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\begin{center}
{\footnotesize Available at:
{\tt http://publications.ictp.it}}\hfill IC/2010/024\\
\vspace{1cm}
United Nations Educational, Scientific and Cultural Organization\\
and\\
International Atomic Energy Agency\\
\medskip
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS\\
\vspace{2.5cm}
{\bf ON DIMENSION THEORY FOR A CERTAIN CLASS \\
OF SIMPLE AH ALGEBRAS}\\
\vspace{2cm}
Toan M. Ho\footnote{hmtoan@math.ac.vn}\\
{\it Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Ha Noi, Vietnam\\
and\\
The Abdus Salam International Centre for Theoretical Physics,
Trieste, Italy.}
\end{center}
\vspace{1cm}
\centerline{\bf Abstract}
\baselineskip=18pt
\bigskip
A class of unital diagonal AH algebras will be studied in this paper. The density property of the set of all elements which are nilpotent up to (left and right multiple) unitaries is presented. As a consequence, these algebras have stable rank one. Section 3 also shows that an algebra in this class has the property LP (i.e., the linear span of projections is dense) provided a certain condition. Finally, restricting our attention to a special subclass which includes Villadsen algebras of the first type, we give the necessary and sufficient condition of real rank zero.
\vfill
\begin{center}
MIRAMARE -- TRIESTE\\
June 2010\\
\end{center}
\vfill
%\subjclass[2000]{Primary 46L05; secondary 46L80.}
%\keywords{ diagonal AH-algebras, stable rank, real rank, linear span of projections}
\newpage
\section{Introduction}
The classification program of Elliott, the goal of which is to classify amenable $C^*$-algebras by their K-theoretical data, has been successful for many classes of $C^*$-algebras, in particular for simple AH algebras with slow dimension growth (\cite{Ell1}, \cite{Ell}, \cite{EG},\cite{EGL}, \cite{lin}). Note that all these simple AH algebras have stable rank one (\cite{dnnp}, \cite{bdr}). Therefore, the AH algebras which may be of most interest nowadays are the ones with higher dimension growth (than zero).
In \cite{Goodearl}, Goodearl introduced an interesting class of simple AH algebras without any condition on the dimension growth as follows. Let $X$ be a nonempty separable compact Hausdorff space, and choose a subset $S = \{x_1,x_2,x_3, \ldots\}$ such that $\{x_i, x_{i+1}, \ldots\}$ is dense in $X$ for every positive integer $i$. Let $k_1, k_2, \ldots$ be a sequence of positive integers such that $k_i | k_{i+1}$ for all $i$ and denote by $A_i$ the matrix algebra $M_{k_i}(C(X))$. Define unital block diagonal $\ast$-homomorphisms $\phi_i : A_{i} \rightarrow A_{i+1}$ by
\[\phi_i (a) = \textrm{diag} (a, \ldots, a, a(x_i), \ldots, a(x_i)), \ \forall a \in A_i \]
Let $\alpha_i$ be the number copies of $a$ appearing on the right side and we also assume that $k_{i+1}/k_i > \alpha_i >0$ (i.e., there is at least one block $a$ and at least one block $a(x_i)$ in $\phi_i(a)$). Goodearl showed that such an algebra has stable rank one and the property SP. Moreover, it has real rank zero if, and only if either $\lim_{i\rightarrow \infty} \omega_{1,j}$ is zero or $X$ is totally disconnected (\cite{Goodearl}).
%A class of algebras which generalizes Goodearl's algebras, including examples of Villadsen (\cite{jes1}) and Villadsen algebras of the first type (\cite{tom}, \cite{TW}) will be studied in this paper.
Let $X$ and $Y$ be compact metric spaces. A $\ast$-homomorphism $\phi$ from $M_{n}(C(X))$ to $M_{nm}(C(Y))$ is said to be {\it diagonal} if there are continuous maps $\la_i$ from $Y$ to $X$ such that
\[\phi (f) = \left( \begin{array}{cccc}
f \circ \la_1 & 0 & \ldots &0 \\
0 & f \circ \la_2 & \ldots &0\\
\vdots & \vdots &\ddots & \vdots\\
0 & 0 & \ldots & f \circ \la_n
\end{array} \right), \quad \forall f \in M_{n}(C(X)). \]
The $\la_i$ are called the {\it eigenvalue maps} (or simply {\it eigenvalues}) of $\phi$. The family $ \{ \la_1, \la_2, \ldots, \la_m \} $ is called {\it the eigenvalue pattern} of $\phi$ and is denoted by $ep(\phi)$. In addition, let $P$, respectively $Q$, be projections in $M_n(C(X))$ and $M_{nm}(C(Y))$. An $\ast$-homomorphism $\psi$ from $PM_{n}(C(X))P$ to $QM_{nm}(C(Y))Q$ is called diagonal if there exists a diagonal $\ast$-homomorphism $\phi$ from $M_{n}(C(X))$ to $M_{nm}(C(Y))$ such that $\psi$ is reduced from $\phi$ on $PM_n(C(X))P$ and $\phi (P) = Q$. This definition can also be extended to $\ast$-homomorphisms
\[ \phi : \oplus_{i=1}^n P_i M_{n_i}(C(X_i))P_i \rightarrow \oplus_{j=1}^m Q_j M_{m_j}(C(Y_j))Q_j \]
by requiring that each partial map
\[ \phi^{i,j}: P_i M_{n_i}(C(X_i))P_i \rightarrow Q_j M_{m_j}(C(Y_j))Q_j \]
induced by $\phi$ be diagonal. The $C^*$-algebras which can be represented by inductive limits of homogeneous $C^*$-algebras is called an AH algebra. An AH algebra is called {\it diagonal} if it can be written as an inductive limit of homogeneous $C^*$-algebras with diagonal connecting maps.
There are many concepts which are borrowed from dimension theory of topological spaces, such as stable rank (\cite{rie}), real rank (\cite{bp}), covering dimension (\cite{winter}). In this paper, we will study stable rank, real rank and some relative (weaker or stronger) properties (approximately nilpotent elements, the property SP) of a class of diagonal AH-algebras which can be written as inductive limits of homogeneous algebras of the forms
$\oplus_{i=1}^n P_i M_{n_i}(C(X_i))P_i$ {\it without any condition on dimension}. This class includes interesting algebras: AF algebras, AI and AT algebras (\cite{thom}, \cite{EEv}), Goodearl algebras (\cite{Goodearl}) and Villadsen algebras of the first type (\cite{tom}, \cite{TW}, \cite{jes1}).
Note that the algebras constructed by Toms in \cite{tom} have the same K-groups and tracial data but different Cuntz semigroups. This means that the class of $C^{*}$-algebras considered in this paper cannot be classified by K-theory and their tracial data.
The paper consists of five sections. Section 2 introduces the class of AH-algebras which will be studied in this paper. Section 3 discusses about the density of nilpotent elements relative to some unitaries, as a consequence, the algebras have stable rank one. Section 4 concerns the density of the linear span of projections. And finally, section 5 generalizes the sufficient and necessary condition of real rank zero which is a generalization of [\cite{Goodearl}, Theorem 9].
\section{Preliminaries and Definitions}
{\bf Notation:} Denote by $M_n$ the $C^*$-algebra of all $n\times n$ matrices with complex coefficients while by $M_n(A)$ the algebra of all $n\times n$ matrices with coefficients in $A$. By a standard basis $\{e_{ij}\}_{1\le i,j\le n}$ of $M_n$ we mean the set of matrices $e_{ij}$ whose $(i,j)^{\textrm{th}}$-entry is equal to $1$ and zero elsewhere.
We always assume the space $X, Y$ are compact metric spaces with finitely many connected components. If $X$ is connected and $p\in M_n(C(X))$ is a projection, then the rank function rank($p(x)$) is constant for every $x\in X$. Hence we can say rank($p$) to be the rank of $p(x)$ for any $x\in X$.
For convenience, let us denote by $\tilde{A}$ the unitalization of $A$ when $A$ does not have the unit and $A$ otherwise. \\
Denote by $\textrm{diag}(a_1,a_2,\ldots,a_n)$ the block diagonal matrix
\[ \left( \begin{array}{cccc}
a_1 & 0 & \ldots &0 \\
0 & a_2 & \ldots &0\\
\vdots & \vdots &\ddots & \vdots\\
0 & 0 & \ldots & a_n
\end{array} \right) \]
Let us consider a $C^{*}$ algebra of the form $PM_{n}(C(X))P$, where $P$ is a projection in $M_{n}(C(X))$. If $X$ is not connected and if $X$ has finitely connected components $X_i$ ($X = \bigsqcup_{i=1}^k X_i$), then
\[ PM_{n}(C(X))P = \oplus_{i=1}^k P_i M_n (C(X_i))P_i, \]
where $P_i$ is the restriction of $P$ on $X_i$. Therefore, for any homogeneous algebra of the form (\ref{equation1}) below, it can be assumed that each $X_i$ is connected.
We will study AH algebras which can be written as inductive limits of homogeneous algebras of the form:
\begin{equation}\label{equation1}
%\[
\oplus_{i=1}^k P_i M_{n_i}(C(X_i)) P_i,
%\]
\end{equation}
where the $X_i$ are connected \cms and the $P_i$ are projections in the corresponding $M_{n_i}(C(X_i))$ with diagonal connecting maps defined below. The projections $P_i$ are called the cutdown projections.
\begin{defn}\label{Proj_diag}
Let $\phi$ be a unital $\ast$-homomorphism \[ \phi: \oplus_{i=1}^k P_i M_{n_i}(C(X_i)) P_i \longrightarrow \oplus_{j=1}^t Q_j M_{m_j}(C(Y_j)) Q_j, \]
where the $P_i$ and the $Q_j$ are projections in the $M_{n_i}(C(X_i))$ and $M_{m_j}(C(X_j))$, respectively.
Then, $\phi$ is said to be {\it diagonal} if there exists a diagonal $\ast$-homomorphism $\tilde{\phi}$ from $\oplus_{i=1}^k M_{n_i}(C(X_i)) $ to $\oplus_{j=1}^t M_{m_j}(C(Y_j))$ such that $\phi$ is the restriction of $\tilde{\phi}$ on the subalgebra $\oplus_{i=1}^k P_i M_{n_i}(C(X_i)) P_i$.
\end{defn}
{\bf Notation:} Let us denote by $\mathcal{D}$ the class of $C^*$-algebras which can be written as an inductive limit of homogeneous $C^*$-algebras of the form (\ref{equation1}) with diagonal connecting maps.
We also denote by $\mathcal{D}_1$ a subset of $\mathcal{D}$ consisting of $C^*$-algebras which can be written as inductive limits $\underrightarrow{\lim} (A_i,\phi_i)$, where the $\phi_i$ are diagonal and the $A_i$ are of the form
\begin{equation} \label{equa1.1}
{A}_{i} = \oplus_{t=1}^{k_{i}} M_{n_{it}}(C(X_{it})), \ X_i = \sqcup_{t=1}^{k_{it}} X_{it},
\end{equation}
the $X_{it}$ are compact connected metric spaces and the $n_{it}, k_{i}$ are positive integers. It means that an algebra $\underrightarrow{\lim}(A_i,\phi_i) \in \mathcal{D}$ belongs to $\mathcal{D}_1$ if all the cutdown projections of the block algebras $A_i$ of the form ($\ref{equation1}$) are the identity. The algebras in $\mathcal{D}_1$ form a rich class. They include AF algebras, AI and AT algebras (hence the rotation algebras) (see \cite{EEv}).
\begin{defn}\label{slowdimgrowth}
A $C^*$-algebra $A$ is said to have {\it slow dimension growth} if $A$ can be written as an inductive limit of a sequence $(A_i, \phi_i)$, where
\begin{displaymath}
{A}_{i} = {\oplus}_{t=1}^{k_{i}}P_{it} C(X_{it})\otimes M_{n_{it}}P_{it}, \textrm{ for } i = 1, 2, 3, \ldots,
\end{displaymath}
such that
\begin{displaymath}
\lim_{i \rightarrow \infty} \max_{t}{\frac{\dim X_{it}}{\textrm{rank} P_{it}}} = 0.
\end{displaymath}
\end{defn}
%Note that since we can assume the $X_{it}$ are connected, $\textrm{rank} (P_{it}(x))$ is a constant for every $x\in X_{it}$.
\begin{prop}
Let $\phi$ be a diagonal $\ast$-homomorphism as stated in Definition \ref{Proj_diag}. Then, the following hold.
\begin{enumerate}
\item $\phi$ is unital. %and $\phi (P_1,P_2,\ldots,P_k) = (Q_1, Q_2,\ldots,Q_t)$.
\item The diagonal $\ast$-homomorphism $\tilde{\phi}$ needs not be unique.
\item The composition of diagonal homomorphisms is again diagonal.
\end{enumerate}
\end{prop}
\begin{proof}
(1) $\phi$ is unital since the map $\tilde{\phi}$ is unital and the subalgebras which are the domain and the range of $\phi$ have the units.
(2) For example, let's consider a diagonal $\ast$-homomorphism $\phi$ defined by
\[ \phi: C[0,1] \longrightarrow e_{11} M_2(C[0,1]) e_{11},\ \phi (f) = \textrm{diag}(f, 0), \]
where $e_{11} = \textrm{diag}(1,0)\in M_2$. There exist two diagonal extensions of $\phi$, let us denote by $\psi$ and $\la$, from $C([0,1])$ to $M_2(C[0,1])$ given by
\[ \psi (f) = \textrm{diag}(f, f); \textrm{ and } \la (f) = \textrm{diag}(f, 1) \]
(3) Let $\phi$ and $\psi$ be diagonal homomorphisms such that $\phi \circ \psi$ exists. By Definition \ref{Proj_diag}, these maps are the restriction of some diagonal homomorphisms $\tilde{\phi}$ and $\tilde{\psi}$ whose domains and the target spaces are the corresponding full matrix algebras. Thus $\phi \circ \psi$ is diagonal, since the composition $\tilde{\phi} \circ \tilde{\psi}$ is diagonal.
\end{proof}
\subsection{Approximately Constant Eigenvalue}
In this subsection we wish to quote the characterization of simplicity for diagonal AH algebras from \cite{EHT}.
\begin{defn} \label{defn1}
Let $A = \underrightarrow{\lim}(A_i,\phi_i)$ where the $A_i$ are of the form (\ref{equa1.1}). $A$ is said to have the approximately constant eigenvalue property if for any $i \in \mathbb{N}$, element $f$ in $A_i$, $\epsilon >0$ and $x \in X_i$ there exist $j \ge i$ and unitaries $u_t \in A_{jt}= M_{n_{jt}}(C(X_{jt}))$, $t \in \{1, \ldots, k_j \}$ such that
\begin{displaymath}
|| u_t \phi_{ij}^{lt}(f) u_t^{*} - \textrm{diag}(f(x),b_t) || < \epsilon
\end{displaymath}
for some appropriately sized $b_t$ such that $\textrm{diag}(f(x),b_t) \in A_{jt}$ .
\end{defn}
The Villadsen algebras of the first type have approximately constant eigenvalue property (\cite{tom}, \cite{TW}, \cite{jes1}). These algebras are simple due to the approximately constant eigenvalue property. The following will ensure the class of algebras under our consideration has the approximately constant eigenvalue property.
\begin{thrm} \label{simplicity}
Let $A = \underrightarrow{\lim}(A_i,\phi_i)$ where the $A_i$ are of the form (\ref{equa1.1}). If $A$ has the approximately constant eigenvalue property then $A$ is simple. The converse holds if the $\ast$-homomorphisms ${\phi}_{ij}$ are diagonal.
\end{thrm}
\begin{proof}
Suppose that $A$ has the approximately constant eigenvalue property. As in [\cite{EHT}, Section 2.2], all the connecting $\ast$-homomorphisms $\phi_{ij}$ can be assumed to be injective. Let $a$ be any non-zero element in ${A}_{i}$ for some positive integer $i$. Then one of the components of $a$ is non-zero, hence we can assume that $A_i$ has only one component, i.e., $A_i = M_{n_i}(C(X_i))$. Then there is an $x \in X_{i}$ such that $a(x) \ne 0$. By the definition of the approximately constant eigenvalue property, there is an integer $j>i$, and a unitary $u$ in $A_{j}$ (we can also assume that $ A_j$ has only one component $A_j=M_{n_j}(C(X_j))$) such that
\begin{displaymath}
||u {\phi}_{ij}(a) u^{*} - \textrm{diag}(a(x),b)||<{\epsilon}
\end{displaymath}
We can choose $\epsilon$ small enough so that ${\phi}_{ij}(a)(y)$ is non-zero for every $y \in X_{j}$. By [\cite{dnnp}, Proposition 2.1], $A$ is simple.\\
The converse is due to [\cite{EHT}, Theorem 3.4].
\end{proof}
\section{Nilpotent Elements}
An element $x$ in $A$ is called nilpotent if there exists an $n$ such that $x^n = 0$. \\
{\bf Notation:} Let $A$ be a unital $C^*$-algebra. Denote by $\mathcal{N}(A)$ the set of all elements $x$ in $A$ such that there are unitaries $u,v \in A$, $uxv$ is nilpotent. $ZD(A)$ is denoted by the set of all zero divisors of $A$ and $U(A)$ is the set of all unitaries of $A$\\
The fact is the set of all nilpotent elements can be approximated arbitrarily closely by the invertible. This leads us to the following.
\begin{prop}\label{nilpotentdense}
Let $A$ be a unital $C^*$-algebra. Then the following hold.
\begin{enumerate}
\item Every element in $\mathcal{N}(A)$ can be approximated arbitrarily closely by the invertible.
\item If every non-invertible element in $A$ can be approximated arbitrarily closely by elements in $\mathcal{N}(A)$ then $A$ has stable rank one.
\item Suppose that $A= \underrightarrow{\lim} (A_n,\phi_n)$, where the $A_n$ are unital $C^*$-algebras and the connecting $\ast$-homomorphisms $\phi_{nm}:A_n \rightarrow A_m$ ($m >n$) are unital and injective. Then $A$ has stable rank one if, for every positive integer $n$, every non-invertible element $x$ in $A_n$, there exists an integer $m \ge n$ such that $\phi_{nm}(x)$ belongs to the closure of $\mathcal{N}(A_m)$ in the norm topology.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Let $x\in A$. If $x$ is nilpotent and for any positive number $\epsilon$, $-\epsilon= x^n-\epsilon^n 1$ is in $GL(A)$. Hence, $x$ can be approximated by an invertible, namely $x-\epsilon$, to within $2\epsilon$. Furthermore, for any unitaries $u,v$ in $A$, $uxv$ is also approximated by an invertible element to within $2\epsilon$. Therefore, $uxv$ belongs to $GL(A)$ if and only if $x$ is in $GL(A)$.
(2) For any $x \in A$, a non-invertible element and any positive number $\epsilon$, there exists an element $y$ in $\mathcal{N}(A)$ such that $||x-y|| < \epsilon$ by (2). By (1), there is an invertible element $z$ in $A$ such that $||y-z||< \epsilon$. Hence, $||x-z||<2\epsilon$. Thus, the invertible is dense in $A$.
(3) Since all the connecting homomorphisms $\phi_{nm}$ are injective, $A_n$ can be identified with its image in $A$ for any $n$ and so $A$ is the closure of $\cup_{n=1}^{\infty} A_n$. For any non-invertible element $x$ in $A$ and any positive number $\epsilon$, there is a positive integer $n$ and an element $y$ in $A_n$ such that $||x-y||<\epsilon$. We can also assume that $y$ is not invertible. By the hypothesis (3), there is a positive integer $m \ge n$ such that $\phi_{nm}(y)$ belongs to $\mathcal{N}(A_m)$ and hence, $\phi_{nm}(y)$ is in $\mathcal{N}(A)$. By (2), $A$ has stable rank one.
\end{proof}
Proposition \ref{nilpotentdense} is usually used to prove stable rank one of a $C^{*}$-algebra. To make use of this fact, R{\o}rdam raised a question: {\it If A is a simple unital $C^*$-algebra and $x \in ZD(A)$, can we find $u \in U(A)$ such that $ux$ is nilpotent} (\cite{ro}). This question can be relaxed as follows without losing our purpose.
\begin{question}\label{questionNilpotent}
Let $A$ be a simple unital $C^*$-algebra and $x \in A$ be non-invertible. Does $x$ belongs to the closure of $\mathcal{N}(A)$?
\end{question}
We can also state Question \ref{questionNilpotent} as: {\it is the set $inv(A)\cup \mathcal{N}(A)$ dense in $A$}? where $inv(A)$ is the set of all invertible elements in $A$. It is obvious that stable rank of $A$ is equal to $1$ if and only if $inv(A)\cup \mathcal{N}(A)$ is dense in $A$ by Proposition \ref{nilpotentdense}.\\
{\bf Example:} Question \ref{questionNilpotent} has a positive answer if $A$ is a Goodearl algebra ([\cite{Goodearl}, Lemma 2]).
\begin{prop}\label{nilpotentsize}
Let $A= \underrightarrow{\lim}(A_i,\phi_i)$ be a simple unital AH algebra, where the $A_i$ are of the form (\ref{equa1.1}). If $A$ has the approximately constant eigenvalue property, then for any non invertible element $a\in A$ there are a positive integer $n$ and unitaries $u,v \in U_n(A)$ such that $u(a\otimes 1_n)v$ is a nilpotent element in $M_n(A)$.
(Note that the $\ast$-homomorphism $\phi$ is not necessarily diagonal).
\end{prop}
\begin{proof}
Let $A= \underrightarrow{\lim}(A_i,\phi_i)$, a non-invertible element $a\in A$ and a positive number $\epsilon$. Since $\displaystyle \cup_{i>0} \phi_{i \infty}(A_i)$ is dense in $A$, we can assume that $\displaystyle a \in \cup_{i>0} \phi_{i \infty}(A_i)$. Since the images of nilpotent elements of $A_i$ under $\phi_{i \infty}$ are again nilpotent, we can assume $a\in A_i$ for some $i$. An element of finitely many components can be approximated by nilpotent elements if and only if each of its components can be approximated by nilpotent elements. Hence, we can assume that each $A_j$ has one component, i.e., $ A_j = M_{n_{j}}(C(X_{j})) $. Since $a$ is not invertible, there is a point $x\in X_i$ such that $det(a(x)) = 0$. Since $A$ has the approximately constant eigenvalue property, there is an integer $j \ge i$ and a unitary $u_0$ such that
\[ || u_0 \phi_{i,j}(a) u_0^{*} - \textrm{diag}(a(x),b_1) || < \epsilon , \]
for some appropriately sized $b_1$ such that $\textrm{diag}(a(x),b) \in A_{j}$. Since $a(x) \in M_{n_i}$ is not invertible, there are unitaries $u_1,v_1$ in $M_{n_i}$ such that
\[u_1a(x)v_1 = \textrm{diag}(0,b_2), \ b_2 \in M_{n_i-1} \]
Set $ \displaystyle u_2 = u_0 \textrm{diag}(u_1, 1_{n_j-n_i}),\
v_2 = \textrm{diag}(v_1,1_{n_j-n_i})u_0^*, \textrm{ and } b = \textrm{diag}(b_1,b_2).$
Then, \[||u_2 \phi_{ij} (a) v_2 - \textrm{diag}(0,b)||<\epsilon, \textrm{ where } b \in M_{n_j-1}.\]
Take a positive integer $n$ such that $m = \frac{n}{n_j} > n_j-1$. The number of copies of $0$ in $(\textrm{diag}(0,b)\otimes 1_m )$ is at least $m$, hence there exists a permutation matrix $e \in M_n$ such that $e(\textrm{diag}(0,b)\otimes 1_m ) \in M_n(A)$ is an upper triangular whose main diagonal entries vanish and thus it is nilpotent. Finally, just take $u$ and $v$ to be $e(u_2\otimes 1_m)$ and $v_2\otimes 1_m$, respectively, and the proof ends.\\
\end{proof}
\begin{thrm}\label{nilpotnet}
Let $A$ be a unital simple AH algebra which can be written as an inductive limit $\underrightarrow{\lim}(A_i, \phi_i)$, where the $\phi_i$ are diagonal and the $A_i$ are of the form (\ref{equation1}). Suppose that $A_1 = \oplus_{t=1}^{k} P_{1t}M_{n_{1t}}(C(X_{1t}))P_{1t}$ and the cutdown projections $P_{it} \in A_1$ are Murray-von Neumann equivalent to the constant projections, then for any non-invertible element $a\in A$ and any positive number $\epsilon$, there exists an element $a_1 \in \mathcal{N}(A)$ such that $||a-a_1||<\epsilon$.
\end{thrm}
In order to prove Theorem \ref{nilpotnet}, we need the following lemma.
\begin{lemma} \label{cutdown}
Let $B$ be a $C^*$-algebra, $p \textrm{ and } q$ be projections in $B$. If $p$ and $q$ are Murray-von Neumann equivalent, then $p B p$ is isomorphic to $q B q$.
In particular, if $B = M_n(C(X))$ with connected compact metric spectrum $X$ and if $q$ is the constant projection in $B$ of rank $m$, then $qBq$ is $\ast$-isomorphic to the full matrix algebra $M_m(C(X))$.
\end{lemma}
\begin{proof}
There is a partial isometry $v$ such that $p $ = $v^{*} v $ and $q$ = $v v^{*}$. Define a map $\phi$ by $\phi (x) = v x v^{*}$ for all $ x \in p B p$.
It is straightforward to check that $\phi$ is a $\ast$-homomorphism from $p B p$ to $q B q$. \\
Similarly, we can define a $\ast$-homomorphism $\psi$ from $q B q$ to $p B p$ by
$\psi (x) = v^{*} x v$, for all $ x \in q B q$.
For any $a \in B$,
\begin{displaymath}
\phi \circ \psi (qaq) = v(v^{*} q a q v)v^{*} = q a q, \textrm{ and } \psi \circ \phi (pap)
= v^{*}( v p a p )v^{*} = pap.
\end{displaymath}
By the same argument, we also have that $\psi \circ \phi$ is the identity map on $p B p$.
Thus $\phi$ is an isomorphism and its inverse is $\psi$.
In the case $B = M_n(C(X))$, and $q$ is the constant projection in $B$ of rank $m$, then $pBp$ is $\ast$-isomorphic to $qBq$. Moreover, since $q \in M_n$ is a Hermitian, there is a unitary $u\in M_n$ such that $q = u(e_{11}+ e_{22}+\ldots + e_{mm})u^*$, where $\{e_{ij}\}_{i,j=1}^{n}$ is the standard base of $M_n$. Then
\[qBq = u (\sum_{i=1}^m e_{ii})u^* B u (\sum_{i=1}^m e_{ii})u^* = (\sum_{i=1}^m e_{ii})B (\sum_{i=1}^m e_{ii}) = M_m(C(X)). \]
\end{proof}
Now, we are ready to prove Theorem \ref{nilpotnet}.
\begin{proof} %{\it Proof of Theorem \ref{nilpotnet}}.
Assume $A = \underrightarrow{\lim}(A_i,\phi_i)$, where the $A_i$ are of the form (\ref{equation1}) and the $\phi_i$ are diagonal $\ast$-homomorphisms. Using the same argument as the beginning of the proof of Proposition \ref{nilpotentsize}, we can assume that each $A_i$ has only one component, that is, $A_i = P_i M_{n_i}(C(X_i))P_i$, where $P_i$ is a projection in $A_i$.
By the hypothesis, the cutdown projection $P_1$ of $A_1$ is Murray-von Neumann equivalent to a projection $ Q \in M_{n_1}$. $Q$ is unitary equivalent to $ Q_1$ = $e_{11}+e_{22}+\ldots + e_{m_1}$, where $m_1$ is the rank of $Q$ and $\{e_{ij}\}_{1\le i,j\le n_1}$ is the standard base of $M_{n_1}$. Hence we can say $P_1$ is Murray-von Neumann equivalent to $Q_1$. For $i>1$, define $Q_i = \phi_{i-1}(Q_{i-1})$.
Then $Q_i = \phi_{1i}(Q_1)$ is constant, since $Q_1$ is constant and $\phi_{1i}$ is diagonal. Let us denote by $m_i $ the rank of $Q_i$, then $m_{i+1}|m_i$ and $\frac{m_{i+1}}{m_i}$ = $\frac{n_{i+1}}{n_i}$ for all $i$. By Lemma \ref{cutdown}, there are $\ast$-isomorphisms
\[ \Theta_i: P_i M_{n_i}(C(X_i))P_i \longrightarrow Q_i M_{n_i}(C(X_i))Q_i = M_{m_i}(C(X)), \quad \Theta_i(a) = v_i a v_{i}^*, \]
where $P_1 = v_1^*v_1 \sim v_1v_1^* = Q_1 $ and $v_i = \phi_{1i}(v_1)$.
Since each $\phi_i$ is diagonal, by definition, there exists a diagonal $\ast$-homomorphism $\tilde{\phi_i}: M_{n_i}(C(X_i)) \longrightarrow M_{n_{i+1}}(C(X_{i+1}))$ such that $\phi$ is the restriction of $\tilde{\phi_i}$ on $A_i$ into $A_{i+1}$. Let $\psi_i$ be the restriction of $\tilde{\phi_i}$ on $Q_iM_{n_i}(C(X_i))Q_i$, for all $i$. Then $\psi_i (Q_i) = Q_{i+1}$, therefore, the map $\psi_i$ can be viewed as the map from $Q_iM_{n_i}(C(X_i))Q_i$ to $Q_{i+1}M_{n_{i+1}}(C(X_{i+1}))Q_{i+1}$ and so we have the diagonal AH-algebra $\underrightarrow{\lim}(M_{m_i}(C(X_i)),\psi_i)$. In addition it is straightforward to check that $\Theta_{i+1}\circ \phi_i$ = $\psi_i \circ \Theta_i$ and hence we can assume that $A$ = $\underrightarrow{\lim}(M_{m_i}(C(X_i)),\psi_i)$.
Let $\psi_{i\infty}$ be the $\ast$-homomorphism from $A_i = M_{m_i}(C(X_i))$ into $A$. The union $\cup_{i=1}^{\infty}\psi_{i\infty}(A_i)$ is dense in $A$. An element belonging to $\mathcal{N}(A_i)$ is again in $\mathcal{N}(\phi_{i \infty}(A_i))$ and so in $\mathcal{N}(A)$. Therefore, it is sufficient to show that every non-invertible element in $A_i$ can be approximated by elements in $\mathcal{N}(A_k)$ for some large enough integer $k \ge i$.
Suppose that $a\in A_i$ is not invertible and $\epsilon>0$. By Theorem \ref{simplicity}, $A$ has the approximately constant eigenvalue property. Using the same argument as the proof of Proposition \ref{nilpotentsize}, there are an integer $j \ge i$ and unitaries $u_2,v_2 \in A_j$ such that $||u_2 \phi_{ij} (a) v_2 - \textrm{diag}(0,b)||<\epsilon$, where $b \in M_{m_j-1}$.
Choose a positive integer $k$ such that $m=\frac{m_k}{m_j} > m_j-1$. Let $\{\la_t \}_{t=1}^m$ be the eigenvalue pattern of $\psi_{jk}$. Then $\psi_{jk}(\textrm{diag}(0,b)) = \textrm{diag}(0,b\circ \la_1,0,b\circ \la_2, \ldots, 0,b \circ \la_m)$. Since the number of $0$'s appearing in $\psi_{jk}(\textrm{diag}(0,b))$ is at least $m$ and $m$ is strictly larger than the size of $b \circ \la_t$ for every $t$, we can choose a permutation matrix $e \in M_{n_k}$ such that $e\psi_{jk}(\textrm{diag}(0,b)) \in A_k$ is an upper triangular whose main diagonal entries are zero. Hence, $e\psi_{jk}(\textrm{diag}(0,b))$ is nilpotent. Let $a_1 =\psi_{jk}(u_2^*) \psi_{jk}(\textrm{diag}(0,b))\phi_{jk}(v_2^*)$. Then,
\[e \psi_{jk}(u_2) a_1 \psi_{jk}(v_2^*) = e \psi_{jk}(\textrm{diag}(0,b)) \]
is nilpotent and so $a_1 \in \mathcal{N}(A_k)$. And,
\[ ||\psi_{ik}(a)- a_1|| = ||\psi_{jk}(u_2)\psi_{ik}(a)\psi_{jk}(v_2) - \psi_{jk}(\textrm{diag}(o,b)) ||\le ||u_2 \psi_{ij}(a)v_2 -\textrm{diag}(0,b) ||<\epsilon. \]
\end{proof}
\subsection{Stable Rank}
We will now apply Theorem \ref{nilpotnet} to show that an algebra has stable rank one. Firstly, we recall the notion of stable rank which was introduced by Marc A. Rieffel (\cite{rie}). A unital $C^*$-algebra $A$ is said to have stable rank one (denoted by sr($A$) = 1) if the invertible is dense in $A$. If $A$ is non-unital, stable rank of $A$ (denoted by sr($A$)) is the stable rank of its unitalization $\tilde{A}$. It is shown in \cite{rie} that for a unital $C^*$-algebra $A$, the invertible is dense in $A$ if, and only if the set of all left-sided invertible elements is dense if, and only if the set of right-side invertible elements is dense.
It was shown in \cite{dnnp} and \cite{bdr} that a simple AH algebra has stable rank one if the dimensions of the spectra of the building blocks in the inductive limit decomposition are bounded, respectively slow dimension growth. The following corollary does not have any assumption of dimension growth. %(This corollary is an other version of [\cite{EHT}, Theorem 4.1]).
\begin{cor}\label{stablerank}
Let $A = \underrightarrow{\lim} (A_i,\phi_i)$ be a simple, diagonal AH algebra, where
\[ A_i = \oplus_{t=1}^{k_i} P_{i,t} M_{n_{i,t}} (C(X_{i,t})) P_{i,t}. \]
If the cutdown projections $P_{1,t}$ in $A_1$ are Murray-von Neumann equivalent to constant ones, then $A$ has stable rank one.
\end{cor}
\begin{proof}
It follows immediately from Proposition \ref{nilpotentdense} and Theorem \ref{nilpotnet}.
\end{proof}
{\bf Note} There are examples of stable rank $n$ for each positive integer $n$ (\cite{jes2}) if the hypothesis on `dimension growth' and `diagonal connecting homomorphisms' are ignored.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear Span of Projections}
Let us recall some notations from \cite{pedersen}. Given a $C^*$-algebra $A$, denote by $L(A)$ the closed linear span of projections in $A$. In this section, we will study the density of the linear span of projections in $A$. More information about this problem can be found in \cite{pedersen}.
{\bf Note:} Let $A = \underrightarrow{\lim}(A_i,\phi_i)$ be an AH algebra. We can view the continuous functions on the spectrum of each $A_i$ as elements in $A_i$ by the natural embedding in the following two cases.
%Replacing $A_i$ by $A_i/\textrm{Ker}(\phi_i)$ if necessary, we can assume that the $\phi_i$ are injective.
{\it Case 1.} If $\displaystyle A_i = \oplus_{t=1}^{k_i} M_{n_{it}} (C(X_{it}))$, then there is an embedding from $C(X_{it})$ into $A_i$ defined by the composition of the following injective $\ast$-homomorphisms:
\[ C(X_{it}) \longrightarrow M_{n_{it}}(C(X_{it})), \ f \longmapsto \textrm{diag}(f,0), \ \forall f\in C(X), \]
and the embedding into the $k^{\textrm{th}}$-coordinate
\[ M_{n_{it}}(C(X_{it})) \longrightarrow \oplus_{t=1}^{k_i} M_{n_{it}} (C(X_{it})),
a \longmapsto (0, \ldots, 0, a, 0, \ldots 0 ), \ \forall a\in M_{n_{it}}(C(X_{it})) \]
{\it Case 2.} If $\displaystyle A_i = \oplus_{t=1}^{k_i} P_{it} M_{n_{it}} (C(X_{it})) P_{it}$ and suppose that the projections $P_{it}$ are Murray-von Neumann equivalent to the constant, by Lemma \ref{cutdown}, $A_i$ is isomorphic to the homogeneous algebra of form (\ref{equa1.1}). Hence, we can also have the embedding from $C(X_{it})$ into $A_i$ as Case 1.
\begin{thrm}
Let $A = \underrightarrow{\lim}(A_i,\phi_i)$ be an AH algebra in $\mathcal{D}$, i.e., the $\phi_i$ are diagonal and the $A_i$ are of the form
\[ A_i = \oplus_{t=1}^{k_i} P_{it} M_{n_{it}} (C(X_{it})) P_{it}. \]
Then $L(A)$ is equal to $A$, i.e., the linear span of projections is dense in $A$ provided that every self-adjoint element in $\phi_{i\infty}(C(X_{it}))$ can be approximated arbitrarily closely by self-adjoint elements with finite spectra.
\end{thrm}
\begin{proof}
Every element in $A$ can be closely approximated by elements in $\cup_{i}\phi_{i\infty}(A_i)$. Hence, it is sufficient to show that for any element $a\in A_i$ and any positive number $\epsilon$, there is an $j \ge i$ such that $\phi_{ij}(a)$ can be approximated by a linear combination of projections in $A_j$ to within $\epsilon$. If $\phi_{ij}(a)$ has finitely many components, it is enough to show that each component can be approximated by linear combinations of projections in $A_j$ to within $\epsilon$. Therefore we can assume that each $A_t$ has only one component, that is, $A_t = M_{n_t}(C(X_t))$ for every positive integer $t$.
We can also assume that $a = a^*$ since $a$ can be written as a sum of two self-adjoint elements. Let $a$ = $(a_{st}) \in M_{n_i}(C(X_i))$, where the $a_{st}$ are continuous functions on $X_i$. Since $a_{st}=\bar{a}_{ts}$, the $a_{ss}$ are real valued functions and we can write $a_{st}= \bar{a}_{ts} = f_{st}+\imath g_{st}$ with $\imath^2=-1$ and $f_{st}, g_{st}$ are real valued functions for every $1~~ 0$, there is a $j \ge i$ such that
\begin{displaymath}
TV(\phi_{ij}(a)) < \epsilon.
\end{displaymath}
\item For any self-adjoint element $a$ in $B_{i}$ and any positive number $\epsilon $, there is a $j \ge i$ such that
\begin{displaymath}
EV(\phi_{ij}(a)) < \epsilon.
\end{displaymath}
\item $B$ has real rank zero.
\end{enumerate}
\begin{itemize}
\item[(i)] The following implications hold in general.
\begin{displaymath}
(4) \ \Rightarrow (3)\ \Rightarrow \ (2) \ \Rightarrow \ (1).
\end{displaymath}
\item[(ii)] If $B$ is simple, then the following equivalences hold.
\begin{displaymath}
(3) \ \Leftrightarrow \ (2) \ \Leftrightarrow \ (1).
\end{displaymath}
\item[(iii)] If $B$ is simple and has slow dimension growth, then all the conditions ($1$), ($2$), ($3$) and ($4$) are equivalent.
\end{itemize}
\end{prop}
\begin{proof}
The statements (i) and (ii) are proved in [\cite{bbek}, Theorem 1.3]. The statement (iii) is an immediate consequence of the statement (ii) and [\cite{bdr}, Theorem 2].
\end{proof}
%Some concepts should be recalled before stating the main theorem of this section.
%A partially ordered Abelian group $G$ is said to be {\it weakly unperforated} provided that whenever a positive integer $n$ and an element $g \in G$ satisfy $ng > 0$ implies that $g > 0$. A partially ordered Abelian group $G$ is said to be {\it unperforated} if whenever $g \in G$, a positive integer $n \in \NN$ and $ng \ge 0$, then $g \ge 0$. Obviously, if $G$ is unperforated, then it is weakly unperforated. The converse is true when $G$ is torsion free. If $A$ is an AF algebra, then the group $K_{0}( A)$ is unperforated (see \cite{davidson}, Proposition IV.6.1). If $A$ is a Goodearl algebra, then $K_{0}(A)$ is weakly unperforated. However, there are Goodearl algebras whose $K_{0}$-groups are not unperforated, for examples, when we choose the base space $X$ such that $K_{0}(C(X))$ has $2$-torsion (e.g. $X$ = $\mathbb{RP}_{n}$, see \cite{Goodearl}). In \cite{jes1}, Villadsen constructed algebras with perforated $K_{0}$-groups.
The concept of tracial rank can be found in \cite{lin}. For the convenience of readers, it will be recalled here.
\begin{defn}
Let $A$ be a simple unital $C^*$-algebra. $A$ is said to have tracial topological rank zero (written $TR(A)$ = $0$) if for any positive number $\epsilon$, and finite subset $\mathcal{F}$ of $A$ containing a non-zero positive element $a$, there exists a finite-dimensional $C^*$-subalgebra $B$ of $A$ with $1_{B}$ = $p$ such that
\begin{enumerate}
\item $||px -xp ||< \epsilon $, for all $x \in \mathcal{F}$,
\item $||pxp - b_{x}|| < \epsilon $, for some element $b_{x} \in B$, every $x \in \mathcal{F}$ and
\item $1 - p \preceq q$ for some projection $q$ in the hereditary $C^*$-subalgebra generated by $a$.
\end{enumerate}
\end{defn}
\subsection{Results}
In \cite{Goodearl}, a necessary and sufficient condition for a Goodearl algebra to have real rank zero is given in terms of the limit of the {\it weighted identity ratio} sequence. In this subsection, we will generalize this result to a subclass of $\mathcal{D}_1$ (let us say {\it generalized Goodearl algebras}). This class includes the interesting algebras constructed in \cite{tom} and Villadsen algebras of the first type (\cite{TW}). Define a simple algebra in the class under consideration as $A = \underrightarrow{\lim}(A_i,\phi_i)$, where
\begin{displaymath}
A_{i} = M_{n_{i}}(C(X_{i})) \textrm{ and }
\phi_{i} (a) = \textrm{diag}(a(x_1),\ldots, a(x_{l_i}), a\circ \la_1, \ldots, a\circ \la_{\alpha_i}),
\end{displaymath}
and where $x_{i1}, x_{i2}, \ldots, x_{il_{i}}$ are elements given in $X_{i}$, and $\la_{i1}, \la_{i2}, \ldots, \la_{i \alpha_{i}}$ are nonconstant continuous maps from $X_{j}$ to $X_{i}$.
If all the $X_i$ are the same as $X$, the sequence $x_{i1}, x_{i2}, \ldots$ is dense and all the eigenvalue maps are the identity on $X$, then the algebra under consideration is a Goodearl algebra.\\
If $X_{i}$ is totally disconnected for every positive integer $i$, then $A$ has real rank zero since all $RR(A_i) =0$ and an inductive limit of algebras with real rank zero has real rank zero. For each pair of positive integers $i,j$, where $i < j$, set
\begin{displaymath}
\omega_{i,j} = \alpha_{i} \alpha_{i+1} \ldots \alpha_{j-1} \frac{n_{i}}{n_{j}}.
\end{displaymath}
Note that the number $\alpha_{i} \alpha_{i+1} \ldots \alpha_{j-1}$ is the number of non-constant eigenvalue maps of $\phi_{ij}$. So we might call $\omega_{i,j}$ the {\it weighted non-constant ratio} for $\phi_{ij}$. It is clear that
\begin{displaymath}
\omega_{i,j+1} = \omega_{i,j} \alpha_{j} \frac{n_{i}}{n_{j+1}}
\end{displaymath}
and $0 < \alpha_{j} \frac{n_{i}}{n_{j+1}} < 1$. Hence, the sequence $\{ \omega_{i,j} \}$, for a fixed integer $i$, is decreasing. Thus, it is convergent as $j \rightarrow \infty$.
We are now ready to state the theorem of this section. This theorem is a generalization of [\cite{Goodearl}, Theorem 9].
\begin{thrm}\label{realrank2}
Let $A$ be a simple AH algebra as above. The following statements are true.
\begin{enumerate}
\item If $\displaystyle \lim_{i \rightarrow \infty}{\omega_{1,i}} = 0,$ then $A$ has tracial topological rank zero. In particular, it has real rank zero and weakly unperforated $K_{0}(A)$.
\item If $A$ has real rank zero, then $ \displaystyle \lim_{i \rightarrow \infty}{\omega_{1,i}} = 0 $ provided that $A$ satisfies the following conditions:
\begin{itemize}
\item[(i)] For any positive integer $i$ and any point $x_{0}$ in $X_{i}$, there is an open neighbourhood $U$ of $x_{0}$ such that the union of $\la^{-1} (U)$ for all non-constant eigenvalue maps $\la$ of $\phi_{ij}$ is not the same as $X_{j}$ for any positive integer $j > i$.
\item[(ii)] For any positive integer $i$ and any integer $j_{0} > i$, there is a $j \ge j_{0}$ with a point $x_{j}$ in the spectrum of $A_{j}$ and a point $y_{0}$ in the spectrum of $A_{i}$ such that $\la (x_{j})$ = $y_{0}$, for every non constant eigenvalue map $\la $ of $\phi_{ij}$.
\end{itemize}
\end{enumerate}
\end{thrm}
\begin{proof}
\quad (1). Suppose that $ \displaystyle \lim_{i \rightarrow \infty}{\omega_{1,i}} = 0$. Hence, given any positive integer $i$,
$\displaystyle \lim_{j \rightarrow \infty}{\omega_{i,j}} = 0.$
For any positive integer $\epsilon$, there is a positive integer $j > i$ such that $\omega_{i,j} < \epsilon$. Let $p$ denote the projection in $M_{n_{j}}$ corresponding to the constant eigenvalue maps of $\phi_{ij}$, that is,
\begin{displaymath}
p = \left( \begin{array}{cc}
1 & 0 \\
0 & 0
\end{array} \right),
\end{displaymath}
where the matrix $1$ on the upper left-hand corner is the identity of $M_{n_{j} - \alpha_{i} \alpha_{i+1} \ldots \alpha_{j-1}}$. Then $p$ is the required projection in the definition of tracial topological rank zero.
\quad (2). Suppose that $A$ satisfies the conditions (i) and (ii) of Theorem \ref{realrank2}(2). Without loss of generality, we can assume that $n_{1}$ = $1$, i.e., $A_{1}$ = $C(X_{1})$. Let $i > 1$ be any positive integer. By the condition (ii) of $(2)$, passing to a subsequence if necessary, there is a point $x_{i}$ in $X_{i}$ and a point $y_{0}$ in $X_{1}$ such that $f(x_{i})$ = $y_{0}$ for every non constant eigenvalue map $f$ of $\phi_{1i}$. By (i) there is a neighbourhood $U$ of $y_{0}$ such that the union of the pre-image of $U$ of all non-constant eigenvalue maps of $\phi_{1i}$ is not equal to $X_{i}$. Hence, there is a point $z_{i}$ in $X_{i}$ such that $\la (z_{i})$ does not belong to $U$, for every non-constant eigenvalue map $\la$ of $\phi_{1i}$. Therefore, the variation of the normalized trace of $\phi_{1i}(a)$ is larger than $\omega_{1,i}$, where $a$ is a continuous function defined on $X_{1}$ such that $a(y_{0})$ = $1$ and $a$ vanishes outside the neighbourhood $U$. If
$ \displaystyle \lim_{i \rightarrow \infty}{\omega_{1,i}} \ne 0,$ then the sequence $\{ {TV(\phi_{1i}(a))} \}$ does not converge to zero as $i \rightarrow \infty$. By Proposition \ref{rr1}, the real rank of $A$ is not zero while $A$ has stable rank one by Corollary \ref{stablerank}. Therefore, $A$ has real rank one.
\end{proof}
\begin{cor}\label{correalrank2}
Let $A$ be as in Theorem \ref{realrank2} and such that $A$ also satisfies conditions (i) and (ii) in this theorem. Then the following are equivalent.
\begin{itemize}
\item[(i)] $A$ has real rank zero.
\item[(ii)] $\displaystyle \lim_{i \rightarrow \infty}{\omega_{1,i}} = 0. $
\item[(iii)] $TR(A)$ = $0$.
\item[(iv)] $A$ has slow dimension growth and projections in $A$ separate the traces.
\end{itemize}
If any statement above holds, then $A$ has fundamental comparison property.
\end{cor}
\begin{proof}
(i) $\leftrightarrow$ (ii) $\rightarrow $ (iii) follows from Theorem \ref{realrank2}.
(iii) $\rightarrow$ (iv) $\rightarrow$ (i) follows from Corollary 2.9 of \cite{lin}, which states that given any unital simple AH algebra $A$, $A$ has tracial topological rank zero if, and only if, $A$ has slow dimension growth and projections in $A$ separate the traces if, and only if, $A$ has real rank zero, stable rank one and weakly unperforated $K_{0}(A)$.
\end{proof}
\begin{rem}
When the space $X_{i}$ in Theorem \ref{realrank2} is the same as a connected compact metric space $X$ for every $i$, and supposing that every non-constant eigenvalue map is the identity, then the algebra discussed in Theorem \ref{realrank2} is a Goodearl algebra. The identity map on $X$ obviously satisfies the conditions (i) and (ii) of the theorem. Therefore the sufficient and necessary condition, for a Goodearl algebra to have real rank zero in \cite{Goodearl}, is just a special case of Theorem \ref{realrank2}.
\end{rem}
{\bf Example:} The Villadsen algebras of the first type (\cite{TW}) satisfy the hypothesis of Theorem \ref{realrank2}, since the nonconstant eigenvalue maps are just the coordinate-projections. Moreover, if the spectrum of each building block in the inductive limit decompositions of these algebras are just the products of $[0,1]$, then we obtain the algebras constructed in \cite{tom}. This shows that the class of algebras satisfying the hypotheses of Theorem \ref{realrank2} contains many interesting algebras.
If a Goodearl algebra, or more generally, an algebra that satisfies the hypothesis of Corollary \ref{correalrank2} has real rank zero, then it has slow dimension growth. Therefore, any algebra, which satisfies the hypothesis of Corollary \ref{correalrank2} and does not have slow dimension growth, has real rank one. However, for the algebras under consideration with real rank one, we still do not know if there is any algebra with a unique trace, or more generally, any algebra whose projections separate the traces.
In the absence of the hypothesis of the dimension growth, statement (iii) in Proposition \ref{rr1} fails. The counter-examples can be found in \cite{jes2}. In this paper, for each positive integer $n \ge 2$, Villadsen constructed a simple AH algebra with unique trace and its real rank is either $n$ or $n-1$.
However, the question {\it `Suppose that projections of a simple inductive limit of homogeneous $C^*$-algebras with diagonal $\ast$-homomorphisms between the building blocks separate the traces, does it have real rank zero?'} is still open for simple AH algebras with diagonal morphisms between the building blocks in the inductive limit decompositions but non-zero dimension growth.
{\it Acknowledgments.} This paper was written while the author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The author acknowledges the support. The author is also partially supported by NAFOSTED, Vietnam.
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