This paper extends the classification of ATAI algebras by removing a torsion restriction on K-theory, using an invariant that includes the Hausdorffized algebraic K1-group, and connects to classification of AHD algebras.
Contribution
It generalizes previous classification results of ATAI algebras by incorporating the Hausdorffized algebraic K1-group, removing the torsion restriction on K1.
Findings
01
Classification of all ATAI algebras with the new invariant.
02
Reduction to classification of AHD algebras with ideal property.
03
Generalization of previous main theorems in the field.
Abstract
An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit n→∞lim(An=i=1⨁Ani,ϕnm), where each Ani is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in K1(A) are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic K1-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of AHD algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main…
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TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Algebraic structures and combinatorial models
Full text
On the inductive limit of direct sums of simple TAI algebras
Bo Cui, Chunlan Jiang, and Liangqing Li
Abstract
An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit n→∞lim(An=i=1⨁Ani,ϕnm), where each Ani is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in K1(A) are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic K1-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of AHD algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).
1. Introduction
In [Lin2], Lin gave an abstract description of simple AH algebras (with no dimension growth) classified
in [EGL1]. He described the decomposition property of simple AH algebras in [G3] as TAI property and
proved that all simple separable nuclear TAI algebras with UCT are classifiable and therefore in the class of [EGL1]. For simple AH algebras of real rank zero, the corresponding decomposition property is called TAF
by [Lin1], which is partially inspired by Popa’s paper [Popa]. As proved by Lin ([Lin1-2]), a simple separable nuclear C*-algebra A with UCT property is a TAI (or TAF, respectively) algebra if and only if A is a simple AH algebra (or simple AH algebra of real rank zero) with no dimension growth, which is classified in [EGL1].
As in [EG2], let TII,k be the 2-dimensional connected simplicial complex with H1(TII,k)=0 and
H2(TII,k)=Z/kZ, and let Ik be the subalgebra of Mk(C[0,1])=C([0,1],Mk(C)) consisting of all functions f with the properties f(0)∈C1k and f(1)∈C1k (this algebra is called an Elliott dimension drop interval algebra). Denote HD the class of algebras consisting of direct sums of building blocks of the forms Ml(Ik) and PMn(C(X))P, with X being one of the spaces {pt}, [0,1], S1, and TII,k, and with P∈Mn(C(X)) being a projection. (In [DG], this class is denoted by SH(2), and in [Jiang1], this class is denoted by B). A C*-algebra is called an AHD algebra, if it is an inductive limit of algebras in HD. In [GJL2], the authors classified all AHD algebras with the ideal property.
In [Jiang1], Jiang classified ATAI algebras A under the extra restriction that K1(A)=torK1(A). In this classification, Jiang used the scaled ordered total K-group (from [DG]) and the tracial state spaces T(pAp) of cut-down algebras pAp, with certain compatibility conditions (from [Stev] and [Ji-Jiang]) as the invariant–we will call it Inv0(A). In this paper, we will use the invariant Inv0(A) together with the Hausdorffized algebraic K1-group to deal with the torsion-free part of K1(A), with certain compatibility conditions–we will call this Inv(A). We will prove that this invariant reduces to Jiang’s invariant in the case that K1(A) is a torsion group, that is, we removed the extra restriction that K1(A)=torK1(A) of Jiang’s classification in [Jiang1] and prove our theorem by reduced to the classification of AHD algebra with ideal property, which is done in [GJL1].
2. Preliminaries And Definitions
2.1
Let A be a C*-algebra. Two projections in A are said to be equivalent if
they are Murray–von Neumann equivalent. We write p⪯q if p is equivalent to a
projection in qAq. We denote by [p] the equivalent class of projections equivalent
to p. Let a∈A+, we write p⪯a if p⪯q for some projection q∈aAa.
Definition 2.2
Let G⊂A be a finite set and δ>0. We shall say that ϕ∈Map(A,B) is G−δ multiplicative if ∥ϕ(ab)−ϕ(a)ϕ(b)∥<δ for all a,b∈G.
We also use MapG−δ(A,B) to denote all G−δ multiplicative maps.
If two maps φ,ϕ∈Map(A,B) satisfy the condition ∥ϕ(a)−ψ(a)∥<δ for all
a∈G, then we will write ϕ≈δφ on G.
If G,H⊂A are two subsets of a C*-algebra A and for any g∈G, there is a b∈H
with ∥g−b∥<δ, then we denote G⊂δH.
Definition 2.3
[Jiang1] We denote by I the class of all unital C*-algebras with the form ⨁i=1nBi, where each Bi≅Mki or Bi≅Mki(C[0,1]) for some k(i). Let A∈I, we have the following:
(1) Every C*-algebra in I is of stable rank one.
(2) Two projections p and q in a C*-algebra A∈I are equivalent if and only if
τ(p)=τ(q) for all τ∈T(A), where T(A) denotes the space of all tracial states.
(3) For any ε>0 and any finite subset F⊂A, there exist a number δ>0 and a finite subset G⊂A satisfying the following: If L:A→B is a G−δ multiplicative contractive completely positive linear map, where B is a C*-algebra, then there exists a homomorphism h:A→B such that
[TABLE]
Definition 2.4
([Lin1]) A unital simple C*-algebra in A is said to be tracially AI (TAI) if
for any finite subset F⊂A containing a nonzero element b, ε>0, integer n>0
and any full element a∈A+, there exist nonzero projection p∈A and a C*-algebra
I⊂A with I∈I and 1I=p such that:
(1) ∥xp−px∥<ε for all x∈F;
(2) pxp∈εI for all x∈F;
(3) n[1−p]⪯[p] and 1−p⪯a.
Definition 2.5
[Jiang1] A C*-algebra A (not necessary unital) is said to be ATAI algebra
(approximately TAI algebra) if it is the inductive limit of a sequence of direct sums
of simple unital TAI algebras with UCT.
2.6
Let A and B be two C*-algebras. We use Map(A,B) to denote the space of all completely positive ∗-contractions from A to B. If both A and B are unital, then Map(A,B)1 will denote the subset of Map(A,B) consisting of all such unital maps.
2.7
In the notation for an inductive limit system lim(An,ϕn,m), we understand that
[TABLE]
where all ϕn,m:An→Am are homomorphisms.
We shall assume that, for any summand Ani in the direct sum An=⨁i=1tnAni, necessarily, ϕn,n+1(1Ani)=0, since, otherwise, we could simply delete Ani from An, without changing the limit algebra.
If An=⨁iAni, Am=⨁jAmj, we use ϕn,mi,j to denote the partial map of ϕn,m from the i-th block Ani of An to the j-th block Amj of Am. Also, we use ϕn,m−,j to denote the partial map of ϕn,m from An to Amj. That is, ϕn,m−,j=i⨁ϕn,mi,j=πjϕn,m, where πj:Am→Amj is the canonical projection. We also use ϕn,mi,− to denote ϕn,m∣Ani:Ani→Am.
2.8
An AH algebra is a C*-algebra which is lim(An=⨁i=1knpn,iM[n,i](C(Xn,i))pn,i,ϕn,m), where each Xn,i is a compact metrizable space, and pn,i∈M[n,i](C(Xn,i)) is a projection. Recall in [G1], Gong proved that a simple AH algebra with uniformly bounded dimension of local spectra, i.e. supn,idim(Xn,i)<∞ can be rewritten as AH inductive limit with spaces being [0,1], S1, S2, TII,k, TIII,k, where each TII,k (or TIII,k respectively) is two–dimensional (or three–dimensional respectively) connected simplicial complexes with H2(TII,k)=Z/kZ and H1(TII,k)=0 (or with H3(TIII,k)=Z/kZ and H1(TIII,k)=0, H2(TIII,k)=0).
For any positive k the dimension drop interval algebra Ik is defined as
[TABLE]
2.9
[GJL1] Let X be a compact space and ψ:C(X)→PMk1(C(Y))P(rank(P)=k) be a unital homomorphism. For any point y∈Y, there are k mutually orthogonal rank-1 projections p1,p2,⋯,pk with i=1∑kpi=P(y) and \big{\{}x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\big{\}}\subset X (may be repeat) such that ψ(f)(y)=i=1∑kf(xi(y))pi,∀f∈C(X). We denote the set \big{\{}x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\big{\}} (counting multiplicities), by Spψy. We shall call Spψy the spectrum of ψ at the point y.
For any f∈Ik, let function f:[0,1]→C⨆Mk(C) (disjoint union) be defined by
[TABLE]
That is, f(t) is the value of irreducible representation of f corresponding to the point t. Similarly, for f∈Ml(Ik), we can define f:[0,1]⟶Ml(C)⨆Mlk(C), by
[TABLE]
Suppose that ϕ:Ik→PMn(C(Y))P is a unital homomorphism. Let r=rank(P). For each y∈Y, there are t1,t2,⋯,tm∈[0,1] and a unitary u∈Mn(C) such that
[TABLE]
and
[TABLE]
for all f∈Ik.
We define the set Spϕy to be the points t1,t2,⋯,tm with possible fraction multiplicity. If ti=0 or 1, For example if we assume
[TABLE]
then Sp\phi_{y}=\big{\{}0^{\sim\frac{1}{k}},0^{\sim\frac{1}{k}},0^{\sim\frac{1}{k}},t_{4},t_{5},\cdots,t_{m-2},1^{\sim\frac{1}{k}},1^{\sim\frac{1}{k}}\big{\}},
which can also be written as
[TABLE]
Here we emphasize that, for t∈(0,1), we do not allow the multiplicity of t to be non-integral. Also for [math] or 1, the multiplicity must be multiple of k1 (other fraction numbers are not allowed).
Let ψ:C[0,1]→PMn(C(Y))P be defined by the following composition
[TABLE]
where the first map is the canonical inclusion. Then we have Spψy={Spϕy}∼k–that is, for each element t∈(0,1), its multiplicity in Spψy is exactly k times of the multiplicity in ϕy.
Recall that for A=Ml(Ik), every point t∈(0,1) corresponds to an irreducible representation πt, defined by πt(f)=f(t). The representations π0 and π1 defined by
[TABLE]
are no longer irreducible. We use 0 and 1 to denote the corresponding points for the irreducible representations. That is,
[TABLE]
Or we can also write f(0)≜f(0) and f(1)≜f(1). Then the equation (∗) could be written as
[TABLE]
where some of ti may be 0 or 1. In this notation, f(0) is equal to diag\big{(}\underbrace{f(\underline{0}),f(\underline{0}),\cdots,f(\underline{0})}_{k}\big{)} up to unitary
equivalence.
Under this notation, we can also write 0∼k1 as 0. Then the example of Spϕy can be written as
[TABLE]
2.10
We use HD to denote all C*-algebras C=⨁Ci, where each Ci is
of the forms Ml(Ik) or PMnC(X)P with X being one of the spaces {pt}, [0,1], S1, TII,k. Each block Ci will be called a basic HD block or a basic building block.
By AHD algebra, we mean the inductive limit of
[TABLE]
where An∈HD for each n.
For an AHD inductive limit A=lim(An,ϕn,m), we write An=⨁i=1tnAni, where Ani=Pn,iM[n,i](C(Xn,i))Pn,i of Ani=M[n,i](Ikn,i). For convenience, even for a block Ani=M[n,i](Ikn,i), we still use Xn,i for Sp(Ani)=[0,1]–that is, Ani is regarded as a homogeneous algebra or a sub-homogeneous algebra over Xn,i.
Definition 2.11
([EG2], [DG]) Let X be a compact connected space and let P∈MN(C(X)) be a projection of rank n. The weak variation of a finite set F⊂PMN(C(X))P is defined by
[TABLE]
where π,σ run through the set of irreducible representations of PMNC(X)P into Mn(C).
For F⊂Mr(Ik), we define ω(F)=ω((F)), where :Mr(Ik)⟶Mrk(C[0,1]) is the canonical embedding
and (F) is regarded as a finite subset of Mrk(C[0,1]). Let A be a basic HD block, a finite set F⊂A is said to be weakly approximately constant to within ε if ω(F)<ε.
2.12
Let K(A)=K∗(A)⊕⨁k=2∞K∗(A,Z/k) be as in [DG]. Let Λ be the Bockstein operation between K(A)s (see 4.1 of [DG]). It is well known that
[TABLE]
As in [DG], let
[TABLE]
and let K(A)+ to be the semigroup of K(A) generated by {K∗(A,Z⊕Z/kZ)+,k=2,3,⋯}.
2.13
Let HomΛ(K(A),K(B)) be the homomorphism between K(A) and K(B) compatible with Bockstein operation Λ. Associativity of Kasparov KK-product gives a map
[TABLE]
Recall that every element α∈KK(A,B) defines a map α∗∈HomΛ(K(A),K(B)). That is, it gives a
sequence of homomorphisms
[TABLE]
which are compatible with the Bockstein operation Λ.
2.14
Let K(A)+ be defined in 4.6 of [DG]. Notice that, from [DG, 4.7], K(A)+⊂K(A)+, where for any unital C*-algebra B, let
[TABLE]
2.15
Let A=PM∙C(TII,k)P, where we use ∙ to denote any possible positive integers. From 5.14 of [G3], we know that an element α∈KK(A,B) is completely determined by α0:K0(C(TII,n))⟶K0(B) and αk1:K1(C(TII,k),Z/k)⟶K1(B,Z/k), k=0,1,2,⋯.
For any positive integer k≥2, denote
[TABLE]
Then for A=PM∙C(TII,k)P, an element α∈KK(A,B) is in KK(A,B)+ if and only if α(K(k)(A)+)⊂K(k)(B)+, k=0,1. Note that K(k)(A)+ is finitely generated.
2.16
Note that, for any C*-algebra A, it follows from Künneth Theorem that
[TABLE]
where Wk=TII,k.
One can choose finite set P⊂M∙(A⊗C(Wk×S1)) of projections such that
[TABLE]
generate K(k)(A)=K0(A)⊕K1(A)⊕K0(A,Z/k)⊕K1(A,Z/k), where Wk=TII,k.
If we choose finite set G(P)⊂A large enough and δ(P)>0 small enough, then every G(P)−δ(P) multiplicative contraction ϕ:A→B determines a map ϕ∗:PK(A)→K(B) which is compatible with the Bockstein operation Λ (see [GL]). If A=PM∙C(TII,k)P, then it also defines a KK element [ϕ]∈KK(A,B).
Let A=PM∙C(TII,k)P, recall that for G⊇G(P), δ≤δ(P), a G−δ multiplicative map ϕ:A→B is called a quasi PK homomorphism, if there is a homomorphism ψ:A→B satisfying ϕ∗=ψ∗:PK(A)→K(B). If ϕ is a quasi PK homomorphism, then [ϕ]∈KK(A,B)+.
Suppose that A=PM∙C(TII,k)P and suppose that B=i=1⨁nBi is a direct sum of basic HD bulding blocks. Let α∈KK(A,B), then α∈KK(A,B)+ if and only if for each i∈{1,2,⋯,n}, αi∈KK(A,Bi) is either zero or αi(1C(TII,k))>0 in K0(Bi). Recall that in [DG], a KK element α∈KK(A,B=i=1⨁nBi) is called m-large if rank(αi(1A))≥m⋅rank(1A), for all i∈{1,2,⋯,n}. By 5.5 and 5.6 of [DG] if α∈KK(A,B)+ is 6-large and α([1A])≤[1B], then α can be realized by a homomorphism from A to B.
2.17
Recall that the scale of A, denoted by ∑A, is a subset of K0(A) consisting of [p], where p∈A is a projection. As all the C*-algebras A in this paper have cancellation of projections, if A has unit 1A, then
[TABLE]
For two C*-algebras A,B, by a ”homomorphism” α from (K(A),K(A)+,∑A) to (K(B),K(B)+,∑B), it means a system of maps
[TABLE]
which are compatible with Bockstein operation and α=⨁k,iαki satisfies α(K(A)+)⊆K(B)+ and finally α00(∑A)⊆∑B.
2.18
For a unital C*-algebra A, let TA denote the space of tracial states of A, i.e. τ∈TA, if and only if τ is a positive linear map from A to the complex plane C, with τ(xy)=τ(yx) and τ(1A)=1. AffTA is the Banach space of all the continuous affine maps from TA to C. (In most references, AffTA is defined to be the set of all the affine maps from TA to R. Our AffTA is a complexification of the standard AffTA.) The
element 1∈AffTA, defined by 1(τ)=1, for all τ∈TA, is called the unit of AffTA. AffTA, together with the positive cone AffTA+ and the unit element 1 forms a
scaled ordered complex Banach space. (Notice that for any element x∈AffTA, there are x1,x2,x3,x4∈AffTA+ such that x=x1−x2+ix3−ix4.)
There is a natural homomorphism ρA:K0(A)→AffTA defined by ρA([p])(τ)=∑i=1nτ(pii) for τ∈TA and [p]∈K0(A) represented by projection p=(pi,j)∈Mn(A).
Any unital homomorphism ϕ:A→B induces a continuous affine map Tϕ:TB→TA, It turns out that Tϕ induces a unital positive linear map
[TABLE]
If ϕ:A→B is not unital, we still have the positive linear map
[TABLE]
but it will not preserve the unit 1, only has property AffTϕ(1AffTA)≤1AffTB.
2.19
If α:(K(A),K(A)+,∑A)→(K(B),K(B)+,∑B) is a homomorphism as in 2.17, then for each projection p∈A, there is a projection q∈B such that α[p]=[q].
Notice that for all the C*-algebras A considered in this paper, the following is true: if p1, p2 are projections and [p1]=[p2] in K0(A), then there is u∈A with up1u∗=p2. Therefore both T(pAp) and T(qBq) depend only on the class [p]∈K0(A) and [q]∈K0(B). For the classes [p]∈∑A(⊂K0(A)), T(pAp) is taken as part of the invariant Inv0(A). For two classes [p]∈∑A, [q]∈∑B, with α([p])=[q], we will consider the system of continuous affine maps ξp,q:T(qBq)→T(pAp). Such system of maps is said to be compatible if for any two projections p1≤p2 with α([p1])=[q1], α([p2])=[q2] and q1≤q2, the following diagram is commutative:
[TABLE]
where the horizontal maps are induced by ξp1,q1 and ξp2,q2 respectively, and the
vertical maps are induced by the inclusions p1Ap1→p2Ap2, q1Bq1→q2Bq2.
2.20
We denote \big{(}\underline{K}(A),\underline{K}(A)^{+},\sum A,\{T(pAp)\}_{[p]\in\sum A}\big{)} by Inv0(A). By a “map” between the invariants \big{(}\underline{K}(A),\underline{K}(A)^{+},\sum A,\{T(pAp)\}_{[p]\in\sum A}\big{)} and \big{(}\underline{K}(B),\underline{K}(B)^{+},\sum B,\{T(qBq)\}_{[q]\in\sum B}\big{)}, we mean a map
[TABLE]
as in 2.17 and maps ξp,q:T(qBq)→T(pAp) which are compatible as 2.19. We denote this map by
[TABLE]
or simply
[TABLE]
2.21
Let A be a unital C*-algebra. Let U(A) denote the group of unitaries of A and let U0(A) denote the connected component of 1A in U(A). Let DU(A) and DU0(A) denote the commutator subgroups of U(A) and U0(A), respectively. (Recall that the commutator subgroup of a group G is the subgroup generated by all elements of the form aba−1b−1, where a,b∈G.) One introduces the following metric DA on U(A)/DU(A) (see [NT,§3]). For u,v∈U(A)/DU(A)
[TABLE]
where, on the right hand side of the equation, we use u, v to denote any elements in U(A), which represent the elements u,v∈U(A)/DU(A).
Denote the extended commutator group DU+(A), which is generated by DU(A)⊂U(A) and the set \big{\{}e^{2\pi itp}=e^{2\pi it}p+(\mathbf{1}-p)\in U(A)\big{|}t\in\mathbb{R},p\in A\ is\ a\ projection\big{\}}. Let DU(A) denote the closure of DU+(A). That is, DU(A)=DU+(A)
2.22
Let A be a unital C*-algebra. Let AffTA and ρA:K0(A)→AffTA be as defined as in 2.18.
For simplicity, we will use ρK0(A) to denote the set ρA(K0(A)). The metric dA on AffTA/ρK0(A) is defined as follows (see [NT, §3]).
Let d′ denote the quotient metric on AffTA/ρK0(A), i.e, for f,g∈AffTA/ρK0(A),
[TABLE]
Define dA by
[TABLE]
Obviously, dA(f,g)≤2πd′(f,g).
Let ρK0(A) denote the closed real vector space spanned by ρK0(A). That is,
[TABLE]
For any u,v∈U(A)/DU(A), define
[TABLE]
Let d′ denote the quotient metric on AffTA/ρK0(A), that is,
[TABLE]
Define dA by
[TABLE]
2.23
Let
[TABLE]
and for simpler, denote ρK0(A)≜ρK0(A), DU(A)=DU(A), SU(A)=SU(A).
A unital homomorphism ϕ:A→B induces a contractive group homomorphism
[TABLE]
If ϕ is not unital, then the map ϕ♮:U(A)/SU(A)⟶U(ϕ(1A)Bϕ(1A))/SU(ϕ(1A)Bϕ(1A)) is induced by the corresponding unital homomorphism. In this case, ϕ also induces the map ∗∘ϕ♮:U(A)/SU(A)⟶U(B)/SU(B), which is denoted by ϕ∗ to avoid confusion.
2.24
In [GJL1] and [GJL2], denote
[TABLE]
by Inv(A).
By a map from Inv(A) to Inv(B), one means
[TABLE]
as in 2.17, and for each pair ([p],[p])∈∑A×∑B with α([p])=[p], there are an associate unital positive linear map
[TABLE]
and an associate contractive group homomorphism
[TABLE]
satisfying the following compatibility conditions:
(a) If p<q, then the diagrams
[TABLE]
and
[TABLE]
commutes, where the vertical maps are induced by inclusions.
(b) The following diagram commutes
[TABLE]
and therefore ξp,p induces a map (still denoted by ξp,p):
[TABLE]
(The commutativity of (III) follows from the commutativity of (I), by 1.20 of [Ji-Jiang]. So this is not an
extra requirement.)
(c) The following diagrams
[TABLE]
and
[TABLE]
commute, where α1 is induced by α.
We will denote the map from Inv(A) to Inv(B) by
[TABLE]
Note that Inv0 is part of Inv(A).
3. Local approximation lemma
Proposition 3.1
Let A=PM∙C(TII,k)P or Ml(Ik) and P be as in 2.16. For any finite set F⊂A, ε>0, there exist a finite set G⊂A (G⊃G(P) large enough), a positive number δ>0(δ<δ(P) small enough) such that the following statement is true:
If B∈HD, ϕ,ψ∈Map(A,B) are G−δ multiplicative and ϕ∗=ψ∗:PK(A)→K(B), then there is a homomorphism ν∈Hom(A,ML(B)) defined by point evaluations and there is a unitary u∈ML+1(B) such that ∥(ϕ⊕ν)(a)−u(ψ⊕ν)(a)u∗∥<ε for all a∈F
Proof
For the special case that A=PM∙C(TII,k)P and B is homogeneous, this is [G3, Theorem 5.18]. The proof of the general case is completely same (see 2.16 above). Note that, the calculation of K0(Πn=1∞Bn/⨁n=1∞Bn), K0(Πn=1∞Bn/⨁n=1∞Bn,Z/k) and K1(Πn=1∞Bn/⨁n=1∞Bn,Z/k) in 5.12 of [G3]
works well for the case that some of Bn being of the form Ml(Ik).
∎
Lemma 3.2
Let A=PM∙(C(X))P or Ml(Ik). Let F⊂A be approximately constant to within ε (i.e., ω(F)<ε). Then for any two homomorphisms ϕ,ψ:A→B defined by point evaluations with K0ϕ=K0ψ, there exists a unitary u∈B such that ∥ϕ(f)−uψ(f)u∗∥<2ε∀f∈F.
Proof
The case A=PM∙(C(X))P is Lemma 3.5 of [GJLP2]. For the case A=Ml(Ik), one can also find a homomorphism ϕ′:Ml(C)→B such that
[TABLE]
where f(0) as in 2.9. Then one follows the same argument as the proof of Lemma 3.5 of [GJLP2] to get the result.
∎
Lemma 3.3
Let A=PM∙(C(X))P or Ml(Ik), ε>0, finite set F⊂A with ω(A)<ε. There exist a finite set G⊂A (G⊃G(P)), a number δ>0 (δ<δ(P))and a positive integer L such that the following statement is true. If B∈HD, and ϕ,ψ∈Map(A,B)1 are G−δ multiplicative and ϕ∗=ψ∗:PK(A)→K(B) and ν:A→M∞(B) is a homomorphism defined by point evaluations with ν([1A])≥L⋅[1B]∈K0(B), then there is a unitary u∈(1B⊕ν(1A))M∞(B)(1B⊕ν(1A)) such that
[TABLE]
Proof
Let L1 be as in Proposition 3.1 and L=2L1. Since ν([1A])≥L⋅[1B]=2L1⋅[1B]∈K0(B), there is a projection Q<ν(1A) such that Q is equivalent to 1ML1(B). By Proposition 3.1, there exist a homomorphism ν1:A→QM∞(B)Q defined by point evaluation and a unitary w∈(1B⊕ν1(1A))M∞(B)(1B⊕ν1(1A)) such that ∥(ϕ⊕ν1)(f)−w(ψ⊕ν1)(f)w∗∥<ε∀f∈F.
Since ν1 (and ν respectively) is homotopic to a homomorphism factoring through Mrank(P)(C) (for A=PM∙(C(X))P) or factoring through Ml(C) (for A=Ml(Ik)), there is a unital homomorphism ν2:A→(ν(1A)−ν1(1A))M∞(B)(ν(1A)−ν1(1A)) such that K0(ν1⊕ν2)=K0(ν).
By Lemma 3.2, ν is approximately unitarily equivalent to ν1⊕ν2 to within 2ε on F. Hence ϕ⊕ν is approximately unitarily equivalent to ψ⊕ν on F to within 2ε+ε+2ε=5ε.
∎
Lemma 3.4
Let A=PM∙C(TII,k)P, and P⊂M∙(A⊗C(Wk×S1)) as in 2.16. And let B=n→∞lim(Bn,ψnm) be a unital simple inductive limit of direct sums of HD building blocks. Let ϕ:A→B be a
unital homomorphism. It follows that for any G⊃G(P) and δ<δ(P), there is a Bn and a unital G−δ multiplicative contraction ϕ1:A→Bn which is a quasi PK-homomorphism such that ∥ψn,∞∘ϕ1(g)−ϕ(g)∥<δ
Proof
There is a finite set G1 and δ1>0 such that if a complete positive linear map ψ:A→C (C is a C*-algebra) and a homomorphism ϕ:A→C satisfy that ∥ϕ(g)−ψ(g)∥<δ1 for all g∈G1, then ψ is G−2δ multiplicative (see Lemma 4.40 in [G3]). Now if ϕ′:A→Bl satisfies ∥ψl,∞∘ϕ′(g)−ϕ(g)∥<δ for all g∈G1, then ψl,∞∘ϕ′ is G−2δ multiplicative. Hence for some n>l (large enough), ϕ1=ψi,n∘ϕ′:A→Bn is G−δ multiplicative. By replacing G by G∪G1 and δ by min(δ,δ1), we only need to construct a unital quasi PK-homomorsim ϕ1:A→Bn such that
[TABLE]
Furthermore, if ϕ1 is a G−δ multiplicative map satisfying the above condition, then
[TABLE]
Note that K(k)(A)+ is finitely generated, there is an m>n such that
[TABLE]
Since B is simple, for certain m>m1, KK(ψn,m∘ϕ1) is 6-large and therefore can be realized by a homomorphism. That is, replacing ϕ1 by ψn,m∘ϕ1, we get a quasi PK -homomorphism. Thus the proof of the lemma is reduced to the construction of ϕ1 to satisfy (∗).
Since B=n→∞lim(Bn,ψnm), there is a Bn and finite set F⊂Bn such that G⊂3δψn,∞(F). Since Bn(⊂B) is a nuclear C*-algebra, there are two complete positive contractions λ1:Bn→MN(C) and λ2:MN(C)→Bn such that
[TABLE]
Since Bn is a subalgebra of B, by Arverson s Extension Theorem, one can extend the map λ1:Bn→MN(C) to β1:B→MN(C) such that β1∘ψn,∞=λ1.
One can verify that ϕ1=λ2∘β1∘ϕ:A→Bn satisfies the condition (∗) as below. For g∈G, there is an f∈F such that ∥ϕ(g)−ψn,∞(f)∥<3δ, and therefore
[TABLE]
This ends the proof of the lemma.
∎
Lemma 3.5
For any finite set F⊂PM∙C(TII,k)P≜B with ω(F)<ε, there are a finite set G⊃G(P), a positive number δ<δ(P)
and a positive integer L (this L will be denoted by L(F,ε) later), such that if C is a HD basic building block, p,q∈C are two projections with p+q=1C, and ϕ0:B→pCp and φ1:B→qCq are two maps satisfying the following conditions:
(1) ϕ0 is quasi PK homomorphism and G−δ multiplicative, and ϕ1 is defined by point evaluations (or equivalently, factoring through a finite dimensional C*-algebra), and
(2) rank(q)≥Lrank(p),
then there is a homomorphism ϕ:B→C such that ∥ϕ0⊕ϕ1(f)−ϕ(f)∥<5ε for all f∈F.
Proof
Since ϕ0 is a quasi PK-homomorphism and is G−δ multiplicative, there is a homomorphism ϕ0′:B→pCp such that ϕ0∗′=ϕ0∗:PK(B)→K(C). By Lemma 3.3, there is unitary u∈C such that
[TABLE]
The homomorphism ϕ=Adu∗∘(ϕ0′⊕ϕ1) is as desired.
∎
Lemma 3.6
Let ε1>ε2>⋯>εn>⋯ be a sequence with ∑εi<+∞. Let A be a simple AH algebra with no dimension growth (as in [EGL] and [Li4]). Then A can be written as an AHD inductive limit
[TABLE]
with the unital subalgebras Bn(=⨁Bni)⊂An(=⨁Ani) (that is, Bni⊂Ani) and Fn⊂Bn and Gn⊂An with Fn=⨁Fni⊂Gn=⨁Gni such that the following statements hold:
(1) If Ani is not of type TII, then Bni=Ani and Fni=Fni. If Ani is of type TII, then Bni=PAniP⊕Dni⊂Ani with Fni=π0(Fni)⊕π1(Fni)⊂Gni and ω(π0(Fni))<ε, where Dni is a direct sum of the HD building blocks
other than type TII, and π0:Bni→PAniP≜Bn0,i and π1:Bni→Dni are canonical projections;
(2) Gni generates Ani, ϕn,n+1(An)⊂Bn+1, ϕn,n+1(Gn)⊂Fn+1, and n=1⋃∞ϕn,∞(Gn)=n=1⋃∞ϕn,∞(Fn)=unit ball of A;
(3) Suppose that both Ani and An+1j are of type TII and ϕ≜π0∘ϕn,n+1i,j∣Bn0,i:Bn0,i→Bn+10,j. Then ϕ(1Bn0,i)=p0⊕p1∈Bn+10,j and ϕ=ϕ0⊕ϕ1 with ϕ0∈Hom(Bn0,i,p0Bn+10,jp0)1, ϕ1∈Hom(Bn0,i,p1Bn+10,jp1)1 such that ϕ1 is defined by point evaluations (or equivalently, ϕ1(Bn0,i) is a finite dimensioned sub-algebra of p1Bn+10,jp1) and
[TABLE]
where L(π0(Fni),εn) is as in Lemma 3.5 (note that ω(π0(Fni))<εn).
Proof
From [Li4], we know that A can be written as an inductive limit of the direct sums of HD building blocks A=n→∞lim(An,ψnm). In our construction, we will choose An=Akn and homomorphisms ϕn,n+1:An=Akn→An+1=Akn+1 satisfying KK(ϕn,n+1)=KK(ψkn,kn+1) and AffTϕn,n+1 is close to AffTψkn,kn+1 within any pregiven small number on a pregiven finite set, in such a way that ϕn,n+1 also
satisfies the desired condition as in the lemma with certain choices of subalgebras Bn⊂An and and finite subsets Fn⊂Bn and Gn⊂An.
Suppose that we already have An=Akn with the unital sub-algebra Bn=⊕iBni=⊕i(Bn0,i⊕Dni)⊂An=⊕iAni, and two subsets Fn⊂Bn, Gn⊂An. as required in the lemma. We will construct, roughly, the
next algebra An+1=Akn+1, the sub-algebra Bn+1⊂An+1, two subsets Fn+1⊂Bn+1, Gn+1⊂An+1, and the homomorphism ϕn,n+1:An→An+1 to satisfy all the requirements in the lemma.
Applying the decomposition theorem-[G3, Theorem 4.37], as in the proof of the main theorem in [Li4],
for l>kn large enough, and for each block Ani=Akni of type TII, the homomorphism ψkn,li,j can be decomposed into three parts ϕ01⊕ϕ1⊕ϕ2, roughly described as below: There are mutually orthogonal projections Q0,Q1,Q2∈Alj with ψkn,li,j(1Ani)=Q0+Q1+Q2, there are two homomorphisms ϕk∈Hom(Ani,QkAljQk)1, (k=1,2) and a sufficient multiplicative quasi PK homomorphism ϕk∈Map(Ani,Q0AljQ0)1 possessing the following properties:
(i) ϕ01⊕ϕ1⊕ϕ2 is close to ψkn,li,j to within εn on Gni;
(ii) ϕ01(1Bn0,i) is a projection and
[TABLE]
(iii) ϕ1 is defined by point evaluation at a dense enough finite subset of Sp(Akni) such that for any homomorphism ϕ0∈Hom(Ani,Q0AljQ0)1, we have ω(ϕ0⊕ϕ1(Gni))<εn+1;
(iv) ϕ2 factors through an interval algebra D1i as ϕ2=ξ2i∘ξ1i:Ani⟶ξ1iD1i⟶ξ2iAlj.
Then we can choose kn+1=l and An+1=Akn+1. Note that ϕ01 is a quasi PK homomorphism, one can choose ϕ0:Akni→ϕ01(1Akni)Akn+1iϕ01(1Akni) such that KK(ϕ01⊕ϕ1⊕ϕ2)=KK(ϕ0⊕ϕ1⊕ϕ2) with ϕ01(1Bn0,i)=ϕ0(1Bn0,i). Modify ψkn,kn+1 by replacing ψkn,kn+1i,j by ϕ0⊕ϕ1⊕ϕ2 to define ϕn,n+1:An→An+1.
Fix j with An+1j=Akn+1j being of type TII. Let I0={i∣Ani=AknjisoftypeTII}, I1={i∣Ani=AknjisnotoftypeTII}. Let Pi=(ϕ0⊕ϕ1)(1Ani)∈An+1j, let P=i∈I0⨁Pi∈An+1j and Bn+10,j≜PAn+1jP, and let Dn+1j=i∈I0⨁ξ2i(D1i)⊕i∈I1⨁ϕkn,kn+1i,j(Aknj)⊂Akn+1j which is a direct sum of HD building blocks other than type TII. Let Bn+1j=Bn+10,j⊕Dn+1j⊂An+1j if An+1j is of type TII; and Bn+1j=An+1j if An+1j is not of type TII. Choose Gn+1=⊕Gn+1j(⊃ϕn,n+1(Gn)) sufficiently large. If An+1j is not of type TII, let Fn+1j=Gn+1j. If An+1j is of type TII, let Fn+1j=ϕn,n+1−,j(Gn).
Note that KK(ψkn,kn+1)=KK(ϕn,n+1). Since the rank of ϕ01(1Akni)=ϕ0(1Akni) is much smaller than the ranks of ϕ1(1Akni,j) and ϕ2(1Akni,j), AffTϕn,n+1 is very close to AffTψkn,kn+1. Hence A′=lim(An,n+1,ϕn,m) has the same Elliott invariant as A=lim(Akn,ψkn,km). Hence A′≅A.
∎
Corollary 3.7
Let A be a simple AH algebra with no dimension growth. And let P1,P2,⋯,Pk∈A be a set of mutually orthogonal projections. Then one can write A as inductive limit A=lim(An,ϕn,m) with mutually orthogonal projections P10,P20,⋯,Pk0∈A1 such that for each i, ϕ1,∞(Pi0)=Pi and
[TABLE]
satisfies the properties of Lemma 3.6.
Proof
In the proof of Lemma 3.6, one can assume that there are P1′,P2′,⋯,Pk′∈A1 with ψ1,∞(Pi′)=Pi for all i=1,2,⋯,k. Then the construction can be carried out to get our conclusion. Note that, applying
Lemma 1.6.8 of [G3], one can strengthen [G3, Theorem 4.37] such that the following is true: For a set of
pre-given orthogonal projections p1,p2,⋯,pk∈An, one can further require that ψ0∈Map(An,Q0AmQ0) satisfies that ψ0(pi) are projections and (ψ0⊕ψ1⊕ψ2)(pi)=ϕn,m(pi) for i=1,2,⋯,k.
∎
The following lemma is the main technique lemma of this section.
Lemma 3.8
Let A,A′ be simple AHD inductive limit algebras with A1⟶ϕ1,2A2⟶ϕ2,3⋯⟶An⋯⟶A and A1′⟶ψ1,2A2′⟶ψ2,3⋯⟶An′⋯⟶A′ being described in Lemma 3.6. Let Λ:A→A′ be a homomorphism. Let F⊂Am be a finite set and ε>0. Then there is an Al′ and a homomorphism Λ1:Am→Al′ such that
[TABLE]
Proof
One can choose n>m large enough such that ϕm,n(F)⊂4εFn and 5εn<8ε. Note that ϕm,n(Am)⊂Bn, where Bn is as in Lemma 3.6. We will construct a homomorphism ϕ:Bn→Al′ such that
[TABLE]
Then the homomorphism ϕ∘ϕm,n is as desired.
Let I_{0}=\big{\{}~{}i~{}|~{}B^{i}_{n}~{}is~{}of~{}type~{}T_{II}\big{\}} and I_{1}=\big{\{}~{}i~{}|~{}B^{i}_{n}~{}is~{}not~{}of~{}type~{}T_{II}\big{\}}. Then \big{\{}\mathbf{1}_{B^{0,i}_{n}}\big{\}}_{i\in I_{0}}\bigcup\big{\{}\mathbf{1}_{D^{i}_{n}}\big{\}}_{i\in I_{0}}\bigcup\big{\{}\mathbf{1}_{B^{i}_{n}}\big{\}}_{i\in I_{1}} are mutually orthogonal projections. Hence \big{\{}\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{B^{0,i}_{n}})\big{\}}_{i\in I_{0}}\bigcup\big{\{}\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{D^{i}_{n}})\big{\}}_{i\in I_{0}}\bigcup\big{\{}\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{B^{i}_{n}})\big{\}}_{i\in I_{1}} are mutually orthogonal projections in A. One can choose n1 (large enough) and mutually orthogonal projections \big{\{}P_{i}\big{\}}_{i\in I_{0}}\bigcup\big{\{}Q_{i}\big{\}}_{i\in I_{0}}\bigcup\big{\{}R_{i}\big{\}}_{i\in I_{1}}\subset A^{{}^{\prime}}_{n_{1}} and a unitary u∈A′ such that ∥u−1∥<16ε and ψn1,∞(Pi)=u∗(Λ∘ϕn,∞(1Bn0,i))u, ψn1,∞(Qi)=u∗(Λ∘ϕn,∞(1Dni))u for i∈I0 and ψn,∞(Ri)=u∗(Λ∘ϕn,∞(1Bni))u for i∈I1.
Note that ∥Adu−id∥<8ε. Replacing Λ:A→A′ by Λ′=Adu∘Λ:A⟶ΛA′⟶AduA′ to make (∗) true, it suffices to construct ϕ:Bn→Al′ for certain l>n1 such that
[TABLE]
Such construction can be carried out for each of the blocks Bn0,i and Dni for i∈I0, Bni for i∈I1. Namely, we need to construct
[TABLE]
[TABLE]
[TABLE]
separately, to satisfy the condition
[TABLE]
for all f\in\big{\{}\pi_{0}(F^{i}_{n})\big{\}}_{i\in I_{0}}\cup\big{\{}\pi_{1}(F^{i}_{n})\big{\}}_{i\in I_{0}}\cup\big{\{}F^{i}_{n}\big{\}}_{i\in I_{1}}.
For the blocks \big{\{}D^{i}_{n}\big{\}}_{i\in I_{0}} and \big{\{}B^{i}_{n}\big{\}}_{i\in I_{1}}, the existence of such homomorphisms follows from the fact that the domain algebras are stably generated. So we only need to construct ϕ:Bn0,i⟶ψn1,l(Pi)Al′ψn1,l(Pi) for l large enough.
Let J_{0}=\big{\{}~{}j~{}|~{}B^{j}_{n+1}~{}is~{}of~{}type~{}T_{II}\big{\}} and J_{1}=\big{\{}~{}j~{}|~{}B^{i}_{n+1}~{}is~{}not~{}of~{}type~{}T_{II}\big{\}}. Let Pi,j=π0(ϕn,n+1i,j(1Bn0,i))∈Bn+10,j for j∈J0, Qi,j=π1(ϕn,n+1i,j(1Bn0,i))∈Dn+1j for j∈J0, Ri,j=π1(ϕn,n+1i,j(1Bn0,i))∈Dn+1j for j∈J1. (Here we only consider the case i∈I0).
As in Lemma 3.6, we have the decomposition π0∘ϕn,n+1i,j(1Bn0,i)=p0+p1∈Bn+10,j and π0∘ϕn,n+1i,j∣Bn0,i=ϕ0⊕ϕ1 with ϕ0∈Hom(Bn0,i,p0Bn+10,jp0)1 and ϕ1∈Hom(Bn0,i,p1Bn+10,jp1)1. Denote p0,p1 by p0i,j and p1i,j, then Pi,j=p0i,j⊕p1i,j. It follows that
[TABLE]
is a set of mutually orthogonal projections with sum to be \Lambda^{{}^{\prime}}\big{(}\phi_{n,\infty}(\mathbf{1}_{B^{0,i}_{n}})\big{)}=\psi_{n_{1},\infty}(P_{i})\in A^{{}^{\prime}}. For n2>n1 (large enough), there are mutually orthogonal projections
[TABLE]
and a unitary v∈ψn,∞(Pi)A′ψn,∞(Pi) such that ∥v−1∥<16ε, and
[TABLE]
for all j∈J0; and
[TABLE]
for all j∈J1.
Let Λ=Adv∘Λ′. Then the construction of ϕ∣Bn0,i satisfying (∗∗) for all f∈π0(F0i) is reduced to the construction of homomorphisms
[TABLE]
for all j∈J0; and
[TABLE]
for all j∈J1, such that
[TABLE]
(Warning: The domain of ξ0j is Bn0,i which is a sub-algebra of Bn but not a sub-algebra of Bn+1. On the other hand, ξ1j(j∈J0) and ξj(j∈J1) are homomorphisms from Qi,jDn+1jQi,j(j∈J0) and Ri,jBn+1jRi,j(j∈J1) which are subalgebras of Bn+1).
The existence of the homomorphisms ξ1j(j∈J0) and ξj(j∈J1) follows from the fact that the corresponding domain algebras Qi,jDn+1jQi,j(j∈J0) and Ri,jBn+1jRi,j(j∈J1) are stably generated of course, we need to choose l>n2 large enough.
So we only need to construct ξ0j to satisfy (∗∗∗) above. Let G⊃G(P) and δ<δ(P) be as in Lemma 3.5 for π0(Fni) and εn (note that ω(π0(F0i))<εn). Recall that L(π0(Fni),εn) is also from Lemma 3.5. Recall that ϕ0∈Hom(Bn0,i,p0i,jBn+1jp0i,j), ϕ1∈Hom(Bn0,i,p1i,jBn+1jp1i,j), and that ϕ1(Bn0,i) is a finite dimensional algebra. There is a homomorphism λ1:ϕ1(Bn0,i)→ψn2,l(P1i,j)Al′ψn2,l(P1i,j) (for l large enough) such that
[TABLE]
Applying Lemma 3.4 to the inductive limit, \lim\big{(}\psi_{n_{2},m}(P^{i,j}_{0})A^{{}^{\prime}}_{m}\psi_{n_{2},m}(P^{i,j}_{0}),\psi_{m,l}\big{)}, the finite set ϕ0(π0(Fni))⊂p0i,jBn+10,jp0i,j and the homomorphism Λ∘ϕn+1,∞:p0i,jBn+10,jp0i,j→ψn2,∞(P0i,j)Am′ψn2,∞(P0i,j), for l(>n2) large enough, one can obtain a ϕ0(G)−δ multiplicative quasi-PK homomorphism λ0:p0i,jBn+10,jp0i,j→ψn2,l(P0i,j)Am′ψn2,l(P0i,j) such that
[TABLE]
Let ξ′=λ0∘ϕ0⊕λ1∘ϕ1:Bn0,i⟶ψn2,l(P0i,j⊕P1i,j)Al′ψn2,l(P0i,j⊕P1i,j). Then
[TABLE]
On the other hand, since [P1i,j]≥L(π0(Fni),εn)⋅[P0i,j] in K-theory, and λ1∘ϕ1 is a homomorphism with finite dimensional image, we know that ξ′=λ0∘ϕ0⊕λ1∘ϕ1 satisfies the condition of Lemma 3.6–note that λ0 is ϕ0(G)−δ multiplicative implies that λ0∘ϕ0 is G−δ multiplicative. By Lemma 3.6, there is a homomorphism ξ0j:Bn0,j⟶ψn2,l(P0i,j⊕P1i,j)Al′ψn2,l(P0i,j⊕P1i,j) such that ∥ξ0j(f)−ξ′(f)∥≤5εn<8ε. Combining with (∗∗∗∗) we know that ξ0j satisfies (∗∗∗) as desired, and therefore the lemma is proved.
∎
Definition 3.9
A C*-algebra A is said to have the ideal property if each closed two-sided ideal in A is generated by its projections.
Remark 3.10
All simple, unital C*-algebras have the ideal property. The direct sum of TAI algebras have the ideal property. The inductive limit of C*-algebras with the ideal property have the ideal property.
The following results is due to Gong, Jiang, Li.
Proposition 3.11
([GJL1]) Suppose that A=lim(An,ϕn,m) and B=lim(Bn,ψn,m) are two (not necessarily unital) AHD inductive limit algebras with the ideal property. Suppose that there is an isomorphism
[TABLE]
which is compatible with Bockstein operations. Suppose that for each projection p∈A and p∈B with α([p])=[p], there exist a unital positive linear isomorphism
[TABLE]
and an isometric group isomorphism
[TABLE]
satisfying the following compatibility conditions:
(1) For each pair of projections p<q<∈A and p<q<∈B with α([p])=[p], α([q])=[q], the diagrams
[TABLE]
and
[TABLE]
commute, where the vertical maps are induced by the inclusion homomorphisms.
(2) The maps α0 and ξp,p are compatible, that is, the diagram
[TABLE]
commutes (this is not an extra requirement, since it follows from the commutativity of the first diagram in
(1) above by [Ji-Jiang]), and then we have the map (still denoted by ξp,p):
[TABLE]
(3) The map ξp,p and γp,p are compatible, that is, the diagram
[TABLE]
commutes.
(4) The map α1:K1(pAp)/torK1(pAp)⟶K1(pBp)/torK1(pBp) (note that α keeps the positive cone of K(A)+ and therefore takes K1(pAp)⊂K1(A) to K1(pBp)⊂K1(B) is compatible with γp,p, that is, the diagram
[TABLE]
commutes.
Then there is an isomorphism Γ:A→B such that
(a)K(Γ)=α, and
(b)If Γp:pAp→Γ(p)BΓ(p) is the restriction of Γ, then AffT(Γp)=ξp,p and Γp♮=γp,p, where [p]=[Γ(p)].
Proposition 3.10
([GJL2, Proposition 2.38]) Let A,B∈HD or AHD be unital C*-algebras. Suppose that K1(A)=tor(K1(A)) and K1(B)=tor(K1(B)). It follows that Inv0(A)≅Inv0(B) implies that Inv(A)≅Inv(B).
4. Main Theorems
Theorem 4.1
If A is an ATAI C*-algebra, then A is an AHD algebra with ideal property.
Proof
By remark 3.10, A has the ideal property. So we will only prove A is an AHD algebra.
Suppose that A is the inductive limit (An=i=1⨁tnAni,ϕn,m), where Ani are simple AHD algebras. Since all Ani are simple, without lose of generality, we can assume that all the homorphisms ϕn,m are injective. We will construct a sequence of sub-C*-algebras Bni⊂Ani which are direct sums of HD building blocks and homomorphisms ψn,n+1:Bn=⨁Bni→Bn+1=⨁Bn+1i such that the diagram
[TABLE]
is approximately commutative in the sense of Elliott and n=1⋃∞ϕn,∞(n(Bn))=A.
Let {an,k}k=1∞ be a dense subset of the unit ball of An and εn=2n1. Let G1⊂A1 be defined by G1=i=1⨁t1{πi(a11)}⊂i=1⨁t1A1i=A1 where π1:A1→A1i are canonical projections.
Fix i∈{1,2,⋯,t1}, one can write A1i=n→∞lim(Cn,ξn,m) as in Lemma 3.6 with injective homomorphisms ξn,m.
For πi(G1)⊂A1i, we choose n large enough such that πi(G1)⊂ε1ξn,∞(Cn). And let B1i=Cn and F1i a finite set with πi(G1)⊂ε1(F1i), where is the inclusion homomorphism ξn,∞. Let B=i=1⨁t1B1i↪i=1⨁t1A1i, and F1=⨁F1i.
The construction will be carried out by induction. Suppose that we have the diagram until Fn⊂Bn↪An⊃Gn. We will construct the next piece of the diagram.
Let Pi,j=ϕn,n+1i,j(1Ani)∈An+1j. Then {Pi,j}i=1tn is a set of mutually orthogonal projections in An+1j. Apply Corollary 3.7 for An+1j in place of A and {Pi,j}i=1tn in places of {P1,P2,⋯,Pk}, we can write An+1j=lim(Ck,λk,l) with Pi,jAn+1jPi,j=klim(Qki,jCkQki,j,λk,l∣Qki,jCkQki,j) (where λk,l(Qki,j)=Qli,j∈Cl) being AHD inductive limit algebra as described in Lemma 3.6. (Warning: Do not confuse these Ck with the Cn in the expression A1i=n→∞lim(Cn,ξn,m).)
For each pair i,j, we apply Lemma 3.8 to the homomorphism ϕn,n+1i,j:Ani→Pi,jAn+1jPi,j in place of Λ:A→A′, and Fni⊂Bni in place of F∈Am, there is an l (large enough) and a homomorphism ψn,n+1i,j:Bni→Qli,jClQli,j such that
[TABLE]
where n+1∣Qli,jClQli,j=λl,∞∣Qli,jClQli,j.
Finally choose G^{j}_{n+1}\supset\pi_{j}\big{[}(G_{n}\cup\{a_{n,n+1}\})\cup\{a_{n+1,j}\}^{n+1}_{j=1}\cup\imath_{n+1}(\bigcup\limits_{k}\psi^{i,j}_{n,n+1}(F^{i}_{n}))\big{]} and Gn+1=⨁Gn+1j. By increasing l, we can assume that there is an Fn+1′j⊂Cl such that Gn+1j⊂εn+1Fn+1′j. Define Bn+1j to be Cl which is a subalgebra of An+1j. Let Fn+1j⊂Bn+1j be defined by F^{j}_{n+1}=F^{{}^{\prime}j}_{n+1}\cup\big{(}\bigoplus\limits_{i}\phi^{i,j}_{n,n+1}(F^{i}_{n})\big{)} and let Fn+1=⨁Fn+1j. This ends our inductive construction. By the Elliott intertwining argument, lim(Bn,ψn,m)=lim(An,ϕn,m). This ends the proof.
∎
Theorem 4.2
Let A=lim(An,ϕn,m) and B=lim(Bn,ψn,m) be inductive limit algebras with Ani, Bni being unital separable nuclear simple TAI algebras with UCT. Suppose that there is an isomorphism
[TABLE]
which is compatible with Bockstein operations. Suppose that for each projection p∈A and p∈B with α([p])=[p], there exist a unital positive linear isomorphism
[TABLE]
and an isometric group isomorphism
[TABLE]
satisfying the following compatibility conditions:
(1) For each pair of projections p<q<∈A and p<q<∈B with α([p])=[p], α([q])=[q], the diagrams
[TABLE]
and
[TABLE]
commute, where the vertical maps are induced by the inclusion homomorphisms.
(2) The maps α0 and ξp,p are compatible, that is, the diagram
[TABLE]
commutes (this is not an extra requirement, since it follows from the commutativity of the first diagram in
(1) above by [Ji-Jiang]), and then we have the map (still denoted by ξp,p):
[TABLE]
(3) The map ξp,p and γp,p are compatible, that is, the diagram
[TABLE]
commutes.
(4) The map α1:K1(pAp)/torK1(pAp)⟶K1(pBp)/torK1(pBp) (note that α keeps the positive cone of K(A)+ and therefore takes K1(pAp)⊂K1(A) to K1(pBp)⊂K1(B) is compatible with γp,p, that is, the diagram
[TABLE]
commutes.
Then there is an isomorphism Γ:A→B such that
(a)K(Γ)=α, and
(b)If Γp:pAp→Γ(p)BΓ(p) is the restriction of Γ, then AffT(Γp)=ξp,p and Γp♮=γp,p, where [p]=[Γ(p)].
Proof
It follows from the fact of Theorem 4.1 that all ATAI algebras are AHD algebras with ideal property and Proposition 3.11 ([GJL1]) that all AHD algebras with ideal property are classified by Inv(A).
∎
Corollary 4.3
(A)([Jiang1], Theorem 3.11) Let A=limn→∞(An=⨁Ani,ϕn,m) and B=limn→∞(Bn=⨁Bni,ψn,m) be inductive limits with Ani, Bni being unital separable nuclear simple TAI algebras with UCT and torsion K1-group. Assume that there is an isomorphism α∈HomΛ(K(A),K(B)) such that
[TABLE]
and for each pair of projections p∈A, q∈B with α([p])=[q], there is a continuous
affine homomorphism
[TABLE]
which is compatible in the sense of 2.19, then there is an isomorphism Λ:A→B which induces α and ξ above.
(B) ([Fa]) Let A and B be two ATAF algebras. Suppose that is an isomorphism of ordered groups
[TABLE]
which preserves the action of the Bockstein operations. Then there is a ∗–isomorphism φ:A→B with φ∗=α.
Proof
(A) It follows from the fact that K1(A)=torK1(A), K1(B)=torK1(B) and Proposition 3.10 ([GJL2, Proposition 2.38]).
(B) From Theorem 4.1, if A is an ATAF C*-algebra, then A is an AHD algebra of real rank zero which is classified in [DG] (Note that AHD algebra in [DG] is denoted by ASH algebra).
∎
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