aa r X i v : . [ m a t h . OA ] M a y COMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS ANDREW S. TOMSA
BSTRACT . We prove that the C ∗ -algebra of a minimal diffeomorphism satisfies Blackadar’s FundamentalComparability Property for positive elements. This leads to the classification, in terms of K -theory and traces,of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar clas-sification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for theCuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontin-uous dimension functions are weakly dense in the space of all dimension functions. These results continue tohold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one.Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C ∗ -algebra. Thisis also used to construct uncountably many non-Morita-equivalent simple separable amenable C ∗ -algebraswith the same K -theory and tracial state space, providing a C ∗ -algebraic analogue of McDuff’s uncountablefamily of II factors. We prove in passing that the range of the radius of comparison is exhausted by simpleC ∗ -algebras.
1. I
NTRODUCTION
The comparison theory of projections is fundamental to the theory of von Neumann algebras, andis the basis for the type classification of factors. For a general C ∗ -algebra this theory is vastly morecomplicated, but but no less central. Blackadar opined in [1] that “the most important general structurequestion concerning simple C ∗ -algebras is the extent to which the Murray-von Neumann comparisontheory for factors is valid in arbitrary simple C ∗ -algebras.” In this article we answer Blackadar’s questionfor the C ∗ -algebras associated to smooth minimal dynamical systems, among others, and give severalapplications.Tellingly, Blackadar’s quote makes no mention of projections. A C ∗ -algebra may have few or no projec-tions, in which case their comparison theory says little about the structure of the algebra. The appropriatereplacement for projections is positive elements, along with a notion of comparison for the latter whichgeneralises Murray-von Neumann comparison for projections. This idea was first introduced by Cuntz in[9] with a view to studying dimension functions on simple C ∗ -algebras. His comparison relation is con-veniently encoded in what is now known as the Cuntz semigroup , a positively ordered Abelian monoidwhose construction is analogous to that of the Murray-von Neumann semigroup. When the natural par-tial order on this semigroup is governed by traces, then we say that the C ∗ -algebra has strict comparisonof positive elements (see Subsection 2.2 for a precise definition); this property, first introduced in [1], isalso known as Blackadar’s Fundamental Comparability Property for positive elements . It is the best availableanalogue among simple C ∗ -algebras for the comparison theory of projections in a factor, and a power-ful regularity property necessary for the confirmation of G. A. Elliott’s K -theoretic rigidity conjecture(see [11] and [27]). Its connection with the comparison theory of projections in a von Neumann algebra is Mathematics Subject Classification.
Primary 46L35, Secondary 46L80.
Key words and phrases.
Hilbert modules, Cuntz semigroup, ASH algebras, C ∗ -dynamical systems.This research was supported in part by an NSERC Discovery Grant. quite explicit: if a unital simple stably finite C ∗ -algebra A has strict comparison of positive elements, thenCuntz comparison for those positive elements with zero in their spectrum is synonymous with Murray-von Neumann comparison of the corresponding support projections in the bidual; the remaining positiveelements have support projections which are contained in A , and Cuntz comparison for these elementsreduces to Murray-von Neumann comparison of their support projections in A , as opposed to A ∗∗ .Our main result applies to a class of C ∗ -algebras which contains properly the C ∗ -algebras associatedto minimal diffeomorphisms. Recall that a C ∗ -algebra is subhomogeneous if there is a uniform bound onthe dimensions of its irreducible representations, and approximately subhomogeneous (ASH) if it is the limitof a direct system of subhomogeneous C ∗ -algebras. There are no known examples of simple separableamenable stably finite C ∗ -algebras which are not ASH. Every unital separable ASH algebra is the limit ofa direct sequence of recursive subhomogeneous C ∗ -algebras , a particularly tractable kind of subhomogeneousC ∗ -algebra ([19]). Theorem 1.1.
Let ( A i , φ i ) be a direct sequence of recursive subhomogeneous C ∗ -algebras with slow dimensiongrowth. Suppose that the limit algebra A is unital and simple. It follows that A has strict comparison of positiveelements. We note that the hypothesis of slow dimension growth is necessary, as was shown by Villadsen in [31].The relationship between Theorem 1.1 and the C ∗ -algebras of minimal dynamical systems is derivedfrom the following theorem: Theorem 1.2 (Lin-Phillips, [17]) . Let M be a compact smooth connected manifold, and let h : M → M be aminimal diffeomorphism. It follows that the transformation group C ∗ -algebra C ∗ ( M, Z , h ) is a unital simple directlimit of recursive subhomogeneous C ∗ -algebras with slow dimension growth (indeed, no dimension growth). K -theoretic considerations show the class of C ∗ -algebras covered by Theorem 1.1 to be considerablylarger than the class covered by Theorem 1.2.Let us describe briefly the applications of our main result. In a C ∗ -algebra A of stable rank one, theCuntz semigroup can be identified with the semigroup of isomorphism classes of countably generatedHilbert A -modules—addition corresponds to the direct sum, and the partial order is given by inclusionof modules ([7]). It is also known that positive elements a, b ∈ A are approximately unitarily equivalentif and only if the canonical maps from C (0 , into A induced by a and b agree at the level of the Cuntzsemigroup ([5]). Thus, to the extent that one knows the structure of the Cuntz semigroup, one also knowswhat the isomorphism classes of Hilbert A -modules and the closures of unitary orbits of positive oper-ators look like. If A is in addition unital, simple, exact, and has strict comparison of positive elements,then its Cuntz semigroup can be described in terms of K -theory and traces (see [4, Theorem 2.6]), and theCiuperca-Elliott classification of orbits of positive operators extends to self-adjoint elements. Thus, forthe algebras of Theorem 1.1, under the additional assumption of stable rank one, we have a description ofthe countably generated Hilbert A -modules and of the closures of unitary orbits of self-adjoints in termsof K -theory and traces. (In fact, this description also captures the inclusion relation for the said modules,and the structure of their direct sums.) This result applies to the C ∗ -algebras of minimal diffeomorphismsas these were shown to have stable rank one by N. C. Phillips ([23]). Our classification is quite practical,as the K -theory of these algebras is accessible through the Pimsner-Voiculescu sequence and their traceshave a nice description as the invariant measures on the manifold M . The classification of Hilbert mod-ules obtained is analogous to the classification of W ∗ -modules over a II factor. (See [4] and Subsections5.4 and 5.5.) Finally, we note that Jacob has recently obtained a description of the natural metric on the OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 3 space of unitary orbits of self-adjoint elements in a unital simple ASH algebra under certain assumptions,one of which is strict comparison. This gives another application of Theorem 1.1 ([15]).It was shown in [3, Theorem 6.4] that if the structure theorem for the Cuntz semigroup alluded toabove holds for A , then the lower semicontinuous dimension functions on A are weakly dense in thespace of all dimension functions on A , confirming a conjecture of Blackadar and Handelman from theearly 1980s. This conjecture therefore holds for the algebras of Theorem 1.1. (See Subsections 5.2 and 5.3.)If A is a unital stably finite C ∗ -algebra, then one can define a nonnegative real-valued invariant calledthe radius of comparison which measures the extent to which the order structure on the Cuntz semigroupof A is determined by (quasi-)traces. This invariant has proved useful in the matter of distinguishingsimple separable amenable C ∗ -algebras both in general ([29]) and in the particular case of minimal C ∗ -dynamical systems ([12]). The proof of Theorem 1.1 follows from a sharp upper bound that we obtainfor the radius of comparison of a recursive subhomogeneous C ∗ -algebra. This bound generalises andimproves substantially upon our earlier bound for homogeneous C ∗ -algebras ([30]). In addtion to beingcrucial for the proof of Theorem 1.1, this bound has other applications. We use it to prove that the rangeof the radius of comparison is exhausted by simple C ∗ -algebras, and that there are uncountably manynon-Morita-equivalent simple separable amenable C ∗ -algebras which all have the same K -theory andtracial state space (Theorem 5.11). This last result is proved using approximately homogenenous (AH)algebras of unbounded dimension growth, and so may be viewed as a strong converse to the Elliott-Gong-Li classification of simple AH algebras with no dimension growth ([10]). It can also be viewed asa C ∗ -algebraic analogue of McDuff’s uncountable family of pairwise non-isomorphic II factors ([18]).(See Subsections 5.1 and 5.6.)W. Winter has recently announced a proof of Z -stability for a class of C ∗ -algebras which includes unitalsimple direct limits of recursive subhomogeneous C ∗ -algebras with no dimension growth, leading to analternative proof of Theorem 1.1 under the stronger hypothesis of no dimension growth. Those workingon G. A. Elliott’s classification program for separable amenable C ∗ -algebras suspect that the conditionsof slow dimension growth and no dimension growth are equivalent, but this problem remains open evenfor AH C ∗ -algebras. Gong has shown that no dimension growth and a strengthened version of slowdimension growth are equivalent for unital simple AH algebras, an already difficult result (see [13]).The paper is organised as follows: Section 2 collects our basic definitions and preparatory results;Section 3 establishes a relative comparison theorem in the Cuntz semigroup of a commutative C ∗ -algebra;Section 4 applies the said comparison theorem to obtain sharp bounds on the radius of comparison of arecursive subhomogeneous algebra; Section 5 describes our applications in detail. Acknowledgements.
Part of this work was carried out at the Fields Institute during its Thematic Programon Operator Algebras in the fall of 2007. We are grateful to that institution for its support. We would alsolike to thank N. P. Brown and N. C. Phillips for several helpful conversations.2. P
RELIMINARIES
The Cuntz semigroup.
Let A be a C ∗ -algebra, and let M n ( A ) denote the n × n matrices whose entriesare elements of A . If A = C , then we may simply write M n .Let M ∞ ( A ) denote the algebraic limit of the direct system (M n ( A ) , φ n ) , where φ n : M n ( A ) → M n +1 ( A ) is given by a (cid:18) a
00 0 (cid:19) . ANDREW S. TOMS
Let M ∞ ( A ) + (resp. M n ( A ) + ) denote the positive elements in M ∞ ( A ) (resp. M n ( A ) ). Given a, b ∈ M ∞ ( A ) + , we say that a is Cuntz subequivalent to b (written a - b ) if there is a sequence ( v n ) ∞ n =1 of el-ements of M ∞ ( A ) such that || v n bv ∗ n − a || n →∞ −→ . We say that a and b are Cuntz equivalent (written a ∼ b ) if a - b and b - a . This relation is an equivalencerelation, and we write h a i for the equivalence class of a . The set W ( A ) := M ∞ ( A ) + / ∼ becomes a positively ordered Abelian monoid when equipped with the operation h a i + h b i = h a ⊕ b i and the partial order h a i ≤ h b i ⇔ a - b. In the sequel, we refer to this object as the
Cuntz semigroup of A . (It was originally introduce by Cuntz in[9].) The Grothendieck enveloping group of W ( A ) is denoted by K ∗ ( A ) .Given a ∈ M ∞ ( A ) + and ǫ > , we denote by ( a − ǫ ) + the element of C ∗ ( a ) corresponding (via thefunctional calculus) to the function f ( t ) = max { , t − ǫ } , t ∈ σ ( a ) . (Here σ ( a ) denotes the spectrum of a .) The proposition below collects some facts about Cuntz subequiv-alence due to Kirchberg and Rørdam. Proposition 2.1 (Kirchberg-Rørdam ([16]), Rørdam ([26])) . Let A be a C ∗ -algebra, and a, b ∈ A + . (i) ( a − ǫ ) + - a for every ǫ > . (ii) The following are equivalent: (a) a - b ; (b) for all ǫ > , ( a − ǫ ) + - b ; (c) for all ǫ > , there exists δ > such that ( a − ǫ ) + - ( b − δ ) + . (iii) If ǫ > and || a − b || < ǫ , then ( a − ǫ ) + - b . Dimension functions and strict comparison.
Now suppose that A is unital and stably finite, anddenote by QT( A ) the space of normalised 2-quasitraces on A (v. [2, Definition II.1.1]). Let S ( W ( A )) denote the set of additive and order preserving maps d : W ( A ) → R + having the property that d ( h A i ) =1 . Such maps are called states . Given τ ∈ QT( A ) , one may define a map d τ : M ∞ ( A ) + → R + by(1) d τ ( a ) = lim n →∞ τ ( a /n ) . This map is lower semicontinous, and depends only on the Cuntz equivalence class of a . It moreover hasthe following properties:(i) if a - b , then d τ ( a ) ≤ d τ ( b ) ;(ii) if a and b are mutually orthogonal, then d τ ( a + b ) = d τ ( a ) + d τ ( b ) ;(iii) d τ (( a − ǫ ) + ) ր d τ ( a ) as ǫ → .Thus, d τ defines a state on W ( A ) . Such states are called lower semicontinuous dimension functions , and theset of them is denoted LDF( A ) . QT( A ) is a simplex ([2, Theorem II.4.4]), and the map from QT( A ) to LDF( A ) defined by (1) is bijective and affine ([2, Theorem II.2.2]). A dimension function on A is a state on OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 5 K ∗ ( A ) , assuming that the latter has been equipped with the usual order coming from the Grothendieckmap. The set of dimension functions is denoted DF( A ) . LDF( A ) is a (generally proper) face of DF( A ) . If A has the property that a - b whenever d ( a ) < d ( b ) for every d ∈ LDF( A ) , then we say that A has strictcomparison of positive elements .2.3. Preparatory results.
We now recall and improve upon some results that will be required in thesequel.
Definition 2.2 (cf. Definition 3.4 of [30]) . Let X be a compact Hausdorff space, and let a ∈ M n (C( X )) bepositive with (lower semicontinuous) rank function f : X → Z + taking values in { n , . . . , n k } , n < n < · · ·
In the statement of Theorem 3.9 of [30], X is required to be a finite simplicial complex, butthis is only to ensure that some further conclusions about the approximant ˜ a can be drawn. The proofof this theorem, followed verbatim, also proves Theorem 2.3—one simply ignores all statements whichconcern the simplicial structure of X . An alternative proof can be found in [17].For our purposes, we require a different and in some ways strengthened version of Theorem 2.3. Itsays that the well-supported approximant ˜ a can be obtained as a cut-down of a , at the possible expenseof condition (i). Lemma 2.5.
Let X , a , and ǫ be as in the statement of Theorem 2.3. Suppose further that a has norm at most one.It follows that there is a positive element h of M n (C( X )) of norm at most one such that the following statementshold: (i) || hah − a || < ǫ ; (ii) || ha − a || < ǫ/ and || ah − a || < ǫ/ ; (iii) hah is well-supported.Proof. Apply Theorem 2.3 to a with the tolerance ǫ/ to obtain the approximant ˜ a . This approximant canbe described as follows (the details can be found in the proof of ?? [Theorem 3.9], which is constructive).At every x ∈ X there are mutually orthogonal positive elements a ( x ) , . . . , a k ( x ) of M n ( C ) such that a ( x ) = a ( x ) ⊕ a ( x ) ⊕ · · · ⊕ a k ( x ) . Note that k varies with x , and that we make no claims about the continuity of the a i s. Our approximantthen has the form ˜ a ( x ) = λ a ( x ) ⊕ λ a ( x ) ⊕ · · · ⊕ λ k a k ( x ) , ANDREW S. TOMS where λ i ∈ [0 , . We also have that || a i ( x ) || < ǫ/ whenever λ i = 1 , and that there is an η > , indepen-dent of x , such that the spectrum of a i ( x ) is contained in [ η, whenever λ i = 1 .Let f : [0 , → [0 , be the continuous map given by f ( t ) = (cid:26) t/η, t ≤ η , t > η . Set h ( x ) = f (˜ a ( x )) = f ( λ a ( x )) ⊕ f ( λ a ( x )) ⊕ · · · ⊕ f ( λ k a k ( x )) , and note that h : X → M n ( C ) is indeed a positive element of M n (C( X )) since ˜ a is.Let us first verify that || ha − a || < ǫ/ ; the proof that || ah − a || < ǫ/ is similar. For every x ∈ X wehave h ( x ) a ( x ) − a ( x ) = k M i =1 ( f ( λ i a i ( x )) a i ( x ) − a i ( x )) . If λ i = 1 , then f ( λ i a i ( x )) = p i ( x ) , where p i ( x ) is the support projection of a i ( x ) in M n ( C ) . Thus, f ( λ i a i ( x )) a i ( x ) − a i ( x ) = p i ( x ) a i ( x ) − a i ( x ) = a i ( x ) − a i ( x ) = 0 . Otherwise, || a i ( x ) || < ǫ/ and || f ( λ i a i ( x )) || ≤ , whence || f ( λ i a i ( x )) a i ( x ) − a i ( x ) || < ǫ/ ǫ/ ǫ/ . We have shown that || f ( λ i a i ( x )) a i ( x ) − a i ( x ) || < ǫ/ for each i ∈ { , . . . , k } , so that || ha − a || < ǫ/ ,proving (ii). For (i), we have k hah − a k = k hah − ha + ha − a k≤ k h k · k ah − a k + k ha − a k < ǫ/ ǫ/ ǫ. To complete the proof, we must show that hah is well-supported. The property of being well-supporteddepends only on the support projection of hah ( x ) as x ranges over X . It will thus suffice for us toshow that the support projection of hah ( x ) is the same as that of ˜ a ( x ) , since ˜ a is well-supported. If λ i is zero, then so is f ( λ i a i ( x )) a i ( x ) f ( λ i a i ( x )) , whence both it and λ i a i ( x ) have the same support pro-jection, namely, zero. If λ i = 0 , then f ( λ i a i ( x )) a i ( x ) f ( λ i a i ( x )) is the image of λ i a i ( x ) under the map t f ( t )( t/λ i ) f ( t ) . This map is nonzero on (0 , , and it follows that λ i a i ( x ) and f ( λ i a i ( x )) a i ( x ) f ( λ i a i ( x )) again have the same support projection. Since these statements hold for each i ∈ { , . . . , k } , we concludethat the support projections of ˜ a ( x ) and hah ( x ) agree for each x ∈ X . (cid:3) Proposition 2.6 (Phillips, Proposition 4.2 (1) of [22]) . Let X be a compact Hausdorff space of finite coveringdimension d , and let E ⊂ X be closed. Let p, q ∈ M n (C( X )) be projections with the property that rank( q ( x )) + 12 ( d − ≤ rank( p ( x )) , ∀ x ∈ X. Let s ∈ M n (C( E )) be such that s ∗ s = q | E and s s ∗ ≤ p | E . It follows that there is s ∈ M n (C( X )) such that s ∗ s = q, ss ∗ ≤ p, and s = s | E . We record a corollary of Proposition 2.6 for use in the sequel.
OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 7 Corollary 2.7.
Let X be a compact Hausdorff space of covering dimension d ∈ N , and let E , . . . , E k be a cover of X by closed sets. Let p ∈ M n (C( X )) and q i ∈ M n (C( E i )) be projections of constant rank for each i ∈ { , . . . , k } .Set n i = rank( q i ) , and assume that n < n < · · · < n k . Assume that q i ( x ) ≤ q j ( x ) whenever x ∈ E i ∩ E j and i ≤ j . Finally, suppose that n i − rank( p ) ≥ (1 / d − for every i .The following statements hold: (i) there is a partial isometry w ∈ M n (C( X )) such that w ∗ w = p and ( ww ∗ )( x ) ≤ ^ { i | x ∈ E i } q i ( x ) , ∀ x ∈ X ; (ii) if Y ⊆ X is closed, p | Y corresponds to a trivial vector bundle, and p ( y ) ≤ ^ { i | y ∈ E i } q i ( y ) , ∀ y ∈ Y, then p | Y can be extended to a projection ˜ p on X which also corresponds to a trivial vector bundle andsatisfies ˜ p ( x ) ≤ ^ { i | x ∈ E i } q i ( x ) , ∀ x ∈ X. Proof. (i)
The rank inequality hypothesis and the stability properties of vector bundles imply that thereis a partial isometry w ∈ M n (C( E )) such that w ∗ w = p and w w ∗ ≤ q . Since q ( x ) ≤ q j ( x ) whenever x ∈ E ∩ E j , we have ( w w ∗ )( x ) ≤ ^ { j | x ∈ E j } q j ( x ) , ∀ x ∈ E . Suppose now that we have found a partial isometry w i ∈ M n (C( E ∪ · · · ∪ E i )) such that w ∗ i w i = p and(2) ( w i w ∗ i )( x ) ≤ ^ { j | x ∈ E j } q j ( x ) , ∀ x ∈ E ∪ · · · ∪ E i . We may now apply Proposition 2.6 with X = E i +1 , E = E i +1 ∩ ( E ∪· · ·∪ E i ) , and s = w i | E i +1 ∩ ( E ∪···∪ E i ) to extend w i to a partial isometry w i +1 ∈ M n (C( E ∪ · · · ∪ E i +1 )) which satisfies (2) with i + 1 in place of i . Continuing inductively yields the desired result. (ii) We will explain how to extend p | Y to ˜ p defined on Y ∪ E . The desired result then follows fromiteration of this procedure.The projection p | Y ∩ E corresponds to a trivial vector bundle, and is subordinate to q . Let ˜ q be aprojection over Y which corresponds to a trivial vector bundle and has the same rank as p . Since both p | Y ∩ E and ˜ q | Y ∩ E correspond to trivial vector bundles, there is a partial isometry w ∈ M n (C( Y ∩ E )) such that ww ∗ = p | Y ∩ E ≤ q | Y ∩ E and w ∗ w = ˜ q | Y ∩ E . We may assume that this partial isometry, viewedas an isomorphism between trivial vector bundles, respects the decomposition of both bundles into aprescribed direct (Whitney) sum of trivial line bundles; we moreover assume that these decompositionsare the restrictions of similar decompositions for ˜ q and p | Y .Apply Proposition 2.6 with X = E , E = Y ∩ E , s = w , q = ˜ q and p = q . The resulting partialisometry s has the following properties: ss ∗ is a projection corresponding to a trivial vector bundle over E , ss ∗ agrees with p on Y ∩ E , and upon viewing s as an isomorphism of vector bundles, the image ofthe given decomposition of ˜ q into trivial line bundles extends the similar decomposition of p . It followsthat the projection ˜ p ( x ) = p ( x ) ∨ ss ∗ ( x ) , x ∈ Y ∪ E , corresponds to a trivial vector bundle, and extends p | Y . ANDREW S. TOMS (cid:3)
The proof of the next lemma is contained in the proof of [30, Proposition 3.7].
Lemma 2.8.
Let X be a compact Hausdorff space, and let a, b ∈ M n (C( X )) be positive. Suppose that there is anon-negative integer k such that rank( a ( x )) + k ≤ rank( b ( x )) , ∀ x ∈ X. It follows that for every ǫ > there is δ > with the property that rank(( a − ǫ ) + ( x )) + k ≤ rank(( b − δ ) + ( x )) , ∀ x ∈ X.
3. A
RELATIVE COMPARISON RESULT IN M n (C( X )) The goal of this section is to prove the following Lemma.
Lemma 3.1.
There is a natural number N such that the following statement holds: Let X be a compact metrisableHausdorff space of finite covering dimension d , and let Y ⊆ X be closed. Let a, b ∈ M n (C( X ) be positive and, fora given tolerance > ǫ > , satisfy (i) || a ( x ) − b ( x ) || < ǫ for each x ∈ Y , and (ii) rank( a ( x )) + ( d − / ≤ rank( b ( x )) for each x / ∈ Y .It follows that there are positive elements c, d and a unitary element u in M n (C( X )) whose restrictions to Y are all equal to ∈ M n (C( Y )) , and which, upon viewing a and b as elements of the upper left n × n corner of M n (C( X )) , satisfy the inequality || ( duc ) b ( duc ) ∗ − a || < N √ ǫ. The proof of Lemma 3.1 proceeds in several steps.
Lemma 3.2.
Let X be a compact metrisable Hausdorff space, and let Y be a closed subset of X . Suppose that wehave positive elements a, b ∈ M n (C( X )) , a tolerance ǫ > , and a natural number k satisfying (i) || a | Y − b | Y || < ǫ , and (ii) rank( a ( x )) + k ≤ rank( b ( x ) for each x / ∈ Y .It follows that there are a positive element ˜ a ∈ M n (C( X )) and open neighbourhoods U ⊆ U of Y with thefollowing properties: (a) || a − ˜ a || < ǫ ; (b) U ⊆ U ; (c) ˜ a ( x ) = ( b ( x ) − ǫ ) + for every x ∈ U \ U ; (d) rank(˜ a ( x )) + k ≤ rank( b ( x )) for each x ∈ X \ U .Proof. By the continuity of a and b we can find open neighbourhoods U ⊆ U ⊆ U of Y such that U ⊆ U , U ⊆ U , and || a | U − b | U || < (3 / ǫ . Let f : X → [0 , be a continuous map which is equal tozero on Y ∪ ( X \ U ) and equal to one on U \ U . As a first approximation to our desired element ˜ a , wedefine a ( x ) = (1 − f ( x )) a ( x ) + f ( x ) b ( x ) . We then have || a | U − b | U || < ǫ and || a − a || < ǫ . Now find a continuous function g : X → [0 , which is zero on Y , and equal to one on X \ U . Set ˜ a ( x ) = ( a ( x ) − ǫg ( x )) + . Thus, conclusions (a) and(b) are satisfied. OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 9 For each x ∈ U \ U we have f ( x ) = g ( x ) = 1 , so that a ( x ) = b ( x ) and ˜ a ( x ) = ( b ( x ) − ǫ ) + . Thisestablishes part (c) of the conclusion.For part (d) of the conclusion we treat two cases. For x ∈ U \ U we have the estimate || a ( x ) − a ( x ) || < ǫ and the fact that ˜ a ( x ) = ( a ( x ) − ǫ ) + . Proposition 2.1 (iii) then implies that ˜ a ( x ) - a ( x ) , whence rank(˜ a ( x )) ≤ rank( a ( x )) ≤ rank( b ( x )) − k. For x ∈ X \ U we have a ( x ) = a ( x ) and ˜ a ( x ) = ( a ( x ) − ǫ ) + . Thus, ˜ a ( x ) - a ( x ) and we proceed asbefore. (cid:3) We can now make our first reduction.
Lemma 3.3.
In order to prove Lemma 3.1, it will suffice to prove the following statement, hereafter referred to as (S) : Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N , and let Y ⊆ X be closed. Let > ǫ > be given. Suppose that ˆ a, ˆ b ∈ M n (C( X )) + have the following properties: (i) || (ˆ a − ˆ b ) | U || < ǫ for some open set U ⊇ Y ; (ii) ˆ b | X \ U is well-supported; (iii) there are an open set V ⊇ U and γ > such that ˆ a ( x ) = (ˆ b ( x ) − γ ) + , ∀ x ∈ V \ U ; (iv) rank(ˆ a ( x )) + ( d − / ≤ rank(ˆ b ( x )) , ∀ x ∈ X \ U. It follows that there are positive elements ˆ c, ˆ d and a unitary element v in M n (C( X )) whose restrictions to U areall equal to ∈ M n (C( U )) , and which, upon viewing ˆ a and ˆ b as elements of the upper left n × n corner of M n (C( X )) , satisfy the inequality || ( ˆ dv ˆ c )ˆ b ( ˆ dv ˆ c ) ∗ − ˆ a || < √ ǫ. Proof.
Let a and b be as in the hypotheses of Lemma 3.1. One can immediately find an open set U ⊇ Y such that || a ( x ) − b ( x ) || ≤ ǫ < ǫ for every x ∈ U . By Lemma 2.8, there is a δ > such that rank( a ( x ) − ǫ ) + ) + ( d − / ≤ rank( b ( x ) − δ ) + , ∀ x ∈ X. Set η = min { ǫ − ǫ , δ } .Fix an open set W ⊇ Y such that W ⊆ U . Apply Lemma 2.5 to b | X \ W with the tolerance η to producea positive element ˆ h ∈ M n (C( X \ W )) with the properties listed in the conclusion of that lemma. Fix acontinuous map f : X → [0 , which is equal to one on W and equal to zero on X \ U . Set h ( x ) = f ( x ) M n + (1 − f ( x ))ˆ h ( x ) , ∀ x ∈ X, and ˆ b ( x ) = h ( x ) b ( x ) h ( x ) . For each x ∈ X \ U , we have f ( x ) = 0 . It follows that ˆ b | X \ U = (ˆ h | X \ U )( b | X \ U )(ˆ h | X \ U ) ,whence, by part (i) of the conclusion of Lemma 2.5, ˆ b | X \ U is well-supported.We have || hbh − b || = sup x ∈ X || h ( x ) b ( x ) h ( x ) − b ( x ) || = sup x ∈ X || [ f ( x ) + (1 − f ( x ))ˆ h ( x ))] b ( x )[( f ( x ) + (1 − f ( x ))ˆ h ( x ))] − b ( x ) || < η, where the last inequality follows from part (ii) of the conclusion of Lemma 2.5. Since η ≤ ǫ − ǫ , we have || a | U − ˆ b | U || < ǫ . The inequality η ≤ δ implies that || ˆ b − b || < δ . Combining this fact with part (iii) ofProposition 2.1 yields rank(( a ( x ) − ǫ ) + ) + ( d − / ≤ rank( b ( x ) − δ ) + ) ≤ rank(ˆ b ( x )) , ∀ x ∈ X. We will now apply Lemma 3.2 with ˆ b , ( a − ǫ ) + , and ǫ substituted for b , a , and ǫ , respectively. Notethat by shrinking U and W above, we may assume that they will serve as the sets U and U of Lemma3.2, respectively. Form the approximant ˜ a to ( a − ǫ ) + provided in the conclusion of Lemma 3.2, and set ˆ a = ˜ a . Note that k ˆ a − ( a − ǫ ) + k < ǫ . We have || (ˆ a − ˆ b ) | U || ≤ || (ˆ a − ( a − ǫ ) + ) | U || + || (( a − ǫ ) + − ˆ b ) | U || < ǫ ) + 2 ǫ = 10 ǫ and ˜ a ( x ) = (ˆ b ( x ) − ǫ ) + , ∀ x ∈ X. Our ˆ a and ˆ b now satisfy the hypotheses of statement (S) with ǫ and ǫ substituted for ǫ and γ ,respectively. Let ˆ c , ˆ d , and v be as in the conclusion of statement (S) . Set u = v , d = ˆ d , and c = ˆ ch . Itfollows that || ( duc ) b ( duc ) ∗ − a || = || ( ˆ dv ˆ c )( hbh )( ˆ dv ˆ c ) ∗ − a ||≤ || (( ˆ dv ˆ c )ˆ b ( ˆ dv ˆ c ) ∗ − ˆ a || + || ˆ a − a || < √ ǫ k + k ˆ a − ( a − ǫ ) + k + k a − ( a − ǫ ) + k < √ ǫ + 9 ǫ < √ ǫ. This shows that if (S) holds, then Lemma 3.1 holds (with N = 49 ). (cid:3) The next Lemma constructs the unitary u of Lemma 3.1. Lemma 3.4.
Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N , and let a, b ∈ M n (C( X )) be well-supported positive elements with the property that rank( a ( x )) + 12 ( d − ≤ rank(( b − ǫ ) + ( x )) for some ǫ > and every x ∈ X . Suppose further that a ( y ) ≤ ( b ( y ) − ǫ ) + for each y in the closure of an opensubset Y of X , and that a and b have norm at most one.For each k ∈ { , , . . . , n } , set E k = { x ∈ Z | rank( a ( x )) = k } ; F k = { x ∈ Z | rank( b ( x )) = k } . For each x ∈ E k , let p k ( x ) be the support projection of a ( x ) ; for each x ∈ F k let q k ( x ) be the support projection of b ( x ) . Since a and b are well-supported, the continuous projection-valued maps x p k ( x ) and x q k ( x ) can beextended to E k and F k , respectively. We also denote these extended maps by p k and q k .View M n (C( X )) as the upper-left n × n corner of M n (C( X )) , and let Z ⊆ Y be closed. It follows that there isa unitary u ∈ M n (C( X )) with the following properties: (i) u ( z ) = n ∈ M n ( C ) for each z ∈ Z ; (ii) ( u ∗ p k u )( x ) ≤ ^ { j | x ∈ F j } q j ( x ) , ∀ x ∈ E k , ∀ k ∈ { , . . . , n } ; (iii) u is homotopic to the unit of M n (C( X )) . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 11 Proof.
Step 1.
For each y ∈ Y , set v ( y ) = n . Let us verify conclusion (ii) above with v in place of u foreach x ∈ Y ∩ E k . For each y ∈ Y we have a ( y ) ≤ b ( y ) , and so a ( y ) / n ≤ b ( y ) / n , ∀ n ∈ N . It follows that p k ( y ) ≤ q j ( y ) for each y ∈ Y ∩ E k ∩ F j , and so ( v ∗ p k v )( y ) = p k ( y ) ≤ ^ { j | y ∈ F j } q j ( y ) for each y ∈ Y ∩ E k and k ∈ { , . . . , n } . It remains to prove that the inequality above holds when y ∈ Y ∩ E k .Set r ( x ) = χ [ ǫ/ , ( b ( x )) for each x ∈ X , so that r ( x ) dominates the support projection of ( b − ǫ ) + at x .It follows that p k ( x ) ≤ r ( x ) for each x ∈ E k . In fact,(3) p k ( x ) ≤ r ( x ) ≤ ^ { j | x ∈ F j } q j ( x ) for each k ∈ { , . . . , n } and x ∈ E k , where the second inequality follows from the fact that ^ { j | x ∈ F j } q j ( x ) is the support projection of b ( x ) for each x ∈ E k . It will suffice to prove that the first inequality holds for y ∈ Y ∩ E k . It is well known that r ( x ) is an upper semicontinuous projection-valued map from X into M n (C( X )) . Fix y ∈ Y ∩ E k , and let ( y n ) be a sequence in Y ∩ E k converging to y . Since p k ( y n ) ≤ r ( y n ) foreach n ∈ N we have ( p k ( y n ) ξ | ξ ) ≤ ( r ( y n ) ξ | ξ ) , ∀ ξ ∈ C n , n ∈ N . Now ( p k ( y ) ξ | ξ ) = lim n →∞ ( p k ( y n ) ξ | ξ ) ≤ lim sup n →∞ ( r ( y n ) ξ | ξ ) ≤ ( r ( y ) ξ | ξ ) , ∀ ξ ∈ C n , where the last inequaltiy follows from the upper semicontinuity of r . It follows that p k ( y ) ≤ r ( y ) , asrequired. Step 2.
We will construct partial isometries v k ∈ M n (C( E k \ Y )) with the following properties:(a) ( v ∗ k p k v k )( x ) ≤ ^ { j | x ∈ F j \ Y } q j ( x ) , ∀ x ∈ E k \ Y, ∀ k ∈ { , . . . , n } ; (b) the v k s are compatible in the sense that for each x ∈ E i ∩ E j \ Y with i ≤ j , ( v ∗ i p i v i )( x ) = ( v ∗ j p i v j )( x ); (c) for each x ∈ E k ∩ ∂Y , v k ( x ) = p k ( x ) = v ( x ) p k ( x ) .In the third step of the proof, we will extend the v from Step 1 and the v k s above to produce the unitary u required by the lemma.We will prove the existence of the required v k s by induction on the number of rank values taken by a . Let us first address the case where a has constant rank equal to k . In this case E k = E k = X , and a is Cuntz equivalent to the projection p k ∈ M n (C( X )) . We set v k ( y ) = p k ( y ) for each y ∈ ∂Y , thussatisfying requirements (a) and (c) for these y . (Note that condition (b) is met trivially in the presentcase.) Let j < j < · · · < j l be the indices for which F j i = ∅ . The existence of the required partial isometry extending the definition of v k on ∂Y now follows from repeated application of Proposition 2.6:one substitutes p k and q j i for q and p , respectively, in the hypotheses of the said Proposition.Now let us suppose that we have found partial isometries v , . . . , v k satisfying (a), (b), and (c) above.We must construct v k +1 , assuming k < n . We will first construct v k +1 on the boundary E k +1 ∩ ( E ∪ E ∪ · · · ∪ E k ) ∩ Y c . For x ∈ E k +1 ∩ E k ∩ Y c , we have ( v ∗ k p k v k )( x ) ≤ ^ { j | x ∈ F j \ Y } q j ( x ) . From (3) on E k +1 we also have that the rank of the right-hand side exceeds that of the left hand side byat least rank( p k +1 ( x ) − p k ( x )) + 12 ( d − . Working over E k +1 ∩ E k ∩ Y c , we have that ( p k +1 − p k ) is Murray-von Neumann equilvalent to a projection f k which is orthogonal to v ∗ k p k v k and satisfies f k ( x ) ≤ ^ { j | x ∈ F j \ Y } q j ( x ) . (This follows from part (i) of Corollary 2.7.) Let w k be a partial isometry defined over E k +1 ∩ E k ∩ Y c such that w ∗ k ( p k +1 − p k ) w k = f k , and set v k +1 = v k + w k . With this definition we have ( v ∗ k +1 p k +1 v k +1 )( x ) ≤ ^ { j | x ∈ F j \ Y } q j ( x ) , and ( v ∗ k p k v k )( x ) = ( v ∗ k +1 p k v k +1 )( x ) for each x ∈ E k +1 ∩ E k ∩ Y c .Let us now show how to extend v k +1 one step further, to E k +1 ∩ ( E k ∪ E k − ) ∩ Y c ; its successiveextensions to the various E k +1 ∩ ( E k ∪ · · · ∪ E k − j ) ∩ Y c , j ∈ { , . . . , k − } , are similar, and the details are omitted.In this paragraph we work over the set E k +1 ∩ ( E k ∪ E k − ) ∩ Y c . We will suppose that this set contains E k +1 ∩ E k ∩ Y c strictly, for there is otherwise no extension of v k +1 to be made. Over ( E k +1 ∩ E k ∩ Y c ) ∩ E k − ,we set w k − = v k +1 ( p k +1 − p k ) . Thus, w k − is a partial isometry carrying (the restriction of) p k +1 − p k − to a subprojection of Q ( x ) def = ^ { j | x ∈ F j \ Y } q j ( x ) − ( v k − p k − v ∗ k − )( x ) , x ∈ ( E k +1 ∩ E k ∩ Y c ) ∩ E k − . We moreover have the rank inequality [rank( Q ( x )) − rank(( v k − p k − v ∗ k − )( x ))] − rank(( w k − w ∗ k − )( x )) ≥
12 ( d − . Applying part (i) of Corollary 2.7, we extend w k − to a partial isometry defined on all of E k +1 ∩ E k − ∩ Y c which has the property that ( w k − w ∗ k − ) ≤ Q ( x ) . Finally, set v k +1 = v k − + w k − on this set. It isstraightforward to check that v k +1 has the required properties. OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 13 Iterating the arguments above, we have an appropriate definition of v k +1 on E k +1 ∩ ( E ∪ E ∪ · · · ∪ E k ) ∩ Y c . To extend the definition of v k +1 from the set above to all of E k +1 ∩ Y c , we simply apply part (i) of Corollary2.7. Step 3.
Set H − = Y and H k = E k \ Y , so that H − , . . . , H n is a closed cover of X . For each k ∈{− , , . . . , n } we have a partial isometry v k ∈ M n (C( H k )) from Steps 1 and 2 (assuming that v − = v = ). Let r k denote the source projection of v k . Notice that r k agrees with p k off Y . In this final step ofour proof, we will construct the required unitary u in a manner which extends the v k : ( u | H k ) r k = v k .Suppose that we have found a partial isometry w k ∈ M n (C( H − ∪ · · · ∪ H k )) with source projectionequal to n (i.e., the unit of the upper-left n × n corner) and satisfying ( w k | H j ) r j = v j for each j ∈{ , . . . , k } . Let us show that k can be replaced with k + 1 , and that w k +1 may moreover be chosen to bean extension of w k .Over ( H − ∪· · ·∪ H k ) ∩ H k +1 , w k carries the projection n − r k +1 into a subprojection of n − v k +1 v ∗ k +1 .The rank of the latter projection exceeds that of the former by at least ( d − / , and so the partial isometry w k ( n − r k +1 ) defined over ( H − ∪· · ·∪ H k ) ∩ H k +1 can be extended to a partial isometry w ′ k +1 defined over H k +1 which carries n − r k +1 into a subprojection of n − v k +1 v ∗ k +1 (cf. Proposition 2.6). Setting w k +1 = v k +1 + w ′ k +1 on H k +1 and w k +1 = w k otherwise gives the desired extension. Iterating this extensionprocess yields a partial isometry w ∈ M n (C( X )) with source projection n satisfying ( w | H k ) r k = v k .To complete the proof, it will suffice to find a unitary u ∈ M n (C( X )) which is homotopic to theidentity (for conclusion (iii)), satisfies u n = w (for conclusion (ii)), and is equal to ∈ M n ( C ) over Z (for conclusion (i)). We will find a unitary s satisfying (ii) and (iii), and then modify it to obtain u .The complement of n in M n (C( X )) is Murray-von Neumann equivalent to the complement of ww ∗ ,as both projections have the same K -class and are of rank at least ( d − / . Let w ′ be a partial isometryimplementing this equivalence. It follows that w + w ′ ∈ M n (C( X )) is unitary. Setting s = ( w + w ′ ) ⊕ ( w + w ′ ) ∗ yields our precursor to the required unitary u ∈ M n (C( X )) —the K -class of s is zero, so it ishomotopic to n by virtue of its rank ([24, Theorem 10.12]). The unitary s | Y ∈ M n (C( Y )) has the form n ⊕ ˜ s , where ˜ s is a n × n unitary homotopic to the identity. (This follows from two facts: the K -classof n ⊕ ˜ s is zero, and the natural map ι : U (M n (C( Y )) → U (M n (C( Y )) given by x n ⊕ x is injectiveby [24, Theorem 10.12].) Let H : Y × [0 , → U (M n ( C )) be a homotopy such that H ( y,
0) = ˜ s ( y ) and H ( y,
1) = n ∈ M n ( C ) . Let h : Y → [0 , be a continuousmap equal to one on Z and equal to zero on ∂Y . Finally, define u ( x ) = (cid:26) s ( x ) , x / ∈ Y n ⊕ H ( x, f ( x )) , x ∈ Y .
The unitary u is clearly homotopic to s , and so satisfies conclusion (iii). Conclusion (i) holds for u byconstruction, and conclusion (ii) holds since u n = s n = w . (cid:3) Lemma 3.5.
The statement (S) (cf. Lemma 3.3) holds.Proof.
Step 1.
To avoid cumbersome notation, we use a , b , c , and d in place of their “hatted” versions inthe hypotheses and conclusion of (S) . We will first find the unitary v and the positive elements c and d required by the conclusion of (S) with two failings: c and d are not necessarily equal to ∈ M n ( C ) at each point of U , and the estimate || ( cvd ) b ( cvd ) ∗ − a || < √ ǫ only holds on X \ U . Both of these failings will be attributable to c and d alone, and will be repaired inlater Steps 2. and 3.By combining the hypotheses (i) and (iii) of (S) , we may, after perhaps shrinking the set V , assumethat γ < ǫ . With this choice of V we also have that hypothesis (i) holds with V in place of U . We will alsoweaken hypothesis (iii) to an inequality. This has two advantages. First, by replacing a with ( a − δ ) + forsome small δ > , we can assume that (iv) holds with b replaced by ( b − η ) + for some γ > η > . Second,we can assume that a | X \ U is well-supported by using the following procedure: let W ⊇ Y be an openset whose closure is contained in U ; replace a with a suitably close approximant ˜ a on X \ W , as providedby Lemma 2.5; choose a continuous map f : X → [0 , which is equal to one on W and equal to zero on X \ U ; replace the original a with the positive element equal to f ( x ) a ( x ) + (1 − f ( x ))˜ a ( x ) at each x ∈ X .Let us summarise our assumptions:(i) || ( a − b ) | V || < ǫ for some open set V ⊇ Y ;(ii) b | X \ U and a | X \ U are well-supported (and U ⊆ V );(iii) there is < γ < ǫ such that a ( x ) ≤ ( b ( x ) − η ) + , ∀ x ∈ V \ U ; (iv) rank( a ( x )) + ( d − / ≤ rank(( b − η ) + ( x )) , ∀ x ∈ X \ U. Set Z = X \ U and W = V \ U .For each k ∈ { , , . . . , n } , set E k = { x ∈ Z | rank( a ( x )) = k } ; F k = { x ∈ Z | rank( b ( x )) = k } . For each x ∈ E k , let p k ( x ) be the support projection of a ( x ) . Similarly, define q k ( x ) to be the support pro-jection of b ( x ) for each x ∈ F k . Since (the restrictions of) a and b are well-supported on Z , the continuousprojection-valued maps x p k ( x ) and x q k ( x ) can be extended to E k and F k , respectively. We alsodenote these extended maps by p k and q k . Let f V be an open subset of X such that U ⊆ f V ⊆ f V ⊆ V ,and set V = f V ∩ Z . Apply Lemma 3.4 with b | Z , a | Z , Z , W , V , and η substituted for the variables b , a , X , Y , Z , and ǫ in the hypotheses of the lemma, respectively. Let u be the unitary in M n (C( Z )) provided bythe conclusion of the said lemma. Define v ∈ M n (C( X )) to be the unitary which is equal to u on Z andequal to ∈ M n ( C ) at each point of U . This v will serve as the unitary required in the conclusion of (S) .We will simply use v in place of v | Z whenever it is clear that we are working over Z .From conclusion (ii) of Lemma 3.4 we have(4) p k ( x ) ≤ v ( x ) ^ { j | x ∈ F j } q j ( x ) v ( x ) ∗ , ∀ x ∈ E k , ∀ k ∈ { , . . . , n } . For each δ > let f δ , g δ : [0 , → [0 , be given by the formulas f δ ( t ) = , t ∈ [0 , δ/ t − δ ) /δ, t ∈ ( δ/ , δ )1 , t ∈ [ δ, , OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 15 and g δ ( t ) = (cid:26) , t ∈ [0 , δ/ f δ ( t ) /t, t ∈ ( δ/ , . Note that f δ ( t ) and g δ ( t ) are continuous, and that tg δ ( t ) = f δ ( t ) .Consider the following product in M n (C( Z )) :(5) ( √ av p g δ ( b )) b ( √ av p g δ ( b )) ∗ = ( √ av p g δ ( b )) b ( p g δ ( b ) v ∗ √ a ) . As δ → we have [ p g δ ( b ) b p g δ ( b )]( x ) = f δ ( b )( x ) → ^ { j | x ∈ F j } q j ( x ) , ∀ x ∈ Z. Thus, by (4), [ v p g δ ( b ) b p g δ ( b ) v ∗ ]( x ) converges to a projection which dominates the support projection of a ( x ) . It follows that the product (5), evaluated at x ∈ Z , converges to a ( x ) as δ → . We will prove thatthis convergence is uniform in norm on Z .If δ < κ , then f δ ( b ) ≥ f κ ( b ) . It follows that(6) √ avf δ ( b ) v ∗ √ a ≥ √ avf κ ( b ) v ∗ √ a. Since b ≤ , we have √ avf δ ( b ) v ∗ √ a ≤ √ avv ∗ √ a = a, and similarly for f κ ( b ) . Combining this with (6) yields ≤ a − √ avf δ ( b ) v ∗ √ a ≤ a − √ avf κ ( b ) v ∗ √ a. By positivity,(7) (cid:12)(cid:12)(cid:12)(cid:12) a − √ avf δ ( b ) v ∗ √ a (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) a − √ avf κ ( b ) v ∗ √ a (cid:12)(cid:12)(cid:12)(cid:12) . Let ( δ n ) be a sequence of strictly positive tolerances converging to zero. By (7), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a − √ av p g δ n ( b ) b p g δ n ( b ) v ∗ √ a ]( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) [ a − √ avf δ n ( b ) v ∗ √ a ]( x ) (cid:12)(cid:12)(cid:12)(cid:12) is a monotone decreasing sequence converging to zero for each x ∈ Z . By Dini’s Theorem, this sequenceconverges uniformly to zero on Z . For the remainder of the proof we fix ǫ > δ > with the property that(8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − √ av p g δ ( b ) b p g δ ( b ) v ∗ √ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ. Extend √ a and p g δ ( b ) to positive elements c and d in M n (C( X )) , respectively. This choice of c and d completes Step 1. Step 2.
We must now modify our choice of c and d to address their failings, outlined at the beginning ofStep 1. This modification will be made in three smaller steps. In a slight abuse of notation, we will use c and d to denote the successive modifications of the present c and d .For each x ∈ W we have b ( x ) − a ( x ) ≥ and || b ( x ) − a ( x ) || < ǫ . It is a straightforward exercise to showthat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p b ( x ) − p a ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ ǫ. Choose a continuous map f : Z → [0 , which is equal to one on Z \ W and equal to zero on V . Set a ( x ) = f ( x ) p a ( x ) + (1 − f ( x )) p b ( x ) for each x ∈ Z , and set s = v p g δ ( b ) b p g δ ( b ) v ∗ for brevity. Notethat || s || ≤ . Now for each x ∈ W we have || [ a sa ]( x ) || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ √ a + (1 − f )( √ b − √ a )]( x ) s ( x )[ √ a + (1 − f )( √ b − √ a )]( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) [ √ as √ a ]( x ) + r ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , where || r ( x ) || < √ ǫ + ǫ . We revise our definition of c by setting it equal to a on X \ U and extending itin an arbitrary fashion to a positive element of M n (C( X )) . Combining this new definition of c with (8)above we have the estimate(9) || [( cvd ) b ( cvd ) ∗ ]( x ) − a ( x ) || < √ ǫ + ǫ ) , ∀ x ∈ X \ U. Choose an open subset V of Z such that U ⊆ V ⊆ V ⊆ V , and a continuous map f : Z → [0 , equal to zero on Z \ V and equal to one on V . For each x ∈ V we have c ( x ) = p b ( x ) , d ( x ) = p g δ ( b ) ,and v ( x ) = , whence || [( cvd ) b ( cvd ) ∗ ]( x ) − a ( x ) || = (cid:12)(cid:12)(cid:12)(cid:12) b ( x ) g δ ( b )( x ) − a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (10) = || b ( x ) f δ ( b )( x ) − a ( x ) || (11) ≤ || b ( x ) f δ ( b )( x ) − b ( x ) || + || b ( x ) − a ( x ) || < ǫ. (12)For each s ∈ [0 , define h s ( t ) = (cid:26) ts/δ, t ∈ [0 , δ/ t − δ )[(1 − s ) / (2 − δ )] + s, t ∈ ( δ/ , . It straightforward to verify that h s ( t ) is a homotopy of maps such that h ( t ) = t ; h ( t ) = (cid:26) t/δ, t ∈ [0 , δ/ , t ∈ ( δ/ , . Set g δ,s ( t ) = (cid:26) , t ∈ [0 , δ/ f δ ( t ) /h s ( t ) , t ∈ ( δ/ , . With these definitions we have h s ( t ) g δ,s ( t ) = f δ ( t ) , ∀ s, t ∈ [0 , . For each x ∈ V , we adjust our definitionsof c ( x ) and d ( x ) as follows: c ( x ) = q h f ( x ) ( b ( x )); d ( x ) = q g δ,f ( x ) ( b ( x )) . Since f ( x ) = 0 on ∂V , the definitions of c ( x ) and d ( x ) are not altered on ∂V . Thus, our modifiedversions of c and d are still positive elements of M n (C( Z )) , and the estimate (9) still holds on Z \ V . For x ∈ V we have || [( cvd ) b ( cvd ) ∗ ]( x ) − a ( x ) || = (cid:12)(cid:12)(cid:12)(cid:12) h f ( x ) ( b ( x )) g δ,f ( x ) ( b ( x )) b ( x ) − a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (13) = || f δ ( b ( x )) b ( x ) − a ( x ) || < ǫ, (14)where the last inequality follows from (10) above. Thus, (9) continues to hold with our new definitionsof c and d . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 17 Choose an open subset V of Z such that U ⊆ V ⊆ V ⊆ V , and a continuous map f : Z → [0 , equalto zero on Z \ V and equal to one on V . For each s ∈ [0 , define continuous maps r s , w s : [0 , → [0 , by r s ( t ) = max n s, p h ( t ) o ; w s ( t ) = max (cid:26) s, q g δ, ( t ) (cid:27) . Thus, r s and w s define homotopies of self-maps of [0 , such that r = h , w = g δ, , and r = w = 1 .For each x ∈ V we adjust our definitions of c ( x ) and d ( x ) as follows: c ( x ) = q r f ( x ) ( b ( x )); d ( x ) = q w f ( x ) ( b ( x )) . Since f = 0 on ∂V , the definitions of c ( x ) and d ( x ) are not altered on ∂V . Thus, our modified versionsof c and d are still positive elements of M n (C( Z )) , and the estimate (9) still holds on Z \ V . For x ∈ V we have || [( cvd ) b ( cvd ) ∗ ]( x ) − a ( x ) || = (cid:12)(cid:12)(cid:12)(cid:12) r f ( x ) ( b ( x )) w f ( x ) ( b ( x )) b ( x ) − a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ by a functional calculus argument similar to (13) above—one need only observe that f δ ( t ) ≤ r s ( t ) w s ( t ) ≤ , ∀ s, t ∈ [0 , . Thus, (9) continues to hold with our new definitions of c and d . Moreover, we have c ( x ) = d ( x ) = ∈ M n ( C ) for each x ∈ V . We may thus extend our definitions of c and d to all of X by setting c ( x ) = d ( x ) = ∈ M n ( C ) for every x ∈ U ∪ V . With this final definition of c and d , we see that || [( cvd ) b ( cvd ) ∗ ]( x ) − a ( x ) || = || b ( x ) − a ( x ) || < ǫ, ∀ x ∈ U ∪ V . We conclude that the estimate (9) holds on all of X , whence (S) holds. (cid:3) With (S) in hand, we have completed the proof of Lemma 3.1.4. A
COMPARISON THEOREM FOR RECURSIVE SUBHOMOGENEOUS C ∗ - ALGEBRAS
Background and notation.
Let us recall some of the terminology and results from [22].
Definition 4.1.
A recursive subhomogeneous algebra (RSH algebra) is given by the following recursive definition. (i) If X is a compact Hausdorff space and n ∈ N , then M n (C( X )) is a recursive subhomogeneous algebra. (ii) If A is a recursive subhomogeneous algebra, X is a compact Hausdorff space, X (0) ⊆ X is closed, φ : A → M k (C( X (0) )) is a unital ∗ -homomorphism, and ρ : M k (C( X )) → M k (C( X (0) )) is the restrictionhomomorphism, then the pullback A ⊕ M k (C( X (0) )) M k (C( X )) = { ( a, f ) ∈ A ⊕ M k (C( X )) | φ ( a ) = ρ ( f ) } is a recursive subhomogeneous algebra. It is clear from the definition above that a C ∗ -algebra R is an RSH algebra if and only if it can be writtenin the form(15) R = h · · · hh C ⊕ C (0)1 C i ⊕ C (0)2 C i · · · i ⊕ C (0) l C l , with C k = M n ( k ) (C( X k )) for compact Hausdorff spaces X k and integers n ( k ) , with C (0) k = M n ( k ) (C( X (0) k )) for compact subsets X (0) k ⊆ X (possibly empty), and where the maps C k → C (0) k are always the restric-tion maps. We refer to the expression in (15) as a decomposition for R . Decompositions for RSH algebrasare not unique.Associated with the decomposition (15) are: (i) its length l ;(ii) its k th stage algebra R k = h · · · hh C ⊕ C (0)1 C i ⊕ C (0)2 C i · · · i ⊕ C (0) k C k ; (iii) its base spaces X , X , . . . , X l and total space ⊔ lk =0 X k ;(iv) its matrix sizes n (0) , n (1) , . . . , n ( l ) and matrix size function m : X → N given by m ( x ) = n ( k ) when x ∈ X k (this is called the matrix size of R at x );(v) its minimum matrix size min k n ( k ) and maximum matrix size max k n ( k ) ;(vi) its topological dimension dim( X ) and topological dimension function d : X → N ∪ { } given by d ( x ) = dim( X k ) when x ∈ X k ;(vii) its standard representation σ R : R → ⊕ lk =0 M n ( k ) (C( X k )) defined to be the obvious inclusion;(viii) the evaluation maps ev x : R → M n ( k ) for x ∈ X k , defined to be the composition of evaluation at x on ⊕ lk =0 M n ( k ) (C( X k )) and σ R . Remark 4.2. If R is separable, then the X k can be taken to be metrisable ([22, Proposition 2.13]). If R hasno irreducible representations of dimension less than or equal to N , then we may assume that n ( k ) > N .It is clear from the construction of R k +1 as a pullback of R k and C k +1 that there is a canonical surjective ∗ -homomorphism λ k : R k +1 → R k . By composing several such, one has also a canonical surjective ∗ -homomorphism from R j to R k for any j > k . Abusing notation slightly, we denote these maps by λ k aswell. Remark 4.3.
The C ∗ -algebra M m ( R ) ∼ = R ⊗ M m ( C ) is an RSH algebra in a canonical way: C k and C (0) k arereplaced with C k ⊗ M m ( C ) and C (0) k ⊗ M m ( C ) , respectively, and the clutching maps φ k : R k → C (0) k +1 arereplaced with the amplifications φ k ⊗ id m : C k ⊗ M m ( C ) → C (0) k +1 ⊗ M m ( C ) . From here on we assume that M m ( R ) is equipped with this canonical decomposition whenever R is givenwith a decomposition. We will abuse notation by using φ k to denote both the original clutching map inthe given decomposition for R and its amplified versions.4.2. A comparison theorem.Lemma 4.4.
Let X be a compact metrisable Hausdorff space, and Y a closed subset of X . If a ∈ M n (C( Y )) ispositive, then a can be extended to ˜ a ∈ M n (C( X )) with the property that ˜ a ( x ) is invertible for every x ∈ X \ Y . If u = v ⊕ v ∗ for a unitary v ∈ M n (C( Y )) , then u can be extended to a unitary ˜ u ∈ M n (C( X )) .Proof. By the semiprojectivity of the C ∗ -algebras they generate, both a and u can be extended to theclosure of an open neighbourhood V of Y . We will also denote these extensions by a and u . Fix acontinuous map f : X → [0 , which is equal to zero on Y , equal to one on X \ V , and nonzero at every x ∈ X \ Y .Define ˜ a ( x ) = (cid:26) a ( x ) + f ( x )( || a || − a ( x )) , x ∈ V || a || , x ∈ X \ V Clearly, ˜ a belongs to M n (C( X )) and extends a . It follows that for each x ∈ X \ Y , either ˜ a ( x ) = || a || ∈ GL n ( C ) , or ˜ a ( x ) = a ( x ) + f ( x )( || a || − a ( x )) = f ( x ) || a || + (1 − f ( x )) a ( x ) ≥ f ( x ) || a || > . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 19 In the latter case we conclude that the rank of ˜ a ( x ) is n , whence ˜ a ( x ) ∈ GL n ( C ) as desired.Now let us turn to u . We have u | V \ Y = v | V \ Y ⊕ v ∗ | V \ Y ∼ h ∈ M n (cid:16) C (cid:16) V \ Y (cid:17)(cid:17) by the Whitehead Lemma, where ∼ h denotes homotopy within the unitary group. Let H ( x, t ) : V \ Y × [0 , → U (M n ( C )) be an implementing homotopy, with H ( x,
0) = u | V \ Y and H ( x,
1) = . Define ˜ u ( x ) = u ( x ) , x ∈ YH ( x, f ( x )) , x ∈ V \ Y , x ∈ X \ V It is straightforward to check that ˜ u is a unitary in M n (C( X )) , and ˜ u extends u by definition. (cid:3) Lemma 4.5.
Let A be a separable RSH algebra with a fixed decomposition as above. Let a, b ∈ A be positive, andsuppose that || λ k ( b − a ) || < ǫ inside the k th stage algebra A k , k < l . Suppose further that rank( a ( x )) + ( d ( x ) − / ≤ rank( b ( x )) , ∀ x ∈ X j \ X (0) j , j > k. It follows that there are m ∈ N and v ∈ M m ( A ) such that, upon considering A as the upper-left × corner of M m ( A ) we have || λ k +1 ( vbv ∗ − a ) || < N √ ǫ for the constant N of Lemma 3.1 and rank( a ( x )) + ( d ( x ) − / ≤ rank(( vbv ∗ )( x )) , ∀ x ∈ X j \ X (0) j , j > k + 1 . Proof.
Let φ k : A k → C (0) k +1 be the k th clutching map. Our hypotheses imply that φ k ( b ) , φ k ( a ) ∈ C (0) k +1 =M n ( k +1) (C( X (0) k +1 ) satisfy || φ k ( b ) − φ k ( a ) || < ǫ . Apply Lemma 3.1 with φ k ( a ) , φ k ( b ) , X k +1 , X (0) k +1 , and ǫ inplace of a, b, X, Y , and ǫ , respectively. The conclusion of Lemma 3.1 provides us with positive elements c, d and a unitary element u in M n ( k +1) (C( X k +1 )) such that(i) || ( cud ) φ k ( b )( cud ) ∗ − φ k ( a ) || < N √ ǫ , and(ii) c ( x ) = d ( x ) = u ( x ) = ∈ M n ( k +1) ( C ) for every x ∈ X (0) k +1 .Using (ii) we extend c, d , and u to M ( A k +1 ) (keeping the same notation) by setting λ k ( c ) = λ k ( d ) = λ k ( u ) = ∈ M ( A k ) . Set v k +1 = cud ∈ M ( A k +1 ) . We claim that || v k +1 λ k +1 ( b ) v ∗ k +1 − λ k +1 ( a ) || < N √ ǫ. It will suffice to prove that the image of v k +1 λ k +1 ( b ) v ∗ k +1 − λ k +1 ( a ) under the standard representation σ M ( A k +1 ) : M ( A k +1 ) → k +1 M j =0 M n ( j ) (C( X j )) is of norm at most N √ ǫ . This in turn need only be checked in each of the direct summands of thecodomain. In the summand ⊕ kj =0 M n ( j ) (C( X j )) the desired estimate follows from two facts: σ M ( A k +1 ) ( v k +1 ) is equal to the unit of the said summand (see (ii) above), and the images of a and b in this summand areat distance strictly less than ǫ < N √ ǫ . In the summand M n ( k +1) (C( X k +1 )) the desired estimate followsfrom (i) above.If m ≥ , then any v ∈ M m ( A ) which, upon viewing M ( A ) as the upper-left × corner of M m ( A ) ,has the property that λ k +1 ( v ) = v k +1 will at least satisfy || λ k +1 ( vbv ∗ − a ) || < N √ ǫ . It remains, then, tofind such a v , while ensuring that rank( a ( x )) + ( d ( x ) − / ≤ rank(( vbv ∗ )( x )) , ∀ x ∈ X j \ X (0) j , j > k + 1 . If k + 1 = l , then there is nothing to prove. Suppose that k + 1 < l . Let us first construct an element v k +2 of M ( A k +2 ) with the following properties: λ k +1 ( v k +2 ) = v k +1 , and rank( a ( x )) + ( d ( x ) − / ≤ rank(( v k +2 bv ∗ k +2 )( x )) , ∀ x ∈ X k +2 \ X (0) k +2 . Define c k +1 = c ⊕ , d k +1 = d ⊕ , and u k +1 = u ⊕ u ∗ . Use Lemma 4.4 to extend φ k +1 ( c k +1 ) , φ k +1 ( d k +1 ) , and φ k +1 ( u k +1 ) to positive elements ˜ c k +2 , ˜ d k +2 and a unitary element ˜ u k +2 , respectively,in M n ( k +2) (C( X k +2 )) , all of which are invertible at every x ∈ X k +2 \ X (0) k +2 . Consider M ( A k +2 ) as a sub-algebra of ⊕ k +2 j =0 M n ( j ) (C( X j )) via its standard representation, and define c k +2 to be equal to c k +1 in thefirst k + 1 summands, and equal to ˜ c k +2 in the last summand; define d k +2 and u k +2 similarly. Setting v k +2 = c k +2 u k +2 d k +2 we have that λ k +1 ( v k +2 ) = λ k +1 ( c k +2 u k +2 d k +2 )= c k +1 u k +1 d k +1 = ( c ⊕ u ⊕ u ∗ )( d ⊕ cud ⊕ v k +1 . Moreover, for each x ∈ X k +2 \ X (0) k +2 , we have v k +2 ( x ) = ˜ c k +2 ( x )˜ u k +2 ( x ) ˜ d k +2 ( x ) ∈ GL n ( k +2) ( C ) . It follows that rank(( vbv ∗ )( x )) = rank( b ( x )) ≥ ( d ( x ) − / a ( x )) , ∀ x ∈ X k +2 \ X (0) k +2 , as required.If k + 2 = l then we set v = v k +2 to complete the proof. Otherwise, we repeat the arguments inthe paragraph above using c k +2 , d k +2 , and u k +2 in place of c, d, and u , respectively, to obtain v k +3 ∈ M ( A k +3 ) such that λ k +2 ( v k +3 ) = v k +2 and rank( a ( x )) + ( d ( x ) − / ≤ rank(( v k +3 bv ∗ k +3 )( x )) , ∀ x ∈ X k +3 \ X (0) k +3 . Continuing this process until we arrive at v k +( l − k ) = v l and setting v = v l yields the Lemma in full. (cid:3) Theorem 4.6.
Let A be a separable RSH algebra with a fixed decomposition as above. Let a, b ∈ A be positive, andsuppose that rank( a ( x )) + ( d ( x ) − / ≤ rank( b ( x )) , ∀ x ∈ X k \ X (0) k , k ∈ { , , . . . , l } . It follows that a - b .Proof. We view A as the upper-left × corner of M m ( A ) , and adopt the standard notation for the de-compositions of A and M m ( A ) . Let ǫ > be given; we must find m ∈ N and v ∈ M m ( A ) such that || vbv ∗ − a || < ǫ .Let l be the length of the fixed decomposition for A . Given δ > , we define δ k = N p δ k − for each k ∈ { , . . . , l } , where N is the constant of Lemma 3.1. It follows that δ k = δ / k k − Y j =0 N / j . Assume that δ has been chosen so that δ l < ǫ . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 21 Apply Lemma 3.1 with λ ( a ) , λ ( b ) , X , and ∅ , in place of a, b, X, and Y . Since Y is empty, we canarrange to have any value of ǫ appear in the conclusion of Lemma 3.1. We choose ǫ = δ /N , so thatthe norm estimate in the conclusion of Lemma 3.1 is strictly less than N p δ /N = δ . Let c , d , and u denote the positive elements and the unitary element, respectively, of M n (0) (C( X )) produced byLemma 3.1. Apply the arguments of the second-to-last paragraph in the proof of Lemma 4.5 with c , d , and u in place of c, d, and u , respectively, to produce an element v of M ( A ) such that || λ ( v bv ∗ − a ) || <δ , and rank( a ( x )) + ( d ( x ) − / ≤ rank(( v bv ∗ )( x )) , ∀ x ∈ X j \ X (0) j , j > . Suppose that we have found m k ∈ N and v k ∈ M m k ( A ) such that || λ k ( v k bv ∗ k − a ) || < δ k and rank( a ( x )) + ( d ( x ) − / ≤ rank(( v k bv ∗ k )( x )) , ∀ x ∈ X j \ X (0) j , j > k. An application of Lemma 4.5 yields v k +1 ∈ M m k ( A ) such that || λ k +1 ( v k +1 v k bv ∗ k v ∗ k +1 − a ) || < N √ δ k = δ k +1 and rank( a ( x )) + ( d ( x ) − / ≤ rank(( v k +1 v k bv ∗ k v ∗ k +1 )( x )) , ∀ x ∈ X j \ X (0) j , j > k + 1 . Starting with v , we use the fact above to find, successively, v , . . . , v l . With v = v l v l − · · · v we have || vbv ∗ − a || < δ l < ǫ, as desired. (cid:3)
5. A
PPLICATIONS
The radius of comparison and strict comparison.
Let A be a unital stably finite C ∗ -algebra, and let a, b ∈ M ∞ ( A ) be positive. We say that A has r -strict comparison if a - b whenever d ( a ) + r < d ( b ) , ∀ d ∈ LDF( A ) . The radius of comparison of A , denoted by rc( A ) , is defined to be the infimum of the set { r ∈ R + | A has r − strict comparison } whenever this set is nonempty; if the set is empty then we set rc( A ) = ∞ ([28]). The condition rc( A ) = 0 is equivalent to A having strict comparison (see Subsection 2.2).The radius of comparison should be thought of as the ratio of the topological dimension of A to itsmatricial size, despite the fact that both may be infinite. It has been useful in distinguishing C ∗ -algebraswhich are not K -theoretically rigid in the sense of G. A. Elliott ([12], [29]). Here we give sharp upperbounds on the radius of comparison of a recursive subhomogeneous algebra. These improve significantlyupon the upper bounds established in the homogeneous case by [30, Theorem 3.15]. Theorem 5.1.
Let A be a separable RSH algebra with a fixed decomposition of length l and matrix sizes n (0) , . . . , n ( l ) .It follows that rc( A ) ≤ max ≤ k ≤ l dim( X k ) − n ( k ) . Proof.
Use r to denote the upper bound in the statement of the theorem, and suppose that we are given a, b ∈ M ∞ ( A ) + such that d τ ( a ) + r < d τ ( b ) for every τ ∈ T( A ) . Associated to each x ∈ X k \ X (0) k , ≤ k ≤ l , is an extreme point of T( A ) , denoted by τ x , obtained by composing ev x with the normalisedtrace on M n ( k ) . For any a ∈ M ∞ ( A ) + we have d τ x ( a ) = [rank( ev x ( a ))] /n ( k ) , and so rank( ev x ( a )) n ( k ) + dim( X k ) − n ( k ) ≤ rank( ev x ( a )) n ( k ) + r < rank( ev x ( b )) n ( k ) . Multiplying through by n ( k ) we have rank( a ( x )) + dim( X k ) − < rank( b ( x )) for every x ∈ X k \ X (0) k and k ∈ { , . . . , l } , whence a - b by Theorem 4.6, as desired. (cid:3) Specialising to the homogeneous case we have the following corollary.
Corollary 5.2.
Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N , and p ∈ C( X ) ⊗ K a projection. If follows that rc( p (C( X ) ⊗ K ) p ) ≤ d − p ) . Proof.
The algebra p (C( X ) ⊗ K ) p admits a recursive subhomogeneous decomposition in which everymatrix size is equal to rank( p ) and each X k has covering dimension at most d . (This decompositioncomes from the fact that p corresponds to a vector bundle of finite type—see Section 2 of [22].) TheCorollary now follows from Theorem 5.1. (cid:3) Corollary 5.2 improves upon [30, Theorem 3.15], or rather, the upper bound on the radius of comparisonthat can be derived from it: the latter result leads to an upper bound of (9 d ) / rank( p ) .The property of strict comparison is a powerful regularity property with agreeable consequences. Wewill see some examples of this in Subsections 5.2, 5.3, and 5.4; a fuller treatment of this topic can be foundin [11]. Theorem 5.3.
Let ( A i , φ i ) be a unital direct sequence of recursive subhomogeneous algebras with slow dimensiongrowth. If A = lim i →∞ ( A i , φ i ) is simple, then A has strict comparison of positive elements.Proof. Let us first show that lim inf i →∞ rc( A i ) = 0 . We assume that each A i is equipped with a fixeddecomposition. Let Y i = ⊔ l i k =0 X i,k denote the total space of A i , d i : Y i → { }∪ N its topological dimensionfunction, and n i (0) , . . . , n i ( l i ) its matrix sizes. From [23, Definition 1.1], ( A i , φ i ) has slow dimensiongrowth if the following statement holds: for every i ∈ N , projection p ∈ M ∞ ( A i ) , and N ∈ N , there exists j > i such that for every j ≥ j and y ∈ Y i we have ev y ( φ i,j ( p )) = 0 or rank( ev y ( φ i,j ( p ))) ≥ N d j ( y ); if p = A i , then only the latter statement can hold. If y ∈ X j,k \ X (0) j,k , then rank( ev y ( φ i,j ( A i ))) = rank( ev y ( A j )) = n j ( k ) ≥ N dim( X j,k ) . It now follows from Theorem 5.1 that lim inf i →∞ rc( A i ) = 0 .Theorem 4.5 of [30] would give us strict comparison for A if only each φ i were injective. The origin ofthis injectivity hypothesis lies in [30, Lemma 4.4]—the proof of [30, Theorem 4.5] only uses injectivity ofthe φ i in its appeal to this Lemma. Thus, we must drop injectivity from the assumptions of [30, Lemma4.4]; we must prove the following claim: Claim:
Let B be the limit of an inductive sequence ( B i , ψ i ) of C ∗ -algebras, and let a, b ∈ M ∞ ( B ) bepositive. If ψ i, ∞ ( a ) - ψ i, ∞ ( b ) , then for every ǫ > there is a j > i such that ( ψ i,j ( a ) − ǫ ) + - ψ i,j ( b ) . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 23 Proof of claim.
If will suffice to prove the claim for a, b ∈ B . By assumption, there is a sequence ( v k ) in B such that v k bv ∗ k → a . We may assume that the v k lie in the dense local C ∗ -algebra ∪ i ψ i, ∞ ( B i ) (see theproof of [30, Lemma 4.4]). In fact, by compressing our inductive sequence, we may as well assume that v k = φ k, ∞ ( w k ) for some w k ∈ B k . The statement that v k bv ∗ k → a can now amounts to k ψ k, ∞ ( w k ψ i,k ( b ) w ∗ k − ψ i,k ( a )) k n →∞ −→ . Fix k large enough that the left hand side above is < ǫ . Since k ψ k ,j ( x ) k → k ψ k , ∞ ( x ) k for any x ∈ A k we may find j > i such that k ψ k ,j ( w k ψ i,k ( b ) w ∗ k − ψ i,k ( a )) k < ǫ. Setting r j = ψ k ,j ( w k ) and appealing to part (iii) of Proposition 2.1 we have ( ψ i,j ( a ) − ǫ ) + - r j ψ i,j ( b ) r ∗ j - ψ i,j ( b ) , as desired. This proves the claim, and hence the theorem. (cid:3) We collect an improvement of [30, Theorem 4.5] as a corollary.
Corollary 5.4.
Let A be the limit of an inductive sequence of stably finite C ∗ -algebras ( A i , φ i ) , with each A i and φ i unital. Suppose that A is simple, and that lim inf i →∞ rc( A i ) = 0 . It follows that A has strict comparison of positive elements.Proof. Follow the proof of [30, Theorem 4.5] but use the claim in the proof of Theorem 5.3 instead of [30,Lemma 4.4]. (cid:3)
Corollary 5.5.
Let M be a compact smooth connected manifold and h : M → M a minimal diffeomorphism. Itfollows that the transformation group C ∗ -algebra C ∗ ( M, Z , h ) has strict comparison of positive elements.Proof. By the main result of [17], C ∗ ( M, Z , h ) can be written as the limit of an inductive sequence ofrecursive subhomogeneous C ∗ -algebras with slow dimension growth. Apply Theorem 5.3. (cid:3) The structure of the Cuntz semigroup.
The Cuntz semigroup is a sensitive invariant in the matterof distinguishing simple separable amenable C ∗ -algebras, and has recently received considerable atten-tion (see [3], [4], [5], [7], [8], [11], [27], and [30]). It is, however, very difficult to compute in general—see[27, Lemma 5.1]. This situation improves dramatically in the case of simple C ∗ -algebras with strict com-parison of positive elements.Let A be a unital, simple, exact, stably finite C ∗ -algebra. In this case we may write W ( A ) = V ( A ) ⊔ W ( A ) + (as sets), where V ( A ) denotes the semigroup of Murray-von Neumann equivalence classes of pro-jections in M ∞ ( A ) —here interpreted as the those Cuntz equivalence classes represented by a projection—and W ( A ) + denotes the subsemigroup of W ( A ) consisting of Cuntz classes represented by positive ele-ments having zero as an accumulation point of their spectrum (cf. [21]). Let LAff b (T( A )) ++ denote theset of lower semicontinuous, affine, bounded, strictly positive functions on the tracial state space of A ,and define a map ι : W ( A ) → LAff b (T( A )) ++ by ι ( h a i )( τ ) = d τ ( a ) . We endow the set V ( A ) ⊔ LAff b (T( A )) ++ with an Abelian binary operation + W which restricts to the usual semigroup operation in each compo-nent and is given by x + W f = ι ( x ) + f for x ∈ V ( A ) and f ∈ LAff b (T( A )) ++ . We also define a partialorder ≤ W on this set which restricts to the usual partial orders in each component and satisfies (i) x ≤ W f if and only if ι ( x ) < f , and(ii) x ≥ W f if and only if ι ( x ) ≥ f . Theorem 5.6 (Brown-Perera-T [3], Coward-Elliott-Ivanescu [7]) . Let A be a simple, unital, exact, and stablyfinite C ∗ -algebra with strict comparison of positive elements. It follows that the map V ( A ) ⊔ W ( A ) + id ⊔ ι −→ V ( A ) ⊔ LAff b (T( A )) ++ is a semigroup order embedding. If A is infinite-dimensional and monotracial, then the embedding of Theorem 5.6 is an isomorphism.We suspect that the monotracial assumption is unneccessary. Theorem 5.6 applies to ASH algebras as inTheorem 5.3, and so to the minimal diffeomorphism C ∗ -algebras C ∗ ( M, Z , h ) considered above.5.3. A conjecture of Blackadar-Handelman.
Blackadar and Handelman conjectured in 1982 that thelower semicontinuous dimension functions on a C ∗ -algebra should be dense in the set of all dimensionfunctions. This conjecture was proved for C ∗ -algebras as in Theorem 5.6 in [3, Theorem 6.4]. Thus, wehave the following result. Theorem 5.7.
Let A be a C ∗ -algebra as in Theorem 5.6 (in particular, A could be the C ∗ -algebra of a minimaldiffeomorphism). It follows that the lower semicontinuous dimension functions on A are weakly dense in the set ofall dimension functions on A . Classifying Hilbert modules.
In [7], Coward, Elliott, and Ivanescu gave a new presentation of theCuntz semigroup. Given a C ∗ -algebra A , they considered positive elements in A ⊗ K (as opposed to M ∞ ( A ) , as we have done—the difference is ultimately immaterial). If A is separable, then the hereditarysubalgebras of A ⊗ K are singly generated, and any two generators of a fixed hereditary subalgebra areCuntz equivalent. Thus, Cuntz equivalence factors through the passage from a positive element to thehereditary subalgebra it generates. These hereditary subalgebras are in one-to-one correspondence withcountably generated Hilbert A -modules, and in [7] the notion of Cuntz equivalence, considered as arelation on hereditary subalgebras, is translated into a relation on Hilbert modules. Thus, we may speakof Cuntz equivalence between countably generated Hilbert A -modules. Theorem 5.8 (Coward-Elliott-Ivanescu, [7]) . Let A be a C ∗ -algebra of stable rank one. It follows that countablygenerated Hilbert A -modules X and Y are Cuntz equivalent if and only if they are isomorphic. Corollary 5.9.
Let A be as in Theorem 5.3. Suppose further that A has stable rank one. (In particular, A couldby the C ∗ -algebra of a minimal diffeomorphism, as these have stable rank one by the main result of [23] .) It followsthat countably generated Hilbert A -modules X and Y are isomorphic if and only if they are Cuntz equivalent. If X and Y as in Corollary 5.9 are finitely generated and projective, then they are Cuntz equivalent ifand only if the projections in A ⊗ K which generate them as closed right ideals have the same K -class.Otherwise, X has associated to it an affine function on the tracial state space of A : one extends the map ι of Subsection 5.2 to have domain A ⊗ K , applies it to any positive element of A ⊗ K which generates X as a closed right ideal. This function determines non-finitely generated X up to isomorphism. Thisclassification of Hilbert A -modules is analogous to the classification of W ∗ -modules over a II factor. Werefer the reader to Section 3 of [4] for further details. OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 25 Classifying self-adjoints.
We say that self-adjoint elements a and b in a unital C ∗ -algebra A areapproximately unitarily equivalent if there is a sequence ( u n ) ∞ n =1 of unitaries in A such that u n au ∗ n → b .For a ∈ A + we let φ a : C ∗ ( a, ) ֒ → A denote the canonical embedding. Denote by Ell ( a ) the followingpair of induced maps: K ( φ a ) : K (C ∗ ( a, )) → K ( A ); φ ♯a : T( A ) → T(C ∗ ( a, . Theorem 5.10 (Brown-T, [4]) . Let A be a unital simple exact C ∗ -algebra of stable rank one and strict comparison(in particular, A could have stable rank one and satisfy the hypotheses of Theorem 5.3). If a, b ∈ A + , then a and b are approximately unitarily equivalent if and only if σ ( a ) = σ ( b ) and Ell ( a ) = Ell ( b ) . The range of the radius of comparison, with applications.
The classification theory of operator al-gebras is a rich field. It was begun by Murray and von Neumann with their type classification of factorsin the 1930s, and has been active ever since. In the presence of certain regularising assumptions, thetheory is well-behaved. For instance, there is a complete classification of injective factors with separablepredual (due to Connes and Haagerup—see [6] and [14]), and a similarly successful classification pro-gram for simple C ∗ -algebras upon replacing injectivity and separability of the predual with amenabilityand norm-separability, respectively (see [11] and [25]).Without these regularising assumptions, the theory is fractious, but nonetheless interesting. One of thelandmarks on this side of the theory is McDuff’s construction of uncountably many non-isomorphic fac-tors of type II ([18]). (More recently there is Popa’s work on II factors with Betti numbers invariants—see [20].) One might view McDuff’s result as saying that there are uncountably many non-isomorphicfactors which all have the same naive invariant, namely, the mere fact that they are II factors. (Connesproved that there is only one injective II factor with separable predual.) Here we prove an analogue ofMcDuff’s theorem for simple, separable, amenable C ∗ -algebras, where the corresponding naive invariantconsists of Banach algebra K -theory and positive traces. We even obtain a somewhat stronger result, re-placing non-isomorphism with non-Morita-equivalence. In passing we prove that the range of the radiusof comparison is exhausted by simple C ∗ -algebras, a result which represents the first exact calculationsof the radius of comparison for any simple C ∗ -algebra.Recall that the Elliott invariant of a C ∗ -algebra A is the 4-tuple(16) Ell( A ) := (cid:0) (K A, K A + , Σ A ) , K A, T + A, ρ A (cid:1) , where the K -groups are the Banach algebra ones, K A + is the image of the Murray-von Neumann semi-group V( A ) under the Grothendieck map, Σ A is the subset of K A corresponding to projections in A , T + A is the space of positive tracial linear functionals on A , and ρ A is the natural pairing of T + A and K A given by evaluating a trace at a K -class. Theorem 5.11.
There is a family { A ( r ) } r ∈ R + \{ } of simple, separable, amenable C ∗ -algebras such that rc( A r ) = r and Ell( A r ) ∼ = Ell( A s ) for every s, r ∈ R + \{ } . In particular, A r ≇ A s whenever r = s . If A s and A r are Moritaequivalent, then s/r ∈ Q .Proof. The general framework for the construction of A ( r ) follows [31]. Find sequences of natural num-bers ( n i ) and ( l i ) and a natural number m with the following properties:(i) n i → ∞ ;(ii) n m · n n · · · n i ( n + l )( n + l ) · · · ( n i + l i ) i →∞ −→ r ; (iii) l i = 0 for infinitely many i ;(iv) every natural number divides some m i := m ( n + l )( n + l ) · · · ( n i + l i ) Set X = [0 , n and set X i +1 = ( X i ) n i +1 . Let π ji : X i +1 → X i , ≤ j ≤ n i +1 be the co-ordinateprojections. Let A i be the homogeneous C ∗ -algebra M m i (C( X i )) , and let φ i : A i → A i +1 be the ∗ -homomorphism given by φ i ( f )( x ) = diag (cid:16) f ◦ π i ( x ) , . . . , f ◦ π n i +1 i ( x ) , a ( x i ) , . . . , a ( x l i i ) (cid:17) , ∀ x ∈ X i +1 , where x i , . . . , x l i i ∈ X i are to be specified. Set A ( r ) = lim i →∞ ( A i , φ i ) , and define φ i,j := φ j − ◦ · · · ◦ φ i . Let φ i, ∞ : A i → A be the canonical map. We note that the x i , . . . , x l i i ∈ X i may be chosen to ensure that A is simple (cf. [31]); we assume that they have been so chosen, whence A ( r ) is unital, simple, separable,and amenable.By Theorem 5.1, we have lim i →∞ rc( A i ) = lim i →∞ n n · · · n i − m ( n + l )( n + l ) · · · ( n i + l i ) = r. Since the construction of A ( r ) is the same as that of [29, Theorem 4.1], we conclude that rc( A ( r ) ) ≤ r by[29, Proposition 3.3].Let η > be given. We will exhibit positive elements a, b ∈ M ∞ ( A ( r ) ) with the property that d τ ( a ) + r − η < d τ ( b ) , ∀ τ ∈ T( A ( r ) ) , and yet h a i (cid:2) h b i in W ( A ( r ) ) . This will show that rc( A ( r ) ) ≥ r − η for every η > , whence rc( A ( r ) ) = r ,as desired.Choose i large enough that ⌊ dim( X i ) / ⌋ − m i > r − η/ . It follows from [28, Theorem 6.6] that there are a, b ∈ M ∞ ( A i ) + such that h a i (cid:2) h b i in W ( A i ) and yet d τ ( a ) + r − η < d τ ( b ) , ∀ τ ∈ T( A i ) . Assumption (ii) above ensures that n i = 0 , whence each φ i is injective. We may thus identify a and b withtheir images in A ( r ) so that d τ ( a ) + r − η < d τ ( b ) , ∀ τ ∈ T( A ( r ) ) . We need only prove that h a i (cid:2) h b i in W ( A ( r ) ) . The technique for proving this is an adaptation of Villad-sen’s Chern class obstruction argument from [31].With N i := n n · · · n i , we have A i = M m i (C([0 , N i )) . The element b of M ∞ ( A i ) has the followingproperties: there is a closed subset Y of [0 , N i homeomorphic to S k , N i − ≤ k ≤ N i , such that therestriction of b to Y is a projection of rank k corresponding to the k -dimensional Bott bundle ξ over S k ;and the rank of b is at most k over any point in X i = [0 , N i . The element a has constant rank—it is aprojection corresponding to a trivial line bundle over X i —and need only have normalised rank strictlyless than η/ . By increasing i , and hence m i , if necessary, we may assume that the normalised rank of a is at least η/ . This leads to(17) d τ ( a ) > η r (cid:16) η r (cid:17) ≥ d τ ( b ) (cid:16) η r (cid:17) , ∀ τ ∈ T( A i ) . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 27 The map φ i,j : A i → A j has the form φ i,j ( f ) = diag (cid:16) f ◦ π i,j ( x ) , . . . , f ◦ π k i,j i,j ( x ) , f ( x i ) , . . . , f ( x l i,j i ) (cid:17) , ∀ x ∈ X j , where k i,j = n i +1 n i +2 · · · n j and l i,j = m j /m i − k i,j . Following [31], we have that φ i,j ( a ) is a projectionof rank rank( a ) m j /m i corresponding to the trivial vector bundle θ rank( a ) m j /m i , while the restriction of φ i,j ( b ) to Y k i,j ⊆ X k i,j i = X j is of the form ξ × k i,j ⊕ f b , where f b is a constant positive element of rankat most kl i,j . If p is the image of ∈ A i under the eigenvalue maps of φ i,j which are co-ordinateprojections, then pφ i,j ( b ) p = ξ × k i,j . Let x ∈ A j . Restricting to Y k i,j (and using the same notation for therestriction of x ) we have k x ( ξ × k i,j ⊕ f b ) x ∗ − θ rank( a ) m j /m i k = k [ x ( p ⊕ f / b )]( ξ × k i,j ⊕ θ rank( f b ) )[ x ( p ⊕ f / b )] ∗ − θ rank( a ) m j /m i k . If we can show that θ rank( a ) m j /m i is not Murray-von Neumann equivalent to a subprojection of ξ × k i,j ⊕ θ rank( f b ) , then we will have that the last quantity above is ≥ / (cf. [30, Lemma 2.1]). It will then followthat for every j > i and every x ∈ A j , k xφ i,j ( b ) x ∗ − φ i,j ( a ) k ≥ / in particular, h a i (cid:2) h b i , as desired.By a straightforward adaptation of [31, Lemma 2.1] (using the fact that the top Chern class of ξ is notzero), θ rank( a ) m j /m i will fail to be equivalent to a subprojection of ξ × k i,j ⊕ θ rank( f b ) if rank( a ) m j /m i > rank( f b ) . We have rank( f b ) − rank( a ) · m j m i ≤ kl i,j − rank( a ) · m j m i ≤ N i (cid:18) m j m i − k i,j (cid:19) − rank( a ) · m j m i = ( N i − rank( a )) · m j m i − n n · · · n j , so it will be enough to prove that n n · · · n j > ( N i − rank( a )) · m j m i . Rearranging and using the definitions of m i and N i we must show that ( n i +1 + l i +1 ) · · · ( n j + l j ) n i +1 · · · n j · (1 − rank( a ) /N i ) < . Now rank( a ) > ( η/ m i , so the right hand side above is less than(18) ( n i +1 + l i +1 ) · · · ( n j + l j ) n i +1 · · · n j · (cid:18) − m i η N i (cid:19) . The sequence ( m i η )(2 N i ) is convergent to a nonzero limit, so for some γ > , for all i sufficiently large,the expression in (18) is strictly less than(19) ( n i +1 + l i +1 ) · · · ( n j + l j ) n i +1 · · · n j · (1 − γ ) . Increasing i if necessary we may assume that ( n i +1 + l i +1 ) · · · ( n j + l j ) n i +1 · · · n j < − γ , whence the expression in (19) is strictly less than one, as required. This completes the proof that rc( A ( r ) = r . Since each natural number divides some m i and each X i is contractible, we have K ( A ( r ) ) ∼ = Q , withthe usual order structure and order unit. The contractibility of X i also implies that K ( A i ) = 0 for every i ,whence K ( A ( r ) ) = 0 , too. The pairing ρ between traces and K is determined uniquely since there is onlyone state on K ( A ( r ) ) . In order to complete the proof that Ell( A ( r ) ) ∼ = Ell( A ( s ) ) for every r, s ∈ R + \{ } ,we must prove that T( A ( r ) ) ∼ = T( A ( s ) ) .Recall that the tracial state space of M k (C( X )) is homeomorphic to the space P ( X ) of regular positiveBorel probability measures on X . Let ( A ( r ) i , φ i ) and ( A ( s ) i , ψ i ) be inductive sequences as above, withsimple limits A ( r ) and A ( s ) , respectively. We have Spec( A ( r ) i ) = [0 , N i and Spec( A ( s ) i ) = [0 , M i . Usingthe superscript ♯ to denote the map induced on traces by a ∗ -homomorphism, we have T( A ( r ) ) ∼ = lim ←− ( P ([0 , N i , φ ♯i ); T( A ( s ) ) ∼ = lim ←− ( P ([0 , M i , ψ ♯i ) . We require sequences ( γ i ) and ( δ i ) of continuous affine maps making the triangles in the diagram(20) P ([0 , N ) P ([0 , N ) φ ♯ o o δ x x ppppppppppp P ([0 , N ) φ ♯ o o δ x x ppppppppppp · · · φ ♯ o o δ z z uuuuuuuuuu P ([0 , M ) γ O O P ([0 , M ) ψ ♯ o o γ O O P ([0 , M ) ψ ♯ o o γ O O · · · ψ ♯ o o commute ever more closely on ever larger finite sets as i → ∞ . We will in fact be able to arrange fornear-commutation on the entire source space in each triangle.Let µ be a probability measure on X N , and K a subset of { , . . . , N } . We use µ K to denote the mea-sure on X | K | defined by integrating out those co-ordinates of X N not contained in K . Straightforwardcalculation shows that upon viewing X i +1 as X N i +1 /N i i we have φ ♯i ( µ ) = n i +1 n i +1 + l i +1 N i N i +1 N i +1 /N i M l =1 µ { l } + l i +1 n i +1 + l i +1 λ i , where λ i is a convex combination of finitely many point masses. A similar statement holds for ψ ♯i . Since l i +1 / ( n i +1 + l i +1 ) is negligible for large i , we may in fact assume that φ ♯i ( µ ) = N i N i +1 N i +1 /N i M l =1 µ { l } ; ψ ♯i ( µ ) = M i M i +1 M i +1 /M i M k =1 µ { k } for the purposes of our intertwining argument. We may also assume, by compressing our sequences ifnecessary, that N ≪ M ≪ N ≪ M ≪ · · · . Define γ ( µ ) = 1 ⌊ M /N ⌋ ⌊ M /N ⌋ M l =1 µ { ( l − N +1 ,...,lN } . OMPARISON THEORY AND SMOOTH MINIMAL C ∗ -DYNAMICS 29 Now set B k = { ( k − M + 1 , . . . , kM } for each ≤ k ≤ ⌊ N /M ⌋ , and D t = { ( t − N , . . . , tN } foreach ≤ t ≤ N /N . Define δ ( µ ) = 1 ⌊ N /M ⌋ ⌊ N /M ⌋ M k =1 σ ∗ k ( µ B k ) , where σ ∗ k is the map induced on measures by the homeomorphism σ k : B k → B k defined by the followingproperty: if j is the first co-ordinate of B k contained in a D t which is itself contained in B k , then σ k is thepermutation which subtracts j − | B k | ) from each co-ordinate. (The idea is that σ k moves all of the D t s contained in B k “to the beginning”.)Let L be the number of D t s which are contained in some B k . Since N ≪ M ≪ N , we have that ( N − N L ) /N is (arbitrarily) small. Now γ ◦ δ ( µ ) = 1 L M { t | D t ⊆ B k , for some k } µ D t , while φ ♯ ( µ ) = 1 N /N N /N M k =1 µ D k . The difference [( γ ◦ δ ) − φ ♯ ]( µ ) is a measure of total mass at most N − N L ) /N , and so the first trianglefrom the diagram (20) commutes to within this tolerance on all of P ([0 , N ) . The subsequent γ i s and δ i sare defined in a manner analogous to our definition of δ , and this leads to the desired intertwining. Weconclude that Ell( A ( r ) ) ∼ = Ell( A ( s ) ) , as desired.It remains to prove that if A ( r ) and A ( s ) are Morita equivalent, then r/s ∈ Q . Suppose that they are so.By the Brown-Green-Rieffel Theorem, A ( r ) and A ( s ) are stably isomorphic, and so there are projections p, q ∈ A ( r ) ⊗ K such that A ( r ) ∼ = p ( A ( r ) ⊗ K ) p and A ( s ) ∼ = q ( A ( r ) ⊗ K ) q . Since K ( A ( r ) ⊗ K ) = K ( A ( r ) ) = Q ,there are natural numbers n and m such that n [ p ] = m [ q ] in K . It is proved in [31] that the constructionused to arrive at A ( r ) and A ( s ) always produces C ∗ -algebras of stable rank one, whence A ( r ) ⊗ K hasstable rank one. Thus, ⊕ ni =1 p and ⊕ mj =1 q are Murray-von Neumann equivalent, and M n ( A ( r ) ) ∼ = ( ⊕ ni =1 p )( A ( r ) ⊗ K )( ⊕ ni =1 p ) ∼ = ( ⊕ mi =1 q )( A ( r ) ⊗ K )( ⊕ mi =1 q ) ∼ = M m ( A ( s ) ) . By [28, Proposition 6.2 (ii)] we have r/n = rc (cid:16) M n ( A ( r ) ) (cid:17) = rc (cid:16) M m ( A ( s ) ) (cid:17) = s/m, whence r/s ∈ Q , as required. (cid:3) Remark 5.12. If r/s / ∈ Q , then A ( r ) ⊗ K ≇ A ( s ) ⊗ K . This, to our knowledge, is the first example of simpleseparable amenable stable C ∗ -algebras which are not isomorphic yet have the same Elliott invariant.R EFERENCES[1] Blackadar, B.:
Comparison theory for simple C ∗ -algebras , Operator Algebras and Applications, Vol. 1, 21-54, London Math. Soc.Lecture Note Ser., , Cambridge Univ. Press, Cambridge, 1988[2] Blackadar, B., and Handelman, D.: Dimension Functions and Traces on C ∗ -algebras , J. Funct. Anal. (1982), 297-340[3] Brown, N. P., Perera, F., and Toms, A. S.: The Cuntz semigroup, the Elliott conjecture, and dimension functions on C ∗ -algebras , J.Reine Angew. Math., to appear[4] Brown, N. P., and Toms, A. S.: Three applications of the Cuntz semigroup , Int. Math. Res. Not. (2007), Article ID rnm068, 14 pages[5] Ciuperca, A., and Elliott, G. A.:
A remark on invariants for C ∗ -algebras of stable rank one , Int. Math. Res. Not., to appear[6] Connes, A.: Classification of injective factors. Cases II , II ∞ , III λ , λ = 1 . , Ann. of Math. (2) (1976), 73-115 [7] Coward, K., Elliott, G. A., and Ivanescu, C.: The Cuntz semigroup as an invariant for C ∗ -algebras , J. Reine Angew. Math., toappear[8] Ciuperca, A., Santiago, L., and Robert, L.: The Cuntz semigroup of ideals and quotients, and a generalized Kasparov StabilizationTheorem , J. Op. Th., to appear[9] Cuntz, J.:
Dimension Functions on Simple C ∗ -algebras , Math. Ann. (1978), 145-153[10] Elliott, G. A., Gong, G. and Li, L.: On the classification of simple inductive limit C ∗ -algebras, II: The isomorphism theorem , Invent.Math. (2007), 249-320[11] Elliott, G. A., and Toms, A. S.: Regularity properties in the classification program for separable amenable C ∗ -algebras , Bull. Amer.Math. Soc. (2008), 229-245[12] Giol, J., and Kerr, D.: Subshifts and perforation , preprint (2007)[13] Gong, G.:
On the classification of simple inductive limit C ∗ -algebras I. The reduction theorem. , Doc. Math.7 (2002), 255-461[14] Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type
III , Acta. Math. (1987), 95-148[15] Jacob, B.: A remark on the distance between unitary orbits in ASH algebras with unique trace , preprint (2008)[16] Kirchberg, E. and Rørdam, M.:
Non-simple purely infinite C ∗ -algebras , Amer. J. Math. (2000), 637-666[17] Lin, Q., and Phillips, N. C.: The structure of C ∗ -algebras of minimal diffeomorphisms , in preparation[18] McDuff, D.: Uncountably many II factors Ann. of Math. (2) (1969), 372-377[19] Ng, P. W., and Winter, W.: A note on Subhomogeneous C ∗ -algebras , C. R. Math. Acad. Sci. Soc. R. Can. (2006), 91-96[20] Popa, S.: On a class of II factors with Betti numbers invariants , Ann. of Math. (2) (2006), 809-899[21] Perera, F. and Toms, A. S.: Recasting the Elliott conjecture , Math. Ann. (2007), 669-702[22] Phillips, N. C.:
Recursive subhomogeneous algebras , Trans. Amer. Math. Soc. (2007), 4595-4623[23] Phillips, N. C.:
Cancellation and stable rank for direct limits of recursive subhomogeneous algebras , Trans. Amer. Math. Soc. (2007), 4625-4652[24] Rieffel, M. A.:
Dimension and stable rank in the K -theory of C ∗ -algebras , Proc. London Math. Soc. (3) (1983), 301-333[25] Rørdam, M.: Classification of Nuclear C ∗ -Algebras, Encyclopaedia of Mathematical Sciences , Springer-Verlag, Berlin,Heidelberg 2002[26] Rørdam, M.: The stable and the real rank of Z -absorbing C ∗ -algebras , Int. J. Math. (2004), 1065-1084[27] Toms, A. S.: On the classification problem for nuclear C ∗ -algebras , Ann. of Math. (2) (2008), 1059-1074[28] Toms, A. S.: Flat dimension growth for C ∗ -algebras , J. Funct. Anal. (2006), 678-708[29] Toms, A. S.: An infinite family of non-isomorphic C ∗ -algebras with identical K -theory , Trans. Amer. Math. Soc., to appear[30] Toms, A. S.: Stability in the Cuntz semigroup of a commutative C ∗ -algebra , Proc. London Math. Soc. (3) (2008), 1-25[31] Villadsen, J.: Simple C ∗ -algebras with perforation , J. Funct. Anal. (1998), 110-116D EPARTMENT OF M ATHEMATICS AND S TATISTICS , Y
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