On the inductive limit of direct sums of simple TAI algebras

TL;DR

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

Classification of all ATAI algebras with the new invariant.

Reduction to classification of AHD algebras with ideal property.

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 $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},\phi_{nm})$, where each $A_{n}^{i}$ 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 $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).

Equations

$$\Vert h(a)-L(a)\Vert \le \epsilon ,\forall a\in F$$

$${\varphi }_{n,m}={\varphi }_{m-1,m}\circ {\varphi }_{m-2,m-1}\circ \cdots \circ {\varphi }_{n,n+1}$$

$$I_{k}=\big{\{}f\in M_{k}(C[0,1])\big{|}f(0)=\lambda\mathbf{1}_{k},f(1)=\mu\mathbf{1}_{k},\lambda,\mu\in\mathbb{C}\big{\}}.$$

$$\underline{f}(t)=\{\begin{array}{ll}\lambda ,& ift=0andf(0)=\lambda 1_k\\ \mu ,& ift=1andf(1)=\mu 1_k\\ f(t),& if0<t<1\end{array}$$

$$\underline{f}(t)=\{\begin{array}{ll}a,& ift=0andf(0)=a\otimes 1_k\\ b,& ift=1andf(1)=b\otimes 1_k\\ f(t),& if0<t<1\end{array}$$

$$P(y)=u\left(\begin{array}[]{cc}\mathbf{1}_{rank(P)}&0\\ 0&0\\ \end{array}\right)u^{*}$$

$$(*)\qquad\qquad\ \ \phi(f)(y)=u\left(\begin{array}[]{ccccc}\underline{f}(t_{1})&&&&\\ &\underline{f}(t_{2})&&&\\ &&\ddots&&\\ &&&\underline{f}(t_{m})&\\ &&&&\mathbf{0}_{n-r}\\ \end{array}\right)u^{*}\in P(y)M_{n}(\mathbb{C})P(y)\subset M_{n}(\mathbb{C})$$

$$t_1=t_2=t_3=0<t_4\le t_5\le \cdots \le t_{m-2}<1=t_{m-1}=t_m,$$

$$Sp\phi_{y}=\big{\{}0^{\sim\frac{3}{k}},t_{4},t_{5},\cdots,t_{m-2},1^{\sim\frac{2}{k}}\big{\}}$$

$$\psi :C[0,1]\hookrightarrow I_k\stackrel{\varphi }{⟶}P{M}_n(C(Y))P,$$

$${\pi }_0=f(0)and{\pi }_1=f(1)$$

$${\pi }_{\underline{0}}(f)=\underline{f}(0)and{\pi }_{\underline{1}}(f)=\underline{f}(1).$$

$$\phi(f)(y)=u\left(\begin{array}[]{ccccc}f(t_{1})&&&&\\ &f(t_{2})&&&\\ &&\ddots&&\\ &&&f(t_{m})&\\ &&&&\mathbf{0}_{n-r}\\ \end{array}\right)u^{*}$$

$$Sp\phi_{y}=\big{\{}0^{\sim\frac{1}{k}},0^{\sim\frac{1}{k}},0^{\sim\frac{1}{k}},t_{4},t_{5},\cdots,t_{m-2},1^{\sim\frac{1}{k}},1^{\sim\frac{1}{k}}\big{\}}=\big{\{}\underline{0},\underline{0},\underline{0},t_{4},t_{5},\cdots,t_{m-2},\underline{1},\underline{1}\big{\}}$$

$$A_1\stackrel{{\varphi }_{1,2}}{⟶}{A}_2\stackrel{{\varphi }_{2,3}}{⟶}\cdots ⟶A_n\cdots ⟶\cdots ,$$

$$\omega (F)=\underset{\pi ,\sigma }{\sup }\underset{u\in U(n)}{\inf }\underset{a\in F}{\max }\Vert u\pi (a)u^{\ast }-\sigma (a)\Vert ,$$

$$K_{\ast }(A,\mathbb{Z}\oplus \mathbb{Z}/k)\cong K_0(A\otimes C(T_{II,k}\times S^1)).$$

$$K_{\ast }(A,\mathbb{Z}\oplus \mathbb{Z}/k{)}^{+}\cong K_0(A\otimes C(T_{II,k}\times S^1){)}^{+}.$$

$$\Gamma :KK(A,B)\to Ho{m}_{\Lambda }(\underline{K}(A),\underline{K}(B))$$

$${\alpha }^i:K_i(A)⟶K_i(B)fori=0,1,and{\alpha }_k^i:K_i(A,\mathbb{Z}/k)⟶K_i(B,\mathbb{Z}/k)fori=0,1,$$

$$KK(A,B{)}_{+}=\{\alpha \in KK(A,B),\alpha (\underline{K}(A{)}_{+})\subseteq \underline{K}(B{)}_{+}\},KK(A,B{)}_{D,+}=\{\alpha \in KK(A,B{)}_{+},\alpha [1_A]\le [1_A]\}.$$

$$K_{(k)}(A)=K_0(A)\oplus K_1(A)\oplus K_0(A,\mathbb{Z}/k)\oplus K_1(A,\mathbb{Z}/k)and{K}_{(k)}(A{)}_{+}=\underline{K}(A{)}_{+}\cap K_{(k)}(A)$$

$$K_0(A\otimes C(W_k\times S^1))=K_0(A)\oplus K_1(A)\oplus K_0(A,\mathbb{Z}/k)\oplus K_1(A,\mathbb{Z}/k)$$

$$\{[P]\in K_0(A)\oplus K_1(A)\oplus K_0(A,\mathbb{Z}/k)\oplus K_1(A,\mathbb{Z}/k)|P\in \mathcal{P}(A)\}\subset \underline{K}(A)$$

$$\sum A=\big{\{}x\in K_{0}(A);0\leq x\leq[\mathbf{1}_{A}]\big{\}}.$$

$${\alpha }_k^i:K_i(A,\mathbb{Z}/k)\to K_i(B,\mathbb{Z}/k);i=0,1,k=0,2,3,4,\cdots$$

$$AffT\varphi :AffTA\to AffTB.$$

$$\begin{array}{ccc}AffT(p_{1}A{p}_1)& \stackrel{{\xi }^{p_1,q_1}}{\to }& AffT(q_{1}B{q}_1)\\ {ı}_T\downarrow & & {ı}_T\downarrow \\ AffT(p_{2}A{p}_2)& \stackrel{{\xi }^{p_2,q_2}}{\to }& AffT(q_{2}B{q}_2)\end{array}$$

$$\alpha:\Big{(}\underline{K}(A),\underline{K}(A)^{+},\sum A\Big{)}\rightarrow\Big{(}\underline{K}(B),\underline{K}(B)^{+},\sum B\Big{)}$$

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 $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},\phi_{nm})$, where each $A_{n}^{i}$ 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 $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ 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 $T_{II,k}$ be the 2-dimensional connected simplicial complex with $H_{1}(T_{II,k})=0$ and $H_{2}(T_{II,k})=\mathbb{Z}/k\mathbb{Z}$, and let $I_{k}$ be the subalgebra of $M_{k}(C[0,1])=C([0,1],M_{k}(\mathbb{C}))$ consisting of all functions $f$ with the properties $f(0)\in\mathbb{C}\mathbf{1}_{k}$ and $f(1)\in\mathbb{C}\mathbf{1}_{k}$ (this algebra is called an Elliott dimension drop interval algebra). Denote $\mathcal{HD}$ the class of algebras consisting of direct sums of building blocks of the forms $M_{l}(I_{k})$ and $PM_{n}(C(X))P$, with $X$ being one of the spaces $\{pt\}$, $[0,1]$, $S^{1}$, and $T_{II,k}$, and with $P\in M_{n}(C(X))$ being a projection. (In [DG], this class is denoted by $SH(2)$, and in [Jiang1], this class is denoted by $\mathcal{B}$). A C*-algebra is called an $\mathcal{AHD}$ algebra, if it is an inductive limit of algebras in $\mathcal{HD}$. In [GJL2], the authors classified all $\mathcal{AHD}$ algebras with the ideal property.

In [Jiang1], Jiang classified ATAI algebras $A$ under the extra restriction that $K_{1}(A)=torK_{1}(A)$. 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 $Inv^{0}(A)$. In this paper, we will use the invariant $Inv^{0}(A)$ together with the Hausdorffized algebraic $K_{1}$-group to deal with the torsion-free part of $K_{1}(A)$, 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 $K_{1}(A)$ is a torsion group, that is, we removed the extra restriction that $K_{1}(A)=torK_{1}(A)$ of Jiang’s classification in [Jiang1] and prove our theorem by reduced to the classification of $\mathcal{AHD}$ 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 $p\preceq q$ if $p$ is equivalent to a projection in $qAq$. We denote by $[p]$ the equivalent class of projections equivalent to $p$. Let $a\in A_{+}$, we write $p\preceq a$ if $p\preceq q$ for some projection $q\in aAa$.

Definition 2.2

Let $G\subset A$ be a finite set and $\delta>0$. We shall say that $\phi\in Map(A,B)$ is $G-\delta$ multiplicative if $\|\phi(ab)-\phi(a)\phi(b)\|<\delta$ for all $a,b\in G$.

We also use $Map_{G-\delta}(A,B)$ to denote all $G-\delta$ multiplicative maps.

If two maps $\varphi,\phi\in Map(A,B)$ satisfy the condition $\|\phi(a)-\psi(a)\|<\delta$ for all $a\in G$, then we will write $\phi\approx_{\delta}\varphi$ on $G$.

If $G,H\subset A$ are two subsets of a C*-algebra $A$ and for any $g\in G$, there is a $b\in H$ with $\|g-b\|<\delta$, then we denote $G\subset_{\delta}H$.

Definition 2.3

[Jiang1] We denote by $\mathcal{I}$ the class of all unital C*-algebras with the form $\bigoplus_{i=1}^{n}B_{i}$, where each $B_{i}\cong M_{k_{i}}$ or $B_{i}\cong M_{k_{i}}(C[0,1])$ for some $k_{(i)}$. Let $A\in\mathcal{I}$, we have the following:

(1) Every C*-algebra in $\mathcal{I}$ is of stable rank one.

(2) Two projections $p$ and $q$ in a C*-algebra $A\in\mathcal{I}$ are equivalent if and only if $\tau(p)=\tau(q)$ for all $\tau\in T(A)$, where $T(A)$ denotes the space of all tracial states.

(3) For any $\varepsilon>0$ and any finite subset $F\subset A$, there exist a number $\delta>0$ and a finite subset $G\subset A$ satisfying the following: If $L:A\rightarrow B$ is a $G-\delta$ multiplicative contractive completely positive linear map, where $B$ is a C*-algebra, then there exists a homomorphism $h:A\rightarrow B$ 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 $F\subset A$ containing a nonzero element $b$, $\varepsilon>0$, integer $n>0$ and any full element $a\in A_{+}$, there exist nonzero projection $p\in A$ and a C*-algebra $I\subset A$ with $I\in\mathcal{I}$ and $1_{I}=p$ such that:

(1) $\|xp-px\|<\varepsilon$ for all $x\in F$;

(2) $pxp\in_{\varepsilon}I$ for all $x\in F$;

(3) $n[1-p]\preceq[p]$ and $1-p\preceq a$.

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 $\lim(A_{n},\phi_{n,m})$, we understand that

[TABLE]

where all $\phi_{n,m}:A_{n}\rightarrow A_{m}$ are homomorphisms.

We shall assume that, for any summand $A^{i}_{n}$ in the direct sum $A_{n}=\bigoplus_{i=1}^{t_{n}}A^{i}_{n}$, necessarily, $\phi_{n,n+1}(\mathbf{1}_{A^{i}_{n}})\neq 0$, since, otherwise, we could simply delete $A^{i}_{n}$ from $A_{n}$, without changing the limit algebra.

If $A_{n}=\bigoplus_{i}A^{i}_{n}$, $A_{m}=\bigoplus_{j}A^{j}_{m}$, we use $\phi^{i,j}_{n,m}$ to denote the partial map of $\phi_{n,m}$ from the i-th block $A^{i}_{n}$ of $A_{n}$ to the j-th block $A^{j}_{m}$ of $A_{m}$. Also, we use $\phi^{-,j}_{n,m}$ to denote the partial map of $\phi_{n,m}$ from $A_{n}$ to $A^{j}_{m}$. That is, $\phi^{-,j}_{n,m}=\bigoplus\limits_{i}\phi^{i,j}_{n,m}=\pi_{j}\phi_{n,m}$, where $\pi_{j}:A_{m}\rightarrow A^{j}_{m}$ is the canonical projection. We also use $\phi^{i,-}_{n,m}$ to denote $\phi_{n,m}|_{A^{i}_{n}}:A^{i}_{n}\rightarrow A_{m}$.

2.8

An AH algebra is a C*-algebra which is $\lim(A_{n}=\bigoplus^{k_{n}}_{i=1}p_{n,i}M_{[n,i]}(C(X_{n,i}))p_{n,i},\phi_{n,m})$, where each $X_{n,i}$ is a compact metrizable space, and $p_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ is a projection. Recall in [G1], Gong proved that a simple AH algebra with uniformly bounded dimension of local spectra, i.e. $\sup_{n,i}\dim(X_{n,i})<\infty$ can be rewritten as AH inductive limit with spaces being $[0,1]$, $S^{1}$, $S^{2}$, $T_{II,k}$, $T_{III,k}$, where each $T_{II,k}$ (or $T_{III,k}$ respectively) is two–dimensional (or three–dimensional respectively) connected simplicial complexes with $H^{2}(T_{II,k})=\mathbb{Z}/k\mathbb{Z}$ and $H^{1}(T_{II,k})=0$ (or with $H^{3}(T_{III,k})=\mathbb{Z}/k\mathbb{Z}$ and $H^{1}(T_{III,k})=0$, $H^{2}(T_{III,k})=0$).

For any positive $k$ the dimension drop interval algebra $I_{k}$ is defined as

[TABLE]

2.9

[GJL1] Let $X$ be a compact space and $\psi:C(X)\rightarrow PM_{k_{1}}(C(Y))P$ $(rank(P)=k)$ be a unital homomorphism. For any point $y\in Y$, there are $k$ mutually orthogonal rank-1 projections $p_{1},p_{2},\cdots,p_{k}$ with $\sum\limits_{i=1}^{k}p_{i}=P(y)$ and \big{\{}x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\big{\}}\subset X (may be repeat) such that $\psi(f)(y)=\sum\limits_{i=1}^{k}f(x_{i}(y))p_{i},\ \forall f\in C(X)$. We denote the set \big{\{}x_{1}(y),x_{2}(y),\cdots,x_{k}(y)\big{\}} (counting multiplicities), by $Sp\psi_{y}$. We shall call $Sp\psi_{y}$ the spectrum of $\psi$ at the point $y$.

For any $f\in I_{k}$, let function $\underline{f}:[0,1]\rightarrow\mathbb{C}\bigsqcup M_{k}(\mathbb{C})$ (disjoint union) be defined by

[TABLE]

That is, $\underline{f}(t)$ is the value of irreducible representation of $f$ corresponding to the point $t$. Similarly, for $f\in M_{l}(I_{k})$, we can define $\underline{f}:[0,1]\longrightarrow M_{l}(\mathbb{C})\bigsqcup M_{lk}(\mathbb{C})$, by

[TABLE]

Suppose that $\phi:I_{k}\rightarrow PM_{n}(C(Y))P$ is a unital homomorphism. Let $r=rank(P)$. For each $y\in Y$, there are $t_{1},t_{2},\cdots,t_{m}\in[0,1]$ and a unitary $u\in M_{n}(\mathbb{C})$ such that

[TABLE]

and

[TABLE]

for all $f\in I_{k}$.

We define the set $Sp\phi_{y}$ to be the points $t_{1},t_{2},\cdots,t_{m}$ with possible fraction multiplicity. If $t_{i}=0$ 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 $t\in(0,1)$, we do not allow the multiplicity of $t$ to be non-integral. Also for [math] or $1$, the multiplicity must be multiple of $\frac{1}{k}$ (other fraction numbers are not allowed).

Let $\psi:C[0,1]\rightarrow PM_{n}(C(Y))P$ be defined by the following composition

[TABLE]

where the first map is the canonical inclusion. Then we have $Sp\psi_{y}=\{Sp\phi_{y}\}^{\sim k}$–that is, for each element $t\in(0,1)$, its multiplicity in $Sp\psi_{y}$ is exactly $k$ times of the multiplicity in $\phi_{y}$.

Recall that for $A=M_{l}(I_{k})$, every point $t\in(0,1)$ corresponds to an irreducible representation $\pi_{t}$, defined by $\pi_{t}(f)=f(t)$. The representations $\pi_{0}$ and $\pi_{1}$ defined by

[TABLE]

are no longer irreducible. We use $\underline{0}$ and $\underline{1}$ to denote the corresponding points for the irreducible representations. That is,

[TABLE]

Or we can also write $\underline{f}(0)\triangleq f(\underline{0})$ and $\underline{f}(1)\triangleq f(\underline{1})$. Then the equation $(*)$ could be written as

[TABLE]

where some of $t_{i}$ may be $\underline{0}$ or $\underline{1}$. 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 $0^{\sim\frac{1}{k}}$ as $\underline{0}$. Then the example of $Sp\phi_{y}$ can be written as

[TABLE]

2.10

We use $\mathcal{HD}$ to denote all C*-algebras $C=\bigoplus C^{i}$, where each $C^{i}$ is of the forms $M_{l}(I_{k})$ or $PM_{n}C(X)P$ with $X$ being one of the spaces $\{pt\}$, $[0,1]$, $S^{1}$, $T_{II,k}$. Each block $C^{i}$ will be called a basic $\mathcal{HD}$ block or a basic building block.

By $\mathcal{AHD}$ algebra, we mean the inductive limit of

[TABLE]

where $A_{n}\in\mathcal{HD}$ for each $n$.

For an $\mathcal{AHD}$ inductive limit $A=\lim(A_{n},\phi_{n,m})$, we write $A_{n}=\bigoplus^{t_{n}}_{i=1}A^{i}_{n}$, where $A^{i}_{n}=P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$ of $A^{i}_{n}=M_{[n,i]}(I_{k_{n,i}})$. For convenience, even for a block $A^{i}_{n}=M_{[n,i]}(I_{k_{n,i}})$, we still use $X_{n,i}$ for $Sp(A^{i}_{n})=[0,1]$–that is, $A^{i}_{n}$ is regarded as a homogeneous algebra or a sub-homogeneous algebra over $X_{n,i}$.

Definition 2.11

([EG2], [DG]) Let $X$ be a compact connected space and let $P\in M_{N}(C(X))$ be a projection of rank $n$. The weak variation of a finite set $F\subset PM_{N}(C(X))P$ is defined by

[TABLE]

where $\pi,\sigma$ run through the set of irreducible representations of $PM_{N}C(X)P$ into $M_{n}(\mathbb{C})$.

For $F\subset M_{r}(I_{k})$, we define $\omega(F)=\omega(\imath(F))$, where $\imath:M_{r}(I_{k})\longrightarrow M_{rk}(C[0,1])$ is the canonical embedding and $\imath(F)$ is regarded as a finite subset of $M_{rk}(C[0,1])$. Let $A$ be a basic $\mathcal{HD}$ block, a finite set $F\subset A$ is said to be weakly approximately constant to within $\varepsilon$ if $\omega(F)<\varepsilon$.

2.12

Let $\underline{K}(A)=K_{*}(A)\oplus\bigoplus_{k=2}^{\infty}K_{*}(A,\mathbb{Z}/k)$ be as in [DG]. Let $\Lambda$ be the Bockstein operation between $\underline{K}(A)$s (see 4.1 of [DG]). It is well known that

[TABLE]

As in [DG], let

[TABLE]

and let $\underline{K}(A)^{+}$ to be the semigroup of $\underline{K}(A)$ generated by $\{K_{*}(A,\mathbb{Z}\oplus\mathbb{Z}/k\mathbb{Z})^{+},k=2,3,\cdots\}$.

2.13

Let $Hom_{\Lambda}(\underline{K}(A),\underline{K}(B))$ be the homomorphism between $\underline{K}(A)$ and $\underline{K}(B)$ compatible with Bockstein operation $\Lambda$. Associativity of Kasparov $KK$-product gives a map

[TABLE]

Recall that every element $\alpha\in KK(A,B)$ defines a map $\alpha_{*}\in Hom_{\Lambda}(\underline{K}(A),\underline{K}(B))$. That is, it gives a sequence of homomorphisms

[TABLE]

which are compatible with the Bockstein operation $\Lambda$.

2.14

Let $\underline{K}(A)_{+}$ be defined in 4.6 of [DG]. Notice that, from [DG, 4.7], $\underline{K}(A)^{+}\subset\underline{K}(A)_{+}$, where for any unital C*-algebra $B$, let

[TABLE]

2.15

Let $A=PM_{\bullet}C(T_{II,k})P$, where we use $\bullet$ to denote any possible positive integers. From 5.14 of [G3], we know that an element $\alpha\in KK(A,B)$ is completely determined by $\alpha^{0}:K_{0}(C(T_{II,n}))\longrightarrow K_{0}(B)$ and $\alpha^{1}_{k}:K_{1}(C(T_{II,k}),\mathbb{Z}/k)\longrightarrow K_{1}(B,\mathbb{Z}/k)$, $k=0,1,2,\cdots.$

For any positive integer $k\geq 2$, denote

[TABLE]

Then for $A=PM_{\bullet}C(T_{II,k})P$, an element $\alpha\in KK(A,B)$ is in $KK(A,B)_{+}$ if and only if $\alpha(K_{(k)}(A)_{+})\subset K_{(k)}(B)_{+}$, $k=0,1$. 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 $W_{k}=T_{II,k}$.

One can choose finite set $\mathcal{P}\subset M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$ of projections such that

[TABLE]

generate $K_{(k)}(A)=K_{0}(A)\oplus K_{1}(A)\oplus K_{0}(A,\mathbb{Z}/k)\oplus K_{1}(A,\mathbb{Z}/k)$, where $W_{k}=T_{II,k}$.

If we choose finite set $G(\mathcal{P})\subset A$ large enough and $\delta(\mathcal{P})>0$ small enough, then every $G(\mathcal{P})-\delta(\mathcal{P})$ multiplicative contraction $\phi:A\rightarrow B$ determines a map $\phi_{*}:\mathcal{P}\underline{K}(A)\rightarrow\underline{K}(B)$ which is compatible with the Bockstein operation $\Lambda$ (see [GL]). If $A=PM_{\bullet}C(T_{II,k})P$, then it also defines a $KK$ element $[\phi]\in KK(A,B)$.

Let $A=PM_{\bullet}C(T_{II,k})P$, recall that for $G\supseteq G(\mathcal{P})$, $\delta\leq\delta(\mathcal{P})$, a $G-\delta$ multiplicative map $\phi:A\rightarrow B$ is called a quasi $\mathcal{P}\underline{K}$ homomorphism, if there is a homomorphism $\psi:A\rightarrow B$ satisfying $\phi_{*}=\psi_{*}:\mathcal{P}\underline{K}(A)\rightarrow\underline{K}(B)$. If $\phi$ is a quasi $\mathcal{P}\underline{K}$ homomorphism, then $[\phi]\in KK(A,B)_{+}$.

Suppose that $A=PM_{\bullet}C(T_{II,k})P$ and suppose that $B=\bigoplus\limits_{i=1}^{n}B^{i}$ is a direct sum of basic $\mathcal{HD}$ bulding blocks. Let $\alpha\in KK(A,B)$, then $\alpha\in KK(A,B)_{+}$ if and only if for each $i\in\{1,2,\cdots,n\}$, $\alpha^{i}\in KK(A,B^{i})$ is either zero or $\alpha^{i}(\mathbf{1}_{C(T_{II,k})})>0$ in $K_{0}(B^{i})$. Recall that in [DG], a $KK$ element $\alpha\in KK(A,B=\bigoplus\limits_{i=1}^{n}B^{i})$ is called m-large if $rank(\alpha^{i}(\mathbf{1}_{A}))\geq m\cdot rank(\mathbf{1}_{A})$, for all $i\in\{1,2,\cdots,n\}$. By 5.5 and 5.6 of [DG] if $\alpha\in KK(A,B)_{+}$ is 6-large and $\alpha([\mathbf{1}_{A}])\leq[\mathbf{1}_{B}]$, then $\alpha$ can be realized by a homomorphism from $A$ to $B$.

2.17

Recall that the scale of A, denoted by $\sum A$, is a subset of $K_{0}(A)$ consisting of $[p]$, where $p\in A$ is a projection. As all the C*-algebras $A$ in this paper have cancellation of projections, if $A$ has unit $\mathbf{1}_{A}$, then

[TABLE]

For two C*-algebras $A,B$, by a ”homomorphism” $\alpha$ from $(\underline{K}(A),\underline{K}(A)^{+},\sum A)$ to $(\underline{K}(B),\underline{K}(B)^{+},\sum B)$, it means a system of maps

[TABLE]

which are compatible with Bockstein operation and $\alpha=\bigoplus_{k,i}\alpha_{k}^{i}$ satisfies $\alpha(\underline{K}(A)^{+})\subseteq\underline{K}(B)^{+}$ and finally $\alpha_{0}^{0}(\sum A)\subseteq\sum B$.

2.18

For a unital C*-algebra $A$, let $TA$ denote the space of tracial states of $A$, i.e. $\tau\in TA$, if and only if $\tau$ is a positive linear map from $A$ to the complex plane $\mathbb{C}$, with $\tau(xy)=\tau(yx)$ and $\tau(\mathbf{1}_{A})=1$. $AffTA$ is the Banach space of all the continuous affine maps from $TA$ to $\mathbb{C}$. (In most references, $AffTA$ is defined to be the set of all the affine maps from $TA$ to $\mathbb{R}$. Our $AffTA$ is a complexification of the standard $AffTA$.) The element $\mathbf{1}\in AffTA$, defined by $\mathbf{1}(\tau)=1$, for all $\tau\in TA$, is called the unit of $AffTA$. $AffTA$, together with the positive cone $AffTA_{+}$ and the unit element $\mathbf{1}$ forms a scaled ordered complex Banach space. (Notice that for any element $x\in AffTA$, there are $x_{1},x_{2},x_{3},x_{4}\in AffTA_{+}$ such that $x=x_{1}-x_{2}+ix_{3}-ix_{4}$.)

There is a natural homomorphism $\rho_{A}:K_{0}(A)\rightarrow AffTA$ defined by $\rho_{A}([p])(\tau)=\sum_{i=1}^{n}\tau(p_{ii})$ for $\tau\in TA$ and $[p]\in K_{0}(A)$ represented by projection $p=(p_{i,j})\in M_{n}(A)$.

Any unital homomorphism $\phi:A\rightarrow B$ induces a continuous affine map $T\phi:TB\rightarrow TA$, It turns out that $T\phi$ induces a unital positive linear map

[TABLE]

If $\phi:A\rightarrow B$ is not unital, we still have the positive linear map

[TABLE]

but it will not preserve the unit $\mathbf{1}$, only has property $AffT\phi(\mathbf{1}_{AffTA})\leq\mathbf{1}_{AffTB}$.

2.19

If $\alpha:(\underline{K}(A),\underline{K}(A)^{+},\sum A)\rightarrow(\underline{K}(B),\underline{K}(B)^{+},\sum B)$ is a homomorphism as in 2.17, then for each projection $p\in A$, there is a projection $q\in B$ such that $\alpha[p]=[q]$.

Notice that for all the C*-algebras $A$ considered in this paper, the following is true: if $p_{1}$, $p_{2}$ are projections and $[p_{1}]=[p_{2}]$ in $K_{0}(A)$, then there is $u\in A$ with $up_{1}u^{*}=p_{2}$. Therefore both $T(pAp)$ and $T(qBq)$ depend only on the class $[p]\in K_{0}(A)$ and $[q]\in K_{0}(B)$. For the classes $[p]\in\sum A(\subset K_{0}(A))$, $T(pAp)$ is taken as part of the invariant $Inv^{0}(A)$. For two classes $[p]\in\sum A$, $[q]\in\sum B$, with $\alpha([p])=[q]$, we will consider the system of continuous affine maps $\xi_{p,q}:T(qBq)\rightarrow T(pAp)$. Such system of maps is said to be compatible if for any two projections $p_{1}\leq p_{2}$ with $\alpha([p_{1}])=[q_{1}]$, $\alpha([p_{2}])=[q_{2}]$ and $q_{1}\leq q_{2}$, the following diagram is commutative:

[TABLE]

where the horizontal maps are induced by $\xi^{p_{1},q_{1}}$ and $\xi^{p_{2},q_{2}}$ respectively, and the vertical maps are induced by the inclusions $p_{1}Ap_{1}\rightarrow p_{2}Ap_{2}$, $q_{1}Bq_{1}\rightarrow q_{2}Bq_{2}$.

2.20

We denote \big{(}\underline{K}(A),\underline{K}(A)^{+},\sum A,\{T(pAp)\}_{[p]\in\sum A}\big{)} by $Inv^{0}(A)$. 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 $\xi^{p,q}:T(qBq)\rightarrow T(pAp)$ 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 $U_{0}(A)$ denote the connected component of $\mathbf{1}_{A}$ in $U(A)$. Let $DU(A)$ and $DU_{0}(A)$ denote the commutator subgroups of $U(A)$ and $U_{0}(A)$, respectively. (Recall that the commutator subgroup of a group $G$ is the subgroup generated by all elements of the form $aba^{-1}b^{-1}$, where $a,b\in G$.) One introduces the following metric $D_{A}$ on $U(A)/\overline{DU(A)}$ (see [NT,§3]). For $u,v\in U(A)/\overline{DU(A)}$

[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 $u,v\in U(A)/\overline{DU(A)}$.

Denote the extended commutator group $DU^{+}(A)$, which is generated by $DU(A)\subset U(A)$ 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 $\widetilde{DU(A)}$ denote the closure of $DU^{+}(A)$. That is, $\widetilde{DU(A)}=\overline{DU^{+}(A)}$

2.22

Let $A$ be a unital C*-algebra. Let $AffTA$ and $\rho_{A}:K_{0}(A)\rightarrow AffTA$ be as defined as in 2.18.

For simplicity, we will use $\rho K_{0}(A)$ to denote the set $\rho_{A}(K_{0}(A))$. The metric $d_{A}$ on $AffTA/\overline{\rho K_{0}(A)}$ is defined as follows (see [NT, §3]).

Let $d^{{}^{\prime}}$ denote the quotient metric on $AffTA/\overline{\rho K_{0}(A)}$, i.e, for $f,g\in AffTA/\overline{\rho K_{0}(A)}$,

[TABLE]

Define $d_{A}$ by

[TABLE]

Obviously, $d_{A}(f,g)\leq 2\pi d^{{}^{\prime}}(f,g)$.

Let $\widetilde{\rho K_{0}(A)}$ denote the closed real vector space spanned by $\overline{\rho K_{0}(A)}$. That is,

[TABLE]

For any $u,v\in U(A)/\widetilde{DU(A)}$, define

[TABLE]

Let $\widetilde{d}^{{}^{\prime}}$ denote the quotient metric on $AffTA/\widetilde{\rho K_{0}(A)}$, that is,

[TABLE]

Define $\widetilde{d}_{A}$ by

[TABLE]

2.23

Let

[TABLE]

and for simpler, denote $\widetilde{\rho K_{0}}(A)\triangleq\widetilde{\rho K_{0}(A)}$, $\widetilde{DU}(A)=\widetilde{DU(A)}$, $\widetilde{SU}(A)=\widetilde{SU(A)}$.

A unital homomorphism $\phi:A\rightarrow B$ induces a contractive group homomorphism

[TABLE]

If $\phi$ is not unital, then the map $\phi^{\natural}:U(A)/\widetilde{SU}(A)\longrightarrow U(\phi(\mathbf{1}_{A})B\phi(\mathbf{1}_{A}))/\widetilde{SU}(\phi(\mathbf{1}_{A})B\phi(\mathbf{1}_{A}))$ is induced by the corresponding unital homomorphism. In this case, $\phi$ also induces the map $\imath_{*}\circ\phi^{\natural}:U(A)/\widetilde{SU}(A)\longrightarrow U(B)/\widetilde{SU}(B)$, which is denoted by $\phi_{*}$ 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 $([p],[\overline{p}])\in\sum A\times\sum B$ with $\alpha([p])=[\overline{p}]$, 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 $\xi^{p,\overline{p}}$ induces a map (still denoted by $\xi^{p,\overline{p}}$):

[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 $\alpha_{1}$ is induced by $\alpha$.

We will denote the map from $Inv(A)$ to $Inv(B)$ by

[TABLE]

Note that $Inv^{0}$ is part of $Inv(A)$.

3. Local approximation lemma

Proposition 3.1

Let $A=PM_{\bullet}C(T_{II,k})P$ or $M_{l}(I_{k})$ and $\mathcal{P}$ be as in 2.16. For any finite set $F\subset A$, $\varepsilon>0$, there exist a finite set $G\subset A$ ($G\supset G(\mathcal{P})$ large enough), a positive number $\delta>0$($\delta<\delta(\mathcal{P})$ small enough) such that the following statement is true:

If $B\in\mathcal{HD}$, $\phi,\psi\in Map(A,B)$ are $G-\delta$ multiplicative and $\phi_{*}=\psi_{*}:\mathcal{P}\underline{K}(A)\rightarrow\underline{K}(B)$, then there is a homomorphism $\nu\in Hom(A,M_{L}(B))$ defined by point evaluations and there is a unitary $u\in M_{L+1}(B)$ such that $\|(\phi\oplus\nu)(a)-u(\psi\oplus\nu)(a)u^{*}\|<\varepsilon$ for all $a\in F$

Proof

For the special case that $A=PM_{\bullet}C(T_{II,k})P$ 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 $K_{0}(\Pi^{\infty}_{n=1}B_{n}/\bigoplus^{\infty}_{n=1}B_{n})$, $K_{0}(\Pi^{\infty}_{n=1}B_{n}/\bigoplus^{\infty}_{n=1}B_{n},\mathbb{Z}/k)$ and $K_{1}(\Pi^{\infty}_{n=1}B_{n}/\bigoplus^{\infty}_{n=1}B_{n},\mathbb{Z}/k)$ in 5.12 of [G3] works well for the case that some of $B_{n}$ being of the form $M_{l}(I_{k})$.

∎

Lemma 3.2

Let $A=PM_{\bullet}(C(X))P$ or $M_{l}(I_{k})$. Let $F\subset A$ be approximately constant to within $\varepsilon$ (i.e., $\omega(F)<\varepsilon$). Then for any two homomorphisms $\phi,\psi:A\rightarrow B$ defined by point evaluations with $K_{0}\phi=K_{0}\psi$, there exists a unitary $u\in B$ such that $\|\phi(f)-u\psi(f)u^{*}\|<2\varepsilon$ $\forall f\in F$.

Proof

The case $A=PM_{\bullet}(C(X))P$ is Lemma 3.5 of [GJLP2]. For the case $A=M_{l}(I_{k})$, one can also find a homomorphism $\phi^{{}^{\prime}}:M_{l}(\mathbb{C})\rightarrow B$ such that

[TABLE]

where $f(\underline{0})$ 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 $A=PM_{\bullet}(C(X))P$ or $M_{l}(I_{k})$, $\varepsilon>0$, finite set $F\subset A$ with $\omega(A)<\varepsilon$. There exist a finite set $G\subset A$ ($G\supset G(\mathcal{P})$), a number $\delta>0$ ($\delta<\delta(\mathcal{P})$)and a positive integer $L$ such that the following statement is true. If $B\in\mathcal{HD}$, and $\phi,\psi\in Map(A,B)_{1}$ are $G-\delta$ multiplicative and $\phi_{*}=\psi_{*}:\mathcal{P}\underline{K}(A)\rightarrow\underline{K}(B)$ and $\nu:A\rightarrow M_{\infty}(B)$ is a homomorphism defined by point evaluations with $\nu([\mathbf{1}_{A}])\geq L\cdot[\mathbf{1}_{B}]\in K_{0}(B)$, then there is a unitary $u\in(\mathbf{1}_{B}\oplus\nu(\mathbf{1}_{A}))M_{\infty}(B)(\mathbf{1}_{B}\oplus\nu(\mathbf{1}_{A}))$ such that

[TABLE]

Proof

Let $L_{1}$ be as in Proposition 3.1 and $L=2L_{1}$. Since $\nu([\mathbf{1}_{A}])\geq L\cdot[\mathbf{1}_{B}]=2L_{1}\cdot[\mathbf{1}_{B}]\in K_{0}(B)$, there is a projection $Q<\nu(\mathbf{1}_{A})$ such that $Q$ is equivalent to $\mathbf{1}_{M_{L_{1}}(B)}$. By Proposition 3.1, there exist a homomorphism $\nu_{1}:A\rightarrow QM_{\infty}(B)Q$ defined by point evaluation and a unitary $w\in(\mathbf{1}_{B}\oplus\nu_{1}(\mathbf{1}_{A}))M_{\infty}(B)(\mathbf{1}_{B}\oplus\nu_{1}(\mathbf{1}_{A}))$ such that $\|(\phi\oplus\nu_{1})(f)-w(\psi\oplus\nu_{1})(f)w^{*}\|<\varepsilon\quad\forall f\in F.$

Since $\nu_{1}$ (and $\nu$ respectively) is homotopic to a homomorphism factoring through $M_{rank(P)}(\mathbb{C})$ (for $A=PM_{\bullet}(C(X))P$) or factoring through $M_{l}(\mathbb{C})$ (for $A=M_{l}(I_{k})$), there is a unital homomorphism $\nu_{2}:A\rightarrow(\nu(\mathbf{1}_{A})-\nu_{1}(\mathbf{1}_{A}))M_{\infty}(B)(\nu(\mathbf{1}_{A})-\nu_{1}(\mathbf{1}_{A}))$ such that $K_{0}(\nu_{1}\oplus\nu_{2})=K_{0}(\nu)$.

By Lemma 3.2, $\nu$ is approximately unitarily equivalent to $\nu_{1}\oplus\nu_{2}$ to within $2\varepsilon$ on $F$. Hence $\phi\oplus\nu$ is approximately unitarily equivalent to $\psi\oplus\nu$ on $F$ to within $2\varepsilon+\varepsilon+2\varepsilon=5\varepsilon$.

∎

Lemma 3.4

Let $A=PM_{\bullet}C(T_{II,k})P$, and $\mathcal{P}\subset M_{\bullet}(A\otimes C(W_{k}\times S^{1}))$ as in 2.16. And let $B=\lim\limits_{n\rightarrow\infty}(B_{n},\psi_{nm})$ be a unital simple inductive limit of direct sums of $\mathcal{HD}$ building blocks. Let $\phi:A\rightarrow B$ be a unital homomorphism. It follows that for any $G\supset G(\mathcal{P})$ and $\delta<\delta(\mathcal{P})$, there is a $B_{n}$ and a unital $G-\delta$ multiplicative contraction $\phi_{1}:A\rightarrow B_{n}$ which is a quasi $\mathcal{P}\underline{K}$-homomorphism such that $\|\psi_{n,\infty}\circ\phi_{1}(g)-\phi(g)\|<\delta$

Proof

There is a finite set $G_{1}$ and $\delta_{1}>0$ such that if a complete positive linear map $\psi:A\rightarrow C$ ($C$ is a C*-algebra) and a homomorphism $\phi:A\rightarrow C$ satisfy that $\|\phi(g)-\psi(g)\|<\delta_{1}$ for all $g\in G_{1}$, then $\psi$ is $G-\frac{\delta}{2}$ multiplicative (see Lemma 4.40 in [G3]). Now if $\phi^{{}^{\prime}}:A\rightarrow B_{l}$ satisfies $\|\psi_{l,\infty}\circ\phi^{{}^{\prime}}(g)-\phi(g)\|<\delta$ for all $g\in G_{1}$, then $\psi_{l,\infty}\circ\phi^{{}^{\prime}}$ is $G-\frac{\delta}{2}$ multiplicative. Hence for some $n>l$ (large enough), $\phi_{1}=\psi_{i,n}\circ\phi^{{}^{\prime}}:A\rightarrow B_{n}$ is $G-\delta$ multiplicative. By replacing $G$ by $G\cup G_{1}$ and $\delta$ by $min(\delta,\delta_{1})$, we only need to construct a unital quasi $\mathcal{P}\underline{K}$-homomorsim $\phi_{1}:A\rightarrow B_{n}$ such that

[TABLE]

Furthermore, if $\phi_{1}$ is a $G-\delta$ 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 $m>m_{1}$, $KK(\psi_{n,m}\circ\phi_{1})$ is 6-large and therefore can be realized by a homomorphism. That is, replacing $\phi_{1}$ by $\psi_{n,m}\circ\phi_{1}$, we get a quasi $\mathcal{P}\underline{K}$ -homomorphism. Thus the proof of the lemma is reduced to the construction of $\phi_{1}$ to satisfy $(*)$.

Since $B=\lim\limits_{n\rightarrow\infty}(B_{n},\psi_{nm})$, there is a $B_{n}$ and finite set $F\subset B_{n}$ such that $G\subset_{\frac{\delta}{3}}\psi_{n,\infty}(F)$. Since $B_{n}(\subset B)$ is a nuclear C*-algebra, there are two complete positive contractions $\lambda_{1}:B_{n}\rightarrow M_{N}(\mathbb{C})$ and $\lambda_{2}:M_{N}(\mathbb{C})\rightarrow B_{n}$ such that

[TABLE]

Since $B_{n}$ is a subalgebra of $B$, by Arverson s Extension Theorem, one can extend the map $\lambda_{1}:B_{n}\rightarrow M_{N}(\mathbb{C})$ to $\beta_{1}:B\rightarrow M_{N}(\mathbb{C})$ such that $\beta_{1}\circ\psi_{n,\infty}=\lambda_{1}$.

One can verify that $\phi_{1}=\lambda_{2}\circ\beta_{1}\circ\phi:A\rightarrow B_{n}$ satisfies the condition $(*)$ as below. For $g\in G$, there is an $f\in F$ such that $\|\phi(g)-\psi_{n,\infty}(f)\|<\frac{\delta}{3}$, and therefore

[TABLE]

This ends the proof of the lemma.

∎

Lemma 3.5

For any finite set $F\subset PM_{\bullet}C(T_{II,k})P\triangleq B$ with $\omega(F)<\varepsilon$, there are a finite set $G\supset G(\mathcal{P})$, a positive number $\delta<\delta(\mathcal{P})$ and a positive integer $L$ (this $L$ will be denoted by $L(F,\varepsilon)$ later), such that if $C$ is a $\mathcal{HD}$ basic building block, $p,q\in C$ are two projections with $p+q=\mathbf{1}_{C}$, and $\phi_{0}:B\rightarrow pCp$ and $\varphi_{1}:B\rightarrow qCq$ are two maps satisfying the following conditions:

(1) $\phi_{0}$ is quasi $\mathcal{P}\underline{K}$ homomorphism and $G-\delta$ multiplicative, and $\phi_{1}$ is defined by point evaluations (or equivalently, factoring through a finite dimensional C*-algebra), and

(2) $rank(q)\geq L~{}rank(p)$,

then there is a homomorphism $\phi:B\rightarrow C$ such that $\|\phi_{0}\oplus\phi_{1}(f)-\phi(f)\|<5\varepsilon$ for all $f\in F$.

Proof

Since $\phi_{0}$ is a quasi $\mathcal{P}\underline{K}$-homomorphism and is $G-\delta$ multiplicative, there is a homomorphism $\phi_{0}^{{}^{\prime}}:B\rightarrow pCp$ such that $\phi^{{}^{\prime}}_{0*}=\phi_{0*}:\mathcal{P}\underline{K}(B)\rightarrow\underline{K}(C)$. By Lemma 3.3, there is unitary $u\in C$ such that

[TABLE]

The homomorphism $\phi=Adu^{*}\circ(\phi_{0}^{{}^{\prime}}\oplus\phi_{1})$ is as desired.

∎

Lemma 3.6

Let $\varepsilon_{1}>\varepsilon_{2}>\cdots>\varepsilon_{n}>\cdots$ be a sequence with $\sum\varepsilon_{i}<+\infty$. Let $A$ be a simple AH algebra with no dimension growth (as in [EGL] and [Li4]). Then $A$ can be written as an $\mathcal{AHD}$ inductive limit

[TABLE]

with the unital subalgebras $B_{n}(=\bigoplus B^{i}_{n})\subset A_{n}(=\bigoplus A^{i}_{n})$ (that is, $B^{i}_{n}\subset A^{i}_{n}$) and $F_{n}\subset B_{n}$ and $G_{n}\subset A_{n}$ with $F_{n}=\bigoplus F^{i}_{n}\subset G_{n}=\bigoplus G^{i}_{n}$ such that the following statements hold:

(1) If $A^{i}_{n}$ is not of type $T_{II}$, then $B^{i}_{n}=A^{i}_{n}$ and $F^{i}_{n}=F^{i}_{n}$. If $A^{i}_{n}$ is of type $T_{II}$, then $B^{i}_{n}=PA^{i}_{n}P\oplus D^{i}_{n}\subset A^{i}_{n}$ with $F^{i}_{n}=\pi_{0}(F^{i}_{n})\oplus\pi_{1}(F^{i}_{n})\subset G^{i}_{n}$ and $\omega(\pi_{0}(F^{i}_{n}))<\varepsilon$, where $D^{i}_{n}$ is a direct sum of the $\mathcal{HD}$ building blocks other than type $T_{II}$, and $\pi_{0}:B^{i}_{n}\rightarrow PA^{i}_{n}P\triangleq B^{0,i}_{n}$ and $\pi_{1}:B^{i}_{n}\rightarrow D^{i}_{n}$ are canonical projections;

(2) $G^{i}_{n}$ generates $A^{i}_{n}$, $\phi_{n,n+1}(A_{n})\subset B_{n+1}$, $\phi_{n,n+1}(G_{n})\subset F_{n+1}$, and $\overline{\bigcup\limits_{n=1}^{\infty}\phi_{n,\infty}(G_{n})}=\overline{\bigcup\limits_{n=1}^{\infty}\phi_{n,\infty}(F_{n})}=$unit ball of $A$;

(3) Suppose that both $A^{i}_{n}$ and $A^{j}_{n+1}$ are of type $T_{II}$ and $\phi\triangleq\pi_{0}\circ\phi^{i,j}_{n,n+1}|_{B^{0,i}_{n}}:B^{0,i}_{n}\rightarrow B^{0,j}_{n+1}$. Then $\phi(\mathbf{1}_{B^{0,i}_{n}})=p_{0}\oplus p_{1}\in B^{0,j}_{n+1}$ and $\phi=\phi_{0}\oplus\phi_{1}$ with $\phi_{0}\in Hom(B^{0,i}_{n},p_{0}B^{0,j}_{n+1}p_{0})_{1}$, $\phi_{1}\in Hom(B^{0,i}_{n},p_{1}B^{0,j}_{n+1}p_{1})_{1}$ such that $\phi_{1}$ is defined by point evaluations (or equivalently, $\phi_{1}(B^{0,i}_{n})$ is a finite dimensioned sub-algebra of $p_{1}B^{0,j}_{n+1}p_{1}$) and

[TABLE]

where $L(\pi_{0}(F^{i}_{n}),\varepsilon_{n})$ is as in Lemma 3.5 (note that $\omega(\pi_{0}(F^{i}_{n}))<\varepsilon_{n}$).

Proof

From [Li4], we know that $A$ can be written as an inductive limit of the direct sums of $\mathcal{HD}$ building blocks $A=\lim\limits_{n\rightarrow\infty}(\widetilde{A_{n}},\psi_{nm})$. In our construction, we will choose $A_{n}=\widetilde{A_{k_{n}}}$ and homomorphisms $\phi_{n,n+1}:A_{n}=\widetilde{A_{k_{n}}}\rightarrow A_{n+1}=\widetilde{A_{k_{n+1}}}$ satisfying $KK(\phi_{n,n+1})=KK(\psi_{k_{n},k_{n+1}})$ and $AffT\phi_{n,n+1}$ is close to $AffT\psi_{k_{n},k_{n+1}}$ within any pregiven small number on a pregiven finite set, in such a way that $\phi_{n,n+1}$ also satisfies the desired condition as in the lemma with certain choices of subalgebras $B_{n}\subset A_{n}$ and and finite subsets $F_{n}\subset B_{n}$ and $G_{n}\subset A_{n}$.

Suppose that we already have $A_{n}=\widetilde{A_{k_{n}}}$ with the unital sub-algebra $B_{n}=\oplus_{i}B^{i}_{n}=\oplus_{i}(B^{0,i}_{n}\oplus D^{i}_{n})\subset A_{n}=\oplus_{i}A^{i}_{n}$, and two subsets $F_{n}\subset B_{n}$, $G_{n}\subset A_{n}$. as required in the lemma. We will construct, roughly, the next algebra $A_{n+1}=\widetilde{A_{k_{n+1}}}$, the sub-algebra $B_{n+1}\subset A_{n+1}$, two subsets $F_{n+1}\subset B_{n+1}$, $G_{n+1}\subset A_{n+1}$, and the homomorphism $\phi_{n,n+1}:A_{n}\rightarrow A_{n+1}$ 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 $l>k_{n}$ large enough, and for each block $A_{n}^{i}=\widetilde{A^{i}_{k_{n}}}$ of type $T_{II}$, the homomorphism $\psi^{i,j}_{k_{n},l}$ can be decomposed into three parts $\phi^{1}_{0}\oplus\phi_{1}\oplus\phi_{2}$, roughly described as below: There are mutually orthogonal projections $Q_{0}$,$Q_{1}$,$Q_{2}\in\widetilde{A^{j}_{l}}$ with $\psi^{i,j}_{k_{n},l}(\mathbf{1}_{A^{i}_{n}})=Q_{0}+Q_{1}+Q_{2}$, there are two homomorphisms $\phi_{k}\in Hom(A^{i}_{n},Q_{k}\widetilde{A^{j}_{l}}Q_{k})_{1}$, $(k=1,2)$ and a sufficient multiplicative quasi $\mathcal{P}\underline{K}$ homomorphism $\phi_{k}\in Map(A^{i}_{n},Q_{0}\widetilde{A^{j}_{l}}Q_{0})_{1}$ possessing the following properties:

(i) $\phi^{1}_{0}\oplus\phi_{1}\oplus\phi_{2}$ is close to $\psi^{i,j}_{k_{n},l}$ to within $\varepsilon_{n}$ on $G_{n}^{i}$;

(ii) $\phi^{1}_{0}(\mathbf{1}_{B^{0,i}_{n}})$ is a projection and

[TABLE]

(iii) $\phi_{1}$ is defined by point evaluation at a dense enough finite subset of $Sp(\widetilde{A^{i}_{k_{n}}})$ such that for any homomorphism $\phi_{0}\in Hom(A^{i}_{n},Q_{0}\widetilde{A^{j}_{l}}Q_{0})_{1}$, we have $\omega(\phi_{0}\oplus\phi_{1}(G^{i}_{n}))<\varepsilon_{n+1}$;

(iv) $\phi_{2}$ factors through an interval algebra $D_{1}^{i}$ as $\phi_{2}=\xi_{2}^{i}\circ\xi_{1}^{i}:A^{i}_{n}\stackrel{{\scriptstyle\xi_{1}^{i}}}{{\longrightarrow}}D_{1}^{i}\stackrel{{\scriptstyle\xi_{2}^{i}}}{{\longrightarrow}}\widetilde{A^{j}_{l}}$.

Then we can choose $k_{n+1}=l$ and $A_{n+1}=\widetilde{A_{k_{n+1}}}$. Note that $\phi_{0}^{1}$ is a quasi $\mathcal{P}\underline{K}$ homomorphism, one can choose $\phi_{0}:\widetilde{A^{i}_{k_{n}}}\rightarrow\phi_{0}^{1}(\mathbf{1}_{\widetilde{A^{i}_{k_{n}}}})\widetilde{A^{i}_{k_{n+1}}}\phi_{0}^{1}(\mathbf{1}_{\widetilde{A^{i}_{k_{n}}}})$ such that $KK(\phi^{1}_{0}\oplus\phi_{1}\oplus\phi_{2})=KK(\phi_{0}\oplus\phi_{1}\oplus\phi_{2})$ with $\phi^{1}_{0}(\mathbf{1}_{B^{0,i}_{n}})=\phi_{0}(\mathbf{1}_{B^{0,i}_{n}})$. Modify $\psi_{k_{n},k_{n+1}}$ by replacing $\psi^{i,j}_{k_{n},k_{n+1}}$ by $\phi_{0}\oplus\phi_{1}\oplus\phi_{2}$ to define $\phi_{n,n+1}:A_{n}\rightarrow A_{n+1}$.

Fix $j$ with $A^{j}_{n+1}=A^{j}_{k_{n+1}}$ being of type $T_{II}$. Let $I_{0}=\{~{}i~{}|~{}A^{i}_{n}=\widetilde{A^{j}_{k_{n}}}~{}is~{}of~{}type~{}T_{II}\}$, $I_{1}=\{~{}i~{}|~{}A^{i}_{n}=\widetilde{A^{j}_{k_{n}}}~{}is~{}not~{}of~{}type~{}T_{II}\}$. Let $P_{i}=(\phi_{0}\oplus\phi_{1})(\mathbf{1}_{A^{i}_{n}})\in A^{j}_{n+1}$, let $P=\bigoplus\limits_{i\in I_{0}}P_{i}\in A_{n+1}^{j}$ and $B^{0,j}_{n+1}\triangleq PA^{j}_{n+1}P$, and let $D^{j}_{n+1}=\bigoplus\limits_{i\in I_{0}}\xi^{i}_{2}(D_{1}^{i})\oplus\bigoplus\limits_{i\in I_{1}}\widetilde{\phi^{i,j}_{k_{n},k_{n+1}}}(\widetilde{A^{j}_{k_{n}}})\subset\widetilde{A^{j}_{k_{n+1}}}$ which is a direct sum of $\mathcal{HD}$ building blocks other than type $T_{II}$. Let $B^{j}_{n+1}=B^{0,j}_{n+1}\oplus D^{j}_{n+1}\subset A^{j}_{n+1}$ if $A^{j}_{n+1}$ is of type $T_{II}$; and $B^{j}_{n+1}=A^{j}_{n+1}$ if $A^{j}_{n+1}$ is not of type $T_{II}$. Choose $G_{n+1}=\oplus G^{j}_{n+1}(\supset\phi_{n,n+1}(G_{n}))$ sufficiently large. If $A^{j}_{n+1}$ is not of type $T_{II}$, let $F^{j}_{n+1}=G^{j}_{n+1}$. If $A^{j}_{n+1}$ is of type $T_{II}$, let $F^{j}_{n+1}=\phi^{-,j}_{n,n+1}(G_{n})$.

Note that $KK(\psi_{k_{n},k_{n+1}})=KK(\phi_{n,n+1})$. Since the rank of $\phi^{1}_{0}(\mathbf{1}_{\widetilde{A^{i}_{k_{n}}}})=\phi_{0}(\mathbf{1}_{\widetilde{A^{i}_{k_{n}}}})$ is much smaller than the ranks of $\phi_{1}(\mathbf{1}_{\widetilde{A^{i,j}_{k_{n}}}})$ and $\phi_{2}(\mathbf{1}_{\widetilde{A^{i,j}_{k_{n}}}})$, $AffT\phi_{n,n+1}$ is very close to $AffT\psi_{k_{n},k_{n+1}}$. Hence $A^{{}^{\prime}}=lim(A_{n,n+1},\phi_{n,m})$ has the same Elliott invariant as $A=lim(\widetilde{A_{k_{n}}},\psi_{k_{n},k_{m}})$. Hence $A^{{}^{\prime}}\cong A$.

∎

Corollary 3.7

Let $A$ be a simple $AH$ algebra with no dimension growth. And let $P_{1},P_{2},\cdots,P_{k}\in A$ be a set of mutually orthogonal projections. Then one can write A as inductive limit $A=\lim(A_{n},\phi_{n,m})$ with mutually orthogonal projections $P^{0}_{1},P^{0}_{2},\cdots,P^{0}_{k}\in A_{1}$ such that for each $i$, $\phi_{1,\infty}(P^{0}_{i})=P_{i}$ and

[TABLE]

satisfies the properties of Lemma 3.6.

Proof

In the proof of Lemma 3.6, one can assume that there are $P^{{}^{\prime}}_{1},P^{{}^{\prime}}_{2},\cdots,P^{{}^{\prime}}_{k}\in\widetilde{A_{1}}$ with $\psi_{1,\infty}(P^{{}^{\prime}}_{i})=P_{i}$ for all $i=1,2,\cdots,k$. 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 $p_{1},p_{2},\cdots,p_{k}\in A_{n}$, one can further require that $\psi_{0}\in Map(A_{n},Q_{0}A_{m}Q_{0})$ satisfies that $\psi_{0}(p_{i})$ are projections and $(\psi_{0}\oplus\psi_{1}\oplus\psi_{2})(p_{i})=\phi_{n,m}(p_{i})$ for $i=1,2,\cdots,k$.

∎

The following lemma is the main technique lemma of this section.

Lemma 3.8

Let $A$,$A^{{}^{\prime}}$ be simple $\mathcal{AHD}$ inductive limit algebras with $A_{1}\stackrel{{\scriptstyle\phi_{1,2}}}{{\longrightarrow}}A_{2}\stackrel{{\scriptstyle\phi_{2,3}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle}}{{\longrightarrow}}A_{n}\cdots\stackrel{{\scriptstyle}}{{\longrightarrow}}A$ and $A^{{}^{\prime}}_{1}\stackrel{{\scriptstyle\psi_{1,2}}}{{\longrightarrow}}A^{{}^{\prime}}_{2}\stackrel{{\scriptstyle\psi_{2,3}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle}}{{\longrightarrow}}A^{{}^{\prime}}_{n}\cdots\stackrel{{\scriptstyle}}{{\longrightarrow}}A^{{}^{\prime}}$ being described in Lemma 3.6. Let $\Lambda:A\rightarrow A^{{}^{\prime}}$ be a homomorphism. Let $F\subset A_{m}$ be a finite set and $\varepsilon>0$. Then there is an $A^{{}^{\prime}}_{l}$ and a homomorphism $\Lambda_{1}:A_{m}\rightarrow A^{{}^{\prime}}_{l}$ such that

[TABLE]

Proof

One can choose $n>m$ large enough such that $\phi_{m,n}(F)\subset_{\frac{\varepsilon}{4}}F_{n}$ and $5\varepsilon_{n}<\frac{\varepsilon}{8}$. Note that $\phi_{m,n}(A_{m})\subset B_{n}$, where $B_{n}$ is as in Lemma 3.6. We will construct a homomorphism $\phi:B_{n}\rightarrow A^{{}^{\prime}}_{l}$ such that

[TABLE]

Then the homomorphism $\phi\circ\phi_{m,n}$ 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 $n_{1}$ (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 $u\in A^{{}^{\prime}}$ such that $\|u-1\|<\frac{\varepsilon}{16}$ and $\psi_{n_{1},\infty}(P_{i})=u^{*}(\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{B^{0,i}_{n}}))u$,   $\psi_{n_{1},\infty}(Q_{i})=u^{*}(\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{D^{i}_{n}}))u$ for $i\in I_{0}$ and $\psi_{n,\infty}(R_{i})=u^{*}(\Lambda\circ\phi_{n,\infty}(\mathbf{1}_{B^{i}_{n}}))u$ for $i\in I_{1}$.

Note that $\|Adu-id\|<\frac{\varepsilon}{8}$. Replacing $\Lambda:A\rightarrow A^{{}^{\prime}}$ by $\Lambda^{{}^{\prime}}=Adu\circ\Lambda:A\stackrel{{\scriptstyle\Lambda}}{{\longrightarrow}}A^{{}^{\prime}}\stackrel{{\scriptstyle Adu}}{{\longrightarrow}}A^{{}^{\prime}}$ to make $(*)$ true, it suffices to construct $\phi:B_{n}\rightarrow A^{{}^{\prime}}_{l}$ for certain $l>n_{1}$ such that

[TABLE]

Such construction can be carried out for each of the blocks $B^{0,i}_{n}$ and $D^{i}_{n}$ for $i\in I_{0}$, $B^{i}_{n}$ for $i\in I_{1}$. 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 $\phi:B^{0,i}_{n}\longrightarrow\psi_{n_{1},l}(P_{i})A^{{}^{\prime}}_{l}\psi_{n_{1},l}(P_{i})$ 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 $\widetilde{P}^{i,j}=\pi_{0}(\phi^{i,j}_{n,n+1}(\mathbf{1}_{B^{0,i}_{n}}))\in B^{0,j}_{n+1}$ for $j\in J_{0}$, $\widetilde{Q}^{i,j}=\pi_{1}(\phi^{i,j}_{n,n+1}(\mathbf{1}_{B^{0,i}_{n}}))\in D^{j}_{n+1}$ for $j\in J_{0}$, $\widetilde{R}^{i,j}=\pi_{1}(\phi^{i,j}_{n,n+1}(\mathbf{1}_{B^{0,i}_{n}}))\in D^{j}_{n+1}$ for $j\in J_{1}$. (Here we only consider the case $i\in I_{0}$).

As in Lemma 3.6, we have the decomposition $\pi_{0}\circ\phi^{i,j}_{n,n+1}(\mathbf{1}_{B^{0,i}_{n}})=p_{0}+p_{1}\in B^{0,j}_{n+1}$ and $\pi_{0}\circ\phi^{i,j}_{n,n+1}|_{B^{0,i}_{n}}=\phi_{0}\oplus\phi_{1}$ with $\phi_{0}\in Hom(B^{0,i}_{n},p_{0}B^{0,j}_{n+1}p_{0})_{1}$ and $\phi_{1}\in Hom(B^{0,i}_{n},p_{1}B^{0,j}_{n+1}p_{1})_{1}$. Denote $p_{0},p_{1}$ by $p^{i,j}_{0}$ and $p^{i,j}_{1}$, then $\widetilde{P}^{i,j}=p^{i,j}_{0}\oplus p^{i,j}_{1}$. 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 $n_{2}>n_{1}$ (large enough), there are mutually orthogonal projections

[TABLE]

and a unitary $v\in\psi_{n,\infty}(P_{i})A^{{}^{\prime}}\psi_{n,\infty}(P_{i})$ such that $\|v-1\|<\frac{\varepsilon}{16}$, and

[TABLE]

for all $j\in J_{0}$; and

[TABLE]

for all $j\in J_{1}$.

Let $\widetilde{\Lambda}=Adv\circ\Lambda^{{}^{\prime}}$. Then the construction of $\phi|_{B^{0,i}_{n}}$ satisfying $(**)$ for all $f\in\pi_{0}(F_{0}^{i})$ is reduced to the construction of homomorphisms

[TABLE]

for all $j\in J_{0}$; and

[TABLE]

for all $j\in J_{1}$, such that

[TABLE]

(Warning: The domain of $\xi_{0}^{j}$ is $B^{0,i}_{n}$ which is a sub-algebra of $B_{n}$ but not a sub-algebra of $B_{n+1}$. On the other hand, $\xi^{j}_{1}(j\in J_{0})$ and $\xi^{j}(j\in J_{1})$ are homomorphisms from $\widetilde{Q}^{i,j}D^{j}_{n+1}\widetilde{Q}^{i,j}(j\in J_{0})$ and $\widetilde{R}^{i,j}B^{j}_{n+1}\widetilde{R}^{i,j}(j\in J_{1})$ which are subalgebras of $B_{n+1}$).

The existence of the homomorphisms $\xi^{j}_{1}(j\in J_{0})$ and $\xi^{j}(j\in J_{1})$ follows from the fact that the corresponding domain algebras $\widetilde{Q}^{i,j}D^{j}_{n+1}\widetilde{Q}^{i,j}(j\in J_{0})$ and $\widetilde{R}^{i,j}B^{j}_{n+1}\widetilde{R}^{i,j}(j\in J_{1})$ are stably generated of course, we need to choose $l>n_{2}$ large enough.

So we only need to construct $\xi^{j}_{0}$ to satisfy $(***)$ above. Let $G\supset G(\mathcal{P})$ and $\delta<\delta(\mathcal{P})$ be as in Lemma 3.5 for $\pi_{0}(F_{n}^{i})$ and $\varepsilon_{n}$ (note that $\omega(\pi_{0}(F_{0}^{i}))<\varepsilon_{n}$). Recall that $L(\pi_{0}(F^{i}_{n}),\varepsilon_{n})$ is also from Lemma 3.5. Recall that $\phi_{0}\in Hom(B^{0,i}_{n},p^{i,j}_{0}B^{j}_{n+1}p^{i,j}_{0})$, $\phi_{1}\in Hom(B^{0,i}_{n},p^{i,j}_{1}B^{j}_{n+1}p^{i,j}_{1})$, and that $\phi_{1}(B^{0,i}_{n})$ is a finite dimensional algebra. There is a homomorphism $\lambda_{1}:\phi_{1}(B^{0,i}_{n})\rightarrow\psi_{n_{2},l}(P^{i,j}_{1})A^{{}^{\prime}}_{l}\psi_{n_{2},l}(P^{i,j}_{1})$ (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 $\phi_{0}(\pi_{0}(F_{n}^{i}))\subset p^{i,j}_{0}B^{0,j}_{n+1}p^{i,j}_{0}$ and the homomorphism $\widetilde{\Lambda}\circ\phi_{n+1,\infty}:p^{i,j}_{0}B^{0,j}_{n+1}p^{i,j}_{0}\rightarrow\psi_{n_{2},\infty}(P^{i,j}_{0})A^{{}^{\prime}}_{m}\psi_{n_{2},\infty}(P^{i,j}_{0})$, for $l(>n_{2})$ large enough, one can obtain a $\phi_{0}(G)-\delta$ multiplicative quasi-$\mathcal{P}\underline{K}$ homomorphism $\lambda_{0}:p^{i,j}_{0}B^{0,j}_{n+1}p^{i,j}_{0}\rightarrow\psi_{n_{2},l}(P^{i,j}_{0})A^{{}^{\prime}}_{m}\psi_{n_{2},l}(P^{i,j}_{0})$ such that

[TABLE]

Let $\xi^{{}^{\prime}}=\lambda_{0}\circ\phi_{0}\oplus\lambda_{1}\circ\phi_{1}:B^{0,i}_{n}\longrightarrow\psi_{n_{2},l}(P^{i,j}_{0}\oplus P^{i,j}_{1})A^{{}^{\prime}}_{l}\psi_{n_{2},l}(P^{i,j}_{0}\oplus P^{i,j}_{1})$. Then

[TABLE]

On the other hand, since $[P^{i,j}_{1}]\geq L(\pi_{0}(F^{i}_{n}),\varepsilon_{n})\cdot[P^{i,j}_{0}]$ in K-theory, and $\lambda_{1}\circ\phi_{1}$ is a homomorphism with finite dimensional image, we know that $\xi^{{}^{\prime}}=\lambda_{0}\circ\phi_{0}\oplus\lambda_{1}\circ\phi_{1}$ satisfies the condition of Lemma 3.6–note that $\lambda_{0}$ is $\phi_{0}(G)-\delta$ multiplicative implies that $\lambda_{0}\circ\phi_{0}$ is $G-\delta$ multiplicative. By Lemma 3.6, there is a homomorphism $\xi^{j}_{0}:B^{0,j}_{n}\longrightarrow\psi_{n_{2},l}(P^{i,j}_{0}\oplus P^{i,j}_{1})A^{{}^{\prime}}_{l}\psi_{n_{2},l}(P^{i,j}_{0}\oplus P^{i,j}_{1})$ such that $\|\xi_{0}^{j}(f)-\xi^{{}^{\prime}}(f)\|\leq 5\varepsilon_{n}<\frac{\varepsilon}{8}$. Combining with $(****)$ we know that $\xi^{j}_{0}$ 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 $A=\lim(A_{n},\phi_{n,m})$ and $B=\lim(B_{n},\psi_{n,m})$ are two (not necessarily unital) $\mathcal{AHD}$ 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 $p\in A$ and $\overline{p}\in B$ with $\alpha([p])=[\overline{p}]$, 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 $p<q<\in A$ and $\overline{p}<\overline{q}<\in B$ with $\alpha([p])=[\overline{p}]$, $\alpha([q])=[\overline{q}]$, the diagrams

[TABLE]

and

[TABLE]

commute, where the vertical maps are induced by the inclusion homomorphisms.

(2) The maps $\alpha_{0}$ and $\xi^{p,\overline{p}}$ 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 $\xi^{p,\overline{p}}$):

[TABLE]

(3) The map $\xi^{p,\overline{p}}$ and $\gamma^{p,\overline{p}}$ are compatible, that is, the diagram

[TABLE]

commutes.

(4) The map $\alpha_{1}:K_{1}(pAp)/torK_{1}(pAp)\longrightarrow K_{1}(\overline{p}B\overline{p})/torK_{1}(\overline{p}B\overline{p})$ (note that $\alpha$ keeps the positive cone of $\underline{K}(A)^{+}$ and therefore takes $K_{1}(pAp)\subset K_{1}(A)$ to $K_{1}(\overline{p}B\overline{p})\subset K_{1}(B)$ is compatible with $\gamma^{p,\overline{p}}$, that is, the diagram

[TABLE]

commutes.

Then there is an isomorphism $\Gamma:A\rightarrow B$ such that

(a)$K(\Gamma)=\alpha$, and

(b)If $\Gamma_{p}:pAp\rightarrow\Gamma(p)B\Gamma(p)$ is the restriction of $\Gamma$, then $AffT(\Gamma_{p})=\xi^{p,\overline{p}}$ and $\Gamma^{\natural}_{p}=\gamma^{p,\overline{p}}$, where $[\overline{p}]=[\Gamma(p)]$.

Proposition 3.10

([GJL2, Proposition 2.38]) Let $A,B\in\mathcal{HD}$ or $\mathcal{AHD}$ be unital C*-algebras. Suppose that $K_{1}(A)=tor(K_{1}(A))$ and $K_{1}(B)=tor(K_{1}(B))$. It follows that $Inv^{0}(A)\cong Inv^{0}(B)$ implies that $Inv(A)\cong Inv(B)$.

4. Main Theorems

Theorem 4.1

If $A$ is an ATAI C*-algebra, then $A$ is an $\mathcal{AHD}$ algebra with ideal property.

Proof

By remark 3.10, $A$ has the ideal property. So we will only prove $A$ is an $\mathcal{AHD}$ algebra.

Suppose that $A$ is the inductive limit $(A_{n}=\bigoplus\limits_{i=1}^{t_{n}}A_{n}^{i},\phi_{n,m})$, where $A^{i}_{n}$ are simple $\mathcal{AHD}$ algebras. Since all $A^{i}_{n}$ are simple, without lose of generality, we can assume that all the homorphisms $\phi_{n,m}$ are injective. We will construct a sequence of sub-C*-algebras $B^{i}_{n}\subset A^{i}_{n}$ which are direct sums of $\mathcal{HD}$ building blocks and homomorphisms $\psi_{n,n+1}:B_{n}=\bigoplus B_{n}^{i}\rightarrow B_{n+1}=\bigoplus B_{n+1}^{i}$ such that the diagram

[TABLE]

is approximately commutative in the sense of Elliott and $\overline{\bigcup\limits_{n=1}^{\infty}\phi_{n,\infty}(\imath_{n}(B_{n}))}=A$.

Let $\{a_{n,k}\}^{\infty}_{k=1}$ be a dense subset of the unit ball of $A_{n}$ and $\varepsilon_{n}=\frac{1}{2^{n}}$. Let $G_{1}\subset A_{1}$ be defined by $G_{1}=\bigoplus\limits_{i=1}^{t_{1}}\{\pi_{i}(a_{11})\}\subset\bigoplus\limits_{i=1}^{t_{1}}A_{1}^{i}=A_{1}$ where $\pi_{1}:A_{1}\rightarrow A^{i}_{1}$ are canonical projections.

Fix $i\in\{1,2,\cdots,t_{1}\}$, one can write $A^{i}_{1}=\lim\limits_{n\rightarrow\infty}(C_{n},\xi_{n,m})$ as in Lemma 3.6 with injective homomorphisms $\xi_{n,m}$.

For $\pi_{i}(G_{1})\subset A^{i}_{1}$, we choose $n$ large enough such that $\pi_{i}(G_{1})\subset_{\varepsilon_{1}}\xi_{n,\infty}(C_{n})$. And let $B^{i}_{1}=C_{n}$ and $F^{i}_{1}$ a finite set with $\pi_{i}(G_{1})\subset_{\varepsilon_{1}}\imath(F_{1}^{i})$, where $\imath$ is the inclusion homomorphism $\xi_{n,\infty}$. Let $B=\bigoplus\limits_{i=1}^{t_{1}}B_{1}^{i}\hookrightarrow\bigoplus\limits_{i=1}^{t_{1}}A_{1}^{i}$, and $F_{1}=\bigoplus F_{1}^{i}$.

We will construct the following diagram

[TABLE]

[TABLE]

[TABLE]

such that

(1) $G_{k}\subset_{\varepsilon_{k}}\imath_{k}(F_{k})$, $G_{k}\supset\phi_{k-1,k}(G_{k-1}\cup\{a_{k-1,k}\})\cup\{a_{k,j}\}^{k}_{j=1}$;

(2) $\|\imath_{k}\circ\psi_{k-1,k}(f)-\phi_{k-1,k}\circ\imath_{k-1}(f)\|<\varepsilon_{k-1},\qquad\forall f\in F_{k-1}$.

The construction will be carried out by induction. Suppose that we have the diagram until $F_{n}\subset B_{n}\hookrightarrow A_{n}\supset G_{n}$. We will construct the next piece of the diagram.

Let $P^{i,j}=\phi^{i,j}_{n,n+1}(\mathbf{1}_{A^{i}_{n}})\in A^{j}_{n+1}$. Then $\{P^{i,j}\}^{t_{n}}_{i=1}$ is a set of mutually orthogonal projections in $A^{j}_{n+1}$. Apply Corollary 3.7 for $A^{j}_{n+1}$ in place of $A$ and $\{P^{i,j}\}^{t_{n}}_{i=1}$ in places of $\{P_{1},P_{2},\cdots,P_{k}\}$, we can write $A^{j}_{n+1}=\lim(C_{k},\lambda_{k,l})$ with $P^{i,j}A^{j}_{n+1}P^{i,j}=\lim\limits_{k}(Q^{i,j}_{k}C_{k}Q^{i,j}_{k},\lambda_{k,l}|_{Q^{i,j}_{k}C_{k}Q^{i,j}_{k}})$ (where $\lambda_{k,l}(Q^{i,j}_{k})=Q^{i,j}_{l}\in C_{l}$) being $\mathcal{AHD}$ inductive limit algebra as described in Lemma 3.6. (Warning: Do not confuse these $C_{k}$ with the $C_{n}$ in the expression $A^{i}_{1}=\lim\limits_{n\rightarrow\infty}(C_{n},\xi_{n,m})$.)

For each pair $i,j$, we apply Lemma 3.8 to the homomorphism $\phi^{i,j}_{n,n+1}:A^{i}_{n}\rightarrow P^{i,j}A^{j}_{n+1}P^{i,j}$ in place of $\Lambda:A\rightarrow A^{{}^{\prime}}$, and $F^{i}_{n}\subset B^{i}_{n}$ in place of $F\in A_{m}$, there is an $l$ (large enough) and a homomorphism $\psi^{i,j}_{n,n+1}:B_{n}^{i}\rightarrow Q^{i,j}_{l}C_{l}Q^{i,j}_{l}$ such that

[TABLE]

where $\imath_{n+1}|_{Q^{i,j}_{l}C_{l}Q^{i,j}_{l}}=\lambda_{l,\infty}|_{Q^{i,j}_{l}C_{l}Q^{i,j}_{l}}$.

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 $G_{n+1}=\bigoplus G^{j}_{n+1}$. By increasing $l$, we can assume that there is an $F^{{}^{\prime}j}_{n+1}\subset C_{l}$ such that $G_{n+1}^{j}\subset_{\varepsilon_{n+1}}F^{{}^{\prime}j}_{n+1}$. Define $B^{j}_{n+1}$ to be $C_{l}$ which is a subalgebra of $A^{j}_{n+1}$. Let $F^{j}_{n+1}\subset B^{j}_{n+1}$ 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 $F_{n+1}=\bigoplus F^{j}_{n+1}$. This ends our inductive construction. By the Elliott intertwining argument, $\lim(B_{n},\psi_{n,m})=\lim(A_{n},\phi_{n,m})$. This ends the proof.

∎

Theorem 4.2

Let $A=\lim(A_{n},\phi_{n,m})$ and $B=\lim(B_{n},\psi_{n,m})$ be inductive limit algebras with $A_{n}^{i}$, $B^{i}_{n}$ 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 $p\in A$ and $\overline{p}\in B$ with $\alpha([p])=[\overline{p}]$, 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 $p<q<\in A$ and $\overline{p}<\overline{q}<\in B$ with $\alpha([p])=[\overline{p}]$, $\alpha([q])=[\overline{q}]$, the diagrams

[TABLE]

and

[TABLE]

commute, where the vertical maps are induced by the inclusion homomorphisms.

(2) The maps $\alpha_{0}$ and $\xi^{p,\overline{p}}$ 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 $\xi^{p,\overline{p}}$):

[TABLE]

(3) The map $\xi^{p,\overline{p}}$ and $\gamma^{p,\overline{p}}$ are compatible, that is, the diagram

[TABLE]

commutes.

(4) The map $\alpha_{1}:K_{1}(pAp)/torK_{1}(pAp)\longrightarrow K_{1}(\overline{p}B\overline{p})/torK_{1}(\overline{p}B\overline{p})$ (note that $\alpha$ keeps the positive cone of $\underline{K}(A)^{+}$ and therefore takes $K_{1}(pAp)\subset K_{1}(A)$ to $K_{1}(\overline{p}B\overline{p})\subset K_{1}(B)$ is compatible with $\gamma^{p,\overline{p}}$, that is, the diagram

[TABLE]

commutes.

Then there is an isomorphism $\Gamma:A\rightarrow B$ such that

(a)$K(\Gamma)=\alpha$, and

(b)If $\Gamma_{p}:pAp\rightarrow\Gamma(p)B\Gamma(p)$ is the restriction of $\Gamma$, then $AffT(\Gamma_{p})=\xi^{p,\overline{p}}$ and $\Gamma^{\natural}_{p}=\gamma^{p,\overline{p}}$, where $[\overline{p}]=[\Gamma(p)]$.

Proof

It follows from the fact of Theorem 4.1 that all ATAI algebras are $\mathcal{AHD}$ algebras with ideal property and Proposition 3.11 ([GJL1]) that all $\mathcal{AHD}$ algebras with ideal property are classified by $Inv(A)$.

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Corollary 4.3

(A)([Jiang1], Theorem 3.11) Let $A=\lim_{n\rightarrow\infty}(A_{n}=\bigoplus A_{n}^{i},\phi_{n,m})$ and $B=\lim_{n\rightarrow\infty}(B_{n}=\bigoplus B_{n}^{i},\psi_{n,m})$ be inductive limits with $A_{n}^{i}$, $B_{n}^{i}$ being unital separable nuclear simple TAI algebras with UCT and torsion $K_{1}$-group. Assume that there is an isomorphism $\alpha\in Hom_{\Lambda}(\underline{K}(A),\underline{K}(B))$ such that

[TABLE]

and for each pair of projections $p\in A$, $q\in B$ with $\alpha([p])=[q]$, there is a continuous affine homomorphism

[TABLE]

which is compatible in the sense of 2.19, then there is an isomorphism $\Lambda:A\rightarrow B$ which induces $\alpha$ and $\xi$ 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 $\varphi:A\rightarrow B$ with $\varphi_{*}=\alpha$.

Proof

(A) It follows from the fact that $K_{1}(A)=torK_{1}(A)$, $K_{1}(B)=torK_{1}(B)$ and Proposition 3.10 ([GJL2, Proposition 2.38]).

(B) From Theorem 4.1, if $A$ is an ATAF C*-algebra, then $A$ is an $\mathcal{AHD}$ algebra of real rank zero which is classified in [DG] (Note that $\mathcal{AHD}$ algebra in [DG] is denoted by ASH algebra).

∎