A simple nuclear C*-algebra with an internal asymmetry
aa r X i v : . [ m a t h . OA ] S e p A SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNALASYMMETRY ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
Abstract.
We construct an example of a simple approximately homogeneous C ∗ -algebra such that its Elliott invariant admits an automorphism which isnot induced by an automorphism of the algebra. Classification theory for simple nuclear C ∗ -algebras reached a milestone recently.The results of [EGLN15] and [TWW17], building on decades of work by manyauthors, show that simple nuclear unital C ∗ -algebras satisfying the Universal Co-efficient Theorem are classified via the Elliott invariant, Ell( · ), which consists ofthe ordered K -group along with the class of the identity, the K -group, the tracesimplex, and the pairing between the trace simplex and the K -group. Earliercounterexamples due to Toms and Rørdam ([Tom08, Rør03]), related to ideas ofVilladsen ([Vil98]), show that one cannot expect to be able to extend this clas-sification theorem beyond the case of finite nuclear dimension, at least withouteither extending the invariant or restricting to another class of C ∗ -algebras. Animportant facet of the classification theorems is a form of rigidity. Starting withtwo C ∗ -algebras A and B and an isomorphism Φ : Ell( A ) → Ell( B ), one not onlyshows that A and B are isomorphic, but rather that there exists an isomorphismfrom A to B which induces the given isomorphism Φ on the level of the Elliottinvariant, and furthermore that the isomorphism on the algebra level is unique upto approximate unitary equivalence.The goal of this paper is to illustrate how this existence property may fail inthe infinite nuclear dimension setting, even when restricting to a class consisting ofa single C ∗ -algebra. Namely, we construct an example of a simple unital nuclearseparable AH algebra C , along with an automorphism of Ell( C ), which is notinduced by any automorphism of C . This can be viewed as a companion of sorts to[Tom08, Theorem 1.2], where it was shown that when such automorphisms exist,they need not be unique in the sense described. The mechanism of the example isthat if there were such an automorphism ϕ , there would be projections p, q ∈ C such that ϕ ( p ) = q but such that the corners pCp and qCq have different radiiof comparison ([Tom06]; the definition is recalled at the beginning of Section 1).This further shows that simple unital AH algebras can be quite inhomogeneous.In particular, extending the Elliott invariant by adding something as simple as theradius of comparison will not help for the classification of AH algebras which arenot Jiang-Su stable. Date : 23 September 2019.2010
Mathematics Subject Classification.
We now give an overview of our construction. We start with the counterexamplefrom [Tom08, Theorem 1.1]. We consider two direct systems, described diagram-matically as follows: C ( X ) / / / / / / / / C ( X ) ⊗ M r (1) / / / / / / / / C ( X ) ⊗ M r (2) / / / / / / / / · · · C ([0 , / / / / / / / / C ([0 , ⊗ M r (1) / / / / / / / / C ([0 , ⊗ M r (2) / / / / / / / / · · · The ordinary arrows indicate a large (and rapidly increasing) number of embeddingswhich are carefully chosen, and the dotted arrows indicate a small number of pointevaluation maps, thrown in so as to ensure that the resulting direct limit is simple.The spaces in the upper diagram are contractible CW complexes whose dimensionincreases rapidly compared to the sizes of the matrix algebras. (Toms uses cubes;in our construction we found it easier to use cones over products of spheres, but theunderlying idea is similar.) The direct system is constructed so as to have positiveradius of comparison. We use [Tho94] to choose the lower diagram so as to mimicthe upper diagram, and produce the same Elliott invariant. As the resulting algebraon the bottom is AI, it has strict comparison, and therefore is not isomorphic tothe one on the top. (In [Tom08] it isn’t important for the two diagrams to matchup nicely in terms of the ranks of the matrices involved. However, we will showthat it can be done, as it is important for us.)Our construction involves moving the point evaluations across, so as to mergethe two systems, getting: C ( X ) ' ' / / / / / / C ( X ) ⊗ M r (1) ) ) / / / / / / C ( X ) ⊗ M r (2) & & / / / / / / · · · C ([0 , / / / / / / C ([0 , ⊗ M r (1) / / / / / / C ([0 , ⊗ M r (2) / / / / / / · · · . With care, one can arrange for the flip between the two levels of the diagram tomake sense as an automorphism of the Elliott invariant. The resulting C ∗ -algebrahas positive radius of comparison and behaves roughly as badly as Toms’ example.Nevertheless, we can distinguish a part of it which roughly corresponds to the rapiddimension growth diagram on the top from a part which roughly corresponds tothe AI part on the bottom. Namely, if at the first level C ( X ) ⊕ C ([0 , q the function which is 1 on X and 0 on [0 , q ⊥ = 1 − q , thenthe K -classes of q and q ⊥ will be switched by the automorphism of the Elliottinvariant we construct. However, we can tell apart the corners qCq and q ⊥ Cq ⊥ byconsidering their radii of comparison.Section 1 develops the choices needed to get different radii of comparison indifferent corners of the algebra we construct. Section 2 contains the work needed toassemble the ingredients of the construction into a simple C ∗ -algebra whose Elliottinvariant admits an appropriate automorphism. The main theorem is in Section 3.The second author is grateful to M. Ali Asadi-Vasfi for a careful reading ofSection 1, and in particular finding a number of misprints. SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 3 Upper and lower bounds on the radius of comparison
We recall the required standard definitions and notation related to the Cuntzsemigroup. See [Rør92] for details. For a unital C ∗ -algebra A , we denote its tracialstate space by T( A ). We take M ∞ ( A ) = S ∞ n =1 M n ( A ), using the usual embeddings M n ( A ) ֒ → M n +1 ( A ). For τ ∈ T( A ), we define d τ : M ∞ ( A ) + → [0 , ∞ ) by d τ ( a ) =lim n →∞ τ ( a /n ). If a, b ∈ M ∞ ( A ) + , then a - b ( a is Cuntz subequivalent to b ) ifthere is a sequence ( v n ) ∞ n =1 in M ∞ ( A ) such that lim n →∞ v n bv ∗ n = a .Following [Tom06, Definition 6.1], for ρ ∈ [0 , ∞ ), we say that A has ρ -comparisonif whenever a, b ∈ M ∞ ( A ) + satisfy d τ ( a ) + ρ < d τ ( b ) for all τ ∈ T( A ), then a - b .The radius of comparison of A , denoted rc( A ), isrc( A ) = inf (cid:0)(cid:8) ρ ∈ [0 , ∞ ) | A has ρ -comparison (cid:9)(cid:1) . We take rc( A ) = ∞ if there is no ρ such that A has ρ -comparison. Since AH algebrasare nuclear, all quasitraces on them are traces by [Haa14, Theorem 5.11]. Thus, weignore quasitraces.Our construction uses a specific setup, with a number of parameters of variouskinds which must be chosen to satisfy specific conditions. We list here for referencemany of the objects used in it, and some of the conditions they must satisfy. Furtherobjects and conditions are given in Construction 1.6 and Construction 2.17. Manyof the lemmas use only a few of the objects and their properties, so that the readercan refer back to just the relevant parts of the constructions. In particular, manydetails are used only in this section or only in Section 2. Some of the details areused for just one lemma each.In Construction 1.1(1), the choice k ( n ) = 1 for all n > Construction 1.1.
For much of this paper, we will consider algebras constructedin the following way and using the following notation:(1) ( d ( n )) n =0 , , ,... and ( k ( n )) n =0 , , ,... are sequences in Z ≥ , with d (0) = 1 and k (0) = 0. Moreover, for n ∈ Z > , l ( n ) = d ( n ) + k ( n ) , r ( n ) = n Y j =0 l ( j ) , and s ( n ) = n Y j =0 d ( j ) . Further define t ( n ) inductively as follows. Set t (0) = 0, and t ( n + 1) = d ( n + 1) t ( n ) + k ( n + 1)[ r ( n ) − t ( n )] . (See Lemma 1.14 for the significance of t ( n ).)(2) We will assume that k ( n ) < d ( n ) for all n ∈ Z ≥ .(3) We define κ = inf n ∈ Z > s ( n ) r ( n ) . For estimates involving the radius of comparison, we will assume κ > .(4) The numbers ω, ω ′ ∈ (0 , ∞ ] are defined by ω = k (1) k (1) + d (1) and ω ′ = ∞ X n =2 k ( n ) k ( n ) + d ( n ) . ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
We will require ω ′ < ω < . In particular, ∞ X n =1 k ( n ) k ( n ) + d ( n ) < ∞ . (5) We will also eventually require that κ as in (3) and ω as in (4) are relatedby 2 κ − > ω .(6) ( X n ) n =0 , , ,... and ( Y n ) n =0 , , ,... are sequences of compact metric spaces.(They will be further specified in Construction 1.6.)(7) For n ∈ Z ≥ , the algebra C n is C n = M r ( n ) ⊗ (cid:0) C ( X n ) ⊕ C ( Y n ) (cid:1) . We further make the identifications: C ( X n +1 , M r ( n +1) ) = M l ( n +1) ⊗ C ( X n +1 , M r ( n ) ) ,C ( Y n +1 , M r ( n +1) ) = M l ( n +1) ⊗ C ( Y n +1 , M r ( n ) ) ,C ( X n ) ⊕ C ( Y n ) = C ( X n ∐ Y n ) ,C ( X n , M r ( n ) ) ⊕ C ( Y n , M r ( n ) ) = C ( X n ∐ Y n , M r ( n ) ) . (8) For n ∈ Z > , we are given a unital homomorphism γ n : C ( X n ) ⊕ C ( Y n ) → M l ( n +1) (cid:0) C ( X n +1 ) ⊕ C ( Y n +1 ) (cid:1) , and the homomorphismΓ n +1 , n : C n → C n +1 is given by Γ n +1 , n = id M r ( n ) ⊗ γ n . Moreover, for m, n ∈ Z ≥ with m ≤ n ,Γ n,m = Γ n,n − ◦ Γ n − , n − ◦ · · · ◦ Γ m +1 ,m : C m → C n . In particular, Γ n,n = id C n .(9) We set C = lim −→ n C n , taken with respect to the maps Γ n,m . The mapsassociated with the direct limit will be called Γ ∞ ,m : C m → C for m ∈ Z ≥ .We sometimes use additional objects and conditions in the construction, as fol-lows:(10) For n ∈ Z > , we may be given an additional unital homomorphism γ (0) n : C ( X n ) ⊕ C ( Y n ) → M l ( n +1) (cid:0) C ( X n +1 ) ⊕ C ( Y n +1 ) (cid:1) . Then the maps Γ (0) n +1 , n : C n → C n +1 , Γ (0) n,m : C m → C n are defined analo-gously to (8), the algebra C (0) is given as C (0) = lim −→ n C n , taken with respectto the maps Γ (0) n,m , and the maps Γ (0) ∞ ,m : C m → C are defined analogouslyto (9).(11) We often require that the maps γ n : C ( X n ∐ Y n ) → M l ( n +1) (cid:0) C ( X n +1 ∐ Y n +1 ) (cid:1) in (8) be diagonal, that is, that there exist continuous functions S n, , S n, , . . . , S n,l ( n +1) : X n +1 ∐ Y n +1 → X n ∐ Y n such that for all f ∈ C ( X n ∐ Y n ), we have γ n ( f ) = diag (cid:0) f ◦ S n, , f ◦ S n, , . . . , f ◦ S n,l ( n +1) (cid:1) . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 5 In (10), we may similarly require that there be S (0) n, , S (0) n, , . . . , S (0) n,l ( n +1) : X n +1 ∐ Y n +1 → X n ∐ Y n such that for all f ∈ C ( X n ∐ Y n ), we have γ (0) n ( f ) = diag (cid:0) f ◦ S (0) n, , f ◦ S (0) n, , . . . , f ◦ S (0) n,l ( n +1) (cid:1) . (12) Assuming diagonal maps as in (11), we may require that they agree in thecoordinates 1 , , . . . , d ( n ), that is, for n ∈ Z > and k = 1 , , . . . , d ( n ), wehave S (0) n,k = S n,k . Lemma 1.2.
In Construction 1.1(1), the sequence (cid:16) s ( n ) r ( n ) (cid:17) n =1 , ,... is strictly de-creasing.Proof. The proof is straightforward. (cid:3)
Lemma 1.3.
In Construction 1.1(1), and assuming Construction 1.1(2), we have t (0) r (0) < t (1) r (1) < t (2) r (2) < · · · < . Proof.
We have t (0) = 0 by definition. We prove by induction on n ∈ Z > that(1.1) t ( n − r ( n − < t ( n ) r ( n ) < . This will finish the proof. For n = 1, we have t (1) r (1) = k (1) k (1) + d (1) , which is in (0 , ) by Construction 1.1(2). Now assume (1.1); we prove this relationwith n + 1 in place of n . We have r ( n ) − t ( n ) > t ( n ), so t ( n + 1) r ( n + 1) = d ( n + 1) t ( n ) + k ( n + 1)[ r ( n ) − t ( n )][ d ( n + 1) + k ( n + 1)] r ( n )(1.2) > d ( n + 1) t ( n ) + k ( n + 1) t ( n )[ d ( n + 1) + k ( n + 1)] r ( n ) = t ( n ) r ( n ) . Also, with α = d ( n + 1) d ( n + 1) + k ( n + 1) and β = t ( n ) r ( n ) , starting with the first step in (1.2), and at the end using α > (by Construction1.1(2)) and β < (by the induction hypothesis), we have t ( n + 1) r ( n + 1) = αβ + (1 − α )(1 − β ) = 12 (cid:2) − (2 α − − β ) (cid:3) < . This completes the induction, and the proof. (cid:3)
Lemma 1.4.
With the notation of Construction 1.1(1) and Construction 1.1(4),and assuming the conditions in Construction 1.1(2) and Construction 1.1(4), forall n ∈ Z > we have ω ≤ t ( n ) r ( n ) ≤ ω + ω ′ < ω . ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
Proof.
The third inequality is immediate from Construction 1.1(4).By Lemma 1.3, the sequence (cid:16) t ( n ) r ( n ) (cid:17) n =1 , ,... is strictly increasing. Also,(1.3) t (1) r (1) = k (1) k (1) + d (1) = ω. The first inequality in the statement now follows.Next, we claim that t ( n ) r ( n ) ≤ n X j =1 k ( j ) k ( j ) + d ( j )for all n ∈ Z > . The case n = 1 is (1.3). Assume this inequality is known for n .Then t ( n + 1) r ( n + 1) = (cid:18) d ( n + 1) k ( n + 1) + d ( n + 1) (cid:19) (cid:18) t ( n ) r ( n ) (cid:19) + (cid:18) k ( n + 1) k ( n + 1) + d ( n + 1) (cid:19) (cid:18) r ( n ) − t ( n ) r ( n ) (cid:19) ≤ t ( n ) r ( n ) + k ( n + 1) k ( n + 1) + d ( n + 1) ≤ n +1 X j =1 k ( j ) k ( j ) + d ( j ) , as desired.The second inequality in the statement now follows. (cid:3) Notation 1.5.
For a topological space X , we definecone( X ) = ( X × [0 , / ( X × { } ) . Then cone( X ) is contractible, and cone( · ) is a covariant functor: if T : X → Y is acontinuous map, then it induces a continuous map cone( T ) : cone( X ) → cone( Y ).We identify X with the image of X × { } in cone( X ). Construction 1.6.
We give further details on the spaces X n and Y n in Construc-tion 1.1(6).(13) The space X n is chosen as follows. First set Z = S . Then, with( d ( n )) n =0 , , ,... and ( s ( n )) n =0 , , ,... as in Construction 1.1(1), define induc-tively Z n = Z d ( n ) n − = ( S ) s ( n ) . Finally, set X n = cone( Z n ). (In particular, X n is contractible, and Z n ⊂ X n as in Notation 1.5.) Further, for n ∈ Z ≥ and j = 1 , , . . . , d ( n ),we let P ( n ) j : Z n → Z n − be the j -th coordinate projection, and we set Q ( n ) j = cone (cid:0) P ( n ) j (cid:1) : X n → X n − .(14) Y n = [0 ,
1] for all n ∈ Z > . (In particular, Y n is contractible.)(15) We assume we are given points x m ∈ X m for m ∈ Z ≥ such that, using thenotation in (13), for all n ∈ Z ≥ , the set (cid:8)(cid:0) Q ( n ) ν ◦ Q ( n +1) ν ◦ · · · ◦ Q ( m − ν m − n (cid:1) ( x m ) | m = n + 1 , n + 2 , . . . and ν j = 1 , , . . . , d ( m − j ) for j = 1 , , . . . , m − n (cid:9) is dense in X n .(16) We assume we are given a sequence ( y k ) k =0 , , ,... in [0 ,
1] such that for all n ∈ Z ≥ , the set { y k | k ≥ n } is dense in [0 , SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 7 (17) The maps γ n : C ( X n ∐ Y n ) → M l ( n +1) (cid:0) C ( X n +1 ∐ Y n +1 ) (cid:1) will be as in Construction 1.1(11), with the maps S n,j : X n +1 ∐ Y n +1 → X n ∐ Y n appearing there defined as follows:(a) With Q ( n ) j as in (13), we set S n,j ( x ) = Q ( n +1) j ( x ) for x ∈ X n +1 and j = 1 , , . . . , d ( n + 1).(b) S n,j ( x ) = y n for x ∈ X n +1 and j = d ( n + 1) + 1 , d ( n + 1) + 2 , . . . , l ( n + 1) . (c) There are continuous functions R n, , R n, , . . . , R n,d ( n +1) : Y n +1 → Y n (which will be taken from Proposition 2.14 below) such that S n,j ( y ) = R n,j ( y ) for y ∈ Y n +1 and j = 1 , , . . . , d ( n + 1).(d) S n,j ( y ) = x n for y ∈ Y n +1 and j = d ( n + 1) + 1 , d ( n + 1) + 2 , . . . , l ( n + 1) . With the choices in Construction 1.6(17), the map γ n : C ( X n ) ⊕ C ( Y n ) → C ( X n +1 , M l ( n +1) ) ⊕ C ( Y n +1 , M l ( n +1) )in Construction 1.1(8), as further specified in Construction 1.1(11), is given asfollows. With C d ( n ) viewed as embedded in M d ( n ) as the diagonal matrices, thereis a homomorphism δ n : C ( Y n ) → C ( Y n +1 , C d ( n +1) ) ⊂ C ( Y n +1 , M d ( n +1) )such that γ n ( f, g ) = diag (cid:16) f ◦ Q ( n )1 , f ◦ Q ( n )2 , . . . , f ◦ Q ( n ) d ( n +1) , g ( y n ) , g ( y n ) , . . . , g ( y n ) | {z } k ( n + 1) times (cid:17) , diag (cid:16) δ n ( g ) , f ( x n ) , f ( x n ) , . . . , f ( x n ) | {z } k ( n + 1) times (cid:17)! . (1.4)For the purposes of this section, we need no further information on the maps δ n ,except that they send constant functions to constant functions. Lemma 1.7.
Assume the notation and choices in parts (1), (7), (8), and (9)of Construction 1.1, and in Construction 1.6 and the parts of Construction 1.1referred to there. Then the algebra C is simple. As the proof is a standard argument using Construction 1.6(15), we omit it.
Notation 1.8.
Let p ∈ C ( S , M ) denote the Bott projection, and let L be thetautological line bundle over S ∼ = CP . (Thus, the range of p is the section spaceof L .) Recalling that X = cone( S ), parametrized as in Notation 1.5, define b ∈ C ( X , M ) by b ( λ ) = λ · p for λ ∈ [0 , n ∈ Z ≥ set b n = (id M ⊗ Γ n, )( b, ∈ M ( C n ). ILAN HIRSHBERG AND N. CHRISTOPHER PHILLIPS
We require the following simple lemma concerning characteristic classes. It givesus a way of estimating the radius of comparison which is similar to the one used in[Vil98, Lemma 1], but more suitable for the types of estimates we need here.
Lemma 1.9.
The Cartesian product L × k does not embed in a trivial bundle over ( S ) k of rank less than k .Proof. We refer the reader to [MS74, Section 14] for an account of Chern classes.The Chern character c ( L ) is of the form 1 + ε , where ε is a generator of H ( S , Z ),and the product operation satisfies ε = 0. Let P , P , . . . , P k : ( S ) k → S bethe coordinate projections. For j = 1 , , . . . , k , set ε j = P ∗ j ( ε ). The elements ε , ε , . . . , ε k ∈ H (( S ) k , Z ), along with 1 ∈ H (( S ) k , Z ) (the standard generator)generate the cohomology ring of ( S ) k , and satisfy ε j = 0 for j = 1 , , . . . , k . Bynaturality of the Chern character ([MS74, Lemma 14.2]) and the product theorem([MS74, (14.7) on page 164]), we have c ( L × k ) = Q kj =1 (1 + ε j ). Now, suppose L × k embeds as a subbundle of a trivial bundle E . Let F be the complementary bundle,so that L × k ⊕ F = E . By the product theorem, c ( L × k ) c ( F ) = c ( L × k ⊕ F ) = c ( E ) = 1. Thus, c ( F ) = c ( L × k ) − = Q kj =1 (1 − ε j ). Since c ( F ) has a nonzero termin the top cohomology class H k (( S ) k ), it follows that rank( F ) is at least k . Thus,rank( E ) = rank( L × k ) + rank( F ) ≥ k , as required. (cid:3) Lemma 1.10.
Adopt the assumptions and notation of Notation 1.8. Let n ∈ Z > .Then b n | Z n is the orthogonal sum of a projection p n whose range is isomorphic tothe section space of the Cartesian product bundle L × s ( n ) and a constant functionof rank at most r ( n ) − s ( n ) − t ( n ) . We don’t expect b n | Z n to be a projection, since some of the point evaluationsoccurring in the maps of the direct system will be at points x ∈ cone( Z m ) \ Z m forvalues of m < n , and b m ( x ) is not a projection for such x .We don’t need the estimate on the rank of the second part of the descriptionof b n | Z n ; it is included to make the construction more explicit. If there are noevaluations at the “cone points”( Z m × { } ) / ( Z m × { } ) ∈ ( Z m × [0 , / ( Z m × { } )(following the parametrization in Notation 1.5), then this rank will be exactly r ( n ) − s ( n ) − t ( n ). Proof of Lemma 1.10.
For n ∈ Z ≥ write b n = ( c n , g n ) with c n ∈ M ( C ( X n , M r ( n ) )) and g n ∈ M ( C ( Y n , M r ( n ) )) . Further, for j = 1 , , . . . , s ( n ) let T ( n ) j : ( S ) s ( n ) → S be the j -th coordinate projec-tion. We claim that c n is an orthogonal sum c n, + c n, , in which c n, is the directsum of the functions b ◦ cone (cid:0) T ( n ) j (cid:1) for j = 1 , , . . . , s ( n ) and c n, is a constantfunction of rank at most r ( n ) − s ( n ) − t ( n ), and moreover that g n is a constantfunction of rank at most t ( n ). The statement of the lemma follows from this claim.The proof of the claim is by induction on n . The claim is true for n = 0, by thedefinition of b and since s (0) = 1, t (0) = 0, and r (0) − s (0) − t (0) = 0.Now assume that the claim is known for n , recall that Γ n +1 , n = id M l ( n +1) ⊗ γ n (see Construction 1.1(8)), and examine the summands in the description (1.4) of themap γ n (after Construction 1.6). With this convention, first take ( f, g ) in (1.4) to SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 9 be ( c n, , n +1 ,n ( c n, , is of the form required for c n +1 , ,while Γ n +1 ,n ( c n, , is a constant function of rank k ( n + 1) s ( n ) unless c n ( x n ) = 0,in which case it is zero. In the same manner, we see that: • Γ n +1 ,n ( c n, , is constant of rank at most d ( n + 1)[ r ( n ) − s ( n ) − t ( n )]. • Γ n +1 ,n ( c n, , is constant of rank at most k ( n + 1)[ r ( n ) − s ( n ) − t ( n )]. • Γ n +1 ,n (0 , g n ) is constant of rank at most k ( n + 1) t ( n ). • Γ n +1 ,n (0 , g n ) is constant of rank at most d ( n + 1) t ( n ).Putting these together, we get in the first coordinate of Γ n +1 ,n ( b n ) the direct sumof c n +1 , as described and a constant function of rank at most d ( n + 1)[ r ( n ) − s ( n ) − t ( n )] + k ( n + 1) t ( n ) . A computation shows that this expression is equal to r ( n + 1) − s ( n + 1) − t ( n + 1).In the second coordinate we get a constant function of rank at most k ( n + 1) s ( n ) + k ( n + 1)[ r ( n ) − s ( n ) − t ( n )] + d ( n + 1) t ( n ) = t ( n + 1) . This completes the induction, and the proof. (cid:3)
Corollary 1.11.
Adopt the assumptions and notation of Notation 1.8. Let n ∈ Z ≥ . Let e = ( e , e ) be an element in M ∞ ( C n ) ∼ = M ∞ ( C ( X n ) ⊕ C ( Y n )) suchthat e is a projection which is equivalent to a constant projection. If there exists x ∈ M ∞ ( C n ) such that k xex ∗ − b n k < then rank( e ) ≥ s ( n ) .Proof. Recall from Construction 1.6(13) and Notation 1.5 that Z n = ( S ) s ( n ) and Z n ⊂ cone( Z n ) = X n ⊂ X n ∐ Y n . Also recall the line bundle L and the projection p from Notation 1.8.It follows from Lemma 1.10 that there is a projection q ∈ M r ( n ) ( C ( Z n )) whoserange is isomorphic to the section space of the s ( n )-dimensional vector bundle L × s ( n ) and such that q ( b n | Z n ) q = q . Now k xex ∗ − b n k < implies k q ( xex ∗ | Z n ) q − q k < . Since e | Z n and q | Z n are projections, it follows that q | Z n is Murray-vonNeumann equivalent to a subprojection of e | Z n = e | Z n . Therefore rank( e | Z n ) ≥ s ( n ) by Lemma 1.9. So rank( e ) ≥ s ( n ). (cid:3) Although not strictly needed for the sequel, we record the following.
Corollary 1.12.
Assume the notation and choices in parts (1), (3) (including κ > ), (7), (8), and (9) of Construction 1.1, and in Construction 1.6 and the parts ofConstruction 1.1 referred to there. Then the algebra C satisfies rc( C ) ≥ κ − > .Proof. Suppose ρ < κ −
1. We show that C does not have ρ -comparison. Choose n ∈ Z > such that 1 /r ( n ) < κ − − ρ . Choose M ∈ Z ≥ such that ρ + 1
We assume the notation and choices in parts (1), (6), (7), (8),and (9) of Construction 1.1 and in Construction 1.6(17). In particular, C = C ( X ) ⊕ C ( Y ). Define q = (1 , ∈ C ( X ) ⊕ C ( Y ) and q ⊥ = 1 − q . For n ∈ Z > define q n = Γ n, ( q ) ∈ C n and q ⊥ n = 1 − q n , and finally, define q = Γ ∞ , ( q ) ∈ C and q ⊥ = 1 − q . Lemma 1.14.
Under the assumptions in Notation 1.13, the projection − q n ∈ M l ( n ) ( C ( X n )) ⊕ M l ( n ) ( C ( Y n )) has the form ( e, f ) for a trivial projection e ∈ M l ( n ) ( C ( X n )) of rank t ( n ) and aprojection f ∈ M l ( n ) ( C ( Y n )) of rank r ( n ) − t ( n ) .Proof. The proof is an easy induction argument. (cid:3)
Lemma 1.15.
Assume the notation and choices in parts (1)–(9) of Construction1.1, Construction 1.6, and Notation 1.13, including k ( n ) < d ( n ) for all n ∈ Z ≥ , κ > , ω > ω ′ , and κ − > ω . Then rc( q ⊥ Cq ⊥ ) ≥ κ − ω . Proof.
We proceed as in the proof of Corollary 1.12, although the rank computa-tions are somewhat more involved. The difference is in the definition of d τ . In thiscorner, d τ is normalized so that d τ ( q ⊥ ) = 1 for all τ ∈ T( C ). To avoid redefiningthe notation, we will use τ to denote a tracial state on C , and therefore our dimen-sion functions will be of the form a d τ ( a ) /τ ( q ⊥ ), noting that τ ( q ⊥ ) = d τ ( q ⊥ )since q ⊥ is a projection.It suffices to show that for all ρ ∈ (cid:0) , κ − ω (cid:1) ∩ Q , we have rc( q ⊥ Cq ⊥ ) ≥ ρ .Fix δ ∈ (0 , ω ) such that(1.5) ρ < (1 − δ ) (cid:18) κ − ω (cid:19) . Set(1.6) ε = δ ρ (1 − δ ) > . Since the sequence (cid:16) s ( n ) r ( n ) (cid:17) n =0 , , ,... is nonincreasing and converges to a nonzerolimit κ , there exists n ∈ Z ≥ such that for all n and m with m ≥ n ≥ n , we have0 ≤ − r ( n ) s ( n ) · s ( m ) r ( m ) < ε . This implies that(1.7) r ( m ) r ( n ) − s ( m ) s ( n ) < ε · r ( m ) r ( n ) . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 11 Using (1.5) and δ < ω at the first step, we get1 − ω + 2 ρω < − δ + 2(1 − δ ) (cid:18) κ − ω (cid:19) ω = 2 κ (1 − δ ) . Now write ρ = α/β with α, β ∈ Z > . Choose n ≥ n such that βr ( n ) < κ (1 − δ ) − (1 − ω + 2 ρω ) . Then there exists N ∈ Z > such that ρN ∈ Z > and(1.8) 2 κ (1 − δ ) > N r ( n ) > − ω + 2 ρω . Set(1.9) N = ρN . Using ρ > N r ( n ) = ρN r ( n ) > ρ (1 − ω + 2 ρω ) > ρ (1 − ω ) + 2 ω . Now suppose e ∈ M ∞ ( C n ) = M ∞ (cid:0) C ( X n ) ⊕ C ( Y n ) (cid:1) is an ordered pair whosefirst component is a trivial projection on X n of rank N and whose second com-ponent is a (trivial) projection on Y n of rank N . Let m > n , and let f be thefirst component of Γ m,n ( e ); we estimate rank( f ). (The second component is atrivial projection over Y m whose rank we don’t care about.) Now f is the directsum of r ( m ) /r ( n ) trivial projections, coming from C ( X n , M r ( n ) ) and C ( Y n , M r ( n ) ).At least s ( m ) /s ( n ) of these summands come from C ( X n , M r ( n ) ). So at most r ( m ) /r ( n ) − s ( m ) /s ( n ) of these summands come from C ( Y n , M r ( n ) ). The sum-mands coming from C ( X n , M r ( n ) ) have rank N and the summands coming from C ( Y n , M r ( n ) ) have rank N . Since N > N , we getrank( f ) ≤ (cid:18) r ( m ) r ( n ) − s ( m ) s ( n ) (cid:19) N + s ( m ) s ( n ) · N = r ( m ) r ( n ) · N + (cid:18) r ( m ) r ( n ) − s ( m ) s ( n ) (cid:19) ( N − N ) . Combining this with (1.7) at the first step, and using (1.9) at the second step, (1.6)at the third step, (1.8) at the fifth step, and Construction 1.1(3) at the sixth step,we get rank( f ) < r ( m ) r ( n ) · ( N + εN ) = r ( m ) r ( n ) · (1 + ερ ) · N = r ( m ) r ( n ) · − δ − δ ) · N < r ( m ) r ( n ) · N − δ < κr ( m ) ≤ s ( m ) . So Corollary 1.11 implies that there is no x ∈ M ∞ ( C m ) for which k x Γ n,m ( e ) x ∗ − b m k < . Since m > n is arbitrary,(1.10) Γ ∞ ,n ( e ) - b . Now let τ be a trace on C , and restrict it to C n ∼ = M r ( n ) (cid:0) C ( X n ) ⊕ C ( Y n ) (cid:1) .Denote by tr the normalized trace on M r ( n ) . There is a probability measure µ on X n ∐ Y n such that τ ( a ) = R X n ∐ Y n tr( a ) dµ for all a ∈ C n . Define λ = µ ( X n ), so1 − λ = µ ( Y n ). Then, using (1.9) at the second step τ ( e ) = λN + (1 − λ ) N r ( n ) = [ λ + ρ (1 − λ )] N r ( n ) . Using Lemma 1.14 to calculate the ranks of the components of q ⊥ n , we get(1.11) τ ( q ⊥ n ) = λt ( n ) + (1 − λ )[ r ( n ) − t ( n )] r ( n )and(1.12) τ ( q n ) = 1 − τ ( q ⊥ n ) = λ [ r ( n ) − t ( n )] + (1 − λ ) t ( n ) r ( n ) . It follows from Lemma 1.10 and Lemma 1.14 that d τ ( b n ) ≤ τ ( q n ). Using this atthe first step, and (1.11) and (1.12) at the second step, we get d τ ( b n ) τ ( q ⊥ n ) ≤ τ ( q n ) τ ( q ⊥ n ) = λ [ r ( n ) − t ( n )] + (1 − λ ) t ( n ) λt ( n ) + (1 − λ )[ r ( n ) − t ( n )] . So τ ( e ) − d τ ( b n ) τ ( q ⊥ n ) ≥ (cid:0) λ + ρ (1 − λ ) (cid:1) N − (cid:0) λ [ r ( n ) − t ( n )] + (1 − λ ) t ( n ) (cid:1) λt ( n ) + (1 − λ )[ r ( n ) − t ( n )] . The last expression is a fractional linear function in λ , and is defined for allvalues of λ in the interval [0 , , ω ≤ t ( n ) r ( n ) < ω . If we set λ = 1 and use (1.8), the value we obtain is N /r ( n ) − (1 − t ( n ) /r ( n )) t ( n ) /r ( n ) > (1 − ω + 2 ρω ) − (1 − ω )2 ω = ρ . If we set λ = 0, we get, using (1.8) at the first step and ρ > ρN /r ( n ) − t ( n ) /r ( n )1 − t ( n ) /r ( n ) > ρ (1 − ω + 2 ρω ) − ω − ω = ρ + 2 ρ ω − ω − ω > ρ . Therefore d τ (Γ ∞ ,n ( e )) d τ ( q ⊥ ) > d τ ( b ) d τ ( q ⊥ ) + ρ for all traces τ on C , so rc( q ⊥ Cq ⊥ ) > ρ , as required. (cid:3) We now turn to the issue of finding upper bounds on the radius of comparison.For this, we appeal to results of Niu from [Niu14]. Niu introduced a notion of meandimension for a diagonal AH-system, [Niu14, Definition 3.6]. Suppose we are givena direct system of homogeneous algebras of the form A n = C ( K n, ) ⊗ M j n, ⊕ C ( K n, ) ⊗ M j n, ⊕ · · · ⊕ C ( K n,m ( n ) ) ⊗ M j n,m ( n ) , in which each of the spaces involved is a connected finite CW complex, and theconnecting maps are unital diagonal maps. Let γ denote the mean dimension ofthis system, in the sense of Niu. It follows trivially from [Niu14, Definition 3.6]that γ ≤ lim n →∞ max (cid:18)(cid:26) dim( K n,l ) j n,l | l = 1 , , . . . , m ( n ) (cid:27)(cid:19) . Theorem 6.2 of [Niu14] states that if A is the direct limit of a system as above,then rc( A ) ≤ γ/
2. Since the system we are considering here is of this type, Niu’s
SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 13 theorem applies. With that at hand, we can derive an upper bound for the radiusof comparison of the complementary corner. Lemma 1.16.
Under the same assumptions as in Lemma 1.15, we have rc( qCq ) ≤ − ω . Proof.
By the discussion above,rc( qCq ) ≤
12 lim n →∞ max (cid:18) dim( X n )rank( q n | X n ) , dim( Y n )rank( q n | Y n ) (cid:19) . As dim( Y n ) = 1 for all n , the second term converges to 0. As for the first term, byConstruction 1.6(13), we have dim( X n ) = 2 s ( n )+1. Also, rank( q n | X n ) = r ( n ) − t ( n )by Lemma 1.14. Thus, using Construction 1.1(1), Lemma 1.4, and d ( n ) → ∞ (which follows from Construction 1.1(4)) at the last step,lim n →∞ dim( X n )rank( q n | X n ) = lim n →∞ s ( n ) + 1 r ( n ) − t ( n ) ≤ lim n →∞ r ( n ) + 1 r ( n ) − t ( n ) ≤ − ω . This gives us the required estimate. (cid:3)
Lemma 1.17.
Let the assumptions and notation be as in Notation 1.13, Construc-tion 1.6(13), and Construction 1.6(14). If e ∈ C is a projection which has the samesame K -class as q then e is unitarily equivalent to q . The same holds with q ⊥ inplace of q .Proof. This can be seen directly from the construction. For each n ∈ Z ≥ , since X n and Y n are contractible (Construction 1.6(13) and Construction 1.6(14)), if e ∈ M ∞ ( C n ) is a projection which has the same K -class as q , then e is actuallyunitarily equivalent to q n . The same holds for q ⊥ n . It follows that this is the casein C as well. (cid:3) We point out that this lemma can also be deduced using cancellation. By[EHT09, Theorem 4.1], simple unital AH algebras which arise from AH systemswith diagonal maps have stable rank 1. Rieffel has shown that C ∗ -algebras withstable rank 1 have cancellation; see [Bla98, Theorem 6.5.1].2. The tracial state space
For a compact Hausdorff space X , we will need all of C ( X, R ), the tracial statespace of C ( X ) (and of C ( X, M n )), and the space of affine functions on the tracialstate space. This last space is an order unit space, and much of our work will bedone there.For later reference, we recall some of the definitions, and then describe how tomove between these spaces. We begin with the definition of an order unit spacefrom the discussion before Proposition II.1.3 of [Alf71]. We suppress the order unitin our notation, since (except in several abstract results) our order unit spaces willalways be sets of affine continuous functions on compact convex sets with orderunit the constant function 1. Definition 2.1. An order unit space V is a partially ordered real Banach space(see page 1 of [Goo86] for the axioms of a partially ordered real vector space) whichis Archimedean (if v ∈ V and { λv | λ ∈ (0 , ∞ ) } has an upper bound, then v ≤ with a distinguished element e ∈ V which is an order unit (that is, for every v ∈ V there is λ ∈ (0 , ∞ ) such that − λe ≤ v ≤ λe ), and such that the norm on V satisfies k v k = inf (cid:0)(cid:8) λ ∈ (0 , ∞ ) | − λe ≤ v ≤ λe (cid:9)(cid:1) for all v ∈ V .The morphisms of order unit spaces are the positive linear maps which preservethe order units.The morphisms of compact convex sets (compact convex subsets of locally convextopological vector spaces) are just the continuous affine maps. Definition 2.2. If K is a compact convex set, we denote by Aff( K ) the order unitspace of continuous affine functions f : K → R , with the supremum norm and withorder unit the constant function 1.If K and L are compact convex sets and λ : K → L is continuous and affine, welet λ ∗ : Aff( L ) → Aff( K ) be the positive linear order unit preserving map given by λ ∗ ( f ) = f ◦ λ for f ∈ Aff( L ).This definition makes K Aff( K ) a functor. Definition 2.3. If V is an order unit space with order unit e , we denote by S ( V )(or S ( V, e ) if e is not understood) its state space (the order unit space morphismsto ( R , W is another order unit space and ϕ : V → W is positive, linear, and orderunit preserving, we let S ( ϕ ) : S ( W ) → S ( V ) be the continuous affine map given by S ( ϕ )( ω ) = ω ◦ ϕ for ω ∈ S ( W ).This definition makes V S ( V ) a functor. Theorem 2.4 (Theorem 7.1 of [Goo86]) . There is a natural isomorphism S (Aff( K )) ∼ = K for compact convex sets K , given by sending x ∈ K to the evaluation map ev x : Aff( K ) → R defined by ev x ( f ) = f ( x ) for f ∈ Aff( K ) . Definition 2.5.
For a unital C*-algebra A , we denote its tracial state space byT( A ).If A and B are unital C ∗ -algebras and ϕ : A → B is a unital homomorphism, welet T( ϕ ) : T( B ) → T( A ) be the continuous affine map given by T( ϕ )( τ ) = τ ◦ ϕ for τ ∈ T( B ). We let b ϕ : Aff(T( A )) → Aff(T( B )) be the positive order unit preservingmap given by b ϕ ( f ) = f ◦ T( ϕ ) for f ∈ Aff(T( A )). (Thus, b ϕ = T( ϕ ) ∗ .) Lemma 2.6.
Let X be a compact Hausdorff space. Then C ( X, R ) , with the supre-mum norm and distinguished element the constant function , is a complete orderunit space. Restriction of tracial states on C ( X ) is an affine homeomorphism from T( C ( X )) to S ( C ( X, R )) . The map from X to S ( C ( X, R )) which sends x ∈ X to the point evaluation ev x : C ( X, R ) → R is a homeomorphism onto its image,and the map R X : Aff (cid:0) S ( C ( X, R )) (cid:1) → C ( X, R ) , given by R X ( f )( x ) = f (ev x ) for f ∈ Aff (cid:0) S ( C ( X, R )) (cid:1) and x ∈ X , is an isomorphism of order unit spaces.If Y is another compact Hausdorff space, then the function which sends a positivelinear order unit preserving map Q : C ( X, R ) → C ( Y, R ) to S ( Q ) : S ( C ( Y, R )) → S ( C ( X, R )) is a bijection to the continuous affine maps from S ( C ( Y, R )) to S ( C ( X, R )) .Its inverse is the map E given as follows. For a continuous affine map λ : S ( C ( Y, R )) → S ( C ( X, R )) , define E ( λ ) : C ( X, R ) → C ( Y, R ) by E ( λ ) = R Y ◦ λ ∗ ◦ R − X . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 15 A positive linear order unit preserving map from C ( X, R ) to C ( Y, R ) is called a Markov operator . Proof of Lemma 2.6.
It is immediate that C ( X, R ) is a complete order unit space.The identification of S ( C ( X, R )) is also immediate. The fact that R X is bijectivefollows from [Goo86, Corollary 11.20] using the identification of X with the extremepoints of S ( C ( X, R )).The second paragraph is immediate. (cid:3) Direct limits of direct systems of order unit spaces are constructed at the begin-ning of Section 3 of [Tho94], including Lemma 3.1 there.
Proposition 2.7.
Let (cid:0) ( D n ) n =0 , , ,... , ( ϕ n,m ) ≤ m ≤ n (cid:1) be a direct system of unital C ∗ -algebras and unital homomorphisms. Set D = lim −→ n D n . Then there are anatural homeomorphism T( D ) → lim ←− n T( D n ) and a natural isomorphism Aff(T( D )) → lim −→ n Aff(T( D n )) of order unit spaces.Proof. The first part is Lemma 3.3 of [Tho94].The second part is Lemma 3.2 of [Tho94], combined with the fact (Theorem 2.4)that the state space of Aff( K ) is naturally identified with K . (cid:3) Definition 2.8.
Let V and W be order unit spaces, with order units e ∈ V and f ∈ W . We define the direct sum V ⊕ W to be the vector space direct sum V ⊕ W as a real vector space, with the order ( v , w ) ≤ ( v , w ) for v , v ∈ V and w , w ∈ W if and only if v ≤ v and w ≤ w , with the order unit ( e, f ), and thenorm k ( v, w ) k = max( k v k , k w k ). Lemma 2.9.
Let V and W be order unit spaces. Then V ⊕ W as in Definition 2.8is an order unit space, which is complete if V and W are.Proof. The proof is straightforward. (cid:3)
Lemma 2.10.
Let A and B be unital C ∗ -algebras. Then, taking the direct sum onthe right to be as in Definition 2.8, there is an isomorphism Aff(T( A ⊕ B )) ∼ = Aff(T( A )) ⊕ Aff(T( B )) , given as follows. Identify T( A ) with a subset of T( A ⊕ B ) by, for τ ∈ T( A ) ,defining i ( τ )( a, b ) = τ ( a ) for all a ∈ A and b ∈ B , and similarly identify T( B ) witha subset of T( A ⊕ B ) . Then the map Aff(T( A ⊕ B )) → Aff(T( A )) ⊕ Aff(T( B )) is f ( f | T( A ) , f | T( B ) ) .Proof. It is clear that if f ∈ Aff(T( A ⊕ B )), then f | T( A ) ∈ Aff(T( A )) and f | T( B ) ∈ Aff(T( B )), and moreover that the map of the lemma is linear, positive, and pre-serves the order units. One easily checks that every tracial state on A ⊕ B is aconvex combination of tracial states on A and B , from which it follows that if f | T( A ) = 0 and f | T( B ) = 0 then f = 0. It remains to prove that the map of the lemma is surjective. Let g ∈ Aff(T( A ))and h ∈ Aff(T( B )). Define f : T( A ⊕ B ) → R by, for τ ∈ T( A ⊕ B ), f ( τ ) = τ (1 , g (cid:0) τ (1 , − τ | A (cid:1) + τ (0 , g (cid:0) τ (0 , − τ | B (cid:1) (taking the first summand to be zero if τ (1 ,
0) = 0 and the second summand to bezero if τ (0 ,
1) = 0). Straightforward but somewhat tedious calculations show that f is weak* continuous and affine, and clearly f | T( A ) = g and f | T( B ) = h . (cid:3) The following result generalizes Lemma 3.4 of [Tho94]. It still isn’t the mostgeneral Elliott approximate intertwining result for order unit spaces, because weassume that the underlying order unit spaces of the two direct systems are thesame. The main effect of this assumption is to simplify the notation.
Proposition 2.11.
Let ( V m ) m =0 , , ,... be a sequence of separable complete orderunit spaces, and let (cid:0) ( V m ) m =0 , , ,... , ( ϕ n,m ) ≤ m ≤ n (cid:1) and (cid:0) ( V m ) m =0 , , ,... , ( ϕ ′ n,m ) ≤ m ≤ n (cid:1) be two direct systems of order unit spaces, using the same spaces, and with maps ϕ n,m , ϕ ′ n,m : V m → V n which are linear, positive, and preserve the order units. Let V and V ′ be the direct limits V = lim −→ n (cid:0) ( V m ) m =0 , , ,... , ( ϕ n,m ) ≤ m ≤ n (cid:1) and V ′ = lim −→ n (cid:0) ( V m ) m =0 , , ,... , ( ϕ ′ n,m ) ≤ m ≤ n (cid:1) , with corresponding maps ϕ ∞ ,n : V n → V and ϕ ′∞ ,n : V n → V ′ for n ∈ Z ≥ . For n ∈ Z ≥ further let v ( n )0 , v ( n )1 , . . . ∈ V n be a dense sequence in the closed unit ball of V n , and define F n ⊂ V n to be the finiteset F n = n [ m =0 h(cid:8) ϕ n,m (cid:0) v ( m ) k (cid:1) : 0 ≤ k ≤ n (cid:9) ∪ (cid:8) ϕ ′ n,m (cid:0) v ( m ) k (cid:1) : 0 ≤ k ≤ n (cid:9)i . Suppose that there are δ , δ , . . . ∈ (0 , ∞ ) such that (2.1) ∞ X n =0 δ n < ∞ and for all n ∈ Z ≥ and all v ∈ F n we have k ϕ n +1 , n ( v ) − ϕ ′ n +1 , n ( v ) k < δ n . Then there is a unique isomorphism ρ : V → V ′ such that for all m ∈ Z ≥ and all v ∈ V m we have ρ ( ϕ ∞ ,m ( v )) = lim n →∞ ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) . Its inverse is determined by ρ − ( ϕ ′∞ ,m ( v )) = lim n →∞ ( ϕ ∞ ,n ◦ ϕ ′ n,m )( v ) for m ∈ Z ≥ and v ∈ V m . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 17 Proof.
We first claim that for m ∈ Z ≥ and v ∈ F m , the sequence (cid:0) ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) (cid:1) n ≥ m is a Cauchy sequence in V ′ . For n ≥ m , we estimate, using k ϕ ′∞ ,n +1 k ≤ k v k ≤
1, and ϕ n,m ( v ) ∈ F n at the last step: (cid:13)(cid:13) ( ϕ ′∞ ,n +1 ◦ ϕ n +1 , m )( v ) − ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) (cid:13)(cid:13) = (cid:13)(cid:13) ( ϕ ′∞ ,n +1 ◦ ϕ n +1 , n ◦ ϕ n,m )( v ) − ( ϕ ′∞ ,n +1 ◦ ϕ ′ n +1 , n ◦ ϕ n,m )( v ) (cid:13)(cid:13) ≤ k ϕ ′∞ ,n +1 k (cid:13)(cid:13) ϕ n +1 , n ( ϕ n,m ( v )) − ϕ ′ n +1 , n ( ϕ n,m ( v )) (cid:13)(cid:13) ≤ δ n . The claim now follows from (2.1).Next, we claim that for m ∈ Z ≥ and k ∈ Z > , the sequence (cid:0) ( ϕ ′∞ ,n ◦ ϕ n,m ) (cid:0) v ( m ) k (cid:1)(cid:1) n ≥ m is a Cauchy sequence in V ′ . Indeed, taking m = max( m, k ), this follows from theprevious claim and the fact that ϕ m ,m (cid:0) v ( m ) k (cid:1) ∈ F m .Now we claim that for m ∈ Z ≥ and v ∈ V m , the sequence (( ϕ ′∞ ,n ◦ ϕ n,m )( v )) n ≥ m is a Cauchy sequence in V ′ . Without loss of generality k v k ≤
1. This claim followsfrom a standard ε argument: to show that (cid:13)(cid:13) ( ϕ ′∞ ,n ◦ ϕ n ,m )( v ) − ( ϕ ′∞ ,n ◦ ϕ n ,m )( v ) (cid:13)(cid:13) < ε for all sufficiently large n and n , choose k ∈ Z > such that (cid:13)(cid:13) v − v ( m ) k (cid:13)(cid:13) < ε , anduse the previous claim.Since V ′ is complete, it follows that lim n →∞ ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) exists for all m ∈ Z ≥ and k ∈ Z > . Since k ϕ ′∞ ,n ◦ ϕ n,m k ≤ m, n ∈ Z ≥ satisfy m ≤ n , itfollows that for m ∈ Z > there is a unique bounded linear map ρ m : V m → V ′ suchthat k ρ m k ≤ ρ m ( v ) = lim n →∞ ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) for all k ∈ Z > .It is clear from the construction that ρ n ◦ ϕ n,m = ρ m whenever m, n ∈ Z ≥ satisfy m ≤ n . By the universal property of the direct limit, there is a uniquebounded linear map ρ : V → V ′ such that ρ ◦ ϕ ∞ ,m = ρ m for all m ∈ Z ≥ . It isclearly contractive, order preserving, and preserves the order units, and is uniquelydetermined as in the statement of the proposition.The same argument shows that there is a unique contractive linear map λ : V ′ → V determined in the analogous way. For all m ∈ Z ≥ , we have λ ◦ ρ ◦ ϕ ∞ ,m = λ ◦ ϕ ′∞ ,m = ϕ ∞ ,m , so the universal property of the direct limit implies λ ◦ ρ = id V . Similarly ρ ◦ λ =id V ′ . (cid:3) Proposition 2.12.
The isomorphism of Proposition 2.11 has the following natu-rality property. Let the notation be as there, and suppose that, in addition, we aregiven separable complete order unit spaces W n for n ∈ Z ≥ , direct systems (cid:0) ( W m ) m =0 , , ,... , ( ψ n,m ) ≤ m ≤ n (cid:1) and (cid:0) ( W m ) m =0 , , ,... , ( ψ ′ n,m ) ≤ m ≤ n (cid:1) using the same spaces, with positive linear order unit preserving maps, with directlimits W and W ′ , and with corresponding maps ψ ∞ ,n : W n → W and ψ ′∞ ,n : W n → W ′ for n ∈ Z ≥ . Also suppose that for n ∈ Z > there is a sequence w ( n )0 , w ( n )1 , . . . ∈ W n which is dense in the closed unit ball of W n , and that there is a sequence ( ε n ) n =0 , , ,... in (0 , ∞ ) such that P ∞ n =0 ε n < ∞ and, with G n = n [ m =0 h(cid:8) ψ n,m (cid:0) w ( m ) k (cid:1) | ≤ k ≤ n (cid:9) ∪ (cid:8) ψ ′ n,m (cid:0) w ( m ) k (cid:1) | ≤ k ≤ n (cid:9)i , for all n ∈ Z ≥ and all w ∈ G n we have k ψ n +1 , n ( w ) − ψ ′ n +1 , n ( w ) k < ε n . Let σ : W → W ′ be the isomorphism of Proposition 2.11. Suppose further that wehave positive linear order unit preserving maps µ n , µ ′ n : V n → W n for n ∈ Z ≥ suchthat µ n ◦ ϕ n,m = ψ n,m ◦ µ m and µ ′ n ◦ ϕ ′ n,m = ψ ′ n,m ◦ µ ′ m for all m, n ∈ Z ≥ with m ≤ n . Let µ : V → W and µ ′ : V ′ → W ′ be the inducedmaps of the direct limits. Then µ ′ ◦ ρ = σ ◦ µ .Proof. By construction, ρ : V → V ′ and σ : W → W ′ are determined by(2.2) ρ ( ϕ ∞ ,m ( v )) = lim n →∞ ( ϕ ′∞ ,n ◦ ϕ n,m )( v )for m ∈ Z ≥ and v ∈ V m , and(2.3) σ ( ψ ∞ ,m ( w )) = lim n →∞ ( ψ ′∞ ,n ◦ ψ n,m )( w )for m ∈ Z ≥ and w ∈ W m . Using (2.2) at the first step and (2.3) at the last step,for m ∈ Z ≥ and v ∈ V m we therefore have( µ ′ ◦ ρ )( ϕ ∞ ,m ( v )) = µ ′ (cid:16) lim n →∞ ( ϕ ′∞ ,n ◦ ϕ n,m )( v ) (cid:17) = lim n →∞ ( µ ′ ◦ ϕ ′∞ ,n ◦ ϕ n,m )( v )= lim n →∞ ( ψ ′∞ ,n ◦ ψ n,m ◦ µ m )( v ) = ( σ ◦ µ )( ϕ ∞ ,m ( v )) . Since S ∞ m =0 ϕ ∞ ,m ( V m ) is dense in V , the result follows. (cid:3) Proposition 2.14 below can essentially be extracted from the proof of Lemma 3.7of [Tho94]. We give here a precise formulation which is needed for our purposes.The difference between our formulation and that which appears in [Tho94] is thatwe need to have more control over the matrix sizes in the construction. In theargument, the following result substitutes for Lemma 3.6 there.
Lemma 2.13 (Based on [Tho94]) . Let X and Y be compact Hausdorff spaces,with X path connected. Let λ : T( C ( Y )) → T( C ( X )) be affine and continuous. Let E ( λ ) : C ( X, R ) → C ( Y, R ) be as in Lemma 2.6. Then for every ε > and everyfinite set F ⊂ C ( X, R ) there exists N ∈ Z > such that for every N ∈ Z > with N ≥ N there are continuous functions g , g , . . . , g N : Y → X such that for every f ∈ F we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( λ )( f ) − N N X j =1 f ◦ g j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < ε . Proof.
It suffices to prove the result under the additional assumption that k f k ≤ f ∈ F . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 19 Let ε >
0. Since E ( λ ) is a Markov operator, Theorem 2.1 of [Tho94] provides n ∈ Z > , unital homomorphisms ψ , ψ , . . . , ψ n : C ( X ) → C ( Y ), and α , α , . . . , α n ∈ [0 ,
1] with P nl =1 α l = 1 such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( λ )( f ) − n X l =1 α l ψ l ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < ε f ∈ F . Note that if β , β , . . . , β n ∈ [0 ,
1] satisfy P nl =1 | α l − β l | < ε then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( λ )( f ) − n X l =1 β l ψ l ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < ε for all f ∈ F . Choose N ∈ Z > such that N > n/ε . Let N ∈ Z > satisfy N ≥ N .For l = 1 , , . . . , n − β l ∈ (cid:0) α l − N , α l (cid:3) ∩ N Z , and set β n = 1 − P n − l =1 β l .Then β , β , . . . , β n ∈ N Z ≥ , n X l =1 β l = 1 , and n X l =1 | α l − β l | < ε . Set m l = N β l for l = 1 , , . . . , n . Then for all f ∈ F we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( λ )( f ) − N n X l =1 m l ψ l ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < ε . Now for l = 1 , , . . . , n let h l : Y → X be the continuous function such that ψ l ( f ) = f ◦ h l for all f ∈ C ( X ), and for j = 1 , , . . . , N define g j = h l when l − X k =1 m k < j ≤ l X k =1 m k . Then 1 N n X l =1 m l ψ l ( f ) = 1 N N X j =1 f ◦ g j for all f ∈ C ( X ). (cid:3) Proposition 2.14.
Let K be a metrizable Choquet simplex, and let ( l ( n )) n =0 , , ,... be a sequence of integers such that l ( n ) ≥ for all n > . For n ∈ Z ≥ set r ( n ) = Q nj =1 l ( j ) . Then there exist n < n < n < · · · ∈ Z > and a direct system C ([0 , ⊗ M r ( n ) α , −→ C ([0 , ⊗ M r ( n ) α , −→ C ([0 , ⊗ M r ( n ) α , −→ · · · with injective maps which are diagonal (in the sense analogous to Construction1.1(11)) and such that the direct limit A is simple and T( A ) ∼ = K .Proof. We mostly follow the proof of Lemma 3.7 of [Tho94], using Lemma 2.13in place of Lemma 3.6 of [Tho94], and slightly changing the order of the stepsto accommodate the difference between our conclusion and that of Theorem 3.9of [Tho94]. For convenience, we will use Proposition 2.11 in place of Lemma 3.4of [Tho94].For convenience of notation, and following [Tho94], set P = T( C ([0 , (cid:0) ( P k ) k =0 , ,... , ( λ j,k ) ≤ j ≤ k (cid:1) with continu-ous affine maps λ j,k : P k → P j such that P k = P for all k ∈ Z ≥ and(2.4) lim ←− (cid:0) ( P k ) k =0 , ,... , ( λ j,k ) ≤ j ≤ k (cid:1) ∼ = K. Choose f , f , . . . ∈ C ([0 , , R ) such that { f , f , . . . } is dense in C ([0 , , R ).We now construct numbers n k ∈ Z > for k ∈ Z ≥ , finite subsets F k ⊂ C ([0 , , R )for k ∈ Z ≥ , positive unital linear maps ψ k +1 ,k : C ([0 , , R ) → C ([0 , , R ) for k ∈ Z > , and continuous functions g k, , g k, , . . . , g k, r ( n k +1 ) /r ( n k ) : [0 , → [0 , , such that the following conditions are satisfied:(1) F = { f } and for k ∈ Z ≥ , F k +1 = F k ∪ { f k +1 } ∪ E ( λ k,k +1 )( F k ∪ { f k +1 } ) ∪ ψ k +1 ,k ( F k ∪ { f k +1 } ) . (2) n k +1 > n k and r ( n k +1 ) /r ( n k ) > k for k ∈ Z ≥ .(3) For k ∈ Z ≥ and f ∈ C ([0 , , R ), ψ k +1 ,k ( f ) = r ( n k ) r ( n k +1 ) r ( n k +1 ) /r ( n k ) X l =1 f ◦ g k,l . (4) (cid:13)(cid:13) E ( λ k,k +1 )( f ) − ψ k +1 ,k ( f ) (cid:13)(cid:13) < − k for k ∈ Z ≥ and f ∈ F k .We carry out the construction by induction on k . Define F = { f } and n = 1.Suppose we have F k and n k ; we construct F k +1 , n k +1 , g k, , g k, , . . . , g k, r ( n k +1 ) /r ( n k ) , and ψ k +1 ,k . Apply Lemma 2.13 with λ = λ k,k +1 , with ε = 2 − k , and with F = F k , obtaining N ∈ Z > . Choose n k +1 > n k and so large that r ( n k +1 ) r ( n k ) > max (cid:0) N , k (cid:1) . This gives (2). Apply the conclusion of Lemma 2.13 with N = r ( n k +1 ) /r ( n k ),calling the resulting functions g k, , g k, , . . . , g k, r ( n k +1 ) /r ( n k ) . This gives (4). Thendefine ψ k +1 ,k by (3) and define F k +1 by (1). This completes the induction.For j, k ∈ Z ≥ with j ≤ k , define ψ k,j : C ([0 , , R ) → C ([0 , , R ) by ψ k,j = ψ k, k − ◦ ψ k − , k − ◦ · · · ◦ ψ j +1 , j . An induction argument shows that for j, k ∈ Z ≥ with j ≤ k , we have E ( λ j,k )( f j ) ∈ F k and ψ k,j ( f j ) ∈ F k . This condition, together with Proposition 2.11, allows us to conclude that, as orderunit spaces, we havelim −→ (cid:0) ( C ([0 , , R )) k =0 , ,... , ( E ( λ j,k )) ≤ j ≤ k (cid:1) (2.5) ∼ = lim −→ (cid:0) ( C ([0 , , R )) k =0 , ,... , ( ψ k,j ) ≤ j ≤ k (cid:1) . For k ∈ Z ≥ define h k, , h k, , . . . , h k, r ( n k +1 ) /r ( n k ) : [0 , → [0 , t ∈ [0 , h k,l ( t ) = g k,l ( t ) l = 1 , , . . . , r ( n k +1 ) /r ( n k ) − t/ l = r ( n k +1 ) /r ( n k ) − t + 1) / l = r ( n k +1 ) /r ( n k ) . Then define α k +1 , k : C ([0 , , M r ( n k ) ) → C ([0 , , M r ( n k +1 ) ) = M r ( n k +1 ) /r ( n k ) (cid:0) C ([0 , , M r ( n k ) ) (cid:1) SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 21 by α k +1 , k ( f ) = diag (cid:0) f ◦ h k, , f ◦ h k, , . . . , f ◦ h k, r ( n k +1 ) /r ( n k ) (cid:1) for f ∈ C ([0 , , M r ( n k ) ). Let A be the resulting direct limit C ∗ -algebra. UsingLemma 3.5 of [Tho94] to identify the maps E ( α k +1 ,k ), and recalling Definition 2.5,we get (cid:13)(cid:13) ψ k +1 ,k ( f ) − \ α k +1 ,k ( f ) (cid:13)(cid:13) ∞ ≤ (cid:18) k − (cid:19) k f k ∞ for all f ∈ C ([0 , , R ). Let V and W be the order unit spaces defined by V = lim −→ (cid:0) ( C ([0 , , R )) k =0 , ,... , ( E ( λ j,k )) ≤ j ≤ k (cid:1) and W = lim −→ (cid:0) ( C ([0 , , R )) k =0 , ,... , ( d α k,j ) ≤ j ≤ k (cid:1) . Proposition 2.11 and (2.5) imply V ∼ = W . Lemma 3.2 of [Tho94] and (2.4) implythat V ∼ = Aff( K ). Lemma 3.3 of [Tho94] implies that Aff( T ( A )) ∼ = Aff( K ). So T ( A ) ∼ = K by Theorem 2.4. It is easy to see that A is simple by verifying thehypotheses of Lemma 1.2 of [Tho94]. (cid:3) Proposition 2.15.
Let ( D n ) n =0 , , ,... and ( C n ) n =0 , , ,... be sequences of unital C ∗ -algebras. Let (cid:0) ( D n ) n =0 , , ,... , ( ϕ n,m ) ≤ m ≤ n (cid:1) and (cid:0) ( D n ) n =0 , , ,... , ( ϕ ′ n,m ) ≤ m ≤ n (cid:1) and (cid:0) ( C n ) n =0 , , ,... , ( ψ n,m ) ≤ m ≤ n (cid:1) and (cid:0) ( C n ) n =0 , , ,... , ( ψ ′ n,m ) ≤ m ≤ n (cid:1) be direct systems with unital homomorphisms, and call the direct limits (in order) D , D ′ , C , and C ′ . Suppose further that we have unital homomorphisms µ n , µ ′ n : D n → C n for n ∈ Z ≥ such that µ n ◦ ϕ n,m = ψ n,m ◦ µ n and µ ′ n ◦ ϕ ′ n,m = ψ ′ n,m ◦ µ ′ n for all m, n ∈ Z ≥ with m ≤ n . Let µ : D → C and µ ′ : D ′ → C ′ be the inducedmaps of the direct limits. Assume that for all m ∈ Z ≥ we have ∞ X n = m (cid:13)(cid:13) [ ϕ n,m − [ ϕ ′ n,m (cid:13)(cid:13) < ∞ and ∞ X n = m (cid:13)(cid:13) [ ψ n,m − [ ψ ′ n,m (cid:13)(cid:13) < ∞ . Then there exist isomorphisms ρ : Aff(T( D )) → Aff(T( D ′ )) and σ : Aff(T( C )) → Aff(T( C ′ )) such that b µ ′ ◦ ρ = σ ◦ b µ .Proof. We can apply Proposition 2.11 and Proposition 2.12 using arbitrary count-able dense subsets of the closed unit balls of Aff(T( D n )) and Aff(T( C n )) for n ∈ Z > . (cid:3) Lemma 2.16.
Adopt the notation of Construction 1.1, including (10) (a sec-ond set of maps), and (11) and (12) (diagonal maps, agreeing in the coordinates , , . . . , d ( n ) ). Then (cid:13)(cid:13) \ Γ (0) n +1 , n − \ Γ n +1 , n (cid:13)(cid:13) ≤ k ( n ) d ( n ) + k ( n ) for all n ∈ Z ≥ . Proof.
For a compact metrizable space Z , let M ( Z ) be the real Banach spaceconsisting of all signed Borel measures on Z . (That is, M ( Z ) is the dual space ofthe space C R ( Z ) of real valued continuous functions on Z .) Identify Z with theset of point masses in M ( Z ). For n ∈ Z ≥ , we can identify T( C n ) with the weak*compact convex subset of M ( X n ∐ Y n ) consisting of probability measures. Thus X n ∐ Y n ⊂ T( C n ). For every function f ∈ Aff(T( C n )), the function ι n ( f )( z ) = f ( z ) · M r ( n ) for z ∈ X n ∐ Y n is in C ( X n ∐ Y n , M r ( n ) ) = C n , and τ ( ι n ( f )) = f ( τ )for all τ ∈ X n ∐ Y n ⊂ T( C n ), hence also all τ ∈ T( C n ) by linearity and continuity.For f ∈ Aff(T( C n )) and τ ∈ T( C n +1 ), we can apply the formula in (11) to ι n ( f )and apply τ to everything, to get \ Γ (0) n +1 , n ( f )( τ ) = 1 l ( n ) l ( n ) X k =1 τ ( f ◦ S (0) n, ) and \ Γ n +1 , n ( f )( τ ) = 1 l ( n ) l ( n ) X k =1 τ ( f ◦ S n, ) . Using (12), we get (cid:12)(cid:12)(cid:12)(cid:12) \ Γ (0) n +1 , n ( f )( τ ) − \ Γ n +1 , n ( f )( τ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 l ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ( n ) X k = d ( n )+1 (cid:0) τ ( f ◦ S (0) n, ) − τ ( f ◦ S n, ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ l ( n ) − d ( n ) l ( n ) (cid:0) k f k ∞ ) . The conclusion follows. (cid:3)
We add additional parts to Construction 1.1 and Construction 1.6.
Construction 2.17.
Adopt the assumptions and notation of all parts of Con-struction 1.1 (except (12)), and in addition make the following assumptions anddefinitions:(18) For all m ∈ Z ≥ , the maps S (0) m,j , S m,j : X m +1 ∐ Y m +1 → X m ∐ Y m satisfy: S (0) m,j ( X m +1 ) ⊂ X m and S (0) m,j ( Y m +1 ) ⊂ Y m for j = 1 , , . . . , l ( m ), S m,j ( X m +1 ) ⊂ X m and S m,j ( Y m +1 ) ⊂ Y m for j = 1 , , . . . , d ( m ), and S m,j ( X m +1 ) ⊂ Y m and S m,j ( Y m +1 ) ⊂ X m for j = d ( m ) + 1 , d ( m ) + 2 , . . . , l ( m ).(19) For m ∈ Z ≥ , define D m = M r ( m ) ⊕ M r ( m ) . Define ϕ (0) m +1 ,m , ϕ m +1 ,m : D m → D m +1 by, for a, b ∈ M r ( m ) , ϕ (0) m +1 ,m ( a, b ) = (cid:0) diag( a, a, . . . , a ) , diag( b, b, . . . , b ) (cid:1) and ϕ m +1 ,m ( a, b ) = (cid:0) diag( a, a, . . . , a, b, b, . . . , b ) , diag( b, b, . . . , b, a, a, . . . , a ) (cid:1) , in which a occurs d ( m ) times in the first entry on the right and k ( m ) timesin the second entry, while b occurs k ( m ) times in the first entry and d ( m )times in the second entry. For m, n ∈ Z ≥ with m ≤ n , ϕ n,m = ϕ n,n − ◦ ϕ n − , n − ◦ · · · ◦ ϕ m +1 ,m : D m → D n , SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 23 and define ϕ n,m : D m → D n similarly. Define the following AF algebras: D = lim −→ m ( D m , ϕ m +1 ,m ) and D (0) = lim −→ m ( D m , ϕ (0) m +1 ,m ) , and for m ∈ Z > let ϕ ∞ ,m : D m → D be the associated map.(20) For m ∈ Z ≥ , define µ m : D m → C m as follows. For a, b ∈ M r ( m ) let f ∈ C ( X m , M r ( m ) ) and g ∈ C ( Y m , M r ( m ) ) be the constant functions withvalues a and b . Then set µ m ( a, b ) = ( f, g ). Further, following Lemma2.18(2) below, let µ (0) : D (0) → C (0) and µ : D → C be the direct limits ofthe maps µ m .(21) For m ∈ Z ≥ , define κ m : D m → D m by κ m ( a, b ) = ( b, a ) for a, b ∈ M r ( m ) .Further, following Lemma 2.18(3) below, let κ ∈ Aut( D ) be the directlimits of the maps κ m . Lemma 2.18.
Under the assumptions of Construction 1.1 (except (12)) and Con-struction 2.17, the following hold: (1)
The direct system (cid:0) ( C n ) n =0 , , ,... , (Γ (0) n,m ) ≤ m ≤ n (cid:1) is the direct sum of twodirect systems (cid:0) ( C ( X n , M r ( n ) )) n =0 , , ,... , (Γ (0) n,m | C ( X m , M r ( m ) ) ) ≤ m ≤ n (cid:1) and (cid:0) ( C ( Y n , M r ( n ) )) n =0 , , ,... , (Γ (0) n,m | C ( Y m , M r ( m ) ) ) ≤ m ≤ n (cid:1) , and C (0) is isomorphic to the direct sum of the direct limits A and B ofthese systems. (2) For all m, n ∈ Z ≥ with m ≤ n , Γ (0) n,m ◦ µ m = µ n ◦ ϕ (0) n,m and Γ n,m ◦ µ m = µ n ◦ ϕ n,m . Moreover, the maps µ m induce unital homomorphisms µ (0) : D (0) → C (0) and µ : D → C , and for all m ∈ Z ≥ , Γ (0) ∞ ,m ◦ µ m = µ (0) ◦ ϕ (0) ∞ ,m and Γ ∞ ,m ◦ µ m = µ ◦ ϕ ∞ ,m . (3) For all m, n ∈ Z ≥ with m ≤ n , ϕ (0) n,m ◦ κ m = κ n ◦ ϕ (0) n,m and ϕ n,m ◦ κ m = κ n ◦ ϕ n,m . Moreover, the maps κ m induce an automorphism κ : D → D , and ϕ (0) ∞ ,m ◦ κ m = κ ◦ ϕ (0) ∞ ,m for all m ∈ Z ≥ . (4) For all m ∈ Z ≥ , ( µ m ) ∗ : K ∗ ( D m ) → K ∗ ( C m ) is an isomorphism, and µ ∗ : K ∗ (cid:0) D (0) (cid:1) → K ∗ (cid:0) C (0) (cid:1) and µ ∗ : K ∗ ( D ) → K ∗ ( C ) are isomorphisms.Proof. The fact that all the maps are isomorphisms on K-theory comes from theassumption that the spaces X m and Y m are contractible ((13) and (14) in Con-struction 1.6). Everything else is essentially immediate from the constructions. (cid:3) The main theorem
We now have all the ingredients to deduce the main theorem of this paper,Theorem 3.2.To state the theorem, we first need to define automorphisms of Elliott invariants,so we need a category in which they lie. For convenience, we restrict to unital C ∗ -algebras, and we give a very basic list of conditions. Definition 3.1. An abstract unital Elliott invariant is a tuple G = (cid:0) G , ( G ) + , g, G , K, ρ (cid:1) in which ( G , ( G ) + , g ) is a preordered abelian group with distinguished positiveelement g , G is an abelian group, K is a compact convex set (possibly empty),and ρ : G → Aff( K ) is an order preserving group homomorphism such that ρ ( g )is the constant function 1. (If K = ∅ , we take Aff( K ) = { } , and we take ρ to bethe constant function with value 0.)If G (0) = (cid:0) G (0)0 , (cid:0) G (0)0 (cid:1) + , g (0) , G (0)1 , K (0) , ρ (0) (cid:1) and G (1) = (cid:0) G (1)0 , (cid:0) G (1)0 (cid:1) + , g (1) , G (1)1 , K (1) , ρ (1) (cid:1) are abstract unital Elliott invariants, then a morphism from G (0) to G (1) is a triple F = ( F , F , S ) in which F : G (0)0 → G (1)0 is a group homomorphism satisfying F (( G (0)0 ) + ) ⊂ ( G (1)0 ) + and F ( g (0) ) = g (1) , F : G (0)1 → G (0)1 is a group homomor-phism, and S : K (1) → K (0) is a continuous affine map satisfying(3.1) ρ (1) ( F ( η )) = ρ (0) ( η ) ◦ S for all η ∈ G (0)0 .If F (0) : G (0) → G (1) and F (1) = ( F (1)0 , F (1)1 , S (1) ) : G (1) → G (2) are morphisms of abstract unital Elliott invariants, then F (1) ◦ F (0) = ( F (1)0 ◦ F (0)0 , F (1)1 ◦ F (0)1 , S (0) ◦ S (1) ) . (Note: S (0) ◦ S (1) , not S (1) ◦ S (0) .)The Elliott invariant of a unital C ∗ -algebra A isEll( A ) = (cid:0) K ( A ) , K ( A ) + , [1] , K ( A ) , T( A ) , ρ A (cid:1) , in which ρ A : K ( A ) → Aff(T( A )) is given by ρ A ( η )( τ ) = τ ∗ ( η ) for η ∈ K ( A ) and τ ∈ T( A ).If A and B are unital C ∗ -algebras and ϕ : A → B is a unital homomorphism,then we define ϕ ∗ : Ell( A ) → Ell( B ) to consist of the maps ϕ ∗ from K ( A ) to K ( B )and from K ( A ) to K ( B ), together with the map T( ϕ ) of Definition 2.5. We writeit as ( ϕ ∗ , , ϕ ∗ , , T( ϕ )).Definition 3.1 is enough to make the abstract unital Elliott invariants into acategory such that Ell( · ) is a functor from unital C ∗ -algebras and unital homomor-phisms to abstract unital Elliott invariants. Theorem 3.2.
There exists a simple unital separable AH algebra C with stablerank and with the following property. There exists an automorphism F of Ell( C ) such F ◦ F is the identity morphism of Ell( C ) , but there exists no automorphism α of C such that α ∗ = F .Proof. Let the assumptions and notation be as in Construction 1.1 and Construc-tion 2.17. Let A and B be as in Lemma 2.18(1). Assume that T( A ) ∼ = T( B ). Thehypotheses imply that there is an isomorphism ζ (0)0 : Aff(T( A )) → Aff(T( B )). Thisprovides an automorphism of Aff(T( A )) ⊕ Aff(T( B )), given by( f, g ) (( ζ (0)0 ) − ( g ) , ζ (0)0 ( f )) . SIMPLE NUCLEAR C ∗ -ALGEBRA WITH AN INTERNAL ASYMMETRY 25 Let ζ (0) be the corresponding automorphism of Aff(T( A ⊕ B )) = Aff(T( C )) gottenusing Lemma 2.10. Clearly ζ (0) ◦ ζ (0) = id Aff(T( C )) .Define E = lim −→ n M r ( m ) , with respect to the maps a diag( a, a, . . . , a ), with a repeated l ( n ) times. The direct system defining D (0) is just the direct sum oftwo copies of the direct system just defined, so D (0) ∼ = E ⊕ E and Aff(T( D )) ∼ =Aff(T( E ⊕ E )). Using id Aff(T( E )) in place of ζ (0)0 above, we get an automorphismof Aff(T( D )), but this automorphism is just b κ .We denote by κ (0) ∈ Aut( D (0) ) the automorphism given by κ (0) ( a, b ) = ( b, a ).We claim that ζ (0) ◦ d µ (0) = d µ (0) ◦ d κ (0) . To prove the claim, we work with Aff(T( E )) ⊕ Aff(T( E )) and Aff(T( A )) ⊕ Aff(T( B )) in place of Aff(T( D )) and Aff(T( C )), butkeep the same names for the maps. Observe that E is a UHF algebra, so Aff(T( E )) ∼ = R with the usual order and order unit 1. Since µ (0) : E ⊕ E → A ⊕ B is the direct sumof unital maps from the first summand to A and the second summand to B , the map d µ (0) is similarly a direct sum of maps Aff(T( E )) → Aff(T( A )) and Aff(T( E )) → Aff(T( B )). Let e and f be the order units of Aff(T( A )) and Aff(T( B )). The uniquepositive order unit preserving maps Aff(T( E )) → Aff(T( A )) and Aff(T( E )) → Aff(T( B )) are α αe and β βf for α, β ∈ R . Therefore d µ (0) ( α, β ) = ( αe, βf ).Since ζ (0)0 is order unit preserving, we have ζ (0)0 ( e ) = f , so ζ (0) ( αe, βf ) = ( βe, αf ) = d µ (0) ( β, α ) = ( d µ (0) ◦ d κ (0) )( α, β ) . The claim follows.Using condition (4) in Construction 1.1, apply Proposition 2.15 to get isomor-phisms ρ : Aff(T( D (0) )) → Aff(T( D )) and σ : Aff(T( C (0) )) → Aff(T( C ))such that b µ ◦ ρ = σ ◦ d µ (0) . Then define η = ρ ◦ d κ (0) ◦ ρ − ∈ Aut (cid:0)
Aff(T( C )) (cid:1) and ζ = σ ◦ ζ (0) ◦ σ − ∈ Aut (cid:0)
Aff(T( D )) (cid:1) . The claim above now implies that(3.2) ζ ◦ b µ = b µ ◦ η. Also ζ ◦ ζ = id Aff(T( C )) .We claim that η = b κ . To prove this claim, observe that Aff(T( D )) ∼ = Aff(T( D (0) )) ∼ = R , with order ( α, β ) ≥ α ≥ β ≥ , S ( R ) of R with this order unit space structure is an in-terval, and automorphisms of order unit spaces preserve the extreme points of thestate space, there is only one possible action of a nontrivial automorphism of R on S ( R ). Theorem 2.4 implies that R ∼ = Aff( S ( R )), so there is only one nontrivialautomorphism of R . Since η and b κ are both nontrivial automorphisms, the claimfollows.The claim and (3.2) imply(3.3) ζ ◦ b µ = b µ ◦ b κ. It follows from Theorem 2.4 that there is an affine homeomorphism S : T( C ) → T( C ) such that ζ ( f ) = f ◦ S for all f ∈ Aff(T( C )), and moreover that S ◦ S = id T( C ) .By Lemma 2.18(4), the expression µ ∗ ◦ κ ∗ ◦ ( µ ∗ ) − is a well defined automorphism of K ∗ ( C ), of order 2. We claim that F = (cid:0) µ ∗ ◦ κ ∗ ◦ ( µ ∗ ) − , S ) is an order 2 automor-phism of Ell( C ). The only part needing work is the compatibility condition (3.1)in Definition 3.1, which amounts to showing that ρ C ◦ µ ∗ ◦ κ ∗ ◦ ( µ ∗ ) − = ζ ◦ ρ C . To see this, we calculate, using at the second and last steps the fact that themorphisms of Elliott invariants defined by µ and κ satisfy (3.1) in Definition 3.1,and using (3.3) at the third step, ζ ◦ ρ C = ζ ◦ ρ C ◦ µ ∗ ◦ ( µ ∗ ) − = ζ ◦ b µ ◦ ρ D ◦ ( µ ∗ ) − = b µ ◦ b κ ◦ ρ D ◦ ( µ ∗ ) − = ρ C ◦ µ ∗ ◦ κ ∗ ◦ ( µ ∗ ) − . Thus, we have constructed an automorphism F of Ell( C ) of order 2, whichfurthermore satisfies F ◦ µ ∗ = µ ∗ ◦ κ ∗ . It remains to show that F is not induced byany automorphism of C .Let q and q ⊥ be as in Notation 1.13. In the construction of D as in Construction2.17, set e = ϕ ∞ , ((1 , e ⊥ = 1 − e = ϕ ∞ , ((0 , κ ( e ) = e ⊥ , µ ( e ) = q and µ ( e ⊥ ) = q ⊥ . Therefore, F ([ q ]) = [ q ⊥ ].Suppose now that there exists an automorphism α such that α ∗ = F . Then[ α ( q )] = [ q ⊥ ]. By Lemma 1.17, α ( q ) is unitarily equivalent to q ⊥ . Let u be aunitary such that uα ( q ) u ∗ = q ⊥ . Thus, since α ( qAq ) = α ( q ) Aα ( q ) = u ∗ q ⊥ Aq ⊥ u ,it follows that the qAq and q ⊥ Aq ⊥ have the same radius of comparison, whichcontradicts Lemmas 1.15 and 1.16. (cid:3) Question 3.3.
Does there exist a compact metric space X and a minimal home-omorphism h : X → X such that the crossed product C ∗ ( Z , X, h ) has the samefeatures as the example we construct here? Question 3.4 (Blackadar) . Does there exist a simple separable stably finite unitalnuclear C ∗ -algebra C and an automorphism F of Ell( C ) such that:(1) F ◦ F is the identity morphism of Ell( C ).(2) There is an automorphism α of C such that α ∗ = F .(3) There is no α as in (2) which in addition satisfies α ◦ α = id C .Can such an algebra be chosen to be AH and have stable rank 1?Our method of proof suggests that, instead of being just a number, the radius ofcomparison should be taken to be a function from V ( A ) to [0 , ∞ ]. If one uses thegeneralization to nonunital algebras in [BRT +
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