W e are in a p osition to construct ˜ a 1 , . . . , ˜ a k +1 ∈ A . Since d τ ( a 1 ) < 1 /n < d τ ( c ) for all τ ∈ T( A ) , it follows that a 1 - c by strict comparison. Therefore there exists w ∈ A su ch that ( a 1 − η ) + = wcw ∗ . S e t ˜ a i := ( a i − η ) + for i ∈ { 1 , . . . , k } and ˜ a k +1 := c 1 / 2 w ∗ wc 1 / 2 ≈ ( a 1 − η ) + . B y Lemma 3.3 we have ˜ a i = ( a i − η ) + ≈ ( a 1 − η ) + , ∀ i ∈ { 1 , . . . , k } . This e stablishes (i). Our choice of η establishes (ii) . The mutual o r t hogonality of t he ˜ a i follows fr om t he ort h o gonality of c to all of ( a i − η ) + , i ∈ { 1 , . . . , k } , and the fact that ˜ a k +1 ∈ cAc .  RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 7 Lemma 3.5. Let A be a sep arable unital C ∗ -algebr a with strict compar ison of positive elements. S uppose that b 1 , . . . , b n , n ≥ 2 , are orthogonal positiv e elements in A w ith the following pr ope rties: (i) b i ≈ b j for each i, j ∈ { 1 , . . . , n } ; (ii) 1 /n − 1 / 3 n 2 < d τ ( b i ) < 1 /n for all τ ∈ T( A ) 6 = ∅ and each i ∈ { 1 , . . . , n } . It follows that there is a unital ∗ -ho momorphis m γ : I n,n +1 → A . Pro of. B y ([R W09, Prop. 5.1]), it w ill suffice to find orthogonal positive elements a 1 , . . . , a n of A such that a i ≈ a j for each i, j ∈ { 1 , . . . , n } and (1 − P n i =1 a i ) - ( a 1 − ǫ ) + for some ǫ > 0 . The latter condition can be repla ced by d τ (1 − P n i =1 a i ) < d τ (( a 1 − ǫ ) + ) , ∀ τ ∈ T( A ) , due to th e strict comparison ass umption. Let b 1 , . . . , b n be as specified in the s tatement of the Lemma and s et b = P n i =1 b i . In view of the mutu al orthogonality of the b i , (ii) implies (3) 1 − 1 / 3 n < d τ ( b ) < 1 , ∀ τ ∈ T( A ) . Apply L emma 3.2 to find η > 0 such that d τ (1 − f η ( b )) < 2 / 3 n where f η is d e fined by (1). Se t a i = f η ( b i ) and note t hat the mutual orthogo nality of the b i implies that f η ( b ) = n X i =1 f η ( b i ) = n X i =1 a i . Since d τ ( a 1 ) = d τ ( b 1 ) , by Lemma 3.1 t here is ǫ > 0 s u ch that 1 /n − 1 / 3 n 2 < d τ (( a 1 − ǫ ) + ) for all τ ∈ T( A ) . Now for each τ ∈ T( A ) we have d τ 1 − n X i a i ! = d τ (1 − f η ( b )) < 2 / 3 n ≤ 1 /n − 1 / 3 n 2 < d τ (( a 1 − ǫ ) + ) . T o complete the proof of the Lemma, we observe that a i ≈ a j for each i, j ∈ { 1 , . . . , n } as a consequence of Lemma 3.3.  Theorem 3.6. Let A be a separable un ital C ∗ -algebr a with strict comparison of positive elements and nonempty tracial state space. The following state ments ar e equival ent: (i) for each m ∈ N , ther e is x ∈ Cu( A ) such that mx ≤ h 1 A i ≤ ( m + 1) x ; (ii) for each m ∈ N , ther e is x ∈ Cu( A ) such that md τ ( x ) ≤ 1 ≤ ( m + 1) d τ ( x ) , ∀ τ ∈ T( A ); (iii) for each m ∈ N , ther e is a unital ∗ -homomorphis m φ m : I m,m +1 → A . Pro of. F or (i) ⇒ (ii) , we apply the or der p reserving st ate d τ to the inequality mx ≤ h 1 A i ≤ ( m + 1) x . The implication (ii) ⇒ (iii) is the combination of Le mmas 3.4 and 3.5. Finally , (iii) ⇒ (i) is due to Rø r dam [Rø 04].  8 MARIUS DADARLA T AND ANDREW S. TOMS 4. R A N K F U N C T I O N S O N T R A C E S P A C E S Lemma 4.1. Let A be a u nital simple separable infinite-dimensional C ∗ -algebr a and τ a normalised trace on A . Let 0 < s < r be given. It follows that there are an open neighbour hoo d U of τ in T( A ) and a positiv e element a in some M k ( a ) such that s < d γ ( a ) < r, ∀ γ ∈ U. Pro of. L et τ and 0 < s < r as in t h e hypothe ses o f t he Lemma be g iven. Since A is infinite- dimensional, the re is b ∈ A + with zero as an accumula tion point of its spectrum. F or each n ∈ N , let f n be a p o sitive continuou s function with support (0 , 1 /n ) and set b n := f n ( b ) . It follows t hat 0 < d τ ( b n ) = µ τ ((0 , 1 /n ) ∩ σ ( b )) n →∞ − → 0 . Choose n ∈ N lar ge enou gh such that d τ ( b n ) < r − s . Then s < k d τ ( b n ) < r for s ome k ∈ N . If we set b := ⊕ k i =1 b n ∈ M k ( A ) , then d τ ( b ) = d τ  ⊕ k i =1 b n  = k d τ ( b n ) and hence s < d τ ( b ) < r. W e conclude the pr oof by applying Lemma 3.1 with X = { τ } .  Lemma 4.2. Let A be a unital C ∗ -algebr a and τ a n ormalise d trace on A . Let x, y be positive elements in M k ( A ) . Then d τ ( y 1 / 2 x y 1 / 2 ) ≥ d τ ( x ) − d τ (1 k − y ) , where 1 k denotes the unit of M k ( A ) . Pro of. W e have x = x 1 / 2 y x 1 / 2 + x 1 / 2 (1 k − y ) x 1 / 2 - x 1 / 2 y x 1 / 2 ⊕ x 1 / 2 (1 k − y ) x 1 / 2 ≈ y 1 / 2 x y 1 / 2 ⊕ x 1 / 2 (1 k − y ) x 1 / 2 - y 1 / 2 x y 1 / 2 ⊕ (1 k − y ) . Applying d τ we obtain d τ ( x ) ≤ d τ ( y 1 / 2 x y 1 / 2 ) + d τ (1 k − y ) .  W e r ecall a r esult from [Li07] based on work of Cuntz and Pedersen: Theorem 4.3. ( [Li07, Thm. 9.3] ) Let A be a u nital simple C ∗ -algebr a with nonempty tracial state space, and let f be a stri ctly positive affine continuous function on T( A ) . It follows that for any ǫ > 0 ther e is a positiv e element a of A such that f ( τ ) = τ ( a ) , ∀ τ ∈ T( A ) , and k a k < k f k + ǫ . Lemma 4.4. Let A be a sep arable unital simple in finite-dimensio nal C ∗ -algebr a whose tracial state space has compact extr eme boundary X = ∂ e T( A ) . It follows that for any δ > 0 and closed set Y ⊆ X ther e is a nonzer o positive element a of A with the pro perty that d τ ( a ) < δ, ∀ τ ∈ Y and d τ ( a ) = 1 , ∀ τ ∈ X \ Y . Pro of. L et δ and Y be given. Let us assume first that Y 6 = X . Fix a d ecreasing s equence { U n } ∞ n =2 of open su bs ets of X with the property that Y = ∩ ∞ n =2 U n and U c n 6 = ∅ . The compactness of X implies t hat the natural restriction map Aff (T( A )) → C ( ∂ e T( A ) , R ) is an isometric isomorphism of B anach sp aces, [Go86, Cor . 11.20]. W e may therefor e use The orem 4.3 to pr oduce, a sequence ( b n ) ∞ n =2 in A + with the following properties: RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 9 (i) τ ( b n ) > 1 − 1 /n, ∀ τ ∈ U c n ; (ii) τ ( b n ) < δ/ 2 n n, ∀ τ ∈ Y ; (iii) k b n k ≤ 1 . For any τ ∈ U c n we have d τ (( b n − 1 /n ) + ) ( iii ) ≥ τ (( b n − 1 /n ) + ) ≥ τ ( b n ) − 1 /n ( i ) > 1 − 2 /n. In particular ( b n − 1 /n ) + 6 = 0 . Moreover , for any τ ∈ Y we have d τ (( b n − 1 /n ) + ) = n Z 1 n χ (1 /n, ∞ ) dµ τ ≤ nτ ( b n ) ( ii ) < δ / 2 n . Set c n := 2 − n ( b n − 1 /n ) + , so that d τ ( c n ) > 1 − 2 /n for each τ ∈ U c n , d τ ( c n ) < δ / 2 n for each τ ∈ Y and k c n k ≤ 2 − n . Now set a := ∞ X n =2 c n ∈ A + . If τ ∈ Y , then, using the lower semicontinuity of d τ , we have d τ ( a ) ≤ lim inf k d τ k X n =2 c n ! ≤ lim inf k k X n =2 d τ ( c n ) < δ. If τ ∈ X \ Y , then τ ∈ U c k for all k sufficiently lar ge. It follows that fo r these same k , d τ ( a ) = d τ ∞ X n =2 c n ! ≥ d τ ( c k ) ≥ 1 − 2 /k . W e conclude that d τ ( a ) ≥ 1 for each such τ . On t he o ther hand, a ∈ A , so d τ ( a ) ≤ 1 for any τ ∈ T( A ) . This completes the proof in the case that Y 6 = X . If Y = X and X is a sing let on, the n the Theo rem follows from L emma 4.1 ; we assume that X contains at least two points, say τ , γ . Since X is Hausdorff we can find open su bsets U and V of X such that τ ∈ U , γ ∈ V , and U ∩ V = ∅ . Use the case of the Theo r em established above twice, with Y r eplaced by U c and V c , to o bt ain po sitive elements b, c of A with the following properties: d ν ( c ) = 1 , ∀ ν ∈ U, and d ν ( c ) < δ, ∀ ν ∈ X \ U ; d ν ( b ) = 1 , ∀ ν ∈ V , and d ν ( b ) < δ, ∀ ν ∈ X \ V . It fo llows by Lemma 3.2 applied to the eleme n t b ∈ A and the compact set { γ } , with k = 1 , β = 1 and α = d γ ( c ) / 2 > 0 , that there is η > 0 such that d γ (1 − f η ( b )) < 2 α = d γ ( c ) , where f η is the function g iven in (1). Since d ν ( b ) = d ν ( f η ( b )) for any ν ∈ T( A ) , we may replac e b with f η ( b ) and hence arrange that (4) d γ (1 − b ) < d τ ( c ) . 10 MARIUS DADARLA T AND ANDREW S. TOMS Set a = b 1 / 2 cb 1 / 2 . Since a - c , we have d ν ( a ) < δ for every ν ∈ X \ U . Also, a ≈ c 1 / 2 bc 1 / 2 - b , so d ν ( a ) < δ for every ν ∈ X \ V . S ince U and V a re disjoint we con clude that d ν ( a ) < δ for every ν ∈ X . I t remains to prove that a 6 = 0 . T o this p urpose we show that d γ ( a ) > 0 . By Lemma 4.2 and (4 ) above we have d γ ( a ) = d γ ( b 1 / 2 c b 1 / 2 ) ≥ d γ ( c ) − d γ (1 − b ) > 0 .  Lemma 4.5. Let A be a unital simple separable infinite-dimensional C ∗ -algebr a. Suppose that the extr eme boundary X of T( A ) is compact and n onempty . Let a ∈ A be positive, and let there be given an open subset U of X and δ > 0 . It follows that there is a positiv e element b of A ⊗ K w ith the following pr opert ies: d τ ( b ) = d τ ( a ) , ∀ τ ∈ U and d τ ( b ) ≤ δ, ∀ τ ∈ X \ U. Pro of. U se Lemma 4.4 to find a p ositive element h of A with the property that (5) d τ ( h ) < δ, ∀ τ ∈ X \ U and d τ ( h ) = 1 , ∀ τ ∈ U. Let V 1 ⊆ V 2 ⊆ V 3 ⊆ · · · be a seque nce of ope n subset s of X s u ch that V i ⊆ U for each i and ∪ ∞ i =1 V i = U . T rivial ly , (6) 1 − 1 / 2 i < d τ ( h ) ≤ 1 , ∀ τ ∈ V i , and s o Lemma 3.2 applied for k = β = 1 and α = 1 / 2 i yields η i > 0 such that (7) d τ (1 − f η i ( h )) < 1 /i, ∀ τ ∈ V i . T o simplify notation in th e remainder of the proof, let us simply r e-label f η i ( h ) as h i . W e may assume that the sequence ( η i ) is decreasing so that the sequence ( h i ) is incr easing. Since d τ ( h ) = d τ ( f η ( h )) for any τ ∈ T( A ) and η > 0 , it follows fr om (5 ) that (8) d τ ( h i ) < δ , ∀ τ ∈ X \ U and d τ ( h i ) = 1 , ∀ τ ∈ U. Set a i := h 1 / 2 i ah 1 / 2 i . Since a i - a , we have d τ ( a i ) ≤ d τ ( a ) , ∀ τ ∈ U. Also, s ince h 1 / 2 i ah 1 / 2 i ≈ a 1 / 2 h i a 1 / 2 - h i , w e have d τ ( a i ) ≤ d τ ( h i ) < δ , ∀ τ ∈ X \ U. For ou r lower bound, we observe that by Lemma 4.2 and (7) we have for any τ ∈ V i : d τ ( a i ) = d τ ( h 1 / 2 i ah 1 / 2 i ) ≥ d τ ( a ) − d τ (1 − h i ) > d τ ( a ) − 1 /i. Therefor e we have (9) d τ ( a i ) < δ, ∀ τ ∈ X \ U and d τ ( a ) − 1 /i < d τ ( a i ) < d τ ( a ) , ∀ τ ∈ V i . Since h i ≤ h i +1 and a i = h 1 / 2 i dh 1 / 2 i - h 1 / 2 i +1 dh 1 / 2 i +1 = a i +1 . RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 11 The increasing se quence ( h a i i ) ∞ i =1 has a supremum y , where y = h b i fo r some po sitive element b of A ⊗ K by [CEI08]. Since e ach d τ is a sup remum preserving s tate on Cu( A ) , w e conclude from (9) t h at d τ ( b ) ≤ δ, ∀ τ ∈ X \ U and d τ ( b ) = d τ ( a ) , ∀ τ ∈ U, as d esired.  Lemma 4.6. Let A be a un ital simple sepa rable C ∗ -algebr a with strict compariso n of positive elements and at least one bounded trace. S uppose that X = ∂ e T( A ) is compact, and let a, b ∈ A be posit ive. Suppose that ther e ar e 0 < α < β < γ ≤ 1 an d open sets U, V ⊆ X w ith the pr op erty that α < d τ ( a ) < β , ∀ τ ∈ U and β < d τ ( b ) < γ , ∀ τ ∈ V . It follows that for any closed set K ⊂ U ∪ V , there is a positive element c of M 2 ( A ) with the pr op erty that α < d τ ( c ) < γ , ∀ τ ∈ K . Pro of. S ince the t opology on X is metrizable and K is compact there e xist closed s ubsets E ⊆ U and F ⊆ V su ch that K ⊆ E ◦ ∪ F ◦ ⊆ E ∪ F and E and F are t he closures of their interiors. By L e mma 4.5 and strict comparison we may assume t hat d τ ( a ) < β for all τ ∈ X \ U . Apply Lemma 3.1 to a to find an ǫ > 0 such that d τ (( a − ǫ ) + ) > α for each τ ∈ E . Like wise find δ > 0 such that d τ (( b − δ ) + ) > β for each τ ∈ F . For τ ∈ X we have d τ (( b − δ ) + ) ≤ µ τ ([ δ , ∞ ) ∩ σ ( b )) ≤ d τ ( b ) < γ . The map τ 7→ µ τ ([ δ , ∞ ) ∩ σ ( b )) is upp er semicontinuou s on the compact set X , and so attains a maximum γ 0 < γ . Use Lemma 4.4 to find a positive eleme n t z ∈ A with the property that d τ ( z ) < γ − γ 0 , ∀ τ ∈ F and d τ ( z ) = 1 , ∀ τ ∈ X \ F . Set y = ( b − δ ) + ⊕ z ∈ M 2 ( A ) , so that β < d τ ( y ) < γ , ∀ τ ∈ F and d τ ( y ) ≥ β , ∀ τ ∈ X \ F. It follows that d τ ( y ) > d τ ( a ) for every τ ∈ X , whence a - y by strict comparison. Using this we can find v ∈ M 2 ( A ) such that ( a − ǫ ) + = v y v ∗ . Set x = y 1 / 2 v ∗ v y 1 / 2 , s o that x ≈ ( a − ǫ ) + and x ∈ y M 2 ( A ) y . Moreover , d τ ( x ) < β for all τ ∈ X . Choose 0 < λ < m in { β − α, γ − β } . Use L emma 4.4 to find a positive e lement h in M 2 ( A ) with the following property: d τ ( h ) = 2 , ∀ τ ∈ F ◦ and d τ ( h ) < λ, ∀ τ ∈ X \ F ◦ . W e can find closed se ts E 1 ⊆ E ◦ and F 1 ⊆ F ◦ such that K ⊆ E 1 ∪ F 1 . Replacing h with f η ( h ) , for sufficiently small η , we may arrange by Lemma 3.2 applied with k = β = 2 and α = λ/ 2 , that d τ (1 2 − h ) < λ, ∀ τ ∈ F 1 . W e define c = x + y 1 / 2 hy 1 / 2 . 12 MARIUS DADARLA T AND ANDREW S. TOMS Let us prove that t his de finition yields the inequality requir ed by the L e mma. Let τ ∈ K be given. If τ ∈ E 1 , t hen d τ ( c ) ≥ d τ ( x ) = d τ (( a − ǫ ) + ) > α. If τ ∈ F 1 , then, by Lemma 4.2, we have d τ ( c ) ≥ d τ ( y 1 / 2 hy 1 / 2 ) = d τ ( h 1 / 2 y h 1 / 2 ) ≥ d τ ( y ) − d τ (1 2 − h ) > β − λ > α. Thus, d τ ( c ) > α for e ach τ ∈ K . For the upper bound, observe first that c ∈ y M 2 ( A ) y , whence d τ ( c ) ≤ d τ ( y ) for every τ ∈ X . I n particular , d τ ( c ) < γ , ∀ τ ∈ F . If τ ∈ K \ F then τ ∈ X \ F ◦ . It follows that d τ ( c ) ≤ d τ ( x ) + d τ ( y 1 / 2 hy 1 / 2 ) ≤ d τ ( x ) + d τ ( h ) < β + λ < γ .  5. T H E P R O O F O F T H E O R E M 1 . 1 For clarity of expo sition we se parate Th e orem 1.1 into three parts, conside ring cond itions (iii), (ii), and (i) in order . The idea for the proof of t h e Theorem 5.1 is contained in [BPT08]. Theorem 5.1. Let A be a unital simple separable C ∗ -algebr a with nonempty tracial stat e space . S uppose that for each positive b ∈ A ⊗ K such that d τ ( b ) < ∞ , ∀ τ ∈ T( A ) and δ > 0 , ther e is a positive c ∈ A ⊗ K such that (10) | 2 d τ ( c ) − d τ ( b ) | < δ, ∀ τ ∈ T( A ) . It follows that for any f ∈ Aff (T( A )) and ǫ > 0 ther e is positiv e h ∈ A ⊗ K with the pr operty that | d τ ( h ) − f ( τ ) | < ǫ, ∀ τ ∈ T( A ) . If A m oreove r has strict compa rison, then for any f ∈ SAff (T ( A )) ther e is positive h ∈ A ⊗ K such that d τ ( h ) = f ( τ ) for each τ ∈ T( A ) . Pro of. L et f ∈ Aff (T( A )) ; we may assume k f k = 1 . It will suffice to consider ǫ = 1 /k for given k = 2 n ∈ N . Us e T heorem 4.3 to find po sitive a ∈ A such that 1 ≤ k a k ≤ 1 + 1 /k and τ ( a ) = f ( τ ) , ∀ τ ∈ T( A ) . Conside r the function g ( t ) = P 2 k + 1 i =1 (1 / 2 k ) χ ( i/ 2 k, ∞ ) ( t ) . Using the Borel functional calculus we have k a − g ( a ) k ≤ 1 / 2 k , and hence (11) | τ ( a ) − τ ( g ( a )) | =      τ ( a ) − 2 k + 1 X i =1 (1 / 2 k ) τ  χ ( i/ 2 k, ∞ ) ( a )       ≤ 1 2 k for each τ ∈ T( A ) . For each i and τ we have d τ (( a − i/ 2 k ) + ) = τ ( χ ( i/ 2 k, ∞ ) ( a )) . It is straightfor- war d t o see that (10) implies the following st atement: for b as in the state ment of the Theo r em, for any l ∈ N and δ > 0 , there is a positive c ∈ A ⊗ K such t h at | ld τ ( c ) − d τ ( b ) | < δ , ∀ τ ∈ T( A ) . RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 13 All t hat is actually ne eded is t h e case when l is a power of two. Apply this w ith l = 2 k , b = ( a − i/ 2 k ) + , and δ < 1 / (4 k + 2) to obtain positive h i ∈ A ⊗ K su ch that | d τ ( h i ) − (1 / 2 k ) d τ (( a − i/ 2 k ) + ) | < 1 / (4 k 2 + 2 k ) , ∀ τ ∈ T( A ) . Thus, s etting h = ⊕ 2 k + 1 i =1 h i , we have | τ ( g ( a )) − d τ ( h ) | =      2 k + 1 X i =1 (1 / 2 k ) τ  χ ( i/ 2 k, ∞ ) ( a )  − d τ  ⊕ 2 k + 1 i =1 h i       =      2 k + 1 X i =1 (1 / 2 k ) d τ (( a − i/ 2 k ) + ) − d τ ( h i )      < 2 k + 1 4 k 2 + 2 k = 1 2 k Using (11) we arrive at | f ( τ ) − d τ ( h ) | = | τ ( a ) − d τ ( h ) | ≤ | τ ( a ) − τ ( g ( a )) | + | τ ( g ( a )) − d τ ( h ) | < 1 2 k + 1 2 k = 1 k = ǫ, ∀ τ ∈ T( A ) , as d esired. Now supp o se that A has strict comparison. T h e final conclusion of the Theorem th e n follows fr om th e p r oof of [BT 07 , The orem 2.5], which sho ws ho w one produces an arbitrary f ∈ SAff (T( A )) by t aking suprema.  Theorem 5.2. Let A be a unital simple separable C ∗ -algebr a. Suppose that the extr eme boundary X of T( A ) is compact and nonempty , and tha t for each m ∈ N ther e is x ∈ Cu( A ) with the pr oper ty that (12) md τ ( x ) ≤ 1 ≤ ( m + 1) d τ ( x ) , ∀ τ ∈ T( A ) . It follows that for any f ∈ Aff (T( A )) and ǫ > 0 ther e is positiv e h ∈ A ⊗ K with the pr operty that (13) | d τ ( h ) − f ( τ ) | < ǫ, ∀ τ ∈ T( A ) . If A moreove r has strict compari son, then we may take f ∈ S Aff (T( A )) and arrange that d τ ( h ) = f ( τ ) for each τ ∈ T( A ) . Pro of. L et f and ǫ be given, and assume k f k = 1 . W e need only establish (13) over X . W e may assume that ǫ = 1 /k for so me k ∈ N . For i ∈ { 0 , . . . , 2 k − 1 } , let U i be the open set { τ ∈ X | f ( τ ) > i/ 2 k } . W e then have (14)      f ( τ ) − 2 k − 1 X i =0 (1 / 2 k ) χ U i ( τ )      ≤ 1 2 k . Let us no w prove the following statement: given 1 > r , η , δ > 0 and an open subset U of X , there is a positive e leme nt a o f A ⊗ K with t he property that r − η ≤ d τ ( a ) ≤ r, ∀ τ ∈ U and d τ ( a ) ≤ δ, ∀ τ ∈ X \ U. Choose m lar ge en o ugh that r − η < k / ( m + 1) < k /m < r for some k ∈ N . B y ass umption there is x ∈ Cu( A ) su ch that d τ ( x ) ∈ [1 / ( m + 1) , 1 /m ] for all τ ∈ T( A ) . It follows that d τ ( r x ) ∈ 14 MARIUS DADARLA T AND ANDREW S. TOMS [ r / ( m + 1) , r /m ] fo r all τ ∈ T( A ) . Let c ∈ ( A ⊗ K ) + repr esenting r x ∈ Cu ( A ) . Then c satisfies satisfies r − η < d τ ( c ) < r, ∀ τ ∈ X. Obtaining th e desired element a fr om c is a straightforward app lication of Lemma 4.5. Apply the s tatement proved above with U = U i , r = 1 / 2 k , and η = δ = 1 / 4 k 2 to obtain positive h i ∈ A ⊗ K such that 1 2 k − 1 4 k 2 ≤ d τ ( h i ) ≤ 1 2 k , ∀ τ ∈ U i and d τ ( h i ) < 1 / 4 k 2 for e ach τ ∈ X \ U i . It is then straightforward to check that (15)      2 k − 1 X i =0 (1 / 2 k ) χ U i ( τ ) − 2 k − 1 X i =0 d τ ( h i )      < 2 k − 1 4 k 2 < 1 k . It follows from (14) and (15) t h at h := ⊕ 2 k − 1 i =0 h i has the required p roperty , since d τ ( h ) = P 2 k − 1 i =0 d τ ( h i ) . If A has strict comparison then t h e final conclusion of the Theo rem fo llows on ce again from the proof of [BT07, The orem 2.5].  Remark 5.3. If A is a unital simple s eparable C ∗ -algebra with th e pr operty that for any f ∈ SAff (T( A )) there is positive h ∈ A ⊗ K such t hat f ( τ ) = d τ ( h ) , ∀ τ ∈ T( A ) , then bo t h (10) and (12) hold. Theorem 5.4. Let A be a u nital simple separabl e C ∗ -algebr a with strict compar ison of positive elements . Suppose further that the extr eme boundary X of T( A ) is nonempty , compact, and of finite cove ring dimension. It follows that for each f ∈ SAff (T( A )) , ther e is a positive element a ∈ A ⊗ K with the pr opert y that d τ ( a ) = f ( τ ) for each τ ∈ T( A ) . Pro of. B y arguing as in the p roof of Theorem 5.2, it will suf fice to find, for each 1 > r, ǫ > 0 , a positive eleme n t a in some M N ( A ) w ith the pr operty th at r − ǫ < d τ ( a ) < r, ∀ τ ∈ X. Set d := dim( X ) , and for i ∈ { 0 , . . . , 2 d + 2 } define r i = r − (2 d + 2 − i ) ǫ 2 d + 2 . Fix τ ∈ X . Use Lemma 4.1 t o find, for each k ∈ { 0 , 1 , . . . , d } , a pos itive elemen t ˜ b k ∈ M N ( A ) and an op e n neighbourhood V k of τ in X with t he property th at (16) r 2 k < d γ ( ˜ b k ) < r 2 k + 1 , ∀ γ ∈ V k . Set U τ = ∩ k V k , s o that U : = { U τ } τ ∈ X is an open cover of X . By the finite-dimens ionality and compactness of X , there are a refinement of a finite subcover of U , say W = { W 1 , . . . , W n } , and a map c : W → { 0 , 1 , . . . , d } with the pr operty that if i 6 = j then c ( W i ) = c ( W j ) ⇒ W i ∩ W j = ∅ . RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 15 Each W i is contained in so me U τ , and so (16) furnishes pos itive elements ˜ b k , k ∈ { 0 , 1 , . . . , d } such that r 2 k < d γ ( ˜ b k ) < r 2 k + 1 , ∀ γ ∈ W i . Set η = ǫ/ ( n (2 d + 2)) , and use Lemma 4.5 to p roduce positive elements b ( i ) k in A ⊗ K with the following properties : r 2 k < d γ ( b ( i ) k ) < r 2 k + 1 , ∀ γ ∈ W i and d γ < η , ∀ γ ∈ X \ W i . Now for each k ∈ { 0 , 1 , . . . , d } define b k = M { i | c ( W i )= k } b ( i ) k ∈ A ⊗ K . The W i s appearing in the sum above are mutually disjoint. Suppos e t hat τ ∈ W s and c ( W s ) = k . W e have the following bounds: d τ ( b k ) = X { i | c ( W i )= k } d τ ( b ( i ) k ) = d τ ( b ( s ) k ) + X { i | c ( W i )= k, i 6 = s } d τ ( b ( i ) k ) < r 2 k + 1 + n η = r 2 k + 1 + ǫ/ (2 d + 2) = r 2 k + 2 and d τ ( b k ) > d τ ( b ( s ) k ) > r 2 k . For e ach k ∈ { 0 , 1 , . . . , d } we define W k = [ { i | c ( W i )= k } W i , so that r 2 k < d τ ( b k ) < r 2 k + 2 , ∀ τ ∈ W k . Note that W 0 , W 1 , . . . , W d is a cover of X . T o complete th e p r oof of the Theo r em w e proceed by induction. First obse rve t hat since X is compact and met rizable, we may find a closed subset K 0 of W 0 with the property that K ◦ 0 , W 1 , . . . , W d is a cover of X . Set c 0 = b 0 , so that r − ǫ = r 0 < d τ ( c 0 ) < r 2 , ∀ τ ∈ K 0 . Now suppo s e t hat we have found a closed s et K k ⊆ W 0 ∪· · ·∪W k , k < d , such that K ◦ k , W k +1 , . . . , W d covers X , and a pos itive element c k in so me M N ( A ) with the property that r − ǫ = r 0 < d τ ( c k ) < r 2 k + 2 , ∀ τ ∈ K k . 16 MARIUS DADARLA T AND ANDREW S. TOMS Since X is compact and metrizable, we can find a closed se t K k +1 ⊆ K ◦ k ∪ W k +1 such that K ◦ k +1 , W k +2 , . . . , W d covers X . Applying L e mma 4.6 to c k and b k +1 we obtain a positive e leme nt c k +1 in so me M N ( A ) with the property that r − ǫ = r 0 < d τ ( c k +1 ) < r 2 k + 4 , ∀ τ ∈ K k +1 . Starting with t he base case k = 0 , applying the inductive step above n times, and noting that we must have K d = X , we arrive at a positive element c n in so me M N ( A ) w ith the pr operty th at r − ǫ = r 0 < d τ ( c n ) < r 2 d +2 = r , ∀ τ ∈ X . Setting a = c n completes the proof.  Theorems 5.1, 5.2, and 5.4 together constitute Theorem 1.1. 6. A P P L I C AT I O N S 6.1. The structure of the Cuntz semigroup. Let A be a unital simple C ∗ -algebra with nonempt y tracial s tate space. Cons ider the d isjoint union V ( A ) ⊔ S Aff (T( A )) , where V ( A ) d enotes the s emigroup of Murray-von Neumann equivalence classes o f p r ojections in A ⊗ K . E quip this set with an addition operation as follows: (i) if x, y ∈ V ( A ) , the n their sum is the usual sum in V ( A ) ; (ii) if x, y ∈ SAff (T( A )) , then their sum is the us ual (pointwise) sum in S Aff (T( A )) ; (iii) if x ∈ V ( A ) and y ∈ S Aff (T( A )) , t hen their sum is the usual (pointwise) sum of ˆ x and y in SAff (T( A )) , where ˆ x ( τ ) = τ ( x ) , ∀ τ ∈ T( A ) . Equip V ( A ) ⊔ SAff (T( A )) with the partial order ≤ which r estricts t o the usual partial o rder o n each of V ( A ) and S Aff (T( A )) , and which satisfie s the following conditions for x ∈ V ( A ) and y ∈ S Aff (T( A )) : (i) x ≤ y if and only if ˆ x ( τ ) < y ( τ ) , ∀ τ ∈ T( A ) ; (ii) y ≤ x if and only if y ( τ ) ≤ ˆ x ( τ ) , ∀ τ ∈ T( A ) . It is shown in [BT 07] t hat the Cuntz semigroup o f A is o rder isomorphic t o the ordered Abelian semigroup V ( A ) ⊔ SAff (T( A )) de fined above wheneve r ι ( A ⊗ K ) = SAff (T( A )) . This structure the orem therefor e applies to the algebras covered by Theorem 1.1 provided that they have st rict comparison. 6.2. T wo conjectures of B lackadar-Handelman. In their 1982 study o f dimension functions on unital t racial C ∗ -algebras—equivalently , additive, unital, and order preserving maps from the Cuntz se migr oup into R + —Blackadar and Handelman made two conjectures ([BH82]): (i) The s pace of lower s e micontinuous dimension functions—dimension functions of the form d τ for a no rmaliz ed 2-quasitrace τ —is weakly dens e among all dimension func- tions. (ii) The af fine space of all dimension functions is a Choque t simplex. RANKS OF OPERA T ORS IN SIMPLE C ∗ -ALGEBRAS 17 It was p roved in [B PT08] that the se conjectures ho ld for a unital simple separable exact C ∗ - algebra who se Cuntz se migr oup has the form de scribed in Subsection 6.1, and so t he conje c- tures hold for the algebras covered by Theorem 1.1 provided that they ar e e xact and have strict comparison. 6.3. Z -sta bility and a stably fi nite G eneva T heorem. At the ICM Satellite Meeting on Operator Algebras in 1994 , Kir chberg announced th at a simple separable nuclear C ∗ -algebra is purely infinite if and only if it absorbs the Cuntz algebra O ∞ tensorially . This r esult is a corners tone of the the ory of purely infinite C ∗ -algebras and th e ir classification. Using results of Rørdam fr om [Rø04] and the de finition of strict comparison, o n e can rephrase Kirchber g’s result: Theorem 6.1. Let A be a simple separ able nuclear traceless C ∗ -algebr a. It follo ws that A ∼ = A ⊗ Z ⇔ A has strict comparison . W inter and the second named author have conjectured that Theorem 6.1 cont inues to hold in the absence of the ”traceless” hypothesis, g iving a stably finite vers ion o f the Geneva Theorem. As in t he purely infinite case, confirmations o f this conjectu re lead to strong classification results for simple C ∗ -algebras. Indeed , we shall give such an application in S ubsection 6.4 be low . An important ste p tow ard the s o lution of this conjecture has r ecently been taken by W inter . Theorem 6.2 (W inte r , [W i09]) . Let A be a unital simple sep arable C ∗ -algebr a with locally finite decom- positio n rank. If Cu( A ) ∼ = Cu( A ⊗ Z ) , then A ∼ = A ⊗ Z . If A is a u n ital simple exact C ∗ -algebra with nonemp t y tracial st ate s pace, the n the statement ”The Cuntz semigroup of A has the form des cribed in Subse ction 6.1” can be neatly summarized by s aying that C u( A ) ∼ = Cu( A ⊗ Z ) . W e therefor e have: Corollary 6.3. Let A be a unital simple sep arable C ∗ -algebr a with locally finite decompos ition rank and nonempty tracial state space. Suppose that A satisfies an y of conditions (i)-(iii ) in Theor em 1.1. It follows that A ∼ = A ⊗ Z ⇔ A has strict comparison . This repr esents a subs t antial confirmation of the conjecture de scribed above. W e comment that locally finite decomposition rank is quite a we ak property , satisfied, for inst ance, by any unital separable A SH algebra ([NW06]). 6.4. Classification of C ∗ -algebras. W inte r and, Lin and Niu, have p roved strong classification theorems und e r the assumption of Z -stabil ity ([W i06 ], [W i07], [LN08]). I n light of th e se, w e have the following confirmation of Elliott’s classification conjecture. Corollary 6.4. Let C denote the class of C ∗ -algebr as which satisfy all of the conditions of 6.3 and the UCT and have enough pr oject ions to separate their traces. It follows that E lliott’ s conjectur e holds for C : if A, B ∈ C and φ : K ∗ ( A ) → K ∗ ( B ) is a graded order isomorp hism with φ ∗ [1 A ] = [1 B ] , then ther e is a ∗ -isomorphism Φ : A → B which induces φ . 18 MARIUS DADARLA T AND ANDREW S. TOMS 6.5. Classification of Hilbert mod ules. Let A be a unital separable C ∗ -algebra with nonempty tracial state space, and let E be a countably generated Hilbert module ove r A . It follows fr om [Pd98] t h at E ∼ = a ( A ⊗ K ) for s ome positive a ∈ A ⊗ K . If A has st able rank one , then the isomorphism class of E de pends only on the Cuntz e quivalence class of a , and we may define d τ ( E ) = d τ (˜ a ) for any ˜ a is this equivalence cla ss. B y Kaspar ov’s stabilization the orem, there is a projection P E ∈ B ( H A ) such that E is isomorphic to P E H A . (Here H A = ℓ 2 ⊗ A is the s t andar d Hilbert mod ule over A .) Corolla ry 6.3 yields Z -stability for an algebra A as in The orem 1.1 pro- vided that it has strict comparison and locally finite decomposition rank. Appealing to [Rø04], this g ives stable rank one, and so a furt her appeal to [BT07] g ives th e following classification result. Corollary 6.5. L et A be a unital simple separable C ∗ -algebr a satisfy ing all of the conditions of Cor olla ry 6.3. Given two countably generated Hilbert modules E , F over A , the foll owing ar e equivalent: (i) E is isomorp hic to F ; (ii) Either h P E i = h P F i ∈ V( A ) (in the case P E , P F ∈ A ⊗ K ), or d τ ( E ) = d τ ( F ) , ∀ τ ∈ T( A ) . In particular , if neither E nor F is a finitely generated project ive module, then E ∼ = F if and only if d τ ( E ) = d τ ( F ) , ∀ τ ∈ T( A ) . R E F E R E N C E S [BH82] Blackadar , B. and Handelman, D.: Dimension function s and traces on C ∗ -algebras , J. Funct. Anal., 45 ( 1982), pp. 297–34 0. [BPT08] Brown, N. P ., Perera, F ., and T oms, A. 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[W i06] W int er , W .: Simple C ∗ -algebras with locally finite decomposition rank , J. Funct. Anal. 243 (2006), 394-425 [W i07] W int er , W .: Localizing the Elliott conjecture at strongly self-absorbi ng C ∗ -algebras , arXi v preprint 0708:0283 (2007) [W i09] W int er , W .: Z -stability and pure finiten ess , in preparation D E PA R T M E N T O F M AT H E M AT I C S , P U R D U E U N I V E R S I T Y , 1 5 0 N . U N I V E R S I T Y S T . , W E S T L A FAY E T T E , I N , 4 7 9 0 7 - 2 0 6 7 , U . S . A . E-mail address : mdd@math.purdue .edu D E PA R T M E N T O F M AT H E M AT I C S A N D S T AT I S T I C S , Y O R K U N I V E R S I T Y , 4 7 0 0 K E E L E S T . , T O R O N T O , O N TA R I O , C A N A D A , M 3 J 1 P 3 E-mail address : atoms@mathstat. yorku.ca