Non-amenable simple C*-algebras with tracial approximation
aa r X i v : . [ m a t h . OA ] J a n Non-amenable simple C ∗ -algebras with tracial approximation Xuanlong Fu and Huaxin Lin
Abstract
We construct two types of unital separable simple C ∗ -algebras A C z and A C z , one is exactbut not amenable, and the other is non-exact. Both have the same Elliott invariant as theJiang-Su algebra, namely, A C i z has a unique tracial state,( K ( A C i z ) , K ( A C i z ) + , [1 A Ciz ]) = ( Z , Z + , K ( A C i z ) = { } ( i = 1 , A C i z ( i = 1 ,
2) is essentially tracially in theclass of separable Z -stable C ∗ -algebras of nuclear dimension 1. A C i z has stable rank one,strict comparison for positive elements and no 2-quasitrace other than the unique tracialstate. We also produce models of unital separable simple non-exact C ∗ -algebras which areessentially tracially in the class of simple separable nuclear Z -stable C ∗ -algebras and themodels exhaust all possible weakly unperforated Elliott invariants. We also discuss somebasic properties of essential tracial approximation. Simple unital projectionless amenable C ∗ -algebras were first constructed by B. Blackadar ([2]).The C ∗ -algebra A constructed by Blackadar has property that K ( A ) = Z with the usual orderbut with non-trivial K ( A ) . The Jiang-Su algebra Z given by X. Jiang and H. Su ([22]) is a unitalinfinite dimensional separable amenable simple C ∗ -algebra with the Elliott invariant exactly thesame as that of the complex field C . Let A be any σ -unital C ∗ -algebra. Then K i ( A ) = K i ( A ⊗Z )( i = 0 ,
1) as abelian groups and T ( A ) ∼ = T ( A ⊗ Z ) . If A is a separable simple C ∗ -algebra then A ⊗ Z has nice regularities. For example, A ⊗ Z is either purely infinite, or stably finite ([37]). Infact, if A ⊗Z is not purely infinite, then it has stable rank one when A is not stably projectionless([37]), or it almost has stable rank one when it is stably projectionless ([33]). Also A ⊗ Z hasweakly unperforated K -group ([18]). Another important regularity is that A ⊗ Z has strictcomparison ([37]) (see also Definition 2.5). If A has weakly unperforated K ( A ) , then A and A ⊗ Z have the same Elliott invariant. In other words, A and A ⊗ Z are not distinguishablefrom the Elliott invariant.The Jiang-Su algebra Z is an inductive limit of one-dimensional non-commutative CW com-plexes. In fact Z is the unique infinite dimensional separable simple C ∗ -algebra with finitenuclear dimension in the UCT class which has the same Elliott invariant as that of the complexfield C (see Corollary 4.12 of [13]). These properties give Z a prominent role in the study ofstructure of C ∗ -algebras, in particular, in the study of classification of amenable simple C ∗ -algebras.Attempts to construct a non-exact Jiang-Su type C ∗ -algebra have been in horizon for overa decade. In particular, after M. D˘ad˘arlat constructed non-amenable models for non type Iseparable unital AF-algebras ([12]), this should be possible. The construction in [12] generalizedsome of earlier constructions of simple C ∗ -algebras of real rank zero such as that of Goodearl([21]). Jiang and Su’s construction has a quite different feature. To avoid producing anynon-trivial projections, Jiang and Su did not use any finite dimensional representations. The1onstruction used prime dimension drop algebras and connecting maps are highly inventive sothat the traces eventually collapse to one. In fact, M. Rørdam and W. Winter had anotherapproach ([38]) using a C ∗ -subalgebra of C ([0 , , M p ⊗ M q ) , where p and q are relatively primesupernatural numbers. One possible attempt to construct a non-amenable Jiang-Su type C ∗ -algebra would use C ([0 , , B p ⊗ B q ) , where B p and B q are non-amenable models for M p and M q constructed in [12], respectively. However, one usually would avoid computation of K -theoryof tensor products of non-exact simple C ∗ -algebras such as B p and B q . Moreover, Rørdam andWinter’s construction depend on knowing the existence of the Jiang-Su algebra Z . On theother hand, if one considers non-exact interval “dimension drop algebras”, aside of controlling K -theory, one has additional issues such as each fiber of the “dimension drop algebra” is notsimple (unlike the usual dimension drop algebras whose fibers are simple matrix algebras).We will present some non-exact (or non-nuclear) unital separable simple C ∗ -algebras A Cz which have the property that their Elliott invariants are the same as that the Jiang-Su algebra Z , namely, ( K ( A Cz ) , K ( A Cz ) + , [1 A Cz ]) = ( Z , Z + , , K ( A Cz ) = { } , and A Cz has a unique tracialstate. Moreover, A Cz has stable rank one and has strict comparison for positive elements. A Cz has no (nonzero) 2-quasitrace other than the unique tracial state. Even though A Cz may notbe exact, it is essentially tracially approximated by Z . In particular, it is essentially traciallyapproximated by unital simple C ∗ -algebras with nuclear dimension 1.In this paper, we will also study the tracial approximation. We will make it precise what wemean by that A Cz is essentially tracially approximated by Z (Definition 3.1 and Lemma 8.1). Webelieve that regular properties such as stable rank one, strictly comparison for positive elements,or almost unperforated Cuntz semigroup, as well as approximate divisibility are preserved bytracial approximation. In fact, we show that if a unital separable simple C ∗ -algebra A whichis essentially tracially in C Z , the class of Z -stable C ∗ -algebras, then, as far as usual regularitiesconcerned, A behaves just like C ∗ -algebras in C Z . More precisely, we show that if A is simpleand essentially tracially in C Z , then A is tracially approximately divisible. If A is not purelyinfinite, then A has stable rank one (or almost has stable rank one, if A is not unital), has strictcomparison, and its Cuntz semigroup is almost unperforated. If A is essentially tracially in theclass of exact C ∗ -algebras, then every 2-quasitrace of aAa, for any a in the Pedersen ideal of A, is in fact a trace.Using A Cz , we present a large class of non-exact unital separable simple C ∗ -algebras whichexhaust all possible weakly unperforated Elliott invariant. Moreover, every C ∗ -algebra in theclass is essentially tracially in the class of unital separable simple C ∗ -algebras which are Z -stableand has nuclear dimension at most 1.The paper is organized as follows: Section 2 serves as preliminaries where some of frequentlyused notations and definitions are listed. Section 3 introduces the notion of essential tracialapproximation for simple C ∗ -algebras. In Section 4, we present some basic properties of essentialtracial approximation. For example, we show that, if A is a simple C ∗ -algebra and is essentiallytracially approximated by C ∗ -algebras whose Cuntz semigroups are almost unperforated, thenthe Cuntz semigroup of A is almost unperforated (Theorem 4.3). In particular, A has strictcomparison for positive elements. In Section 5, we study the separable simple C ∗ -algebras whichare essentially tracially approximated by Z -stable C ∗ -algebras. We show such C ∗ -algebras areeither purely infinite, or almost has stable rank one (or stable rank one if the C ∗ -algebrasare unital). These simple C ∗ -algebras are tracially approximately divisible and have strictcomparison for positive elements. In Section 6, we begin the construction of A Cz . In Section7, we show that the construction in Section 6 can be made simple and the Elliott invariant of A Cz is precisely the same as that of complex filed just as the Jinag-Su algebra Z . In Section 8,we show that A Cz has all expected regularity properties. Moreover, A Cz is essentially traciallyapproximated by Z . Using A Cz , we also produce, for each weakly unperforated Elliott invariant,2 unital separable simple non-exact C ∗ -algebra B which has the said Elliott invariant, has stablerank one, is essentially tracially approximated by C ∗ -algebras with nuclear dimension at most1, has almost unperforated Cuntz semigroup, has strict comparison for positive elements andhas no 2-quasitraces which are not traces. Acknowledgement : The first named author was supported by China Postdoctoral Sci-ence Foundation, grant
In this paper, the set of all positive integers is denoted by N . If A is unital, U ( A ) is the unitarygroup of A. Notation 2.1.
Let A be a C ∗ -algebra and F ⊂ A be a subset. Let ǫ >
0. Let a, b ∈ A, wewrite a ≈ ǫ b if k a − b k < ǫ . We write a ∈ ε F , if there is x ∈ F such that a ≈ ε x. Notation 2.2.
Let A be a C ∗ -algebra and let S ⊂ A be a subset of A. Denote by Her A ( S ) (orjust Her( S ) , when A is clear) the hereditary C ∗ -subalgebra of A generated by S. Denote by A the closed unit ball of A, by A + the set of all positive elements in A, by A := A + ∩ A , and by A sa the set of all self-adjoint elements in A. Denote by e A the minimal unitization of A. When A is unital, denote by GL ( A ) the set of invertible elements of A. Notation 2.3.
Let ǫ > . Define a continuous function f ǫ : [0 , + ∞ ) → [0 ,
1] by f ǫ ( t ) = t ∈ [0 , ǫ/ , t ∈ [ ǫ, ∞ ) , linear t ∈ [ ǫ/ , ǫ ] . Definition 2.4.
Let A be a C ∗ -algebra and let M ∞ ( A ) + := S n ∈ N M n ( A ) + . For x ∈ M n ( A ) , weidentify x with diag( x, ∈ M n + m ( A ) for all m ∈ N . Let a ∈ M n ( A ) + and b ∈ M m ( A ) + . Define a ⊕ b := diag( a, b ) ∈ M n + m ( A ) + . If a, b ∈ M n ( A ) , we write a . A b if there are x i ∈ M n ( A )such that lim i →∞ k a − x ∗ i bx i k = 0. We write a ∼ A b if a . A b and b . A a hold. We alsowrite a . b and a ∼ b, when A is clear. The Cuntz relation ∼ is an equivalence relation. Set W ( A ) := M ∞ ( A ) + / ∼ A . Let h a i denote the equivalence class of a . We write h a i ≤ h b i if a . A b .( W ( A ) , ≤ ) is a partially ordered abelian semigroup. W ( A ) is called almost unperforated, if forany h a i , h b i ∈ W ( A ), and for any k ∈ N , if ( k + 1) h a i ≤ k h b i , then h a i ≤ h b i (see [35]).If B ⊂ A is a hereditary C ∗ -subalgebra, a, b ∈ B + , then a . A b ⇔ a . B b . Definition 2.5.
Denote by QT ( A ) the set of 2-quasitraces of A with k τ k = 1 (see [4, II1.1, II 2.3]) and by T ( A ) the set of all tracial states on A. We will also use T ( A ) as well as QT ( A ) for the extensions on M k ( A ) for each k. In fact T ( A ) and QT ( A ) may be extended tolower semicontinuous traces and lower semicontinuous quasitraces on A ⊗ K (see lines aboveProposition 4.2 of [15] and Remark 2.27 (viii) of [6]).Let A be a C ∗ -algebra. Denote by Ped( A ) the Pedersen ideal of A (see 5.6 of [31]). Supposethat A is a σ -unital simple C ∗ -algebra. Choose b ∈ Ped( A ) + with k b k = 1 . Put B := bAb =Her( b ) . Then, by [7], A ⊗ K ∼ = B ⊗ K . For each τ ∈ T ( B ) , define a lower semi-continuous function3 τ : A ⊗K + → [0 , + ∞ ], x lim n →∞ τ ( f /n ( x )). The function d τ is called the dimension functioninduced by τ .We say A has strict comparison (for positive elements), if for any a, b ∈ A ⊗K + , d τ ( a ) < d τ ( b )for all τ ∈ QT ( B ) implies that a . b. Definition 3.1.
Let P be a class of C ∗ -algebras and let A be a simple C ∗ -algebra. We say A isessentially tracially in P (abbreviated as e. tracially in P ), if for any finite subset F ⊂ A, any ε > , any s ∈ A + \ { } , there exist an element e ∈ A and a non-zero C ∗ -subalgebra B of A which is in P such that(1) k ex − xe k < ε for all x ∈ F , (2) (1 − e ) x ∈ ε B and k (1 − e ) x k ≥ k x k − ε for all x ∈ F , and(3) e . s. Proposition 3.2.
Let P be a class of C ∗ -algebras and let A be a simple C ∗ -algebra. Then A ise. tracially in P if and only if the following hold: For any ε > , any finite subset F ⊂ A, any a ∈ A + \ { } , and any finite subset G ⊂ C ((0 , , there exist an element e ∈ A and a non-zero C ∗ -subalgebra B of A such that B in P , and(1) k ex − xe k < ε for all x ∈ F ,(2) g (1 − e ) x ∈ ε B for all g ∈ G and k (1 − e ) x k ≥ k x k − ε for all x ∈ F , and(3) e . a. Proof.
The “if” part follows easily by taking G = { ι } , where ι ( t ) = t for all t ∈ [0 , . We now show the “only if” part.Suppose that A is e. tracially in P . Let ε >
F ⊂ A be a finite subset and, withoutloss of generality, we may assume that F ⊂ A . Moreover, without loss of generality (omittingan error within ε/ , say), we may further assume that there is e A ∈ A such that e A x = x = xe A . (e 3.1)Let a ∈ A + \ { } , let ε > , and let G = { g , g , ...g n } ⊂ C ((0 , m ∈ N and polynomials p i ( t ) = P mk =1 β ( i ) k t k such that | p i ( t ) − g i ( t ) | < ε/ t ∈ [0 ,
1] and all i ∈ { , , ..., n } . (e 3.2)Let M = 1 + max {| β ( i ) k | : i = 1 , , ..., n, k = 1 , , ..., m } and δ := ε m M . Now, since A is e. tracially in P , there exist an element e ∈ A and a non-zero C ∗ -subalgebra B ⊂ A such that B in P , and(1) k ex − xe k < δ for all x ∈ F ∪ { e A } , (2 ′ ) (1 − e ) x ∈ δ B and k (1 − e ) x ) k ≥ k x k − δ for all x ∈ F ∪ { e A } , and(3) e . a. It remains to show that g i (1 − e ) x ∈ ε/ B for all x ∈ F , i = 1 , , ..., n. Claim: For all x ∈ F and all k ∈ { , , ..., m } , (1 − e ) k x ∈ ε mM B. In fact,(1 − e ) k x (e 3.1) = (1 − e ) k e k − A x (1) ≈ k δ k − z }| { (1 − e ) e A (1 − e ) e A · · · (1 − e ) e A (1 − e ) x (2 ′ ) ∈ kδ B. (e 3.3)Note that 2 k δ ≤ m δ < ε/ mM. The claim follows.4y (e 3.2) and the claim above, for x ∈ F and i ∈ { , , ..., n } , we have g i (1 − e ) x ≈ ε/ p i (1 − e ) x = m X k =1 β ( i ) k (1 − e ) k x ∈ ε/ B. (e 3.4) Remark 3.3. (1) A similar notion as in Definition 3.1 could also be defined for non-simple C ∗ -algebras. However, in the present paper, we will be only interested in the simple case.(2) Note in 3.2, g (1 − e ) is an element in e A. But g (1 − e ) x ∈ A. In the case that A is unital,the condition k (1 − e ) x k ≥ k x k − ε for all x ∈ F in (2) of the definition is redundant for mostcases (we leave the discussion to [17]).(3) The origin of the notion of tracial approximation were first introduced in [25] (see also[26]). Current definition also related to the notion of “centrally large subalgebra” in Definition4.1 of [32] (see also Definition 2.1 of [1]) but not the same.(4) In [16] a notion of asymptotically tracial approximation is introduced which studies thephenomena of asymptotic preserving nature. It also mainly studies the unital simple C ∗ -algebraswith rich structure of projections. It is different from the Definition 3.1. However, if A is aunital simple C ∗ -algebra which is asymptotically tracially in the class C of 1-dimensional non-commutative CW complexes, then A is also essentially tracially in the same class C . Moreover,many classes P of C ∗ -algebras are preserved by asymptotically tracial approximation (see Section4 of [16]). For these classes P , of course, a simple C ∗ -algebra A is asymptotically tracially in P implies that A is also essentially tracially in P . Some more discussion may be found in aforthcoming paper ([17]).
Definition 3.4.
Let P be a class of C ∗ -algebras. The class P is said to have property (H), iffor any nonzero A in P and any nonzero hereditary C ∗ -subalgebra B ⊂ A, B is also in P . Proposition 3.5.
Let P be a class of C ∗ -algebras which has property (H). Suppose that A isa simple C ∗ -algebra which is e. tracially in P . Then every nonzero hereditary C ∗ -subalgebra B ⊂ A is also e. tracially in P . Proof.
Assume P has property (H) and A is e. tracially in P . Let B ⊂ A be a nonzero hereditary C ∗ -subalgebra of A. Let
F ⊂ B and s ∈ B + \ { } , and let ε ∈ (0 , / . Without loss of generality, we may assume that
F ⊂ B . Let d ∈ B be such that dx ≈ ε/ x ≈ ε/ xd and x ≈ ε/ dxd for all x ∈ F . Put ε = ε/ . By Lemma 3.3 of [14], there is δ ∈ (0 , ε ) such that for any C ∗ -algebra E and any x, y ∈ E , if x ≈ δ y, then there is an injective homomorphism ψ : Her E ( f ε / ( x )) → Her E ( y ) satisfying z ≈ ε ψ ( z ) for all z ∈ Her E ( f ε / ( x )) . Note that there is δ ∈ (0 , δ ) such that, for any C ∗ -algebra E, any x, y ∈ E , if xy ≈ δ yx, then x / y ≈ δ / yx / , x / y / ≈ δ / y / x / , and x / y ≈ δ / yx / . Let δ = min { δ / , δ / } . Let G = { t, t / , t / } ⊂ C ((0 , . Since A is e. tracially in P , byProposition 3.2, there exist a positive element a ∈ A and a non-zero C ∗ -subalgebra C ⊂ A which is in P such that(1) k ax − xa k < δ for all x ∈ F ∪ { d, d / , d } , (2) g (1 − a ) x ∈ δ C for all g ∈ G and k (1 − a ) x k ≥ k a k − δ for all x ∈ F ∪ { d, d / , d } , and(3) a . s. By (2), there is c ∈ C such that c ≈ δ / (1 − a ) / d. By (1) and the choice of δ , we have c ≈ δ d / (1 − a ) / d / . Then by Lemma 3.3 of [14] and by the choice of δ , there is a monomorphism ϕ : Her A ( f ε / ( c )) → Her A ( d / (1 − a ) / d / ) ⊂ B k ϕ ( x ) − x k < ε for all x ∈ Her C ( f ε / ( c )) . Define D := ϕ (Her C ( f ε / ( c ))) ⊂ B. Since C is in P and P has property (H), we have D ∼ = Her C ( f ε/ ( c )) is in P . Set b := dad ∈ B . Thenby (1) and by the choice of d, we have k bx − xb k = k dadx − xdad k ≈ ε k adxd − dxda k ≈ ε k ax − xa k < δ for all x ∈ F . (e 3.5)By (2), for any x ∈ F , there is ¯ x ∈ C such that (1 − a ) / x (1 − a ) / ≈ ε ¯ x. Then(1 − b ) x = (1 − dad ) x ≈ ε (1 − a ) dxd ≈ ε (1 − a ) / d (1 − a ) / · (1 − a ) / x (1 − a ) / · (1 − a ) / d (1 − a ) / ≈ ε c ¯ xc ≈ ε ( c − ε ) + ¯ x ( c − ε ) + ≈ ε ϕ (( c − ε ) + ¯ x ( c − ε ) + ) ∈ D. (e 3.6)In other words, (1 − b ) x ∈ ε D. (e 3.7)Therefore, for all x ∈ F , k (1 − b ) x k = k (1 − dad ) x k ≥ k (1 − ad ) x ) k − δ ≥ k (1 − a ) x k − ε ≥ k x k − δ − ε ≥ k x k − ε. (e 3.8)By (3), we have b = dad . A s. Note that b, s ∈ B. Since B is a hereditary SCA we have b . B s. By (e 3.5) and (e 3.7), we see that B is also e. tracially in P . Notation 4.1.
Let W be the class of C ∗ -algebras A such that W ( A ) is almost unperforated.Let Z be the Jiang-Su algebra (see [22]). A C ∗ -algebra A is called Z -stable if A ⊗ Z ∼ = A. Let C Z be the class of separable Z -stable C ∗ -algebras. Lemma 4.2.
Let A be a simple C ∗ -algebra which is e. tracially in W and a, b, c ∈ A + \{ } . Suppose that there exists n ∈ N satisfying ( n + 1) h a i ≤ n h b i . Then, for any ε > , there exist a , a ∈ A + such that(1) a ≈ ǫ a + a ,(2) a . A b , and(3) a . A c .Proof. Without loss of generality, one may assume that a, b, c ∈ A \{ } and ǫ < /
2. Then( n + 1) h a i ≤ n h b i implies that there exists r = P n +1 i,j =1 r i,j ⊗ e i,j ∈ A ⊗ M n +1 such that a ⊗ n +1 X i =1 e i,i ≈ ǫ/ r ∗ b ⊗ n X i =1 e i,i ! r. (e 4.1)Set F := { a, b } ∪ { r i,j , r ∗ i,j : i, j = 1 , , ..., n + 1 } . Let M := 1 + k r k . Let σ = ε M ( n +1) . Since A is e. tracially in W , by Proposition 3.2, for any δ ∈ (0 , ε M ( n +1) ) , there exist f ∈ A \ { } and a C ∗ -subalgebra B ⊂ A which has almost unperforated W ( B ) , such that(1 ′ ) k f x − xf k < δ for x ∈ F , (2 ′ ) (1 − f ) / x, (1 − f ) / a (1 − f ) / , (1 − f ) / x (1 − f ) / ∈ δ B for all x ∈ F , (3 ′ ) f . c. g = 1 − f. Let G := { g / x, g / xg / , g / xg / : x ∈ F } . By (2 ′ ), there is a map α : G → B such that α ( G ∩ A + ) ⊂ B + , and x ≈ δ α ( x ) for all x ∈ G . (e 4.2)From (1 ′ ) and (2 ′ ), one can choose δ sufficiently small such that, a ≈ ε/ g / ag / + (1 − g ) / a (1 − g ) / , and (e 4.3)( g / ag / − ε/ + ≈ ε/ ( α ( g / ag / ) − ǫ/ + . (e 4.4)By (1 ′ ) and (e 4.1) (with δ sufficiently small), one can also assume that g / ag / ⊗ n +1 X i =1 e i,i ≈ ε/ R ∗ g / bg / ⊗ n X i =1 e i,i ! R, (e 4.5)where R := P n +1 i,j =1 ( g / r i,j ) ⊗ e i,j . By (e 4.5), (e 4.2), and δ < ε M ( n +1) , one has α ( g / ag / ) ⊗ n +1 X i =1 e i,i ≈ ε/ ¯ R ∗ α ( g / bg / ) ⊗ n X i =1 e i,i ! ¯ R, (e 4.6)where ¯ R := P n +1 i,j =1 α ( g / r i,j ) ⊗ e i,j . Then by the choice of σ,α ( g / ag / ) ⊗ n +1 X i =1 e i,i ≈ ε/ ¯ R ∗ ( α ( g / bg / ) − σ ) + ⊗ n X i =1 e i,i ! ¯ R. (e 4.7)By (e 4.7) and [35, Proposition 2.2], one has( α ( g / ag / ) − ǫ/ + ⊗ n +1 X i =1 e i,i . ( α ( g / bg / ) − σ ) + ⊗ n X i =1 e i,i . (e 4.8)Since W ( B ) is almost unperforated, one obtains( α ( g / ag / ) − ǫ/ + . ( α ( g / bg / ) − σ ) + . (e 4.9)By (e 4.4), [35, Proposition 2.2], (e 4.9), and (e 4.2), it follows that( g / ag / − ε/ + . ( α ( g / ag / ) − ǫ/ + (e 4.10) . ( α ( g / bg / ) − σ ) + . g / bg / . b. (e 4.11)By (1 ′ ) and the choice of δ,a ≈ ε/ (1 − f ) / a (1 − f ) / + f / af / . (e 4.12)Choose a := ( g / ag / − ε/ + = ((1 − f ) / a (1 − f ) / − ε/ + , and (e 4.13) a := f / af / . (e 4.14)Then, by (e 4.11), one has a . A b . Note that (3 ′ ) above implies a . A c . Thus a and a satisfy(2) and (3) of the lemma. By (e 4.12) a ≈ ε/ (1 − f ) / a (1 − f ) / + f / af / ≈ ǫ/ a + a . So (1) of the lemma also holds and the lemma follows.7 heorem 4.3.
Let A be a simple C ∗ -algebra which is e. tracially in W (see Lemma 4.2). Then A ∈ W . Proof.
We may assume that A is non-elementary. Let a, b ∈ M m ( A ) + \ { } with k a k = 1 = k b k for some integer m ≥ . Let n ∈ N and assume ( n + 1) h a i ≤ n h b i . To prove the theorem, itsuffices to prove that a . b. Note that, if B ∈ W , then, for each integer m, M m ( B ) ∈ W . It follows that M m ( A ) ise. tracially in W . To simplify notation, without loss of generality, one may assume a, b ∈ A + . By Lemma 4.3 of [16], Her( f / ( b )) + contains 2 n + 1 nonzero mutually orthogonal elements b , b , ..., b n such that h b i i = h b i , i = 1 , , ..., n. Without loss of generality, we may assumethat k b k = 1 . If b is a projection, choose e = b . Otherwise, by replacing b by g ( b ) forsome continuous function g ∈ C ((0 , , we may assume that there is a nonzero e ∈ A + such that b e = e b = e . Replacing b by g ( b ) for some g ∈ C ((0 , , one may assume that bb = b b = b . Put c = b − b . Note that ce = ( b − b ) e = be − e = b e − e = 0 = e c. (e 4.15)Keep in mind that b ≥ c + e , c ⊥ e , and 2 n h b i ≤ h c i = h b − b i . One has(2 n + 2) h a i ≤ n h b i ≤ n ( h b − b i + h b i ) ≤ n h c i + h c i = (2 n + 1) h c i . (e 4.16)By Lemma 4.2, for any ε ∈ (0 , / , there exist a , a ∈ A + such that(i) a ≈ ǫ/ a + a ,(ii) a . A c, and(iii) a . A e .By (i), (ii) and (iii), and applying [35, Proposition 2.2], one obtains (recall be = e b = e )( a − ε ) + . a + a . c + e ≤ b. (e 4.17)Since this holds for every ε ∈ (0 , / , one concludes that a . b. Corollary 4.4.
Let A be a simple C ∗ -algebra which is e. tracially in C Z . Then W ( A ) is almostunperforated.Proof. It follows from [37, Theorem 4.5] and Theorem 4.3.
Definition 4.5.
Let A be a C ∗ -algebra. Let T denote the class of C ∗ -algebras A such that, forevery a ∈ Ped( A ) + \ { } , every 2-quasitrace of aAa is a trace. Proposition 4.6.
Let A be a simple C ∗ -algebra which is e. tracially in T . Then A is in T . Proof.
Fix a ∈ Per( A ) and let C = Her( a ) . We will show that every 2-quasitrace of C is a trace.We may assume that C is non-elementary. Let τ ∈ QT ( C ) . Fix x, y ∈ C sa with k x k , k y k ≤ / . Let ε ∈ (0 , / . Let F := { x, y, x + y } . Let n ∈ N such that ε > /n. By [16, Lemma 4.3],there exist mutually orthogonal norm one positive elements c , c , ..., c n ∈ A + \{ } , such that c ∼ c ∼ ... ∼ c n . Then d τ ( c ) ≤ /n < ε. Let δ ∈ (0 , ε ) be such that, for any d ∈ C and z ∈ C sa , if k [ d, z ] k < δ, then z ≈ ε (1 − d ) / z (1 − d ) / + d / zd / (e 4.18)and (see [4, II.2.6], note that k [(1 − d ) / z (1 − d ) / , d / zd / ] k can be sufficiently small dependingon δ ) τ ( z ) ≈ ε τ ((1 − d ) / z (1 − d ) / ) + τ ( d / zd / ) . (e 4.19)8ote that T has property (H). Since A is simple and e. tracially in T , by Proposition 3.5, C isalso e tracially in T . There exist an element e ∈ C and a non-zero C ∗ -subalgebra B ⊂ C suchthat B is in T , and(1) k ez − ze k < δ for all z ∈ F ,(2) (1 − e ) / z (1 − e ) / ∈ δ/ B for all z ∈ F , and(3) e . c . We may choose e B ∈ Ped( B ) such that(2 ′ ) (1 − e ) / z (1 − e ) / ∈ δ B := e B Be B for all z ∈ F . Note that for z ∈ F , e / ze / is self-adjoint. One has ( e / ze / ) + , ( e / ze / ) − ∈ Her A ( e ) . Then | τ ( e / ze / ) | = | τ (( e / ze / ) + ) − τ (( e / ze / ) − ) | (e 4.20) ≤ d τ (( e / ze / ) + ) + d τ (( e / ze / ) − ) ≤ d τ ( e ) ≤ ε. (e 4.21)Then by (1), the choice of δ, (e 4.18), and (e 4.19), for z ∈ F ,τ ( z ) ≈ ε τ ((1 − e ) / z (1 − e ) / ) + τ ( e / ze / ) (e 4.22)(by (e 4.21)) ≈ ε τ ((1 − e ) / z (1 − e ) / ) . (e 4.23)By (2 ′ ), there are ¯ x, ¯ y ∈ ( B ) sa such that(1 − e ) / x (1 − e ) / ≈ δ ¯ x, (1 − e ) / y (1 − e ) / ≈ δ ¯ y. (e 4.24)Then τ ( x + y ) (e 4.23) ≈ ε τ ((1 − e ) / ( x + y )(1 − e ) / ) (e 4.24) ≈ δ τ (¯ x + ¯ y )( τ is a trace on B ) = τ (¯ x ) + τ (¯ y ) (e 4.24) ≈ δ τ ((1 − e ) / x (1 − e ) / ) + τ ((1 − e ) / y (1 − e ) / ) (e 4.23) ≈ ε τ ( x ) + τ ( y ) . Since ε (and δ ) are arbitrary small, we have τ ( x + y ) = τ ( x ) + τ ( y ) . therefore τ is a trace on C. Definition 4.7.
Let A be a C ∗ -algebra. Recall that an element a ∈ Ped( A ) + is said to beinfinite, if there are nonzero elements b, c ∈ Ped( A ) + such that bc = cb = 0 and b + c . c and c . a. A is said to be finite, if every element a ∈ Ped( A ) + is not infinite (see, for example,Definition 1.1 of [28]). A is stably finite, if M n ( A ) is finite for every integer n ≥ . Recall that a simple C ∗ -algebra A is purely infinite if and only if every non-zero elementin Ped( A ) + is infinite (see Condition (vii) and Theorem 2.2 of [28]). Let PI be the class of C ∗ -algebras that every nonzero positive element is infinite. Theorem 4.8.
Let A be a simple C ∗ -algebra which is e. tracially in PI . Then A is purelyinfinite.Proof. Note A has infinite dimension. Let a ∈ Ped( A ) + \ { } with k a k = 1 . Since f / ( a ) Af / ( a ) is infinite dimensional simple C ∗ -algebra, one may choose c, d ∈ f / ( a ) Af / ( a ) + \ { } such that cd = dc = 0 . Since A is e. tracially in PI , there exist a sequence of positive elements e n ∈ A + with k e n k ≤ C ∗ -subalgebra B n ⊂ A such that B in PI , and91) a ≈ / n e / n ae / n + (1 − e n ) / a (1 − e n ) / , and(2) (1 − e n ) / a (1 − e n ) / ∈ / n B n , and(3) e n . c. By (2), there is b n ∈ B n + such that b n ≈ / n (1 − e n ) / a (1 − e n ) / . Then by (1), a ≈ / n b n + e / n ae / n . (e 4.25)Case (i): there exists a subsequence { n k } and 1 > δ > k {k b n k k} ≥ δ > . Choose 0 < ε < δ/ . By [32, Lemma 1.7], for all sufficiently large k, we have0 = ( b n k − ε ) + . ( b n k + e / n k ae / n k − ε ) + . a. (e 4.26)Note ( b n k − ε ) + ∈ Ped( B ) + \ { } . Then there are d , d ∈ Ped( B ) + \ { } such that d ⊥ d ,d + d . d and d + d . ( b n k − ε ) + . a. (e 4.27)It follows that a is infinite.Case (ii): lim n →∞ k b n k = 0 . This happens only when lim n →∞ k (1 − e n ) / a (1 − e n ) / k = 0 . Then, for any 0 < ε < / , by [35, Proposition 2.2], (1) and (2), for all large n,f ε ( a ) . e n ae n . c (e 4.28)Note that f ε ( a ) f / ( a ) = f / ( a ) . Therefore c + d . f ε ( a ) . c . a. (e 4.29)It also follows that a is infinite. Therefore A is purely infinite. Proposition 4.9 (Corollary 5.1 of [37]) . Let A be a σ -unital simple C ∗ -algebra such that W ( A ) is almost unperforated. If A is not purely infinite, then aAa has a nonzero 2-quasitrace for every a ∈ Ped( A ) + \ { } . Consequently A is stably finite.Proof. This is a theorem of M. Rørdam (Corollary 5.1 of [37]). Since we do not assume that A is exact and will use only 2-quasitraces, some more explanation is in order. The explanationfollows of course exactly the same lines of the proof of Corollary 5.1 in [37].Let a ∈ Ped( A ) and B := aAa. Then B is algebraically simple (see, for example, [3,II.5.4.2]). Assume that B has no nonzero 2-quasitraces.Consider W ( B ) . Note that W ( B ) ⊂ W ( A ) and W ( B ) has the property that, if x ∈ W ( B )and y ∈ W ( A ) such that y ≤ x, then y ∈ W ( B ) . It follows that W ( B ) is almost unperforated.Since B is algebraically simple, every element in W ( B ) is a strong order unit.Let t, t ′ ∈ W ( B ) (with t a strong order unit). The statement (and the proof) of Proposition3.1 of [35] imply that if there is no state on W ( B ) (with the strong order unit t ), then, theremust be some integer n ∈ N and u ∈ W ( B ) such that nt ′ + u ≤ nt + u. (e 4.30)Then, Proposition 3.2 of [35] (see the proof also), as W ( B ) is almost unperforated, t ′ ≤ t. (e 4.31)On the other hand, by II. 2.2 of [4], every lower semicontinuous dimension function on W ( B ) isinduced by a 2-quasitrace on B. Since B is assumed to have no nonzero 2-quasitraces, combining10roposition 4.1 (as well as paragraph above it) of [35], there is no state on W ( B ) . Therefore(e 4.31) implies that, for any b, c ∈ B + \ { } , b . c. It follows that B is purely infinite and so is A. To see the last of the statement, suppose that there are b, c ∈ Ped( A ) \ { } such that bc = cb = 0 and b + c . c. Let a = b + c and B = aAa. Note that a ∈ Ped( A ) + . Then B hasnonzero 2-quasitraces.Therefore d τ ( c ) ≥ d τ ( b + c ) for all τ ∈ QT ( B ) . (e 4.32)On the other hand, for any τ ∈ QT ( B ) , for any 1 > ε > ,τ ( f ε ( b + c )) = τ ( f ε ( b ) + f ε ( c )) = τ ( f ε ( b )) + τ ( f ε ( c )) . (e 4.33)Fix 1 > ε > f ε ( b ) = 0 . Since B is algebraically simple, τ ( f ε ( b )) > τ. Fix τ ∈ QT ( B ) . Then, by (e 4.33), d τ ( b + c ) ≥ τ ( f ε ( b )) + d τ ( c ) > d τ ( c ) . (e 4.34)This contradicts with (e 4.32). It follows that no such pairs b and c exist. Thus A is finite.Since M n ( A ) has the same used property as A, we conclude that A is stably finite. Corollary 4.10.
Let A be a σ -unital simple C ∗ -algebra such that A is e. tracially in W . Then A has strict comparison.Proof. Fix a ∈ A + \ { } and let B := Her( a ) . As in the proof of Proposition 4.9, every lowersemicontinuous dimension function on W ( B ) is induced by a 2-quasitrace of B. Therefore thiscorollary follows from [37, Corollary 4.7] and Theorem 4.3 above. Z stable C ∗ -algebras Recall from Notation 4.1 that C Z is the class of separable Z -stable C ∗ -algebras. Theorem 5.1.
Let A be a σ -unital simple C ∗ -algebra which is e. tracially in C Z . Then A iseither purely infinite, or stably finite. Moreover, if A is not purely infinite, then A has strictcomparison for positive elements.Proof. It follows from Theorem 4.5 of [37] that every C ∗ -algebra B in C Z has almost unperforated W ( B ) . It follows from Theorem 4.3 that W ( A ) is almost unperforated. By Proposition 4.9, if A is not purely infinite, then A is stably finite and has strict comparison for positive elements. Definition 5.2.
Let A be a simple C ∗ -algebra. A is said to be tracially approximately divisible,if, for any ε > , any finite subset F ⊂ A, any element e F ∈ A with e F x ≈ ε/ x ≈ ε/ xe F for all x ∈ F , any s ∈ A + \ { } , and any integer n ≥ , there are θ ∈ A , a C ∗ -subalgebra D ⊗ M n ⊂ A and a c.p.c. map β : A → A such that(1) x ≈ ε x + β ( x ) for all x ∈ F , where k x k ≤ k x k , x ∈ Her( θ ) , and(2) β ( x ) ∈ ε D ⊗ n and e F β ( x ) ≈ ε β ( x ) ≈ ε β ( x ) e F for all x ∈ F , and(3) θ . a. roposition 5.3. (cf. [26, 5.3]) Suppose that A is a simple C ∗ -algebra which satisfies thefollowing conditions: for any ε > , any finite subset F ⊂ A, any s ∈ A + \ { } , and any integer n ≥ , there are θ ∈ A and C ∗ -subalgebra D ⊗ M n ⊂ A such that(i) θx ≈ ε xθ for all x ∈ F , (ii) (1 − θ ) x ∈ ε D ⊗ n for all x ∈ F , and(iii) θ . s. Then A is tracially approximately divisible.Proof. Let
F ⊂ A be a finite subset, ε > e F ∈ A such that e F x ≈ ε/ x ≈ ε/ xe F for all x ∈ F , let s ∈ A + \ { } and an integer n be given. Without loss of generality, wemay assume that F ⊂ A . Let δ ∈ (0 , ε/
8) be a positive number such that, for any elements z ∈ A and w ∈ A , k zw − wz k < δ implies that k (1 − w ) / z − z (1 − w ) / k < ε/ . (e 5.1)Put F = F ∪ { e F } . Suppose that there are θ ∈ A and D as in the statement of the propositionsuch that (i), (ii) and (iii) hold for δ (in place of ε ) and F (in place of F ).Thus (3) in Definition 5.2 holds.Define β : A → A by β ( a ) = (1 − θ ) / a (1 − θ ) / for all a ∈ A. It is a c.p.c. map. Define,for each x ∈ F , x := θ / xθ / ∈ Her( θ ) . Then k x k ≤ k x k . Note that, by the choice of δ, forall x ∈ F ,e F β ( x ) = e F (1 − θ ) / x (1 − θ ) / ≈ ε/ (1 − θ ) / e F x (1 − θ ) / ≈ ε/ β ( x ) ≈ ε/ β ( x ) e F . (e 5.2)Moreover, for all x ∈ F ,β ( x ) = (1 − θ ) / x (1 − θ ) / ≈ ε/ (1 − θ ) x ∈ δ D ⊗ n . (e 5.3)So (2) in Definition 5.2 holds. Also, by the choice of δ, for all x ∈ F ,x = θx + (1 − θ ) x ≈ ε/ θ / xθ / + (1 − θ ) / x (1 − θ ) / = x + β ( x ) . (e 5.4)Hence (1) in Definition 5.2 holds. Thus A is tracially approximately divisible.The following lemma is a convenient folklore. Lemma 5.4.
Let δ > . There is an integer N ( δ ) ≥ satisfies the following:For any C ∗ -algebra A, any e ∈ A , and any x ∈ A, if x ∗ x ≤ e and xx ∗ ≤ e, then e /n x ≈ δ x ≈ δ xe /n for all n ≥ N ( δ ) . (e 5.5) Proof.
Let δ > N ( δ ) ≥ {| (1 − t /n ) t | : t ∈ [0 , } < δ for all n ≥ N ( δ ) . (e 5.6)Then for any C ∗ -algebra A, any e ∈ A , and any x ∈ A satisfying x ∗ x ≤ e and xx ∗ ≤ e, k (1 − e /n ) x k = k (1 − e /n ) xx ∗ (1 − e /n ) k / ≤ k (1 − e /n ) e (1 − e /n ) k / < δ (e 5.7)for all n ≥ N ( δ ) . Similarly, we also have k x (1 − e /n ) k < δ for all n ≥ N ( δ ) . Lemma follows.
Theorem 5.5. If A is a simple C ∗ -algebra which is tracially approximately divisible, then everyhereditary C ∗ -subalgebra of A is also tracially approximately divisible. roof. Let B be a hereditary C ∗ -subalgebra of A, F ⊂ B be a finite subset, ε > , s ∈ B + \ { } be a positive element, and let n ≥ b , b ∈ B such that b b = b = b b , b x ≈ ε/ x ≈ ε/ xb for all x ∈ F . (e 5.8)To simplify notation, without loss of generality, we may assume, by replacing x by b xb , that b x = xb = x, b x ∗ = x ∗ b = x ∗ for all x ∈ F . (e 5.9)Let F = { b , b , b / , x ∈ F } . Choose δ > ε/
64 (in placeof ε ) and σ = ε/ . Set η = min { δ/ , ε/ } . We choose N := N ( δ ) ≥ < δ < η/ . Moreover, we choose δ sufficiently small such that, if C ⊂ C be any pairof C ∗ -algebras and c ∈ C with 0 ≤ c ≤ c ∈ δ C , if 0 ≤ c , c ≤ c c ≈ δ c ≈ δ c c , then c /N ∈ η C and c c /N c ≈ η c /N . (e 5.10)Since A is tracially approximately divisible, there are θ a ∈ A , a C ∗ -subalgebra D a ⊗ M n ⊂ A and a c.p.c. map β : A → A such that(1) x ≈ δ x + β ( x ) such that k x k ≤ x ∈ Her( θ a ) for all x ∈ F , (2) for all x ∈ F , β ( x ) ∈ δ D a ⊗ n ,b β ( x ) ≈ δ β ( x ) ≈ δ β ( x ) b , and b / β ( x ) ≈ δ β ( x ) ≈ δ β ( x ) b / , and(3) θ a . s. Choose d ( x ) ∈ ( D a ⊗ n ) such that k β ( x ) − d ( x ) k < δ / x ∈ F . (e 5.11)Let b = β ( b ) /N . By (e 5.9), β ( b ) ≥ β ( x ) ∗ β ( x ) and β ( b ) ≥ β ( x ) β ( x ) ∗ for all x ∈ F (see, forexample, Corollary 4.1.3 of [5]).By (2) above and the choice of N and applying Lemma 5.4, b β ( x ) = β ( b ) /N β ( x ) ≈ δ β ( x ) for all x ∈ F . (e 5.12)Choose d ∈ ( D a ⊗ n ) + such that k d − b k < η/ b := b b b , by also (e 5.10), k d − b k < η/ f ε/ ( d ) b ≈ η/ f ε/ ( d ) d ≈ ε/ d ≈ η/ b. (e 5.15)By the choice of η, applying Lemma 3.3 of [14], there is an isomorphism ϕ : f ε/ ( d )( D a ⊗ M n ) f ε/ ( d ) → bAb ⊂ B such that k ϕ ( y ) − y k < ε/ k y k for all y ∈ f ε/ ( d )( D ⊗ n ) f ε/ ( d ) . (e 5.16)13ote that f ε/ ( d )( D a ⊗ n ) f ε/ ( d ) ∼ = D ⊗ n for some C ∗ -subalgebra D ⊂ D a . Let D b = ϕ ( D ) . Define a c.p.c. map α : B → B by α ( y ) = bβ ( y ) b for all y ∈ B. (e 5.17)Then, for all x ∈ F , by (e 5.15) and (e 5.11), α ( x ) = bβ ( x ) b ≈ η/ ε/ η/ f ε/ ( d ) dβ ( x ) df ε/ ( d ) (e 5.18) ≈ δ / f ε/ ( d ) dd ( x ) df ε/ ( d ) ∈ ε/ D b ⊗ n . (e 5.19)For each x ∈ F , let x ′ = b x b . Then, for all x ∈ F , by (1) and (2) above, and by (e 5.12), x ≈ δ b ( x + β ( x )) b = x ′ + b β ( x ) b (e 5.20) ≈ δ x ′ + β ( x ) ≈ δ x ′ + bβ ( x ) b = x ′ + α ( x ) . (e 5.21)Put θ b = b θ a b . Then x ′ ∈ θ b Bθ b . Moreover θ b . θ a . s. (e 5.22)Theorem follows.Recall that a non-unital C ∗ -algebra is called almost has stable rank one, if for every hereditary C ∗ -subalgebra B ⊂ A, B lies in the closure of invertible elements of e B ([33, Definition 3.1]). Lemma 5.6.
Let A be a C ∗ -algebra and n ∈ N . Let e , ..., e n ∈ A + be mutually orthogonalnon-zero positive elements. Assume d , ...d n ∈ A + such that d i . e i ( i = 1 , ..., n ), and e i d j = 0 whenever i ≤ j and i, j = 1 , ..., n. Then, for any a ∈ d Ad + ... + d n Ad n and any ε > , thereare nilpotent elements x, y ∈ A such that k a − yx k < ε. Proof.
Let a ∈ d Ad + ... + d n Ad n and fix ε > . Then there exist a , ..., a n ∈ A and δ > a ≈ ε f δ ( d ) a f δ ( d ) + ... + f δ ( d n ) a n f δ ( d n ) . Let x , ..., x n ∈ A such that x ∗ i x i = f δ ( d i )and x i x ∗ i ∈ e i Ae i , i = 1 , ..., n (see [35, Proposition 2.4]). For i, j ∈ { , ..., n } and i ≤ j, e i d j = 0implies x ∗ i x i x j x ∗ j = 0 , thus x i x j = 0 ( i ≤ j ) . (e 5.23)Claim 1: ( x + x + · · · + x n ) n +1 = 0 . Proof of Claim 1: Note that ( x + x + · · · + x n ) n +1 is a sum of n n +1 terms with theform x k x k · · · x k n +1 ( k , ..., k n +1 ∈ { , ..., n } ) . Assume x k x k ...x k n +1 = 0 , then x k i x k i +1 = 0( i = 1 , ..., n ). By (e 5.23), it follows that k i +1 ≤ k i − i = 1 , ..., n ). In particular, k n +1 ≤ k n − . Then k n +1 ≤ k n − ≤ k n − − . An induction implies that k n +1 ≤ k − n ≤ n n +1 terms of the form x k x k · · · x n +1 are zero. It follows that( x + x + · · · + x n ) n +1 = 0 . Claim 2: ( f δ ( d ) a x ∗ + ... + f δ ( d n ) a n x ∗ n ) n +1 = 0 . Proof of Claim 2: Let y i = f δ ( d i ) a i x ∗ i ( i = 1 , ..., n ). For i ≥ j, using (e 5.23), we have y i y j = f δ ( d i ) a i x ∗ i f δ ( d j ) a j x ∗ j = f δ ( d i ) a i ( x ∗ i x ∗ j ) x j a j x ∗ j = 0 . (e 5.24)Then, as in the proof of Claim 1, we have ( y + ... + y n ) n +1 = 0 , Claim 2 follows.Let x = x + ... + x n and let y = y + ... + y n = f δ ( d ) a x ∗ + ... + f δ ( d n ) a n x ∗ n . Then by Claim1 and Claim 2, both x and y are nilpotent elements. For i, j ∈ { , ..., n } and i = j, e i e j = 0implies x i x ∗ i x j x ∗ j = 0 , thus x ∗ i x j = 0 . Then yx = f δ ( d ) a f δ ( d ) + ... + f δ ( d n ) a n f δ ( d n ) ≈ ε a. heorem 5.7. Let A be a simple C ∗ -algebra which is tracially approximately divisible. Supposethat A is stably finite and W ( A ) is almost unperforated. Then A has stable rank one if A isunital, or A almost has stable rank one if A is not unital.Proof. We assume that A is infinite dimensional. Fix an element x ∈ A. Fix ε > . We mayassume that x is not invertible. Since A is finite, x is not one sided invertible. To show that x is a norm limit of invertible elements, it suffices to show that ux is a norm limit of invertibleelements for some unitary u ∈ e A. Note that, since A is simple, e A is prime. Thus, by Proposition3.2 and Lemma 3.5 of [34], we may assume that there is a ′ ∈ e A + \{ } and a ′ x = xa ′ = 0 . Thereis e ∈ A + such that a ′ ea ′ = 0 . Put a = a ′ ea ′ . Let B = { z ∈ A : az = za = 0 } . Then x ∈ B , and B is a hereditary C ∗ -subalgebra of A. There is e ′ b ∈ B with k e b k = 1 such that e ′ b xe ′ b ≈ ε/ x. So f ε/ ( e ′ b ) xf ε/ ( e ′ b ) ≈ ε/ x. Put e b = f ε/ ( e ′ b ) . Put B = Her( e b ) . Without loss of generality, we may further assume that x ∈ B. Since we assume that A is infinite dimensional, aAa contains non-zero positive elements a , a such that a a = 0 . Since A is simple, there is c ∈ A such that e b c ( a ) / = 0 (see the proof of [10, 1.8]).Note, since e b ∈ Ped( B ) . Then Ped( B ) = B (see, for example, [3, II.5.4.2]). It follows thatthere are y , y , ..., y m ∈ B such that m X i =1 y ∗ i e b ca c ∗ e b y i = e b . (e 5.25)It follows that h e b i ≤ m h a i . Put n = 2 m. For any z , z , ..., z n ∈ B + which are n mutually orthogonal and mutually equivalent positiveelements, n h z i ≤ h e b i ≤ m h a i . Since W ( A ) is almost unperforated, z . a . (e 5.26)Since B is a hereditary C ∗ -subalgebra of A, by Theorem 5.5, B is also tracially approximatelydivisible. There are b ∈ B , a C ∗ -subalgebra D ⊗ M n ⊂ B, and a c.p.c. map β : A → A suchthat(1) x ≈ ε/ x + β ( x ) , where x ∈ bAb, (2) β ( x ) ∈ ε/ D ⊗ n , and(3) b . a . Thus, there is x ∈ D \ { } such that k x − ( x + x ⊗ n ) k < ε/ . (e 5.27)Choose a positive element d ∈ D such that k dx d − x k < ε/ . (e 5.28)By the choice of n, d ⊗ e , . a (where { e i,j } forms a system of matrix units for M n ).Define g = a , g = a , g i = d ⊗ e i,i ( i = 1 , ..., n − h = b, h i = d ⊗ e i,i ( i = 1 , ..., n ).Note that h i . g i ( i = 1 , ..., n + 1), and g i h j = 0 , if i ≤ j, and i, j = 1 , ..., n + 1 . Note that x + dx d ⊗ n ∈ h Ah + h Ah + ... + h n +1 Ah n +1 . Then, by Lemma 5.6, there are nilpotentelements v, w ∈ A such that x + dx d ⊗ n ≈ ε/ vw. Choose δ > vw ≈ ε/ ( v + δ )( w + δ ) . v, w are nilpotent elements, v + δ and w + δ are invertible. Then, combining (e 5.27) and(e 5.28), x ≈ ε/ x + x ⊗ n ≈ ε/ x + dx d ⊗ n ≈ ε/ ( v + δ )( w + δ ) ∈ GL ( e A ) . (e 5.29)Therefore we have shown that x ∈ GL ( e A ) . Thus, in the case that A is unital, A has stablerank one. Since, by Theorem 5.5, this works for every hereditary C ∗ -subalgebra of A, A almosthas stable rank one in the case that A is not unital. Remark 5.8.
In [17], we show that a separable simple C ∗ -algebra which is tracially approx-imately divisible has strict comparison for positive elements. So there is a redundancy in theassumption of Theorem 5.7. Since we do not use this fact here in this paper, we leave it for [17]. Theorem 5.9.
Let A be a simple C ∗ -algebra. If A is e.tracially in C Z then A is traciallyapproximately divisible.Proof. We assume that A is infinite dimensional. Let A be a simple C ∗ -algebra which is e. tra-cially in C Z . Let ε > , F ⊂ A be a finite subset, a ∈ A + \ { } and n ≥ A is infinite dimensional, choose a , a ∈ Her( a ) + \ { } such that a a = a a = 0 . There is e A ∈ A and δ > f δ ( e A ) x ≈ ε/ x ≈ ε/ xf δ ( e A ) for all x ∈ F . (e 5.30)Note, by Theorem 5.5, A := f δ/ ( e A ) Af δ/ ( e A ) is also a ( σ -unital) simple C ∗ -algebra which ise.tracially in C Z (as C Z has property (H), see Corollary 3.1 of [41]).Note also f δ/ ( e A ) af δ/ ( e A ) . a. To simplify notation, by replacing x by f δ ( e A ) xf δ ( e A ) forall x ∈ F , and by replacing a by f δ/ ( e A ) af δ/ ( e A ) , and a i by f δ/ ( e A ) a i f δ/ ( e A ) ( i = 1 , x, a, a , a ∈ A . We may also assume, withoutloss of generality, e x = x = xe for all x ∈ F (e 5.31)for some strictly positive element e ∈ A . Note that f δ/ ( e A ) ∈ Ped( A ) . Therefore A isalgebraically simple and f δ/ ( e A ) is a strictly positive element of A . There is an integer l ≥ x i ∈ A , i = 1 , , ..., l, such that l X i =1 x ∗ i a x i = e . (e 5.32)Set F = F ∪ { e } . There exists θ ∈ A and a Z -stable C ∗ -subalgebra B of A such that(i) k θ x − xθ k < ε/
64 and k (1 − θ ) / x − x (1 − θ ) / k < ε/
64 for all x ∈ F , (ii) (1 − θ ) / x (1 − θ ) / , (1 − θ ) / x, (1 − θ ) x, x (1 − θ ) , (1 − θ ) x (1 − θ ) ∈ ε/ B for all x ∈ F , and(iii) θ . a . Let F = { (1 − θ ) / x (1 − θ ) / , (1 − θ ) / x, (1 − θ ) x, x (1 − θ ) , (1 − θ ) x (1 − θ ) : x ∈ F } . For each f ∈ F , fix b ( f ) ∈ B such that k b ( f ) k ≤ k f − b ( f ) k < ε/ . (e 5.33)Let G = { b ( f ) : f ∈ F } . We write B = C ⊗ Z . Since Z is strongly self-absorbing, withoutloss of generality, we may assume that there is a finite subset G ⊂ C such that G = { y ⊗ Z :16 ∈ G } ⊂ C ⊗ Z . To further simplify notation, without loss of generality, we may assume thatthere exists a strictly positive element e C ∈ C such that such that e b y = y = ye b for all y ∈ G , (e 5.34)where e b = e C ⊗ Z . For any integer n, choose m > l and n divides m. Let ψ : M m → Z be an oder zero c.p.c mapsuch that 1 Z − ψ (1 m ) . ψ ( e , ) (e 5.35)(see (iv) implies (ii) of Proposition 5.1 of [38]). Define ϕ : M m → B by ϕ ( c ) = e C ⊗ ψ ( c ) for all c ∈ M m . Set θ := e b − ϕ (1 m ) = e C ⊗ Z − e C ⊗ ψ (1 m ) = e C ⊗ (1 Z − ψ (1 m )) . e C ⊗ ψ ( e , ) . (e 5.36)Note that θ g = gθ for all g ∈ G . (e 5.37)Define D = e C Ce C ⊗ ϕ ( e , ) and D ′ the C ∗ -subalgebra generated by { e C ce C ⊗ ψ ( z ) : c ∈ C and z ∈ M m } . (e 5.38)Recall that ψ gives a homomorphism H : C ∗ ( ψ (1 m )) ⊗ M n → Z such that H ( f ⊗ g ) = f ( ψ (1 A )) H ( g ) for all g ∈ M n and f ∈ C (sp( ψ (1 A ))) . It follows D ′ ∼ = D ⊗ M m . Define β : A → A by β ( y ) = (1 e A − θ ) / y (1 e A − θ ) / for all y ∈ A (e 5.39)(where 1 e A denotes the identity of e A when A is not unital, and 1 e A is the identity of A if A has one).Note also (1 − θ ) / is an element which has the form 1+ f ( θ ) for f ( t ) = (1 − t ) / − ∈ C ((0 , . If g = y ⊗ Z ∈ G , then (note y ∈ G ⊂ C and see also (e 5.34)), β ( g ) = (1 − θ ) g = g − e C ⊗ (1 Z − ψ (1 m )) g (e 5.40)= ( e C ⊗ Z ) g − e C ⊗ (1 Z − ψ (1 m )) g (e 5.41)= ( e C ⊗ ψ (1 m )( y ⊗ Z ) = ( e / C ye / C ) ⊗ ψ (1 m ) ∈ D ⊗ m . (e 5.42)Define a c.p.c map β : A → A by β ( x ) = β ((1 − θ ) / x (1 − θ ) / ) for all x ∈ A. (e 5.43)For x ∈ F , let f = (1 − θ ) / x (1 − θ ) / . Then, by (e 5.33), β ( x ) = β ((1 − θ ) / x (1 − θ ) / ) ≈ ε/ β ( b ( f )) ∈ D ⊗ m . (e 5.44)Put θ = θ + (1 − θ ) / θ (1 − θ ) / . We have0 ≤ θ ≤ θ + (1 − θ ) / (1 − θ ) / = 1 . (e 5.45)For x ∈ F , let f ′ = (1 − θ ) x. Recall that we assume that b ( f ′ ) = y ′ ⊗ Z for some y ′ ∈ C . Then, for x ∈ F , applying (e 5.33) and (e 5.37) repeatedly,(1 − θ ) x = (1 − θ ) x − (1 − θ ) / θ (1 − θ ) / x (e 5.46) ≈ ε/ (1 − θ ) x − (1 − θ ) xθ = (1 − θ ) x (1 − θ ) (e 5.47) ≈ ε/ b ( f ′ )(1 − θ ) = (1 − θ ) / b ( f ′ )(1 − θ ) / (e 5.48)= β ( b ( f ′ )) ≈ ε/ β ((1 − θ ) / x (1 − θ ) / ) = β ( x ) . (e 5.49)17herefore x ≈ ε θ / xθ / + β ( x ) for all x ∈ F . (e 5.50)Note that, by (e 5.35) and (e 5.32), in W ( B ) ,m h θ i = m h e C ⊗ (1 Z − ψ (1 m )) i (e 5.51) ≤ m h e C ⊗ ψ ( e , ) i ≤ h e C ⊗ ψ (1 m ) i ≤ h e C ⊗ Z i ≤ l h a i . (e 5.52)By Theorem 4.5 of [37], W ( B ) is almost unperforated. Therefore, since l < m,θ . a . (e 5.53)It follows that (note a a = a a = 0) θ = θ + (1 − θ ) / θ (1 − θ ) / . a + a . a. (e 5.54)Finally, the theorem follows from (e 5.37), (e 5.42) and (e 5.54), and the fact that D ⊗ n embeddedinto D ⊗ m unitally (as n divides m ). Corollary 5.10.
Let A be a simple C ∗ -algebra which is e.tracially in C Z . If A is not purelyinfinite, then A has stable rank one, if A is unital, and A almost has stable rank one, if A isnot unital.Proof. By Theorem 5.9, A is tracially approximately divisible. By Theorem 5.1, if A is not purelyinfinite, then A has strict comparison for positive elements. It follows then form Theorem 5.7that A has stable rank one, if A is unital, and A almost has stable rank one, if A is not unital. Remark 5.11.
For the rest of this paper, we will present non-amenable examples of C ∗ -algebraswhich are possibly stably projectionless and are essentially tracially in the class C Z , the class of Z -stable C ∗ -algebras. A Cz In this section, we first fix a separable residually finite dimensional (RFD) C ∗ -algebra C whichmay not be exact.Let B be the unitization of C ((0 , , C ) . Since C ((0 , , C ) is contractible, V ( B ) = N ∪ { } ,K ( B ) = Z and K ( B ) = { } . Let us make the convention that B includes the case that C = { } , i.e., B = C . Let p = p r · p r · · · · be a supernatural number, where p , p , ..., is a sequence (could befinite) of distinct prime numbers and r i ∈ N ∪ {∞} . Denote by D p the subgroup of Q generatedby finite sums of rational numbers of the form mp ij , where m ∈ Z and i ∈ N ∩ [1 , r j ] . Suppose that q is another supernatural number. Then we may identify D p ⊗ D q with D pq . Denote by M p the UHF-algebra associated with the supernatural number p . The following is a result of M. D˘ad˘arlat [12].
Theorem 6.1.
Fix a supernatural number d . There is a unital simple C ∗ -algebra A d which isan inductive limit of M m ( n ) ( B ) such that ( K ( A d ) , K ( A d ) + , [1 A d ]) = ( D d , D d + , and K ( A d ) = { } , A d has a unique tracial state and A d has tracial rank zero. efinition 6.2. Fix a supernatural number d . One may write A d = lim n →∞ ( M d ′ n ( B ) , δ ′ n ) , where d ′ n +1 = d n · d ′ n , d n , d ′ n > δ ′ n : M d ′ n ( B ) → M d ′ n +1 ( B ) is defined by δ ′ n ( f ) = (cid:18) f γ n ( f ) (cid:19) for all f ∈ M d ′ n ( B ) , (e 6.1)and where γ n : B → M d n − is a d n − γ n for theextension γ n ⊗ id d ′ n : M d ′ n ( B ) → M ( d n − d ′ n ). We may also assume thatlim n →∞ d n = ∞ . (e 6.2)This is possible, as we may choose a subsequence { d n } (and d ′ n +1 ) and the fact that f (0) ∈ C for f ∈ B (see the proof of Proposition 8 of [12]). Also, we assume, for any n, { γ m : m ≥ n } is aseparating sequence of finite dimensional representations. For more specific construction of A d , the reader is referred to [12].It is important that, for any τ ∈ T ( B ) , lim n →∞ | τ ◦ δ ′ n ( a ) − τ ( γ n ( a )) | = 0 for all a ∈ M d ′ n ( B ) . (e 6.3)(Note, by τ, we mean τ ⊗ tr d ′ n , where tr d ′ n is the tracial state of M d ′ n . )Consider δ ′ m,n := δ ′ n − ◦ δ ′ n − ◦ · · · ◦ δ ′ m : M d ′ m ( B ) → M d ′ n ( B ) . Then, we may write δ ′ m,n ( f ) = (cid:18) f γ m,n ( f ) (cid:19) for all f ∈ M d ′ m ( B ) , (e 6.4)where γ m,n : B → M d ′ n /d ′ m − is a finite dimensional representation (in what follows throughoutthe rest of the paper, we also use γ m,n := γ m,n ⊗ id d m : M N ( B ) → M ( d ′ n /d ′ m − N for all integers N ≥ F m ⊂ M d ′ m ( B ) , we may assume that γ m,n ( a ) = 0 forsome large n ≥ m. In fact, by first adding | a | ∈ F m , and then considering f a ( | a | ) for a continuousfunction f a ∈ C ((0 , k a k ) , we may assume that k γ m,n ( a ) k ≥ (1 − /m ) k a k . (e 6.5)In fact, for any fixed m and any nonzero element a ∈ M d ′ m ( B ) \ { } , there is n ≥ m such that k γ m,n ( a ) k 6 = 0 . (e 6.6)In what follows that A d = lim n →∞ ( M d ′ n ( B ) , δ ′ n ) is the C ∗ -algebra in Theorem 6.1 and δ ′ n asdescribed in (e 6.1) such that (e 6.5) holds once F m is chosen.We wish to construct a unital simple C ∗ -algebra A z with a unique tracial state such that K ( A z ) = Z and K ( A z ) = { } . The strategy is to have a Jiang-Su style inductive limit of some C ∗ -subalgebras of C ([0 , , M p ( B ) ⊗ M q ( B )) for some nonnuclear RFD algebra B, or perhaps some C ∗ -subalgebraof C ([0 , , M pq ( B )) . However, there were several difficulties to resolve. One should avoid to use M p ( B ) ⊗ M q ( B ) as building blocks as there are different C ∗ -tensor products and potential diffi-culties to compute the K -theory. Other issues include the fact that, for each t ∈ [0 , , M m ( B )is not simple.We begin with the following building blocks. Definition 6.3.
Define, for a pair of integers m, k ≥ ,E m,k = { f ∈ C ([0 , , M mk ( B )) : f (0) ∈ M m ( B ) ⊗ k and f (1) ∈ m ⊗ M k } . Note here one views M m ( B ) ⊗ k , m ⊗ M k ⊂ M m ( B ) ⊗ M k ⊂ M mk ( B ) as C ∗ -subalgebras.19ix integers m, n ≥ . Let D ( m, k ) = M m ( B ) ⊕ M k . Define ϕ : D ( m, k ) → M m ( B ) ⊗ k by ϕ (( a, b )) = a ⊗ k for all ( a, b ) ∈ D ( m, k ) and ϕ : D ( m, k ) → M k by ϕ (( a, b )) = 1 m ⊗ b. Then E m,k ∼ = { ( f, g ) ∈ C ([0 , , M mk ( B )) ⊕ D ( m, k ) : f (0) = ϕ ( g ) and f (1) = ϕ ( g ) } . (e 6.7)Denote by π e : E m,k → D ( m, k ) the quotient map which maps ( f, g ) to g. Denote by π : E m,k → M m ( B ) ⊗ k the homomorphism defined by π (( f, g )) = ϕ ( g ) = f (0) and π : E m,k → m ⊗ M k the homomorphism defined by π (( f, g )) = ϕ ( g ) = f (1) . Lemma 6.4. If m and k are relatively prime, then E m,k has no proper projections.Proof. Recall that B = C ^ ((0 , , C ), the unitization of C ((0 , , C ) . Let τ B be the tracial stateon M m ( B ) induced by the quotient map B → B/C ((0 , , C ) ∼ = C , and let tr k be the tracialstate of M k . Let τ = τ B ⊗ tr k . Let e ∈ E m,k be a nonzero projection. Note E m,k ⊂ C ([0 , , M mk ( B )) . Note that K ( B ) = Z and 1 B is the only nonzero projection of B. Then, for each x ∈ [0 , , e ( x ) is a nonzero projectionin M mk ( B ) . One easily shows that τ ( e ( x )) is a constant function on [0 , . Let τ ( e ( x )) = r ∈ (0 , . But τ ( e (0)) ∈ { i/m : i = 0 , , ..., m } and τ ( e (1)) ∈ { j/k, i = 0 , , ..., k } . Since m and k arerelatively prime, τ ( e (0)) = τ ( e (1)) = 1 . Hence τ ( e ( x )) = 1 for all x ∈ [0 , . This is possible onlywhen e = 1 m ⊗ k . Lemma 6.5.
Suppose that m and k are relatively prime. Then ( K ( E m,k ) , K ( E m,k ) + , [1 E m,k ]) = ( Z , N ∪ { } ,
1) and K ( E m,k ) = { } . Proof.
Let I = { f ∈ E ( m, k ) : f (0) = f (1) = 0 } . Then I ∼ = C ((0 , ⊗ M mk ( B ) := S ( M mk ( B )) . It follows that K ( I ) = K ( M mk ( B )) = { } and K ( I ) = K ( M mk ( B )) = Z . (e 6.8)Consider the short exact sequence0 → I ι I −→ E m,k π e −→ D ( m, k ) → , (e 6.9)where ι I : I → E m,k is the embedding and π e : E m,k → D ( m, k ) is the quotient map. Oneobtains the following six-term exact sequence K ( I ) ι I ∗ −→ K ( E m,k ) π e ∗ −→ K ( D ( m, k )) ↑ δ ↓ δ K ( D ( m, k ))) π e ∗ ←− K ( E m,k ) ι I ∗ ←− K ( I ) (e 6.10)which becomes 0 ι I ∗ −→ K ( E m,k ) π e ∗ −→ Z ⊕ Z ↑ δ ↓ δ π e ∗ ←− K ( E m,k ) ι I ∗ ←− Z (e 6.11)Note that im( π e ∗ ) = { ( x, y ) ∈ K ( D ( m, k )) : ϕ ∗ ( x ) = ϕ ∗ ( y ) } . Z m,k = { f ∈ C ([0 , , M mk ) : f (0) ∈ M m ⊗ k and f (1) ∈ m ⊗ M k } . (e 6.12)We may view Z m,k ⊂ E m,k by identifying M mk and M m with obvious unital C ∗ -subalgebras of M mk ( B ) and M m ( B ) , respectively.For each s ∈ [0 , , define θ s : M mk ( B ) → M mk ( B ) by θ s ( f )( x ) = f ((1 − s ) x ) for x ∈ (0 , f ∈ M mk ( B ) = M mk ( C ((0 , , C ) e ) . Note that θ ( f )( x ) = f (0) ∈ M mk for all f ∈ M mk ( B ) . For each s ∈ [0 , , define a unital homomorphism Φ s : E m,k → E m,k byΦ s ( a )( t ) = θ s ( a ( t )) for all a ∈ E m,k ⊂ C ([0 , , M mk ( B )) and t ∈ [0 , . (e 6.13)Then Φ = id E m,k and Φ maps E m,k into Z m,k and Φ | Z m,k = id Z m,k . In other words, E m,k ishomotopic to Z m,k . It is known, when m and k are relatively prime (see, for example, Lemma2.3 of [22]), and easy to check that( K ( Z m,k ) , K ( Z m,k ) + , [1 Z m,k ] , K ( Z m,k )) = ( Z , N ∪ { } , , { } ) . Let I be the ideal in the proof of Lemma 6.5. Then I ∼ = C ((0 , ⊗ M mk ( B ) . Let τ ∈ T ( E m,k ) . Then it is well known that τ | I ( f ) = R (0 , σ t ( f ( t )) dµ for all f ∈ C ((0 , ⊗ M mk ( B ) , where σ t is a tracial state of M mk ( B ) and µ is a Borel measure on (0 ,
1) (with k µ k ≤ E m,k /I = M m ( B ) ⊕ M k , as in 2.5 of [27], one may write τ ( f ) = Z σ t ( f ( t )) dν for all f ∈ E m,k , (e 6.14)where σ is a tracial state on M m ( B ) , σ is a tracial state on M k , and ν is a probability Borelmeasure on [0 ,
1] (and ν | (0 , = µ ). Notation 6.6.
Let γ : B → M r be a finite dimensional representation with rank r, i.e., γ isa finite direct sum of irreducible representations γ j : j = 1 , , ..., l, each of which has rank r j (1 ≤ j ≤ l ) such that r = P lj =1 r j . We will also use γ for γ ⊗ id m : M m ( B ) → M rm for allintegers m ≥ . In what follows we may also write M L for M L ( C · B ) for all integers L ≥ . Inthis way γ ( γ ⊗ id m ) is a homomorphism from M m ( B ) into M rm ⊂ M rm ( B ) . Let ξ , ξ , ξ , ..., ξ k − : [0 , → [0 ,
1] be continuous paths. Define a homomorphism H : C ([0 , , M mn ( B )) → C ([0 , , M (( k − r +1) mn ( B )) by H ( f ) = f ◦ ξ · · · γ ( f ◦ ξ ) · · · · · · γ ( f ◦ ξ k − ) for all f ∈ C ([0 , , M mn ( B )) . (e 6.15)Note that H can be also defined on E m,n ⊂ C ([0 , , M mn ( B )) . But, in general, H does notmap E m,n into E m,n . However, with some restrictions on the boundary (restriction on ξ i ’s), itis possible that H maps E m,n into E m,n . For the convenience of the construction in the next lemma, let us add some notation andterminologies.Let f, g ∈ M n ( B ) . We write f = s g, if there is a scalar unitary w ∈ M n such that w ∗ f w = g. Also, if f, g ∈ C ([0 , , M n ( B )) , we write f = s g, if there is a unitary w ∈ C ([0 , , M n ) such that w ∗ f w = g. emma 6.7. There exists an inductive limit A = lim n →∞ ( A n , ϕ n ) , where each A n = E p n ,q n with ( p n , q n ) = 1 , such that each connecting map ϕ m : A m → A m +1 is a unital injective homo-morphism of the form: ϕ m ( f ) = u ∗ Θ m ( f ) 0 · · · γ m ( f ◦ ξ ( t )) ⊗ · · · ... ... ... · · · γ m ( f ◦ ξ k ( t )) ⊗ u for all f ∈ A m , (e 6.16) where u ∈ U ( C ([0 , , M p m +1 q m +1 )) , Θ m : A m → C ([0 , , M R ( m, p m q m ( B )) is a homomorphism, R ( m, ≥ is an integer, t ∈ [0 , , and γ m : B → M R ( m ) is a finite dimensional representation.Moreover,(1) each ξ i : [0 , → [0 , is a continuous map which has one of the following three forms: ξ i ( t ) = ( (2 / t if t ∈ [0 , / , / if t ∈ (3 / , , ξ i ( t ) = 1 / for all t ∈ [0 , , and (e 6.17) ξ i ( t ) = ( / / t if t ∈ [0 , / , if t ∈ (3 / , , (e 6.18) and each type of ξ i appears in (e 6.16) at least once,(2) R ( m, / kR ( m ) < / m , (3) for a fixed finite subset F m ⊂ A m \ { } ⊂ C ([0 , , M p m q m ( B )) , k γ m ( f ( t )) k > (1 − / m ) k f k 6 = 0 for some t ∈ [0 , , (4) Θ m ( f ) = diag( θ (1) ( f ) , · · · , θ ( k ) ( f )) , (e 6.19) where θ ( i ) : E p m ,q m → C ([0 , , M p m q m ( B )) is defined by, if ξ i (3 /
4) = 1 / , for each f ∈ E p m ,q m ,θ ( i ) ( f )( t ) = ( f ( ξ i ( t )) if t ∈ [0 , / ,θ t − / ( f ( ξ i ( t ))) if t ∈ (3 / , , and where θ t : M p m q m ( B ) → M p m q m ( B ) (recall B = ^ C ((0 , , C ) ) is a unital homomorphismdefined by θ t ( f )( x ) = f ((1 − t ) x ) for all x ∈ (0 ,
1] and for all t ∈ [0 , , (e 6.20)and, if ξ i (3 /
4) = 1 , for each f ∈ E p m ,q m ,θ ( i ) ( f )( t ) = f ( ξ i ( t )) t ∈ [0 , . Proof.
The construction will be by induction. Set A = E , . Denote by ¯ the supernatural number 3 ∞ . Write A ¯ = lim n →∞ ( M d ′ n ( B ) , δ ′ n ) (see Theorem6.1), where δ ′ n ( f ) = (cid:18) f γ n ( f ) (cid:19) for all f ∈ M d n ( B ) (e 6.21)22s in (e 6.1) which also has the properties (e 6.2), (e 6.3) (with d n = 1 / l for some integer l ≥ n, d n < / n . (e 6.22)Recall that B = C ( ^ (0 , , C ) . Denote, for each t ∈ [0 , , by θ t : B → B the homomorphismdefined by, for all f ∈ B, θ t ( f )( x ) = f ((1 − t ) x ) for all x ∈ (0 , . (e 6.23)Note also, for any integer l ≥ , we will use θ t for θ t ⊗ id l : M l ( B ) → M l ( B ) . Thus, if f ∈ M l ( B ) ,θ ( f ) = f (0) ∈ M l . (e 6.24)It should be noted that θ = id M l ( B ) . To avoid potential complication of computing relative primitivity of integers, we will havethree-stage construction.Stage 1: Write A m = E p m ,q m , where ( p m , q m ) = 1 . Also (5 , p m ) = 1 and (3 , q m ) = 1 . Fix any finite subset F m ⊂ E p m ,q m \ { } . One can choose a finite subset S ⊂ [0 ,
1] such that,for any f ∈ F m , there is s ∈ S, k f ( s ) k > (1 − / m ) k f k 6 = 0 . Note that F ′ = { f ( s ) : f ∈ F m and s ∈ S } \ { } is a finite subset of M p m q m ( B ) . By passing to asubsequence we may assume (replacing γ m by γ m,n as mentioned in (e 6.5)) that k γ m ( g ) k > (1 − / m ) k g k 6 = 0 for all g ∈ F ′ . (e 6.25)It follows that, for any f ∈ F m , k γ m ( f ( s )) k ≥ (1 − / m ) k f k 6 = 0 for some s ∈ S ⊂ [0 , . (e 6.26)Define ψ ′ m : M p m ( B ) ⊗ M q m → M d m p m ( B ) ⊗ M q m by ψ ′ m = δ m ⊗ s, where δ m ( a ) = (cid:18) a γ m ( a ) (cid:19) for all a ∈ M p m ( B ) , and s ( c ) = c ⊗ for all c ∈ M q m . (e 6.27)Define ψ m : E p m ,q m → E d m p m , q m by ψ m ( f )( t ) = ψ ′ m ( f ( t )) for all f ∈ E p m ,q m and t ∈ [0 , . (e 6.28)Let f ∈ E p m ,q m . Then f (0) = b ⊗ q m , where b ∈ M p m ( B ) . Thus, ψ m ( f )(0) = ψ ′ m ( f (0)) = δ m ( b ) ⊗ (1 q m ⊗ ) ∈ M d m p m ( B ) ⊗ q m . (e 6.29)On the other hand, f (1) = 1 p m ⊗ c, where c ∈ M q m . Thus ψ m ( f )(1) = ψ ′ m ( f (1)) = 1 d m p m ⊗ ( c ⊗ ) ∈ d m p m ⊗ M q m . (e 6.30)So, indeed, ψ m maps E p m ,q m into E d m p m , q m . Note, for t ∈ [0 , , we have, for all f ∈ E p m ,q m ,ψ m ( f )( t ) = ψ ′ m ( f ( t )) = (cid:18) f ( t ) 00 γ m ( f ( t )) (cid:19) ⊗ . (e 6.31)Stage 2: We will use a modified construction of Jiang-Su and define ϕ m on [0 , / . k and k such that k > q m and k > k d m p m . (e 6.32)In particular, k , k = 3 , . Recall (3 , q m ) = 1 , (5 , p m ) = 1 and d m = 3 l m for some l m ≥ . Therefore ( k d m p m , k q m ) =1 . Let p m +1 = k d m p m , q m +1 = k q m , and k = k k . Write k = r + m (0) q m +1 and k = r + m (1) p m +1 , (e 6.33)where m (0) , r , m (1) , r ≥ < r < q m +1 , r ≡ k (mod q m +1 ) , and (e 6.34)0 < r < p m +1 , r ≡ k (mod p m +1 ) . (e 6.35)Moreover, by (e 6.32), k − r − r > k − q m +1 − p m +1 = k − k q m − k d m p m = k ( k − q m ) − k d m p m > k (10 q m ) − k d m p m > . (e 6.36)We will construct paths ξ i . At t = 0 , define ξ i (0) = ( ≤ i ≤ r , / r < i ≤ k . (e 6.37)Note that, since r q m ≡ k q m ≡ k k q m ≡ q m +1 ) , (e 6.38) r q m = t q m +1 for some integer t ≥ . Note also, if f ∈ E d m p m , q m , then f (0) = b ⊗ q m forsome b ∈ M d m p m ( B ) . Hence f (0) ⊗ r = b ⊗ r q m = ( b ⊗ t ) ⊗ q m +1 for any f ∈ E d m p m , q m . On the other hand, for any f ∈ E d m p m , q m , diag( f ( ξ r +1 (0)) , · · · , f ( ξ k (0))) = s ( f (1 / ⊗ m (0) q m +1 . (e 6.39)Therefore there exists a unitary v ∈ U ( M p m +1 q m +1 ) such that, for all f ∈ E d m p m , q m ,ρ ( f ) := v ∗ f ( ξ (0)) 0 · · · f ( ξ (0)) · · · · · · f ( ξ k (0)) v (e 6.40)is in M p m +1 ( B ) ⊗ q m +1 . So ρ defines a homomorphism from E d m p m , q m into M p m +1 ( B ) ⊗ q m +1 . At t = 3 / , define ξ i (3 /
4) = ( / ≤ i ≤ k − r k − r < i ≤ k . (e 6.41)As in the case at 0 , by (e 6.35), r d m p m ≡ kd m p m ≡ k k d m p m ≡ p m +1 ) .
24o one may write r d m p m = t p m +1 for some integer t ≥ . Let f ∈ E d m p m , q m . Then f (1) =1 d m p m ⊗ c for some c ∈ M q m . It follows that 1 r ⊗ f (1) = 1 p m +1 ⊗ (1 t ⊗ c ) . Also,diag( f ( ξ (3 / , · · · , f ( ξ k − r (3 / s m (1) p m +1 ⊗ f (1 / . Thus there is a unitary v / ∈ U ( M p m +1 q m +1 ) such that (for f ∈ E d m p m , q m ) ρ / ( f ) := v ∗ / f ( ξ (3 / · · · f ( ξ (3 / · · · · · · f ( ξ k (3 / v / (e 6.42)defines a homomorphism from E d m p m , q m to 1 p m +1 ⊗ M q m +1 ( B ) . To connect ξ i (0) and ξ i (3 /
4) continuously, on [0 , / , let us define (see (e 6.36)) ξ i ( t ) = t/ ≤ i ≤ r , / r < i ≤ k − r , and1 / t/ k − r < i ≤ k . (e 6.43)Let v ∈ C ([0 , / , M p m +1 q m +1 ) be a unitary such that v (0) = v and v (3 /
4) = v / . Now, on[0 , / , define, for all f ∈ E p m ,q m ,ϕ m ( f )( t ) = v ( t ) ∗ ψ m ( f ) ◦ ξ ( t ) 0 · · · ψ m ( f ) ◦ ξ ( t ) · · · · · · ψ m ( f ) ◦ ξ k ( t ) v ( t ) . (e 6.44)Stage 3. Connecting 3 / π ( E p m +1 ,q m +1 ) = 1 p m +1 ⊗ M q m +1 ).We first extend ξ i by defining ξ i ( t ) = ξ i (3 /
4) for all t ∈ (3 / , , i = 1 , , ..., k. (e 6.45)Recall (e 6.31), at t = 3 / , for each i and for f ∈ E p m ,q m ,ψ m ( f )( ξ i (3 / (cid:18) f ( ξ i (3 / γ m ( f ( ξ i (3 / (cid:19) ⊗ , (e 6.46)For k − r < i ≤ k, define, for t ∈ (3 / ,
1] and for f ∈ E p m ,q m , ˜ ψ m,i ( f )( t ) = (cid:18) f ( ξ i (3 / γ m ( f ( ξ i (3 / (cid:19) ⊗ = (cid:18) f (1) 00 γ m ( f (1)) (cid:19) ⊗ (e 6.47)= s d m ⊗ f (1) ⊗ . (e 6.48)Note f (1) has the form 1 p m ⊗ c for some c ∈ M q m . So, one may writediag( ˜ ψ m,k − r +1 ( f )( t ) , · · · ˜ ψ m,k ( f )( t )) = s r d m p m ⊗ c ⊗ = 1 t p m +1 ⊗ c ⊗ . (e 6.49)Now recall (e 6.23) for the definition of θ t . For 1 ≤ i ≤ k − r , define, for t ∈ (3 / , , ˜ ψ m,i ( f )( t ) = (cid:18) θ t − / ( f ( ξ i ( t )) 00 γ m ( f ( ξ i ( t ))) (cid:19) ⊗ for all f ∈ E p m ,q m . (e 6.50)25ote that θ ( f ( ξ i (3 / θ ( f (1 / ∈ M p m q m (see (e 6.41) and (e 6.24)). Recall that ϕ m ( f )(3 / ∈ p m +1 ⊗ M q m +1 ( B ) . Moreover (see also (e 6.33)),diag( ˜ ψ m, ( f )(1) , · · · , ˜ ψ m,k − r ( f )(1)) = s m (1) p m +1 ⊗ (cid:18) θ ( f (1 / γ m ( f (1 / (cid:19) ⊗ . (e 6.51)Thus, for t = 1 , there is a unitary v ∈ p m +1 ⊗ M q m +1 such that ρ ( f ) = v ∗ ˜ ψ m, ( f )(1) 0 · · ·
00 ˜ ψ m, ( f )(1) · · · · · · ˜ ψ m,k ( f )(1) v (e 6.52)defines a homomorphism from E p m ,q m to 1 p m +1 ⊗ M q m +1 . There is a continuous path of unitaries { v ( t ) : t ∈ [3 / , } ⊂ M p m +1 q m +1 such that v (3 /
4) is as defined and v (1) = v (so now v ∈ C ([0 , , M p m +1 q m +1 ) with v (0) = v and v (1) = v as well as v (3 /
4) is consistent with previousdefinition). Now define, for t ∈ (3 / , ,ϕ m ( f )( t ) = v ( t ) ∗ ˜ ψ m, ( f )( t ) 0 · · ·
00 ˜ ψ m, ( f )( t ) · · · · · · ˜ ψ m,k ( f )( t ) v ( t ) (e 6.53)for all f ∈ E p m ,q m . Note, by (e 6.46), (e 6.50), (e 6.52) and (e 6.42), ϕ m ( f )(1) = ρ ( f ) for all f ∈ E p m ,q m . (e 6.54)Hence ϕ m is a unital injective homomorphism from E p m ,q m to E p m +1 ,q m +1 . (Note that injectivityfollows from the fact that ∪ ki =1 ξ i ([0 , , , as r ≥ k − r > ψ m,i : E p m ,q m → C ([0 , , M d m p m q m ( B )) by˜ ψ m,i ( f )( t ) = ( ψ m ( f ◦ ξ i ( t )) if t ∈ [0 , / ψ m,i ( f )( t ) if t ∈ (3 / ,
1] for all f ∈ E p m ,q m . (e 6.55)Then, we may write, for all t ∈ [0 , , for all f ∈ E p m ,q m ,ϕ m ( f )( t ) = v ( t ) ∗ ˜ ψ m, ( f )( t ) 0 · · ·
00 ˜ ψ m, ( f )( t ) · · · · · · ˜ ψ m,k ( f )( t ) v ( t ) . (e 6.56)Define θ ( i ) : E p m ,q m → C ([0 , , M p m q m ( B )) by, for each f ∈ E p m ,q m ,θ ( i ) ( f )( t ) = ( f ( ξ i ( t )) if t ∈ [0 , / ,θ t − / ( f ( ξ i ( t ))) if t ∈ (3 / , , (e 6.57)if ξ i (3 /
4) = 1 /
2; and, if ξ i (3 /
4) = 1 , define θ ( i ) ( f )( t ) = f ( ξ i ( t )) for all t ∈ [0 , . (e 6.58)26efine Θ m : E p m ,q m → C ([0 , , M kp m q m ( B )) by, for each f ∈ E p m ,q m , Θ m ( f ) = diag( θ (1) ( f ) , · · · , θ ( k ) ( f )) for all t ∈ [0 , . (e 6.59)By (e 6.53), (e 6.31), (e 6.50) as well as the definition of Θ m , and by conjugating another unitaryin C ([0 , , M p m +1 q m +1 ) , we may write ϕ m ( f ) = u ∗ Θ m ( f ) 0 · · · γ m ( f ◦ ξ ) ⊗ · · · · · · γ m ( f ◦ ξ k ) ⊗ u for all f ∈ A m . (e 6.60)So ϕ m does have the form in (e 6.16). Condition (1) follows from the definition of ξ i , (e 6.43),(e 6.45), and (e 6.36). Condition (2) follows from (e 6.22) and (e 6.31). Condition (3) followsfrom (e 6.26). Finally, condition (4) follows from the definition of Θ m . The construction is thencompleted by induction.
Remark 6.8.
It should be noted that, if f (0) , f (1) ∈ M p m q m , then Θ( f (0)) and Θ( f (1)) arealso scalar matrices. Definition 7.1.
Let { ξ i : 1 ≤ i ≤ m } be a collection of maps described in (1) of Lemma6.7. The collection is said to be full, if each of the three types occurs at least once. Let C := { ξ j (1) ◦ ξ j (2) : 1 ≤ j (1) ≤ m , ≤ j (2) ≤ m } be a collection of compositions of two mapsin (1) of Lemma 6.7. The collection is called full, if { ξ j (2) : 1 ≤ j (2) ≤ m } is a full collection,and for each fixed j (2) , { ξ j (1) : ξ j (1) ◦ ξ j (2) ∈ C } is also a full collection. Inductively, a collectionof n composition of maps in (1) of Lemma 6.7 C n = { ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n ) : 1 ≤ j ( i ) ≤ m i : i = 1 , , .., n } is called full, if { ξ j ( n ) : 1 ≤ j ≤ m n } is a full collection, and, for each fixed ξ j ( n ) , the collection { ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n − : ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n − ◦ ξ j ( n ) ∈ C n } is full. Lemma 7.2.
Let ξ j : [0 , → [0 , be one of the three types of continuous maps given by (1) ofLemma 6.7. Let Ξ j = ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n ) : [0 , → [0 , be a composition of n maps of some ξ j ’s. Then, for any x, y ∈ [0 , , | Ξ j ( x ) − Ξ j ( y ) | ≤ (2 / n . (e 7.1) Moreover, if { Ξ j : 1 ≤ j ≤ l } is a full collection of compositions of n maps as above, then ∪ lj =1 Ξ j ([0 , , , (e 7.2) and, for each t ∈ [0 , , { Ξ j ( t ) : 1 ≤ j ≤ l } is (2 / n -dense in [0 , . roof. Note that, for each i, and for any x, y ∈ [0 , , | ξ i ( x ) − ξ i ( y ) | ≤ (2 / | x − y | . Then, byinduction, for all x, y ∈ [0 , , | Ξ j ( x ) − Ξ j ( y ) | = | ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n ) ( x ) − ξ j (1) ◦ ξ j (2) ◦ · · · ◦ ξ j ( n ) ( y ) | (e 7.3) ≤ (2 / | ξ j (2) ◦ · · · ◦ ξ j ( n ) ( x ) − ξ j (2) ◦ · · · ◦ ξ j ( n ) ( y ) | (e 7.4) ≤ ... ≤ (2 / n | x − y | ≤ (2 / n . (e 7.5)One already observes that ∪ j ∈ S ξ j ([0 , , , if { ξ j : j ∈ S } is a full collection. An inductionshows that, if { Ξ j : 1 ≤ j ≤ l } is a full collection, then ∪ lj =1 Ξ j ([0 , , . To show the last statement, fix t ∈ [0 , . Let x ∈ [0 , . Then, for some y ∈ [0 ,
1] and j ∈{ , , ..., l } , Ξ j ( y ) = x. Now, by the first part of the statement, for any t ∈ [0 , , | Ξ j ( t ) − Ξ j ( y ) | ≤ (2 / n . It follows that | Ξ j ( t ) − x | = | Ξ j ( t ) − Ξ j ( y ) | ≤ (2 / n . Theorem 7.3.
The inductive limit A in Lemma 6.7 can be made into a unital simple C ∗ -algebra A Cz such that ( K ( A z ) , K ( A z ) + , [1 A z ] , K ( A z )) = ( Z , Z + , , { } ) . (e 7.6) If C is not exact, then A Cz is not exact.Proof. For convenience, one makes an additional requirement in the construction. Let F m, ⊂F m, , ...., F m,n , ... be an increasing sequence of finite subsets of A m such that ∪ n F m,n is densein A m . One requires ϕ m ( F m,m +1 ) ⊂ F m +1 , and ϕ m ( F m,,m + n ) ⊂ F m +1 ,n , m, n = 1 , , .... This is done inductively as follows: Choose any increasing sequence of finite subsets F , ⊂F , , ..., ⊂ A such that ∪ n F ,n is dense in A . Specify F = F , \ { } . Choose A and define ϕ : A → A as in the construction of Lemma 6.7.Choose an increasing sequence of finite subsets F , , F , , ... of A such that ϕ ( F ,n ) ⊂ F ,n ,n = 1 , , .... Specify F = F , \ { } . Once F m, , F m, , ... are determined. Specified F m = F m, \ { } . Then construct A m +1 and ϕ m : A m → A m +1 as in Lemma 6.7. Choose F m +1 , , F m +1 , , .... so that ϕ m ( F m,m +1 ) ⊂ F m +1 , and ϕ m ( F m,m + n ) ⊂ F m +1 ,n as well as F m +1 ,n ⊂ F m +1 ,n +1 . Moreover ∪ n F m +1 ,n is dense in A m +1 . Choose F m +1 = F m +1 , \ { } . Thus the requirement can be made.Let us now prove that A is simple. For this, we will prove the following claim.Claim: for any fixed i, and g ∈ A i \{ } , there exists n > i such that ϕ i,n ( g ) is full in A n . Without loss of generality, we may assume that k g k = 1 . There is j and f ∈ F i,j such that k f − g k < / . To simplify notation, without loss of generality, we may write i = 1 . Set ϕ j,j ′ = ϕ j ′ − ◦ · · · ϕ j for j ′ > j. Then ϕ ,j ′ ( f ) ∈ ϕ j ′ ( F j ′ − ,j ′ ) ⊂ F j ′ , . Recall also that each ϕ j is unital and injective.28o further simplify the notation, without loss of generality, we may write F i,j \ { } = F m = F m, \ { } . We assume that m > . By the construction, for some t ∈ (0 , , k γ m ( f ( t )) k > (1 − / m ) k f k 6 = 0 . (e 7.7)By the continuity, there is n ( m ) ≥ / n ( m ) − -dense set S of [0 , , k γ m ( f ( s )) k ≥ (1 − / m ) k f k 6 = 0 for some s ∈ S . (e 7.8)For any f ∈ C ([0 , , M p m q m ( B )) and i, denote h ( t ) = γ m ( f ◦ ξ i ( t )) ⊗ (for t ∈ [0 , k > m and j ( t ∈ [0 , γ k ( h ◦ ξ j ( t )) = γ k ( γ m ( f ◦ ξ i ◦ ξ j ( t )) ⊗ ) = γ m ( f ◦ ξ i ◦ ξ j ( t )) ⊗ R ( k ) , (e 7.9)where R ( k ) is the rank of γ k and ξ i and ξ j are as defined in (1) of Lemma 6.7. Denote¯ γ k +1 ( f )( t ) = γ k +1 ( f ( t )) ⊗ for all f ∈ C ([0 , , M p k q k ( B )) (and t ∈ [0 , . (e 7.10)Therefore, from Lemma 6.7 and (e 6.16) (also keep Remark 6.8 in mind), we may write, for each f ∈ A m = E p m ,q m ,ϕ m,m +2 ( f ) (e 7.11)= w ∗ H ( f ) 0¯ γ m ( f ◦ ξ (2)1 ) ⊗ R ( m +1) . . .0 ¯ γ m ( f ◦ ξ (2) l ( m +1) ) ⊗ R ( m +1) w , (e 7.12)where H : A m → C ([0 , , M L p m q m ) is a homomorphism (for some integer L ≥ w ∈ C ([0 , , M p m +2 q m +2 ) is a unitary, R ( m + 1) is the rank of ¯ γ m +1 , and { ξ (2) j : 1 ≤ j ≤ l ( m + 1) } isa full collection of compositions of two ξ i ’s (maps in (1) of Lemma 6.7).Therefore, by the induction, for any n > n ( m ) + m, one may write, from the construction ofLemma 6.7 (see (e 6.16)), for all f ∈ A m = E p m ,q m ,ϕ m,n ( f ) = w ∗ H ( f ) 0¯ γ m ( f ◦ Ξ ) ⊗ R ( n, . . .0 ¯ γ m ( f ◦ Ξ l ) ⊗ R ( n,l ) w, (e 7.13)where H : A m → C ([0 , , M Lp m q m ( B )) is a homomorphism (for some integer L ≥ j is acomposition of n − m maps in (1) of Lemma 6.7 such that the collection { Ξ j : 1 ≤ j ≤ l } is full,and R ( n, j ) ≥ j = 1 , , ..., l, and w ∈ C ([0 , , M p n q n ) is a unitary.It follows from Lemma 7.2 that | Ξ i ( x ) − Ξ i ( y ) | < (2 / m − n for all x, y ∈ [0 , , ≤ i ≤ l, and ∪ li Ξ i ([0 , , . (e 7.14)Fix any t ∈ [0 ,
1] and x ∈ [0 , , by Lemma 7.2, there is y ∈ [0 ,
1] and j ∈ { , , ..., l } such thatΞ j ( y ) = x. Then, | Ξ j ( t ) − x | = | Ξ j ( t ) − Ξ j ( y ) | < (2 / n − m < (2 / n ( m ) . (e 7.15)It follows from the choice of n ( m ) and (e 7.8) that, for f ∈ F m , k γ m ( f ◦ Ξ j ( t )) k ≥ (1 − /m ) k f k ≥ ( 6364 ) for all t ∈ [0 , . (e 7.16)29ince k f (Ξ j ( t )) − g (Ξ j ( t )) k < / , this implies that k γ m ( g (Ξ j ( t ))) k ≥ − for all t ∈ [0 , . (e 7.17)Since, for each t ∈ [0 , , γ m ( g ◦ Ξ i ( t )) ∈ M p m q m , i = 1 , , ..., l, ϕ m,n ( g )( t ) is not in any closedideal of M p n q n ( B ) for each t ∈ [0 , . Therefore ϕ m,n ( g ) is full in E p n ,q n = A n . This proves theclaim.It follows from the claim that A Cz is simple. To see this, let I ⊂ A Cz be an ideal such that I = A Cz and put C n = ϕ n, ∞ ( A n ) . Then C n ⊂ C n +1 for all n. Let a ∈ C m \ { } . By the claim,there is n ′ > m such that a is full in C n ′ and therefore a is full in every C n for n ≥ n ′ . In otherwords, a C n ∩ I for all n. It follows that C m ∩ I = { } as C m ⊂ C n for all n ≥ m. It is thenstandard to show that I = { } . Thus A Cz is simple.Since, for each m, by Lemma 6.5,( K ( A m ) , K ( A m ) + , [1 A m ] , K ( A m )) = ( Z , Z + , , { } ) , one concludes (as each ϕ n is unital) that( K ( A Cz ) , K ( A Cz ) + , [1 A Cz ] , K ( A Cz )) = ( Z , Z + , , { } ) . (e 7.18)Finally if C is not exact, then B is not exact since B has quotients with the form C ⊕ C whichis not exact.Define Φ : B → C ([0 , , M ( B )) byΦ( f )( t ) = θ t ( f ) ⊗ for all f ∈ B and t ∈ [0 , , (e 7.19)where θ t : B → B is defined in (e 6.23). Note, for f ∈ B, Φ( f )(0) = θ ( f ) ⊗ = f ⊗ ∈ M ( B ) ⊗ and Φ( f )(1) = f (0) ⊗ ∈ C · . (e 7.20)One then obtains a unitary u ∈ C ([0 , , M ) such that u ∗ Φ( f ) u ∈ E , . (e 7.21)Define Ψ( f ) = u ∗ Φ( f ) u for all f ∈ B. Then Ψ is a unital injective homomorphism. In otherwords, B is embedded unitally into A = E , . Since each ϕ m : A m → A m +1 is unital and injec-tive, B is embedded into A Cz . Since B is not exact, neither is A Cz (see, for example, Proposition2.6 of [42]). Proposition 7.4. If C is exact but not nuclear, then A Cz is exact and not nuclear.Proof. Note that, since C is non-nuclear and exact, so is B. Note also that A n = E p n ,q n is a C ∗ -subalgebra of exact C ∗ -algebra C ([0 , , M p n q n ( B )) . So each A n is exact. By [42, 2.5.5], A Cz is exact.Let Φ : B → A = E , be as in the end of the proof of Theorem 7.3. Let π (1)0 : A → M ( B ) ⊗ M be the evaluation at 0 , namely π (1)0 ( f ) = f (0) for all f ∈ A . Let η : M ( B ) ⊗ → B bydefining η (( b i,j ) × ⊗ ) = b , , where b i,j ∈ B, ≤ i, j ≤ . Then η is a norm one c.p.c map.Define π (1 , : A → B by π (1 , ( f ) = η ◦ π (1)0 . Note π (1 , ◦ Φ is an isomorphism. In fact, π (1 , ◦ Φ( b ) = θ ( b ) = b (see (e 7.20)) for all b ∈ B. B Φ / / id B ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ A π (1)0 (cid:15) (cid:15) M ( B ) ⊗ η (cid:15) (cid:15) B (e 7.22) We will use the same diagram in the n-stage.In (4) of Lemma 6.7, let us choose ξ so that ξ ( t ) = 2 t/ t ∈ [0 , /
4] and so θ (1) ( f (0)) = f (0) for all f ∈ E , . So, by (e 6.16) and (e 6.19), we may write ϕ ( f )(= ϕ , ( f )) = diag( θ (1 , ( f )) , H ′ ( f )) for all f ∈ A , (e 7.23)where θ (1 , := θ (1) and θ (1 , ( f )(0) = f (0) for f ∈ A and H ′ : A = E , → C ([0 , , M p q ( B ))is a homomorphism (note that the image of H ′ is in a corner of C ([0 , , M p q ( B ))). Similarly,by the formula (e 6.16) and (e 6.19) again, we may also write ϕ , ( f ) = diag( θ (1 , ( f ) , H ′ ( f )) for all f ∈ A , (e 7.24)where θ (1 , ( f )(0) = f (0) for f ∈ A and H ′ : A → C ([0 , , M p q ( B )) is a homomorphism. Byinduction, for any n > , we may write ϕ ,n ( f ) = diag( θ (1 ,n ) ( f ) , H ′ n ( f )) for all f ∈ A , (e 7.25)where θ (1 ,n ) ( f )(0) = f (0) and H ′ n : A → C ([0 , , M p n q n ( B )) is a homomorphism. (One shouldbe warned that diag( θ (1 ,n ) , , ...,
0) is not in A n . )Now we prove that A Cz is not nuclear. We follow the proof of Proposition 6 of [12]. Assumeotherwise, for any finite subset F ⊂ B and ε > , if A Cz were nuclear, then ϕ , ∞ ◦ Φ would benuclear. Therefore there would be a finite dimensional C ∗ -algebra D and c.p.c maps α : B → D and β : D → A Cz such that k ϕ , ∞ ◦ Φ( b ) − β ◦ α ( b ) k < ε/ b ∈ F . (e 7.26)Since A Cz is assumed to be nuclear, by the Effros-Choi lifting theorem [9], there exist an integer n ≥ β n : D → A n such that k β ( x ) − ϕ n, ∞ ◦ β n ( x ) k < ε/ x ∈ α ( F ) . (e 7.27)Thus k ϕ n, ∞ ( ϕ ,n ◦ Φ( b ) − β n ◦ α ( b )) k < ε. (e 7.28)As ϕ n, ∞ is an isometry, this implies that k ϕ ,n ◦ Φ( b ) − β n ◦ α ( b ) k < ε for all b ∈ B. (e 7.29)Let π ( n )0 : E p n ,q n → M p n ( B ) ⊗ q n be the evaluation at 0 defined by π ( n )0 ( a ) = a (0) . We have, by(e 7.25), π ( n )0 ( ϕ ,n ◦ Φ( b )) = diag( θ ( b ) ⊗ , H ′ n (Φ( f ))(0)) for all b ∈ B. (e 7.30)31ecall that θ ( b ) = b. Now a rank 1 projection p corresponding the first (1 ,
1) corner is in M p n ( B ) ⊗ q n . We now use the n -stage diagram of (e 7.22). Define η n : M p n ( B ) ⊗ q n → B definedby η n ( x ) = pxp for all x ∈ M p n ( B ) ⊗ q n which is a unital c.p.c map ( η n (( b i,j ) p n × p n ⊗ q n ) = b , ).Note that η n ◦ π ( n )0 ◦ ϕ ,n ◦ Φ = id B . By (e 7.29), k b − η n ◦ π ( n )0 ◦ β n ◦ α ( b ) k = k η n ◦ π ( n )0 ( ϕ ,n ◦ Φ( b ) − β n ◦ α ( b )) k < ε for all b ∈ B. (e 7.31)This would imply that B is nuclear. Therefore A Cz is not nuclear. The above could be illustratedby the following diagram: M p n ( B ) ⊗ q n η n t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ B B id B o o ϕ ,n ◦ Φ / / α x x rrrrrrrrrrrrr ϕ , ∞ ◦ Φ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ A nϕ n, ∞ (cid:15) (cid:15) π ( n )0 i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ D η n ◦ π ( n )0 ◦ β n O O β / / β n ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ A Cz (which is only approximately commutative below the top triangle). Theorem 7.5.
The inductive limit A Cz in Theorem 7.3 has a unique tracial state.Proof. First we note each unital C ∗ -algebra A m = E p m ,q m has at least one tracial state, say τ m . Note that ϕ m, ∞ is injective homomorphism. So we may view τ m as a tracial state of ϕ m, ∞ ( A m ) . Extend τ m to a state t m on A Cz . Choose a weak*-limit of { t m } , say t. Then t is astate of unital C ∗ -algebra A Cz . Note that ϕ m, ∞ ( A m ) ⊂ ϕ n, ∞ ( A n ) if n > m. Then, for each pair x, y ∈ ϕ m, ∞ ( A m ) , and for any n > m, t n ( xy ) = t n ( yx ) . It follows that t is a tracial state of A Cz . In other words, A Cz has at least one tracial state.Claim: for each k and each a ∈ A k with k a k ≤ , any each ε > , there exists N > k suchthat, for all n ≥ N, | τ ( ϕ k,n ( a )) − τ ( ϕ k,n ( a )) | < ε for all τ , τ ∈ T ( A n ) . (e 7.32)Fix a ∈ A k . To simplify the notation, without loss of generality, we may assume that k = 1 . Choose m > / m − < ε/ . (e 7.33)Put g = ϕ ,m ( a ) . There is δ > k g ( x ) − g ( y ) k < ε/ x, y ∈ [0 ,
1] and | x − y | < δ. (e 7.34)Recall that here we view γ m as a map from M p m q m ( B ) to M R ( m ) p m q m . Note, for each f ∈ A m , since γ m ( f ( t )) is a scalar matrix for all t ∈ [0 , , γ m ( f ( t ))( x ) , as an element in M R ( m ) p m q m ( B ) , is a constant matrix (for x ∈ (0 , M R ( m ) p m q m ( C ( ^ (0 , , C )) . Hence (see (e 7.10) for ¯ γ m ), for t ∈ [3 / , , (recall that, ξ i ( t ) = ξ i (3 /
4) for all t in [3 / , θ t − / (¯ γ m ( f (Ξ j ◦ ξ i ))(3 / γ m ( f (Ξ j ◦ ξ i )(3 / γ m ( f (Ξ j ◦ ξ i ))( t ) . (e 7.35)(Recall the definition of θ t in (e 6.20)). Therefore (see the definition of θ ( i ) in (e 6.57)), for any i with ξ i (3 / = 1 ,θ ( i ) (¯ γ m ( f ◦ Ξ j ))( t ) = ( ¯ γ m ( f ◦ Ξ j ◦ ξ i )( t ) if t ∈ [0 , / ,θ t − / (¯ γ m ( f ◦ Ξ j ◦ ξ i )(3 / t ∈ (3 / ,
1] (e 7.36)= ¯ γ m ( f (Ξ j ◦ ξ i ))( t ) . (e 7.37)32or those i so that ξ i (3 /
4) = 1 , one also has θ ( i ) (¯ γ m ( f ◦ Ξ j )) = ¯ γ m ( f ◦ Ξ j ◦ ξ i ) . (e 7.38)It follows that (recall (4) of Lemma 6.7 for the definition of Θ m +1 and also keep Remark 6.8 inmind)Θ m +1 ( ϕ m ( f )) = u ∗ Θ ′ m +1 ( f ) 0 · · ·
00 ¯ γ m ( f ◦ ξ (2)1 ) · · · · · · ¯ γ m ( f ◦ ξ (2) k ′ ) u for all f ∈ A m , (e 7.39)where u ∈ C ([0 , , M R ( m +1 , p m +1 q m +1 ) is a unitary, Θ ′ m +1 : A m → C ([0 , , M T (0) p m q m ( B )) isa homomorphism for some integer T (0) ≥ , and { ξ (2) j : 1 ≤ j ≤ k ′ } is a full collection ofcompositions of two maps in (1) of Lemma 6.7. Moreover, by (2) of Lemma 6.7, T (0) / k ′ R ( m ) < / m . (e 7.40)Then, combing with (e 7.9), we may write ϕ m,m +2 : A m → A m +2 as ϕ m,m +2 ( f ) = u ∗ H m +1 ( f ) 0 · · ·
00 ¯ γ m ( f ◦ ξ (2)1 ) ⊗ r (1) · · · · · · ¯ γ m ( f ◦ ξ (2) l ( m ) ) ⊗ r ( l ( m )) u (e 7.41)for all f ∈ A m , where u ∈ C ([0 , , M p m +2 q m +2 ) is a unitary, H m +1 : A m → C ([0 , , M T (1) p m q m ( B ))is a homomorphism for some integer T (1) ≥ , { ξ (2) j : 1 ≤ j ≤ l } is a full collection of com-positions of two maps in (1) of Lemma 6.7, and r ( l ( j )) ≥ j = 1 , , ..., l ( m ) . Moreover, T (1) / R ( m )( l ( m ) X j =1 r ( l ( j ))) < / m . (e 7.42)Therefore, by Lemma 6.7, noting (e 6.16), (e 6.19), (e 7.37), and the proof of Theorem 7.3 (see(e 7.9)), as well as (e 7.39), repeatedly, one may write, for each n > m, for all f ∈ A m ,ϕ m,n ( f ) = w ∗ H m,n ( f ) 0¯ γ m ( f ◦ Ξ ) . . .0 ¯ γ m ( f ◦ Ξ L ) w, (e 7.43)where w ∈ C ([0 , , M p n q n ) is a unitary, H m,n : A m → C ([0 , , M L (0) p m q m ( B )) is a homomor-phism for some integer L (0) ≥ , and where Ξ j : [0 , → [0 ,
1] is a composition of n − m many ξ i ’ s, and { Ξ j : 1 ≤ j ≤ L } is a full collection. Moreover, L (0) / LR ( m ) < / m . (e 7.44)We choose N such that (2 / N − m < δ and choose any n ≥ N. Let τ i ∈ T ( A n ) ( i = 1 , f ∈ A n ,τ i ( f ) = Z σ i ( t )( f ( t )) dµ i , i = 1 , , (e 7.45)33here σ i ( t ) is a tracial state of M p n q n ( B ) for all t ∈ (0 , , σ i (0) is a tracial state of M p n ( B ) ⊗ q n ,σ i (1) is a tracial state of 1 p n ⊗ M q n , and µ i is a probability Borel measure on [0 , , i = 1 , . Foreach t ∈ [0 ,
1] and for f ( t ) ∈ M p n q n ⊂ M p n q n ( B ) ,σ i ( t )( f ( t )) = tr( f ( t )) , i = 1 , , (e 7.46)where tr is the normalized trace on M p n q n (see (e 6.14)). For, each j ∈ { , , ..., L } , by Lemma7.2, | Ξ j ( x ) − Ξ j ( y ) | < (2 / n − m < δ for all x, y ∈ [0 , . (e 7.47)By the choice of δ, k g ◦ Ξ j ( x ) − g ◦ Ξ j ( y ) k < ε/ x, y ∈ [0 , . (e 7.48)For each f ∈ A m , write H ′ ( f )( t ) = (cid:18) H m +1 ( f )( t ) 00 0 (cid:19) for all t ∈ [0 , . (e 7.49)Then one has, for each f ∈ A m ,τ i ( ϕ m,n ( f )) = Z σ i ( t )( ϕ m,n ( f )) dµ (e 7.50)= Z σ i ( t )( H ′ ( f )( t )) dµ + Z tr( L M j =1 ( f ◦ Ξ j ( t ))) dµ. (e 7.51)By (e 7.48) (recall that k g k ≤ Z | tr( L M j =1 ( g ◦ Ξ j (1 / − L M j =1 ( g ◦ Ξ j ( t ))) | dµ < ( ε/ Z dµ = ε/ . (e 7.52)By (e 7.44), Z | σ i ( t )( H ′ ( g )( t )) | dµ < (1 / m < ε/ . (e 7.53)Recall ϕ ,n ( a ) = ϕ m,n ( g ) . Thus, by (e 7.50), (e 7.51), (e 7.52), and (e 7.53), | τ i ( ϕ ,n ( a )) − L X j =1 tr( g (Ξ j (1 / | < ε/ , i = 1 , . (e 7.54)Therefore | τ ( ϕ ,n ( a )) − τ ( ϕ ,n ( a )) | < ε. (e 7.55)This proves the claim.To complete the proof, let s , s ∈ T ( A Cz ) . Let a ∈ A Cz and let ε > . Then there is f ∈ A k for some k ≥ k a − ϕ k, ∞ ( f ) k < ε/ . (e 7.56)34et τ i,n = s i ◦ ϕ n, ∞ . Then, by the claim, there exists N ≥ k such that, for all n > N, | τ ,n ( ϕ k,n ( f )) − τ ,n ( ϕ k,n ( f )) | < ε/ . (e 7.57)It follows that | s ( ϕ k, ∞ ( f )) − s ( ϕ k, ∞ ( f )) | ≤ ε/ . (e 7.58)Therefore | s ( a ) − s ( a ) | ≤ | s ( a ) − s ( ϕ k, ∞ ( f )) | + | s ( ϕ k, ∞ ( f )) − s ( ϕ k, ∞ ( f )) | + | s ( a ) − s ( ϕ k, ∞ ( f )) | < ε. It follows that s ( a ) = s ( a ) . Thus A Cz has a unique tracial state. Remark 7.6.
Recall the construction allows B = C (with C = { } ). In that case, of course A Cz = Z . Note that, when B = C , θ t ( b ) = b for all b ∈ M p m q m . In other words, θ t = id B . Let Z p m ,q m = { f ∈ C ([0 , , M p m q m ) : f (0) ∈ M p m ⊗ q m and f (1) ∈ p m ⊗ M q m } . (e 7.59)In general (when C = { } ), one has Z p m ,q m ⊂ E p m ,q m , as we view C ⊂ B and M p m q m ⊂ M p m q m ( B ) . Let ϕ zm = ϕ m | Z pm,qm . Then, since v ∈ C ([0 , , M p m +1 q m +1 ) (see the line above(e 6.53)), ϕ zm ( Z p m ,q m ) ⊂ Z p m +1 ,q m +1 . Thus, one obtains a unital C ∗ -subalgebra (of A Cz ) B z = lim n →∞ ( Z p m ,q m , ϕ zm ) . (e 7.60)Then B z ∼ = Z ([22]). A Cz In this section, let A Cz be the C ∗ -algebra in Theorem 7.3. Lemma 8.1. A Cz has the following property.(1) A Cz has a unital C ∗ -subalgebra B z ∼ = Z , (2) for any finite subset F ⊂ A m and ε > , there is e ∈ ( A m +1 ) \ { } such that(i) e ( t ) ∈ M p m +1 q m +1 for all t ∈ [0 , and e (1) = 0 , (ii) k ex − xe k < ε for all x ∈ ϕ m ( F ) , (iii) ϕ m +1 , ∞ ((1 − e ) β ϕ m ( f )) ∈ ε B z for all f ∈ F , and, for any β ∈ (0 , ∞ ) , k ϕ m +1 , ∞ ((1 − e ) β ϕ m ( y ) k ≥ (1 − ε ) k ϕ m ( y ) k for all y ∈ F m and (e 8.1) (iv) d τ ( e ) < / m for all τ ∈ T ( A m +1 ) . (Recall that F m was constructed in the proof of Theorem 7.3.)Proof. We will keep the notation used in the proof of Lemma 6.7.For (i), we note that the C ∗ -subalgebra B z = lim n →∞ ( Z p m ,q m , ϕ m | Z pm,qm ) has been identifiedin Remark 7.6, where Z p m ,q m = { f ∈ C ([0 , , M p m q m ) : f (0) ∈ M p m ⊗ q m and f (1) ∈ p m ⊗ M q m } . (e 8.2)There is δ ∈ (0 , ε/
2) such that, if | t − t ′ | < δ, k ϕ m ( f )( t ) − ϕ m ( f )( t ′ ) k < ε/ f ∈ F . (e 8.3)35n particular, there is t ∈ (0 ,
1) (1 − t < δ ) such that k ϕ m ( f )( t ) − ϕ m ( f )(1) k < ε/ f ∈ F and t ∈ ( t , . (e 8.4)Choose a continuous function g ∈ C ([0 , ≤ g ≤ , g ( t ) = 1 for all t ∈ [0 , t ]and g ( t ) = (1 − t ) / (1 − t ) for t ∈ ( t , . Let e ( t ) = g ( t ) · A m for all t ∈ [0 , . Note that e (0) = 1 p m q m ∈ M p m ( B ) ⊗ q m and e (1) = 0 ∈ p m ⊗ M q m . So e ∈ A m . Moreover, e is in thecenter of A m . Define σ : M p m ( B ) ⊗ M q m → M d m p m ( B ) ⊗ M q m by σ ′ ⊗ s, where σ ′ ( a ) = (cid:18) θ ( a ) 00 0 (cid:19) for all a ∈ M p m ( B ) and s ( c ) = c ⊗ for all c ∈ M q m , (e 8.5)where θ : M p m ( B ) → M p m ⊂ M p m ( B ) is defined by θ ( c )( x ) = c (0) for c ∈ M p m ( B ) = M p m ( C ( ^ (0 , , C )) , and for all x ∈ [0 , , and the “0” in the lower corner has the size of ( d m − p m × ( d m − p m . Then define σ : A m → C ([0 , , M d m p m q m ( B )) by σ ( f )( t ) = σ ( f ( t )) for all f ∈ E p m ,q m and t ∈ [0 , . (e 8.6)It follows that, for all fixded t ∈ [0 , ,σ ( e )( t ) = σ ( e ( t )) = σ ( g ( t ) · A m ) = σ ( g ( t ) · p m ⊗ q m ) (e 8.7)= (cid:18) ( θ ( g ( t ) · p m ) 00 0 (cid:19) ⊗ q m = (cid:18) g ( t ) · p m
00 0 (cid:19) ⊗ q m = (cid:18) g ( t ) · p m q m
00 0 (cid:19) ⊗ , (e 8.8)where the last “0” has the size ( d m − p m q m ) × ( d m − p m q m in the last matrix above). Thus σ ( e )(0) = b ⊗ q m and σ ( e (1)) = 0 , (e 8.9)where b = (cid:18) p m
00 0 (cid:19) . It follows that σ ( e ) ∈ E d m p m , q m . Note that, for each τ ∈ T ( A m ) , by(e 6.22). τ ( σ ( e )) < / m . (e 8.10)Let us recall the definition of ˜ ψ m,i in the proof of Lemma 6.7, 1 ≤ i ≤ k (see (e 6.55)). Then,for all f ∈ A m , by (e 6.55), (e 6.47), (e 6.50), and (e 6.57), for each t ∈ [0 , , ˜ ψ m,i ( f )( t ) σ ( e )( t ) = (cid:18) θ ( i ) ( f )( t ) 00 γ m ( f ( t )) (cid:19) ⊗ · (cid:18) g ( t ) · p m q m
00 0 (cid:19) ⊗ (e 8.11)= (cid:18) θ ( i ) ( f )( t ) · g ( t ) · p m q m
00 0 (cid:19) ⊗ (e 8.12)= (cid:18) g ( t ) · p m q m
00 0 (cid:19) ⊗ · (cid:18) θ ( i ) ( f )( t ) 00 γ m ( f ( t )) (cid:19) ⊗ (e 8.13)= σ ( e )( t ) ˜ ψ m,i ( f )( t ) . (e 8.14)In other words, for all f ∈ E p m ,q m , ˜ ψ m,i ( f ) σ ( e ) = σ ( e ) ˜ ψ m,i ( f ) , i = 1 , , ..., k. (e 8.15)Define α : [0 , → [0 ,
1] by α ( t ) = ( tt if t ∈ [0 , t ];1 if t ∈ ( t , . (e 8.16)36hen f ◦ α ∈ E p j ,q j , if f ∈ E p j ,q j for all j. Moreover, by (e 8.3), k ϕ m ( f ) ◦ α − ϕ m ( f ) k < ε/ f ∈ F . (e 8.17)Therefore, for each f ∈ A m , and, for each t ∈ [0 , , each β ∈ (0 , ∞ ) , with l = d m p m q m , (1 l − σ ( e )) β ˜ ψ m,i ( f ) ◦ α ( t ) = (cid:18) (1 − g ( t )) β · p m q m · θ ( i ) ( f ) ◦ α ( t ) 00 γ m ( f ( t )) (cid:19) ⊗ , (e 8.18)for i = 1 , , ..., k. For t ∈ [0 , t ] , by the definition of g, (1 − g ( t )) β · p m q m · θ ( i ) ( f )( t ) = 0 . (e 8.19)For t ∈ ( t , , (1 − g ( t )) β · p m q m · θ ( i ) ( f ) ◦ α ( t ) = (1 − g ( t )) β · θ ( i ) ( f )(1) ∈ M p m q m . (e 8.20)Hence (1 l − σ ( e )) β ˜ ψ m,i ( f ◦ α ) ∈ C ([0 , , M d m p m q m ) . (e 8.21)Moreover, by (e 7.8), for f ∈ F m , k (1 l − σ ( e )) β ˜ ψ m,i ( f ) ◦ α k ≥ (1 − /m ) k f k . (e 8.22)Define, using the same v as in (e 6.53) ( σ ( e )( t ) repeats k times), e := v ( t ) ∗ σ ( e )( t ) 0 · · · σ ( e )( t ) · · · · · · σ ( e )( t ) v ( t ) . (e 8.23)With b in the line below (e 8.9), b ⊗ q m ⊗ r = b ⊗ r q m = ( b ⊗ t ) ⊗ q m +1 (see (e 6.38)).Since σ ( e ) ∈ E d m p m , q m , as in (e 8.9), e (0) = v ∗ b ⊗ q m · · · b ⊗ q m · · · · · · b ⊗ q m v ∈ M p m +1 ⊗ q m +1 (e 8.24)(see (e 6.40)). Combining the fact e (1) = 0 , one concludes e ∈ E p m +1 ,q m +1 = A m +1 . Moreover,by (e 8.7) and the fact v ∈ C ([0 , , M p m +1 q m +1 ) (see lines above (e 6.53)), e ( t ) ∈ M p m +1 q m +1 foreach t ∈ [0 ,
1] and e (1) = 0 . So (i) in part (2) of the statement of the lemma holds.By (e 8.15) and (e 6.56), one computes, for all f ∈ A m ,eϕ m ( f ) = v ∗ σ ( e ) ˜ ψ m, ( f ) 0. . .0 σ ( e ) ˜ ψ m,k ( f ) v (e 8.25)= v ∗ ˜ ψ m, ( f ) σ ( e ) 0. . .0 ˜ ψ m,k ( f ) σ ( e ) v = ϕ m ( f ) e (e 8.26)37n other words, (ii) of (2) in the statement holds. Now(1 − e ) β ϕ m ( f ◦ α ) = v ∗ (1 l − σ ( e )) β ˜ ψ m, ( f ) ◦ α
0. . .0 (1 l − σ ( e )) β ˜ ψ m,k ( f ) ◦ α v for all f ∈ A m , where l = d m p m q m . Note that (1 − e ) β ϕ m ( f ) ◦ α ∈ E p m +1 ,q m +1 . It follows from(e 8.21) that (1 − e ) β ϕ m ( f ) ◦ α ∈ Z p m +1 ,q m +1 for all f ∈ F . (e 8.27)Then, by (e 8.17), (1 − e ) β ϕ m ( f ) ∈ ε/ Z p m +1 ,q m +1 for all f ∈ F . (e 8.28)It follows that ϕ m, ∞ ((1 − e ) β f ) ∈ ε B z for all f ∈ F . (e 8.29)Moreover, by (e 8.22), (e 8.1) also holds. So (iii) of part (2) of the statement holds. It followsfrom (e 8.10) that (iv) in the statement of the lemma also holds. Lemma 8.2.
Let E p,q = { ( f, c ) : C ([0 , , M pq ( B )) ⊕ ( M p ( B ) ⊕ M q ) : π ( c ) = f (0) and π ( c ) = f (1) } , (e 8.30) where π : M p ( B ) ⊕ M q → M p ( B ) ⊗ q ⊂ M pq ( B ) defined by π ( c ⊕ c ) = c ⊗ q for all c ∈ M p ( B ) and c ∈ M q , and π : M p ( B ) ⊕ M q → p ⊗ M q ⊂ M pq ( B ) defined by π (( c ⊕ c )) =1 p ⊗ c for all c ∈ M p ( B ) and c ∈ M q (see (e 6.7) ). Let L p,q = { ( f, c ) : C ([0 , , M pq ) ⊕ M p : π | M p ( c ) = f (0) } , (e 8.31) where π | M p ( c ) = c ⊗ q for all c ∈ M p . Suppose that a, b ∈ E p,q + such that(1) a ( t ) ∈ C ([0 , , M pq ) and a (1) = 0 , (2) there is b ∈ C ([0 , , M pq ) + such that b ( t ) ≤ b ( t ) for all t ∈ [0 , and a . b in L p,q , (e 8.32) (i.e., there exists a sequence x n ∈ L p,q such that x ∗ n b x n → a ). Then a . b in E p,q . (e 8.33) Proof.
Let 1 > ε > . Consider continuous function h δ ∈ E p,q ,h δ ( t ) = M pq ( B ) if t ∈ [0 , − δ ] , t ∈ (1 − δ/ , , linear , otherwise . (e 8.34)Since a (1) = 0 , there exists δ > k a − h / δ a · h / δ k < ε. Note that h δ lies in the center of C ([0 , , M pq ( B )) , and for any f ∈ L p,q , any n ∈ N ,h /nδ · f ∈ E p,q . Then since a . b in L p,q , one checks h / δ ah / δ . h / δ b · h / δ ≤ b ≤ b in E p,q . This implies a ≈ ε h / δ a · h / δ . b in E p,q . Since this holds for any 1 > ε > , one concludes a . b in E p,q . efinition 8.3. In the spirit of Definition 3.1, a simple C ∗ -algebra A is said to have essentialtracial nuclear dimension at most n, if A is essentially tracially in N n , the class of C ∗ -algebraswith nuclear dimension at most n, i.e., if, for any ε > , any finite subsets F ⊂ A and a ∈ A + \{ } , there exist an element e ∈ A and a C ∗ -subalgebra B ⊂ A which has nuclear dimension at most n such that(1) k ex − xe k < ε for all x ∈ F , (2) (1 − e ) x ∈ ε B and k (1 − e ) x k ≥ k x k − ε for all x ∈ F , and(3) e . a. Let us denote N Z ,s,s the class of separable nuclear simple Z -stable C ∗ -algebras. Theorem 8.4.
The unital simple C ∗ -algebra A Cz is essentially tracially in N Z ,s,s and has essen-tial tracial nuclear dimension at most 1, has stable rank one, and strict comparison for positiveelements. Moreover, A Cz has a unique tracial state and has no 2-quasitraces other than theunique tracial state, and ( K ( A Cz ) , K ( A Cz ) + , [1 A Cz ] , K ( A Cz )) = ( Z , Z + , , { } ) . (e 8.35)(Recall that, if C is exact and not nuclear, then A Cz is exact and not nuclear (Theorem 7.3),and if A Cz is not exact, then A Cz is not exact (Proposition 7.4)). Proof.
We will first show that A Cz is essentially tracially in N Z ,s,s . We will retain the notationin the construction of A Cz . Fix a finite subset F and an element a ∈ A Cz + with k a k = 1 . To verify A z has the saidproperty, without loss of generality, we may assume that there is a finite subset G ⊂ A suchthat ϕ , ∞ ( G ) = F . By the first few lines of the proof of Theorem 7.3, to further simplify notation,without loss of generality, we may assume that G = F , = F ∪ { } . Without loss of generality,we may assume that there is a ′ ∈ ( A ) with k a ′ k = 1 such that k ϕ , ∞ ( a ′ ) − a k < / . (e 8.36)It follows from Proposition 2.2 of [35] that ϕ , ∞ ( f / ( a ′ )) = f / ( ϕ , ∞ ( a ′ )) . a. (e 8.37)Put a ′ = f / ( a ′ ) ( = 0) . Since A z is simple, there are x , x , ..., x k ∈ A z such that k X i =1 x ∗ i ϕ , ∞ ( a ′ ) x i = 1 . (e 8.38)It follows that, for some large n , there are y , y , ..., y k ∈ A n and n ≥ n such that k k X i =1 ϕ n ,n ( y ∗ i ) ϕ ,n ( a ′ ) ϕ n ,n ( y i ) − A n k < / . (e 8.39)It follows that a := ϕ ,n ( a ′ ) is a full element in A n . Set d = inf { d τ ( a ) : τ ∈ T ( A n ) } . (e 8.40)Since a is full in A n and a ∈ ( A n ) + , d > . Choose m > n such that d/ > / m − . (e 8.41)39y applying Lemma 8.1, we obtain e ∈ ( A m +1 ) \ { } such that(i) e ( t ) ∈ M p m +1 q m +1 for all t ∈ [0 ,
1] and e (1) = 0 , (ii) k ex − xe k < ε for all x ∈ ϕ m ( ϕ ,m ( G )) , (iii) ϕ m +1 , ∞ ((1 − e ) ϕ m ( ϕ ,m ( x ))) ∈ ε B z , and k ϕ m +1 , ∞ ((1 − e ) ϕ m ( ϕ ,m ( x ))) k ≥ (1 − /m ) k ϕ ,m ( x ) k for all x ∈ F , . (iv) d τ ( e ) < / m for all τ ∈ T ( A m +1 ) . Denote a = ϕ n ,m ( a ) . It is full in A m . Write, as in Theorem 6.7 and (e 6.16), ϕ m ( a ) = u ∗ Θ m ( a ) 0 · · · γ m ( a ◦ ξ ) ⊗ · · · · · · γ m ( a ◦ ξ k ) ⊗ u, (e 8.42)where u ∈ U ( C ([0 , , M p m +1 q m +1 ( B ))) , Θ m : A m → C ([0 , , M R ( m, p m q m ( B )) is a homomor-phism, R ( m, ≥ γ m : B → M R ( m ) is a finite dimensional representation.Moreover R ( m, / kR ( m ) < / m . (e 8.43)Let b = u ∗ · · · γ m ( a ◦ ξ ) ⊗ · · · · · · γ m ( a ◦ ξ k ) ⊗ u. (e 8.44)Note that b ∈ C ([0 , , M p m +1 q m +1 ) . Moreover, since a ∈ E p m ,q m , a (0) = a ′ ⊗ q m for some a ′ ∈ M p m ( B ) . Therefore γ m ( a (0)) = γ m ( a ′ ) ⊗ q m . (e 8.45)Put c ′ = (cid:18) γ m ( a ′ ) (cid:19) and (e 8.46) c = (cid:18) γ m ( a (0)) (cid:19) ⊗ = (cid:18) γ m ( a ′ ) (cid:19) ⊗ q m = c ′ ⊗ q m . (e 8.47)Note c ′ ∈ M d m p m . Put c i ( t ) = (cid:18) γ m ( a ◦ ξ i ( t ))) (cid:19) ⊗ , i = r + 1 , r + 2 , ..., k. (e 8.48)Recall that at t = 0 (see (e 6.37)), ξ i (0) = ( ≤ i ≤ r , / r < i ≤ k . (e 8.49)Recall (see the line below (e 6.38)) also that r q m = t q m +1 for some integer t ≥ . Hence( c ′ ⊗ q m ) ⊗ r = c ′ ⊗ r q m = ( c ′ ⊗ t ) ⊗ q m +1 . On the other hand, since k = r + m (0) q m +1 (see (e 6.33)),diag( c r +1 (0) , · · · c k (0)) = s (cid:18)(cid:18) γ m ( a (1 / (cid:19) ⊗ (cid:19) ⊗ m (0) q m +1 . (e 8.50)40ote that “= s ” is implemented by the same scalar unitary as in (e 6.39)(see also the end of 6.6for the notation “= s ”). As in (e 6.40) (and the line next to it), since b ∈ C ([0 , , M p m +1 q m +1 )(mentioned earlier), this implies that b ∈ L p m +1 ,q m +1 (see (e 8.31)).Since a ≥ , b ( t ) ≤ a ( t ) for all t ∈ [0 ,
1] (see (e 8.44)). Since ϕ k is an injective unitalhomomorphism for all k, by (e 8.40), we also haveinf { d τ ( ϕ m ( a ) : τ ∈ T ( A m +1 ) } = inf { d τ ( ϕ n ,m +1 ( a )) : τ ∈ T ( A m ) } ≥ d. (e 8.51)By (e 8.42), (e 8.44), (e 8.43), and (e 8.41), we conclude, for each t ∈ (0 , ,d σ ( e ( t )) < d σ ( b ( t )) for all σ ∈ T ( M p m +1 q m +1 ) , (e 8.52) d τ ( e (0)) < d τ ( b (0)) and d τ ( e (1)) < d τ ( ϕ m ( a )(1)) , (e 8.53)where τ is the unique tracial state of M p m +1 ⊗ q m +1 and τ is the unique tracial state of1 q m +1 ⊗ M q m +1 . Note that e (1) = 0 . It follows that, for all τ ∈ T ( L p m +1 ,q m +1 ) ,d τ ( e ) < d τ ( b ) . (e 8.54)By, for example, Theorem 3.18 of [20], e . b in L p m +1 ,q m +1 . (e 8.55)By Lemma 8.2, e . ϕ m ( a ) in E p m +1 ,q m +1 = A m +1 . (e 8.56)It follows that e . ϕ m, ∞ ( a ) = f / ( ϕ , ∞ ( a ′ )) . a. (e 8.57)Combining this with (ii) and (iii) above, we conclude that A Cz is essentially tracially in N Z ,s,s . Since B z ∼ = Z which has nuclear dimension 1 ([39, Theorem 6.4.]), A Cz has essential tracialnuclear dimension at most 1. Since Z is in T , A Cz is e.tracially in T . By Proposition 4.6, every2-quasitrace of A Cz is a tracial state. By Corollary 5.10, A has stable rank one. Remark 8.5.
Note that Theorem 8.4 actually shows that A Cz is essentially tracially approxi-mated by Z itself, as B z ∼ = Z . Theorem 8.6.
Let ( G, G + , g ) be a countable weakly unperforated simple ordered group, F bea countable abelian group, ∆ be a metrizable Choquet simplex, and λ : G → Aff + (∆) be ahomomorphism.Then, there is a unital simple non-exact C ∗ -algebra A which is e. tracially in N Z ,s,s , hasessential tracial nuclear dimension at most 1, stable rank one, and strict comparison for positiveelements such that ( K ( A ) , K ( A ) + , [1 A ] , K ( A ) , T ( A ) , ρ A ) = ( G, G + , g, F, ∆ , λ ) . (e 8.58) Proof.
It follows from Theorem 13.50 of [20] that there is a unital simple A with finite nucleardimension which satisfies the UCT such that( K ( A ) , K ( A ) + , [1 A ] , K ( A ) , T ( A ) , ρ A ) = ( G, G + , g, F, ∆ , λ ) . (e 8.59)Let A Cz be the C ∗ -algebra in Theorem 7.3 with A Cz is non-exact. Then define A = A ⊗ A Cz . Note that, since A is a separable amenable C ∗ -algebra satisfying the UCT, by [40, Theorem41.14], the K¨unneth formula holds for the tensor product A = A ⊗ A Cz . Since the only normalized2-quasitrace of A Cz is the unique tracial state and( K ( A Cz ) , K ( A Cz ) + , [1 A Cz ] , K ( A Cz )) = ( Z , Z + , , , one computes, applying the K¨unneth formula, that( K ( A ) , K ( A ) + , [1 A ] , K ( A ) , T ( A ) , ρ A ) = ( G, G + , g, F, ∆ , λ ) . (e 8.60)We will show that A is essentially tracially in N Z ,s,s and has e.tracial nuclear dimension at most1. Once this is done, by Definition 8.3, A has essentially tracial nuclear dimension at most 1,and, by Corollary 5.10, has stable rank one and strict comparison for positive elements.To see that A is essentially tracially in N z,s,s , let ε > , F ⊂ A be a finite subset set and let d ∈ A + \ { } . Without loss of generality, we may assume that there are n ∈ N , M > , and a finite subset F ⊂ A and F ⊂ A Cz such that F = { n X i =1 a i ⊗ b i , a i ∈ F , b i ∈ F } and (e 8.61) k f k , k f k ≤ M, if f ∈ F and f ∈ F . (e 8.62)By Kirchberg’s Slice Lemma (see, for example, [36, Lemma 4.1.9]), there are a ∈ ( A ) + \ { } and b ∈ ( A Cz ) + \ { } such that a ⊗ b . d. Let us identity A with A ⊗ Z , see [43, Corollary 7.3]. In A ⊗ Z , choose 1 A ⊗ a z with a z ∈ Z + \ { } such that 1 A ⊗ a z . A a. Choose b z ∈ ( B z ) + \ { } ⊂ A Cz with B z ∼ = Z (seeRemark 8.5) such that b z . A Cz b. Note B z ∼ = Z b ⊗ B z , where Z b ∼ = Z . Put c := σ ( a z ) ⊗ b z ∈ B z , where σ : 1 A ⊗ Z ( ⊂ A ) →Z b ⊗ B z is an isomorphism. Consider D := A ⊗ σ (1 A ⊗ Z ) ⊗ B z ⊂ A ⊗ B z . We may alsowrite D = ( A ⊗ Z ) ⊗ σ (1 A ⊗ Z ) ⊗ B z . There is a sequence of unitaries v n ∈ D such thatlim n →∞ v ∗ n (1 A ⊗ σ ( a z ) ⊗ B z ) v n = 1 A ⊗ a z ⊗ B z . (e 8.63)It follows that 1 A ⊗ c = 1 A ⊗ σ ( a z ) ⊗ b z . a ⊗ b. (e 8.64)By Remark 8.5, there exists e ∈ A Cz with 0 ≤ e ≤ f ∈ F , k e f − f e k < ε/ nM ) , (1 A Cz − e ) f ∈ ε/ nM B z , (e 8.65) k (1 − e ) f k ≥ (1 − ε/ nM ) ) k f k and e . c . (e 8.66)Put B = A ⊗ B z . Then, since A has nuclear dimension at most 1 (see, for example, [8, TheoremB]) and A is Z -stable, B ∼ = A . Therefore B is Z -stable and has nuclear dimension at most 1.Put e = 1 A ⊗ e . Then 0 ≤ e ≤ . For any f ∈ F , f = P ni =1 a i ⊗ b i for some a i ∈ F and b i ∈ F . It follows that k ef − f e k = k e ( n X i =1 a i ⊗ b i ) − n X i =1 a i ⊗ b i ) e k = k ( n X i =1 a i ⊗ ( e b i − b i e ) k < ε. (e 8.67)Also (1 − e ) f = (1 − A ⊗ e )( n X i =1 a i ⊗ b i ) = n X i =1 a i ⊗ (1 A Cz − e ) b i ∈ ε A ⊗ B z . (e 8.68)42oreover, by (e 8.64) and (e 8.66), e . a ⊗ b . d. (e 8.69)These imply that A ⊗ A Cz is essentially tracially in N Z ,s,s . Since B has nuclear dimension nomore than 1 (see [8, Theorem B]).Since A Cz embedded into A ⊗ A Cz and A Cz is not exact, A ⊗ A Cz is also not exact (see, forexample, Proposition 2.6 of [42]). Remark 8.7. (1) Let A be a unital separable nuclear purely infinite simple C ∗ -algebra in theUCT class. Then the proof of Theorem 8.6 also shows that A := A ⊗ A Cz is an non-exact purelyinfinite simple C ∗ -algebra which has essential tracial nuclear dimension 1 and Ell( A ) = Ell( A ) . (2) If the RFD C ∗ -algebra C at the beginning of section 6 is amenable then C ((0 , , C )is a nuclear contractible C ∗ -algebra which satisfies the UCT. It follows that the unitization B of C ((0 , , C ) also satisfies the UCT. Therefore D ( m, k ) and I = C ((0 , , M mk ( B )) in (e 6.9)satisfy the UCT. Thus E m,k is nuclear and satisfies the UCT. It follows that A Cz is a unitalamenable separable simple stable rank one C ∗ -algebra with a unique tracial state which alsohas strict comparison for positive elements and satisfies the UCT. By [29, Theorem 1.1] A Cz is Z -stable. By [30, Theorem 1.1] A Cz has finite nuclear dimension. Then by [13, Corollary 4.11], A Cz is classifiable by the Elliott invariant. Since A Cz has the same Elliott invariant of Z , it followsthat A Cz ∼ = Z . References [1] D. Archey, J. Buck and N. C. Phillips,
Centrally large subalgebras and tracial Z -absorption ,Int. Math. Res. Not. IMRN 2018, no. 6, 1857–1877.[2] B. Blackadar, A simple C ∗ -algebra with no nontrivial projections, Proc. Amer. Math. Soc.,
78 (4) (1980), 504-508.[3] B. Blackadar,
Operator algebras. Theory of C ∗ -algebras and von Neumann algebras, Ency-clopaedia of Mathematical Sciences, . Operator Algebras and Non-commutative Geome-try,
III , Springer-Verlag, Berlin, 2006. xx+517 pp. ISBN: 978-3-540-28486-4; 3-540-28486-9.[4] B. Blackadar and D. Handelman,
Dimension functions and traces on C ∗ -algebra, J. Funct.Anal. (1982), 297-340.[5] B. Blackadar and E. Kirchberg, Generalized inductive limits of finite-dimensional C ∗ -algebras, Math. Ann. (1997), no. 3, 343-380.[6] E. Blanchard and E. Kirchberg,
Non-simple purely infinite C ∗ -algebras: the Hausdorff case, J. Funct. Anal. (2004), no. , 461–513.[7] L. G. Brown, Stable isomorphism of hereditary subalgebras of C ∗ -algebras, Pacific J. Math. (1977), no. , 335–348.[8] J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter, Nuclear dimension ofsimple C ∗ -algebras , preprint, arXiv: 1901.05853v1.[9] M. Choi and E. Effros, The completely positive lifting problem for C ∗ -algebras. Ann. ofMath. (2), (1976) no. 3, 585-609. 4310] J. Cuntz,
The structure of multiplication and addition in simple C ∗ -algebras. Math. Scand. (1977), no. 2, 215-233.[11] J. Cuntz, Dimension functions on simple C ∗ -algebras , Math. Ann. (1978), no. 2, 145-153.[12] M. D˘ad˘arlat, Nonnuclear subalgebras of AF algebras,
Amer. J. Math, (2000), no.3,581-597.[13] G. A. Elliott, G. Gong, H. Lin and Z. Niu,
On the classification of simple amenable C*-algebras with finite decomposition rank, II , preprint. arXiv:1507.03437.[14] G. A. Elliott, G. Gong, H. Lin and Z. Niu,
Simple stably projectionless C ∗ -algebrasof generalized tracial rank one , J. Non-commutative geometry, (2020), 251-347.(arXiv:1711.01240).[15] G. A. Elliott, L. Robert, and L. Santiago, The cone of lower semicontinuous traces on aC*-algebra , Amer. J. Math (2011), 969–1005.[16] X. Fu and H. Lin,
On tracial approximation of simple C ∗ -algebras, preprint, arXiv:2004.10901.[17] X. Fu and H. Lin, On tracial approximation of separable C ∗ -algebras , in preparation.[18] G. Gong, X. Jiang and H. Su, Obstructions to Z -stability for unital simple C ∗ -algebras, Canad. Math. Bull.
43 (4) (2000), 418-426.[19] G. Gong and H. Lin,
On classification of simple non-unital amenable C*-algebras, II , Jour-nal of Geometry and Physics, (2020), https://doi.org/10.1016/j.geomphys.2020.103865.[20] G. Gong, H. Lin, and Z. Niu,
A classification of finite simple amenable Z-stable C ∗ -algebras,I: C ∗ -algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Canada, toappear, arXiv: 1812.11590v3.[21] K. R. Goodearl,
Notes on a class of simple C ∗ -algebras with real rank zero, Publ. Mat. (1992), no. 2A, 637-654.[22] X. Jiang and H. Su, On a simple unital projectionless C ∗ -algebra, Amer. J. Math. (1999), no. 2, 359-413.[23] E. Kirchberg and M. Rørdam,
Non-simple purely infinite C*-algebras , Amer. J. Math., (2000) no. 3, 637-666.[24] H. Lin,
An introduction to the classification of amenable C ∗ -algebras , World Scientific Pub-lishing Co. Inc. River Edge, NJ, 2001, xii+320 pp, ISBN: 981-02-4680-3.[25] H. Lin, Tracially AF C*-algebras , Trans. Amer. Math. Soc. (2001), 693–722.[26] H. Lin,
The tracial topological rank of C ∗ -algebras , Proc. London Math. Soc. (2001),199–234.[27] H. Lin, Traces and simple C ∗ -algebras with tracial topological rank zero , J. Reine Angew.Math. (2004), 99–137.[28] H. Lin and S. Zhang, On infinite simple C ∗ -algebras, J. Funct. Anal. (1991), 221-231.4429] H. Matui and Y. Sato,
Strict comparison and Z -absorption of nuclear C ∗ -algebras, ActaMath. (2012), no. 1, 179-196.[30] H. Matui and Y. Sato,
Decomposition rank of UHF-absorbing C ∗ -algebras, Duke Math. J.,
163 (14) (2014), 2687-2708.[31] G. K. Pedersen. C ∗ -algebras and their automorphism groups. Number in London Math-ematical Society Monographs. Academic Press, London, 1979.[32] N. C. Phillips, Large subalgebras, preprint, arXiv: 1408.5546v1.[33] L. Robert,
Remarks on Z -stable projectionless C ∗ -algebras. Glasg. Math. J. (2016), no. 2,273-277.[34] M. Rørdam, On the structure of simple C ∗ -algebras tensored with a UHF-algebra, J. Funct.Anal. (1991), 1-17.[35] M. Rørdam,
On the structure of simple C ∗ -algebras tensored with a UHF-algebra. II, J.Funct. Anal. (1992), 255-269.[36] M. Rørdam,
Classification of nuclear, simple C ∗ -algebras. Classification of nuclear C ∗ -algebras. Entropy in operator algebras , 1-145, Encyclopaedia Math. Sci. , Oper. Alg.Non-commut. Geom. , Springer, Berlin, 2002.[37] M. Rørdam, The stable and the real rank of Z -absorbing C ∗ -algebras, Internat. J. Math. (2004), no. 10, 1065-1084.[38] M. Rørdam and W. Winter, The Jiang-Su algebra revisited , J. Reine Angew. Math. (2010), 129-155.[39] Y. Sato, S. White, W. Winter
Nuclear dimension and Z -stability. Invent. Math. 202 (2015),no. 2, 893–921.[40] C. Schochet,
Topological methods for C ∗ -algebras, II. Geometric resolution and the K¨unnethformula , Pacific J. Math. (1982), 443–458.[41] A. Toms and W. Winter, Strongly self-absorbing C ∗ -algebras, Trans. Amer. Math. Soc. (2007), no. 8, 3999-4029.[42] S. Wassermann,
Exact C ∗ -algebras and related topics, Lecture Notes Series, . Seoul Na-tional University, Research Institute of Mathematics, Global Analysis Research Center,Seoul, 1994. viii+92 pp.[43] W. Winter, Nuclear dimension and Z -stability of pure C ∗ -algebras , Invent. Math.187