text:
\section{Introduction}
Given a unital C*-algebra $A$, let $\mathcal{U}_n(A)$ denote the group of $n\times n$ unitary matrices over $A$. This is a topological group, and its homotopy groups $\pi_k(\mathcal{U}_n(A))$ are collectively referred to as the \emph{nonstable $K$-theory} groups of $A$. The study of these groups goes back to the work of Kuiper \cite{kuiper}, who proved that $\u(\b(H))$ is contractible whenever $H$ is an infinite dimensional Hilbert space. Since then, these groups were calculated for a variety of individual C*-algebras (over many years). These ideas were eventually placed in the broader context of noncommutative topology by Thomsen \cite{thomsen}. In particular, he introduced the notion of a quasi-unitary, and was thus able to study non-unital and unital C*-algebras on the same footing. \\

If $\widehat{\mathcal{U}}_n(A)$ denotes the group of quasi-unitaries in $M_n(A)$, one is then faced with the problem of computing $\pi_k(\widehat{\mathcal{U}}_n(A))$. Unfortunately, for an arbitrary C*-algebra $A$, such computations are prohibitively difficult. In fact, these groups are largely unknown even for the algebra of complex numbers! In order to alleviate this difficulty, we studied these groups \emph{upto rationalization} in \cite{apurva_pv_af}. Here, tools from rational homotopy theory allowed us to compute $\pi_k(\widehat{\mathcal{U}}_n(A))\otimes {\mathbb{Q}}$ when $A$ is an AF-algebra. Indeed, maps between finite dimensional C*-algebras translated to maps between ${\mathbb{Q}}$-vector spaces in a simple and predictable manner. \\

In this paper, we look to understand A$\mathbb{T}$-algebras along similar lines. Our first result is an explicit calculation of the groups $\pi_k(\widehat{\mathcal{U}}_n(A))\otimes {\mathbb{Q}}$ when $A$ is an A$\mathbb{T}$-algebra (\cref{thm_at_alg_rat_hom}). This calculation is a consequence of the fact that maps between two circle algebras may be modified (upto homotopy) into maps of a particularly nice form. \\

We then use these results to obtain a characterization of A$\mathbb{T}$-algebras that have slow dimension growth (see \cref{defn_slow_dim_growth}). To put it this in context, observe that the groups $\pi_k(\widehat{\mathcal{U}}_n(A))$ depend, in general, on the matrix size $n$. However, for certain classes of C*-algebras, $\pi_k(\widehat{\mathcal{U}}(A))$ is naturally isomorphic to $\pi_k(\widehat{\mathcal{U}}_n(A))$ for all $n\geq 1$ and all $k\geq 0$. In particular, for such an algebra,
\[
\pi_k(\widehat{\mathcal{U}}(A)) \cong \begin{cases}
K_0(A) &: \text{ if } k \text{ is odd, and} \\
K_1(A) &: \text{ if } k \text{ is even}.
\end{cases}
\]
Thomsen \cite{thomsen} had proved that the Cuntz algebras and simple, infinite dimensional AF-algebras have this property, and he termed this phenomenon \emph{$K$-stability}. A variety of interesting C*-algebras are now known to have this property (see \cite[Remark 1.5]{apurva_pv_cx}). Similarly, we say that a C*-algebra $A$ is \emph{rationally} $K$-stable if the groups $\pi_k(\widehat{\mathcal{U}}_n(A))\otimes {\mathbb{Q}}$ are all naturally isomorphic to one another. We had proved in \cite{apurva_pv_af} that these two notions coincide for AF-algebras. Our main result here is that the same is true for all A$\mathbb{T}$-algebras, and that it is equivalent to slow dimension growth.

\begin{mainthm}\label{mainthm_k_stable}
For an A$\mathbb{T}$-algebra $A$, the following are equivalent:
\begin{enumerate}
\item $A$ is $K$-stable.
\item $A$ is rationally $K$-stable.
\item $A$ has slow dimension growth.
\end{enumerate}
\end{mainthm}

The paper is organized as follows. In \cref{pre}, we briefly recall the preliminary results and definitions that will be used throughout the paper. In \cref{sec_hom_circle}, we describe a simple class of $\ast$-homomorphisms between circle algebras, and show that, for our purposes, understanding such maps is sufficient. This reduction in turn results in \cref{thm_at_alg_rat_hom}, which is the main goal of \cref{sec_rat_hom_at}. Finally, \cref{sec_k_stability_at} is devoted to a proof of \cref{mainthm_k_stable}.

\section{Preliminaries}\label{pre}
We begin by reviewing the work of Thomsen of constructing the nonstable K-groups associated to a C*-algebra. For the proofs of all the facts mentioned below, the reader may refer to \cite{thomsen}. \\

Let $A$ be a C*-algebra (not necessarily unital). Define an associative composition $\cdot$ on $A$ by
\[
a\cdot b=a+b-ab
\]
An element $u\in A$ is said to be a quasi-unitary if $u\cdot u^{\ast} = u^{\ast}\cdot u = 0$, and we write $\widehat{\mathcal{U}}(A)$ for the set of all quasi-unitary elements in $A$. Moreover, we write $\widehat{\mathcal{U}}_n(A)$ for the group $\widehat{\mathcal{U}}(M_n(A))$.\\

\begin{defn}
Let $A$ be a C*-algebra, and $k\geq 0$ and $m\geq 1$ be integers. Define
\[
G_k(A) :=\pi_k(\widehat{\mathcal{U}}(A)), \text{ and } F_m(A) := \pi_m(\widehat{\mathcal{U}}(A))\otimes {\mathbb{Q}}.
\]
\end{defn}

Note that if $A$ is unital and $\u(A)$ denotes the group of unitaries in $A$, then the map $\widehat{\mathcal{U}}(A) \to \u(A)$ given by $u \mapsto (1-u)$ induces a natural isomorphism of groups. In particular,
\[
G_k(A) \cong \pi_k(\u(A)) \text{ and } F_m(A) \cong \pi_m(\u(A))\otimes {\mathbb{Q}}
\]

Recall \cite{schochet} that a homology theory on the category of $C^{\ast}$-algebras is a sequence $(h_n)$ of covariant, homotopy invariant functors from the category of C*-algebras to the category of abelian groups such that, if $0 \to J\xrightarrow{\iota} B\xrightarrow{p} A\to 0$ is a short exact sequence of C*-algebras, then for each $n \in {\mathbb{N}}$, there exists a connecting map $\partial : h_n(A)\to h_{n-1}(J)$, making the following sequence exact
\[
\ldots \xrightarrow{\partial} h_n(J)\xrightarrow{h_n(\iota)} h_n(B) \xrightarrow{h_n(p)} h_n(A)\xrightarrow{\partial} h_{n-1}(J)\to \ldots
\]
and furthermore, $\partial$ is natural with respect to morphisms of short exact sequences. Finally, we say that a homology theory $(h_n)$ is continuous if, whenever $A = \lim A_i$ is an inductive limit in the category of C*-algebras, then $h_n(A) = \lim h_n(A_i)$ in the category of abelian groups. The next proposition is a consequence of \cite[Proposition 2.1]{thomsen} and \cite[Theorem 4.4]{handelman}.

\begin{prop}\label{prop: continuous_homology}
Both $(G_k)$ and $(F_m)$ are continuous homology theories.
\end{prop}

Let us now describe the behaviour of the functor $F_m$ on finite dimensional C*-algebras. From now on, we write $M_n$ for the algebra $M_n({\mathbb{C}})$, and ${\mathbb{N}}_0$ for the set of all non-negative integers. Now suppose $A = \bigoplus_{j=1}^K M_{n_j}$ and $B = \bigoplus_{i=1}^L M_{\ell_i}$ are two finite dimensional C*-algebras, and $\varphi:A\to B$ is a $\ast$-homomorphism. Then, $\varphi$ is determined upto unitary equivalence by an $L\times K$ matrix $\Phi \in M_{L\times K}({\mathbb{N}}_0)$, called the multiplicity matrix of $\varphi$ (see, for instance, \cite[Section III.2]{davidson}). One of the main results of \cite{apurva_pv_af} was the following: For each $m\geq 1$,
\[
F_m(A) = \bigoplus_{j=1}^K {\mathbb{Q}}^{d(m,j)} \text{ where } d(m,j) = \begin{cases}
1 &: \text{ if } 1\leq m\leq 2n_j-1, m\text{ odd}, \\
0 &: \text{ otherwise}
\end{cases}
\]
and $F_m(B)$ has an analogous expression. Moreover, $F_m(\varphi):F_m(A)\to F_m(B)$ is represented as multiplication by the matrix $\Phi$. We now use this as the starting point to study A$\mathbb{T}$-algebras.

\section{$\ast$-Homomorphisms between Circle Algebras}\label{sec_hom_circle}

Let $\mathbb{T} := \{z \in {\mathbb{C}} : |z| = 1\}$ denote the unit circle. A circle algebra is an algebra of the form $C(\mathbb{T})\otimes F$ where $F$ is a finite dimensional C*-algebra. In this section, we will describe a particularly tractable class of $\ast$-homomorphisms between two circle algebras, and prove that they are generic in a certain sense. We begin by revisiting a result of Thomsen \cite{thomsen_circle}, where he classified such maps upto approximate unitary equivalence.  \\

Let $\t^n$ denote the set of unordered $n$-tuples in $\mathbb{T}$ (with $\t^0=\emptyset$).  If $\Sigma_n$ denotes the symmetric group on $n$ letters, we may think of $\t^n$ as a closed subset of ${\mathbb{C}}^n/\Sigma_n$. Let
\[
A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}, \text{ and } B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}.
\]
Let $\varphi: A\to B$ be a $\ast$-homomorphism and let $u$ denote the canonical unitary generator of $C(\mathbb{T})$. Fix $1\leq j\leq K$, and choose minimal projections $e_j\in M_{n_j}$. Set $u_j=(0,0,\ldots,0,u\otimes e_j,0,\ldots,0)\in A$ and write
\[
\varphi(u_j)=(w_{1,j}, w_{2,j},\hdots, w_{L,j}) \in B
\]
For each $1\leq i\leq L$, $p_{i,j}=w_{i,j}w_{i,j}^{\ast}=w_{i,j}^{\ast}w_{i,j}$ is a projection in $C(\mathbb{T})\otimes M_{\ell_i}$, so the value of $\tr(w_{ij}w_{ij}^{\ast}(z))$ does not vary with $z\in\mathbb{T}$. Let $a_{i,j} \in {\mathbb{N}}\cup\{0\}$ be that constant value. Define $\widehat{\varphi}_{i,j}:\mathbb{T}\to \t^{a_{i,j}}$ by
\[
\widehat{\varphi}_{i,j}(z)=[\lambda\in \sigma(w_{i,j}(z)): |\lambda|=1].
\]
It is a fact that the maps $\widehat{\varphi}_{i,j}$ are all continuous. The values $\{a_{i,j}\}$ are called the \emph{multiplicity constants} of $\varphi$, and $\{\widehat{\varphi}_{i,j}\}$ are called the \emph{characteristic functions} for $\varphi$. Thomsen's main result is that the $\ast$-homomorphism $\varphi$ is uniquely determined (upto approximate unitary equivalence) by these quantities.

\begin{theorem}[\cite{thomsen_circle}, Theorem 2.1]\label{thomsen_1}
~\begin{enumerate}
\item Two $\ast$-homomorphisms $\varphi,\psi:A\to B$ are approximately unitarily equivalent if and only if $\widehat{\varphi}_{i,j}=\widehat{\psi}_{i,j}$ for all $i=1,2,\ldots, L$ and $j=1,2,\ldots, K$.
\item Let $\{a_{i,j}: 1\leq i\leq L, 1\leq j\leq K\}$ be a set of non-negative integers such that $\sum_{j=1}^{K}a_{ij}n_j\leq m_i$ for each $i \in \{1,2,\ldots, L\}$, and let $\eta_{i,j}:\mathbb{T}\to \t^{a_{i,j}}$ be continuous maps. Then, there exists a $\ast$-homomorphism $\varphi:A\to B$ such that $\widehat{\varphi}_{i,j}=\eta_{i,j}$ for all $i,j$.
\end{enumerate}
\end{theorem}
\begin{proof}
Since the construction is crucial for our needs, we recall the proof of part (2). For clarity, we break it into cases. In what follows we will identify $C(\mathbb{T})$ with $\{f \in C[0,1] : f(0) = f(1)\}$.\\

Suppose first that $A = C(\mathbb{T})\otimes M_n$ and $B = C(\mathbb{T})\otimes M_{\ell}$ so that $na \leq \ell$, and $\eta : \mathbb{T}\to \t^a$ is a continuous function. By \cite[Theorem 5.2 of Chapter II]{kato}, there are continuous functions $\lambda_1, \lambda_2, \ldots, \lambda_a : [0,1]\to \mathbb{T}$ such that
\[
\eta(e^{2\pi it}) = [\lambda_1(t), \lambda_2(t), \ldots, \lambda_a(t)]
\]
for all $t\in [0,1]$, and there is a permutation $\sigma \in \Sigma_a$ such that $\lambda_{\sigma(p)}(0)=\lambda_{p}(1)$ for each $p \in \{1,2,\ldots, a\}$. Let $v_{\sigma} \in \u_a$ be the permutation matrix obtained from the identity matrix by permutating its rows according to $\sigma$. Let $w:[0,1]\to \u_a$ be a continuous path with $w(0) = 1$ and $w(1) = v_{\sigma}$, and let $u : [0,1]\to \u_{\ell}$ be given by $u(t) = \diag(w(t)\otimes I_n, I_{\ell-na})$. We may then define $\varphi : C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_{\ell}$ by
\begin{equation}\label{generic_hom}
\varphi(f)(t)=u(t)\begin{pmatrix}
f(\lambda_1(t))& 0 &\ldots & 0 & 0\\
0 & f(\lambda_2(t))& \ldots & 0 & 0 \\
\vdots &\vdots &\vdots & \vdots & \vdots \\
0& 0 & \ldots & f(\lambda_a(t)) & 0 \\
0 & 0 & \ldots & 0 & 0_{m-na}
\end{pmatrix} u(t)^{\ast}.
\end{equation}
Then, $\varphi$ is a well-defined $\ast$-homomorphism with characteristic function $\eta$.\\

Now suppose $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = C(\mathbb{T})\otimes M_{\ell}$. Then, $\{a_1, a_2, \ldots, a_K\}$ are non-negative integers such that $\sum_{j=1}^K a_jn_j \leq \ell$ and $\eta_j:\mathbb{T}\to \t^{a_j}$ are continuous functions. For $j \in \{1,2,\ldots, K\}$, define a $\ast$-homomorphism $\varphi_j : C(\mathbb{T})\otimes M_{n_j}\to C(\mathbb{T})\otimes M_{a_{1,j}n_j}$ whose characteristic function is $\eta_{j}$ using the recipe from \cref{generic_hom}. We may then define $\varphi : A\to B$ by
\[
(g_1,g_2,\hdots,g_K)\mapsto \diag(\varphi_1(g_1),\varphi_2(g_2),\hdots, \varphi_K(g_K),0,0\hdots,0).
\]
Then, $\varphi$ is a $\ast$-homomorphism with multiplicity constants $\{a_1,a_2,\ldots, a_K\}$ and characteristic functions $\{\eta_{1}, \eta_{2},\ldots, \eta_{K}\}$. \\

Finally, if $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}$, then we may define $\varphi_i : A \to C(\mathbb{T})\otimes M_{\ell_i}$ using the above recipe with multiplicity constants $\{a_{i,1}, a_{i,2}, \ldots, a_{i,K}\}$ and characteristic functions $\{\eta_{i,1}, \eta_{i,2}, \ldots, \eta_{i,K}\}$. Then define $\varphi : A\to B$ by $\varphi(a) = (\varphi_1(a), \varphi_2(a), \ldots, \varphi_L(a))$. Once again, this map satisfies the required properties.
\end{proof}

The proof of this theorem allows us to make the following definition.

\begin{defn}\label{def:gen_hom}
A $\ast$-homomorphism between two circle algebras is said to be of \emph{Type A} if it is in the form described in the preceding proof.
\end{defn}

With this terminology in place, Thomsen's theorem merely states that every $\ast$-homomorphism between circle algebras is approximately unitarily equivalent to a $\ast$-homomorphism of Type A. \\

Now consider the special case of the earlier proof with $A = C(\mathbb{T})\otimes M_n$ and $B = C(\mathbb{T})\otimes M_{\ell}$. Then, a $\ast$-homomorphism $\varphi : A\to B$ of Type A is described by the following data:
\begin{itemize}
\item A non-negative integer $a$ with $an\leq \ell$ (which is the multiplicity of $\varphi$).
\item A permutation $\sigma_{\varphi}\in \Sigma_a$.
\item Continuous maps $\lambda_{p} : [0,1]\to \mathbb{T}$ such that $\lambda_{\sigma_{\varphi}(p)}(0) = \lambda_{p}(1)$ for each $p \in \{1,2,\ldots, a\}$.
\item A continuous path $w : [0,1]\to \u_a$ with $w(0) = I_a$ and $w(1) = v_{\sigma_{\varphi}}$ (where $v_{\sigma_{\varphi}}$ is the permutation matrix associated to $\sigma_{\varphi})$.
\end{itemize}
The tuple $(a, \sigma_{\varphi}, \lambda_1, \lambda_2, \ldots, \lambda_a, w)$ will henceforth referred to as the \emph{data tuple} associated to $\varphi$. For convenience, we will write $\sigma_0$ for the identity permutation, and $w_0$ for the constant path at the identity. A $\ast$-homomorphism $\varphi : A\to B$ is said to be of
\begin{itemize}
\item Type B if it is of Type A and $\sigma_{\varphi} = \sigma_0$.
\item Type C if it is of Type B and $w = w_0$.
\item Type D if it is of Type C and each $\lambda_{p}$ is a loop in $\mathbb{T}$ based at $1$.
\end{itemize}

Now suppose $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}$. Then, a $\ast$-homomorphism $\varphi : A\to B$ of Type A can be built up from $\ast$-homomorphisms $\varphi_{i,j} : C(\mathbb{T})\otimes M_{n_j} \to C(\mathbb{T})\otimes M_{\ell_i}$ as in the proof above. We say that $\varphi$ is of Type B (respectively of Type C or Type D) if each $\varphi_{i,j}$ of Type B (respectively of Type C or Type D). \\

We now wish to show that every $\ast$-homomorphism of Type A is homotopic to a $\ast$-homomorphism of Type B. Before we prove this, we introduce some notations that will be used in the proof. Let $\lambda_1,\lambda_2:[0,1]\to\mathbb{T}$ be two continuous maps such that $\lambda_1(1)=\lambda_2(0)$. Then, for $s\in[0,1]$, define $\lambda_i^s,\lambda_{1,2}^s:[0,1]\to\mathbb{T}$ by
\[
\lambda_{1,2}^s(t)= \left\{
\begin{aligned}
& \lambda_1(2st + 1 -s) &: \text{ if } 0\leq t\leq \frac{1}{2} \\
& \lambda_2(2t-1) &: \text{ if } \frac{1}{2}\leq t\leq 1\\
\end{aligned}
\right\}
\text{ and }\lambda_i^s(t)=\lambda_i(t(1-s)).
\]
We also write $\lambda_{1,2} := \lambda_{1,2}^1$.

\begin{theorem}\label{thm_homotopy_type_ab}
A $\ast$-homomorphism of Type A between two circle algebras is homotopic to a $\ast$-homomorphism of Type B with the same multiplicity constants.
\end{theorem}
\begin{proof}
Let $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}$, and let $\varphi : A\to B$ be a $\ast$-homomorphism of Type A. Then, the components of $\varphi$, given by $\varphi_i : A\to C(\mathbb{T})\otimes M_{\ell_i}, i=1,2,\ldots, L$, are themselves $\ast$-homomorphisms of Type A. Furthermore, each $\varphi_i$ is of the form
\[
\varphi_i(g_1,g_2,\ldots, g_K) = \diag\Big(\varphi_{i,1}(g_1), \varphi_{i,2}(g_2), \ldots, \varphi_{i,K}(g_K), 0, \ldots, 0 \Big).
\]
where each map $\varphi_{i,j} : C(\mathbb{T})\otimes M_{n_j} \to C(\mathbb{T})\otimes M_{\ell_i}$ is of Type A. Hence, to prove the theorem, it suffices to assume that $A = C(\mathbb{T})\otimes M_n, B = C(\mathbb{T})\otimes M_{na}$ and $\varphi : A\to B$ is a unital $\ast$-homomorphism of Type $A$. Let $(a, \sigma_{\varphi}, \lambda_1, \lambda_2, \ldots, \lambda_a, w)$ be the data tuple associated to $\varphi$ so that
\[
\varphi(f)(t)=u(t)\begin{pmatrix}
f(\lambda_1(t))& 0 &\hdots & 0\\
0 & f(\lambda_2(t))& \hdots & 0\\
\hdots &\hdots &\hdots & \hdots\\
0& 0 & \hdots & f(\lambda_a(t))
\end{pmatrix} u(t)^{\ast}.
\]
where $u:[0,1]\to \u_{na}$ is the path $u(t) = w(t)\otimes I_n$. We now break the proof into cases for convenience.
\begin{enumerate}
\item Suppose $\sigma_{\varphi}$ is a transposition in $\Sigma_a$, then for simplicity of notation we assume $\sigma_{\varphi} = (1,2)$. Then $\lambda_1(0) = \lambda_2(1), \lambda_1(1) = \lambda_2(0)$, and the other $\lambda_{p}$ are all loops. Consider $H:[0,1]\times A\to B$ given by
\[
H(s,f)(t) = u(t)\diag\Big(f(\lambda_1^s(t)), f(\lambda_{1,2}^s(t)), f(\lambda_3(t)), \ldots, f(\lambda_a(t)) \Big)u(t)^{\ast}.
\]
Then, $H$ is well-defined because for any $f\in A$ and $s\in [0,1]$,
\begin{eqsplit}
H(s,f)(1) &= v_{\sigma_{\varphi}}\diag\Big(f(\lambda_1(1-s)), f(\lambda_2(1)), f(\lambda_3(1)), \ldots, f(\lambda_a(1)) \Big)v_{\sigma_{\varphi}}^{\ast} \\
&= \diag\Big(f(\lambda_2(1)), f(\lambda_1(1-s)), f(\lambda_3(1)), \ldots, f(\lambda_a(1)) \Big) \\
&= \diag\Big(f(\lambda_1(0)), f(\lambda_1(1-s)), f(\lambda_3(0)), \ldots, f(\lambda_a(0)) \Big) \\
&= H(s,f)(0).
\end{eqsplit}
Furthermore, $\lambda_{1,2}^0 \sim_h \lambda_2$, so
\[
H(0,f)(t) = u(t)\diag\Big(f(\lambda_1(t)), f(\lambda_{1,2}^0(t)), f(\lambda_3(t)), \ldots, f(\lambda_a(t)) \Big)u(t)^{\ast}
\]
defines a $\ast$-homomorphism such that $H(0,\cdot) \sim_h \varphi$. Also,
\[
H(1,f)(t) = u(t)\diag\Big(f(\lambda_1(0)), f(\lambda_{1,2}(t)), f(\lambda_3(t)), \ldots, f(\lambda_a(t)) \Big)u(t)^{\ast}
\]
which is of Type B since $\lambda_{1,2}$ is a loop.
\item Now suppose $\sigma_{\varphi}$ is a cycle of length $n$, and assume by induction that the result is true for any $\ast$-homomorphism of Type A whose associated permutation is a cycle of length $<n$. Once again, for simplicity, we assume $\sigma_{\varphi} = (1,2,\ldots, n)$. Then, $\lambda_1(1) = \lambda_2(0)$, $\lambda_2(1) = \lambda_3(0)$, $\ldots, \lambda_{n-1}(1) = \lambda_n(0)$, $\lambda_n(1) = \lambda_1(0)$. Also, the other $\lambda_{p}$ are all loops. Define $H:[0,1]\times A \to B$ by
\[
H(s,f)(t) = u(t)\diag\Big(f(\lambda_1^s(t)), f(\lambda_{1,2}^s(t)), f(\lambda_3(t)), \ldots, f(\lambda_n(t)), f(\lambda_{n+1}(t)), \ldots, f(\lambda_a(t)) \Big)u(t)^{\ast}.
\]
Then, $H$ is well-defined because for any $f\in A$ and $s\in [0,1]$,
\begin{eqsplit}
H(s,f)(1) &= v_{\sigma_{\varphi}}\diag\Big(f(\lambda_1(1-s)), f(\lambda_2(1)), f(\lambda_3(1)), \ldots, f(\lambda_n(1)), f(\lambda_{n+1}(1)), \ldots, f(\lambda_a(1)) \Big)v_{\sigma_{\varphi}}^{\ast} \\
&= \diag\Big(f(\lambda_n(1)), f(\lambda_1(1-s)), f(\lambda_2(1)), \ldots, f(\lambda_{n-1}(1)), f(\lambda_{n+1}(1)), \ldots, f(\lambda_a(1)) \Big) \\
&= \diag\Big(f(\lambda_1(0)), f(\lambda_1(1-s)), f(\lambda_3(0)), \ldots, f(\lambda_n(0)), f(\lambda_{n+1}(0)), \ldots, f(\lambda_a(0)) \Big) \\
&= H(s,f)(0).
\end{eqsplit}
As before, since $\lambda_{1,2}^0 \sim_h \lambda_2$, it follows that $H(0,\cdot) \sim_h \varphi$. Now note that
\[
H(1,f)(t) = u(t)\diag\Big(f(\lambda_1(0)), f(\lambda_{1,2}(t)), f(\lambda_3(t)), \ldots, f(\lambda_n(t)), f(\lambda_{n+1}(t)), \ldots, f(\lambda_a(t)) \Big)u(t)^{\ast}.
\]
Thus, $H(1,\cdot) : A\to B$ is a $\ast$-homomorphism of Type A whose associated permutation $\tau$ is the cycle $\tau = (2,3,\ldots, n)$. By induction, $H(1,\cdot)$ is homotopic to a $\ast$-homomorphism $\psi : A\to B$ of Type B, and thus $\varphi \sim_h \psi$.
\item Now suppose $\sigma_{\varphi} \in \Sigma_a$ is any permutation, then we may express $\sigma_{\varphi}$ as a product of disjoint cycles $\sigma_{\varphi} = \sigma_1\sigma_2\ldots \sigma_k$. Then each $\sigma_i$ may be reduced to the identity permutation (keeping the multiplicity intact) by part (2). This reduces $\sigma_{\varphi}$ to the identity permutation, proving the result.
\end{enumerate}
\end{proof}

Let $\varphi : C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_{\ell}$ be a $\ast$-homomorphism of Type B, and let $(a, \sigma_0, \lambda_1, \lambda_2, \ldots, \lambda_a, w)$ be a data tuple associated to $\varphi$. Then, by construction $w:[0,1]\to \u_a$ is a path based at the identity, and hence defines a unitary $w \in C(\mathbb{T})\otimes M_a$. Therefore, $\varphi$ takes the form
\[
\varphi(f)(t) = u(t)\diag\Big(f(\lambda_1(t)), f(\lambda_2(t)), \ldots, f(\lambda_a(t))u(t)^{\ast}
\]
where $u$ is now a unitary in $C(\mathbb{T})\otimes M_m$ given by $u(t) = \diag(w(t)\otimes I_n,I_{\ell-na})$. We conclude that

\begin{prop}\label{prop_type_bc}
A $\ast$-homomorphism between two circle algebras of Type B is unitarily equivalent to one of Type C.
\end{prop}

Now let $\varphi : C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_{\ell}$ be a $\ast$-homomorphism of Type C, and let $(a, \sigma_0, \lambda_1, \lambda_2, \ldots, \lambda_a, w_0)$ be a data tuple associated to $\varphi$. Then, by construction, each $\lambda_{p}$ is a loop in $\mathbb{T}$. If $b_{p}$ denotes the winding number of the loop $\lambda_{p}$, then there exists $c_{p} \in \mathbb{T}$ such that
\[
\lambda_{p} \sim_h c_{p}\delta_{b_{p}}
\]
where $\delta_n : \mathbb{T}\to \mathbb{T}$ denotes the map $z \mapsto z^n$. Hence, $\varphi$ is homotopic to a $\ast$-homomorphism of Type C whose data set takes the form $(a, \sigma_0, c_1\delta_{b_1}, c_2\delta_{b_2}, \ldots, c_a\delta_{b_a}, w_0)$. Now fix $1\leq p\leq a$, and choose $d_{p} \in {\mathbb{R}}$ such that $c_{p} = e^{2\pi i d_{p}}$. Then, the function $F_{p}:[0,1]\times C(\mathbb{T}) \to C(\mathbb{T})$ given by
\[
F_{p}(s,f)(z) = f(e^{2\pi i sd_{p}}z^{b_{p}})
\]
is a homotopy from $c_{p}\delta_{b_{p}}$ to $\delta_{b_{p}}$. Applying these homotopies along the diagonal as before, we see that $\varphi$ is homotopic to a $\ast$-homomorphism of Type C whose data set is $(a, \sigma_0, \delta_{b_1}, \delta_{b_2}, \ldots, \delta_{b_a}, w_0)$. Applying this componentwise to an arbitrary $\ast$-homomorphism of Type C, we conclude that

\begin{prop}\label{prop_type_cd}
A $\ast$-homomorphism between two circle algebras of Type C is homotopic to one of Type D.
\end{prop}

Since we are interesting in continuous homology theories applied to inductive limits of circle algebras, it suffices for our purposes to consider homomorphisms of Type D. Therefore, we will henceforth refer to $\ast$-homomorphisms of Type D as \emph{diagonal} maps.

\section{Calculation of $F_m(A)$ when $A$ is an A$\mathbb{T}$-algebra}\label{sec_rat_hom_at}

Recall that, for a C*-algebra $A$ and for $m\geq 1$, $F_m(A) =\pi_m(\widehat{\mathcal{U}}(A))\otimes {\mathbb{Q}}$. The aim of this section is to compute $F_m(\varphi)$ when $\varphi$ is a diagonal $\ast$-homomorphism between circle algebras, and then use it to compute $F_m(A)$ when $A$ is an A$\mathbb{T}$-algebra.

\begin{prop}\label{prop_iso_fm}
If $A= \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$, then for $m\geq 1$, $F_m(A)$ is given by
\begin{eqsplit}
F_m(A) &\cong \bigoplus_{j=1}^{K} \pi_m(\mathcal{U}_{n_j})\otimes{\mathbb{Q}}\oplus\pi_{m+1}(\mathcal{U}_{n_j})\otimes{\mathbb{Q}} \\
&\cong \bigoplus_{j=1}^K {\mathbb{Q}}^{d(m,j)} \text{ where } d(m,j) = \begin{cases}
1 &: \text{ if } 1\leq m \leq 2n_j-1, \\
0 &: \text{ otherwise}.
\end{cases}
\end{eqsplit}
\end{prop}
\begin{proof}
If $D = \bigoplus_{j=1}^K M_{n_j}$, then there is a split exact sequence $0 \to C_{\ast}(\mathbb{T}, D) \to A \to D \to 0$, where $C_{\ast}(\mathbb{T}, D) = \{f \in C(\mathbb{T},D) : f(1) = 0\} \cong SD$, the suspension of $D$. Since $F_m$ is a homology theory, it follows that $F_m(A) \cong F_m(D)\oplus F_m(SD) \cong F_m(D)\oplus F_{m+1}(D)$. The result now follows from additivity of $F_m(\cdot)$ and the fact that
\[
F_m(M_n) = \pi_m(\u_n)\otimes {\mathbb{Q}} \cong \begin{cases}
{\mathbb{Q}} &: \text{ if } 1 \leq m\leq 2n-1, m \text{ odd} \\
0 &: \text{ otherwise}.
\end{cases}
\]
by \cite[Example 1.6]{apurva_pv_af}.
\end{proof}

Since it appears frequently below, we define
\[
V_n^m := \pi_m(\u_n)\otimes {\mathbb{Q}} \oplus \pi_{m+1}(\u_n)\otimes {\mathbb{Q}}.
\]
For convenience of notation, we will often treat entries in $V_n^m$ as pairs $(x,y)$ with $x \in \pi_m(\u_n)\otimes {\mathbb{Q}}$ and $y\in \pi_{m+1}(\u_n)\otimes {\mathbb{Q}}$ (even though both terms cannot be non-zero simultaneously).

\begin{prop}\label{prop_fm_circle}
For $1\leq p \leq a$, let $\lambda_{p} : \mathbb{T}\to \mathbb{T}$ be continuous loops based at $1$ with winding number $w(\lambda_{p})$. Let $\varphi:C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_{\ell}$ be the diagonal $\ast$-homomorphism given by
\[
f\mapsto\diag(f\circ\lambda_1,f\circ\lambda_2,\hdots,f\circ\lambda_a, 0_{\ell-na}).
\]
Then for any $m\geq 1$, $F_m(\varphi): V_n^m\to V_{\ell}^m$ is given by
\[
(x,y) \mapsto \left(ax, by\right)
\]
where $b = \sum_{p=1}^a w(\lambda_{p})$.
\end{prop}
The pair $(a,b)$ will henceforth be referred to as the \emph{signature} of the diagonal map $\varphi$.
\begin{proof}
Let $\iota : C(\mathbb{T})\otimes M_{na}\to C(\mathbb{T})\otimes M_{\ell}$ be the inclusion map $x \mapsto \diag(x,0)$. By the naturality of the isomorphism in \cref{prop_iso_fm} and \cite[Lemma 2.2]{apurva_pv_af},
\[
F_m(\iota) = \begin{cases}
\text{id} &: \text{ if } 1 \leq m\leq 2na-1, \\
0 &: \text{ otherwise}.
\end{cases}
\]
Therefore, it suffices to consider the case when $\ell=na$. Furthermore, we may assume without loss of generality that $a=2$. In that case, for $p \in \{1,2\}$, let $\varphi_{p} : C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_n$ denote the map $f\mapsto f\circ \lambda_{p}$. Then $\varphi(f) = \diag(\varphi_1(f), \varphi_2(f))$ for all $f\in C(\mathbb{T})\otimes M_n$. For any $f\in C(\mathbb{T},\u_n)$, $\varphi(f) = \widehat{\varphi}_1(f)\cdot \widehat{\varphi}_2(f)$, where $\widehat{\varphi}_{p} : C(\mathbb{T},\u_n) \to C(\mathbb{T},\u_{2n})$ are the maps
\[
\widehat{\varphi}_1(f)=\begin{pmatrix}
\varphi_1(f) & 0\\
0 & 1
\end{pmatrix}\text{ and } \widehat{\varphi}_2(f)=\begin{pmatrix}
1 & 0\\
0 & \varphi_2(f)
\end{pmatrix}.
\]
Once again, since inclusion induces the identity map, we conclude that
\[
F_m(\varphi) = F_m(\varphi_1) + F_m(\varphi_2).
\]
Therefore, it remains to compute $F_m(\psi)$ when $\psi : C(\mathbb{T})\otimes M_n\to C(\mathbb{T})\otimes M_n$ is the $\ast$-homomorphism given by
\[
\psi(f) = f\circ \lambda,
\]
where $\lambda : \mathbb{T}\to \mathbb{T}$ is a continuous loop based at $1$. In that case, fix $m\geq 1$, and observe that $F_m$ is a homotopy invariant functor. Therefore, we may assume without loss of generality that $\lambda(z) = z^b$ where $b = w(\lambda)$ is the winding number of $\lambda$. Now let $C_{\ast}(\mathbb{T},M_n) := \{f\in C(\mathbb{T},M_n) : f(1) = 0\}$ and consider the split exact sequence
\[
\xymatrix{
0\ar[r] & C_{\ast}(\mathbb{T},M_n) \ar[d]^{\widehat{\psi}} \ar[r] &C(\mathbb{T},M_n)\ar[d]^{\psi}\ar[r]^{q} & M_n\ar[d]^{\text{id}}\ar[r] & 0 \\
0\ar[r] & C_{\ast}(\mathbb{T},M_n) \ar[r] & C(\mathbb{T},M_n)\ar[r]^{q} & M_n\ar[r] & 0
}
\]
where $q$ is the evaluation map at $1\in\mathbb{T}$ and $\widehat{\psi}$ is the restriction of $\psi$ to $C_{\ast}(\mathbb{T},M_n)$. Under the isomorphism $F_m(C(\mathbb{T})\otimes M_n) \cong V_n^m$, the map $F_m(\psi) : V_n^m\to V_n^m$ is given by
\[
F_m(\psi)(x,y) = (F_m(\text{id})(x), F_m(\widehat{\psi})(y)).
\]
Under the isomorphism $\pi_m(\u(C_{\ast}(\mathbb{T}, M_n))) \cong \pi_{m+1}(\u_n)$, $\widehat{\psi}$ induces the map $\pi_m(\widehat{\psi})([G]) = b[G]$. 	Thus, $F_m(\widehat{\psi})$ is realized as multiplication by $b$, proving the result.
\end{proof}

Now suppose $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}$, and suppose $\varphi : A\to B$ is a diagonal $\ast$-homomorphism. Then, each map
\[
\varphi_{i,j} : C(\mathbb{T})\otimes M_{n_j} \hookrightarrow A \xrightarrow{\varphi} B \to C(\mathbb{T})\otimes M_{\ell_i}
\]
is a diagonal $\ast$-homomorphism as in \cref{prop_fm_circle}. If $(a_{i,j}, b_{i,j})$ is the signature of $\varphi_{i,j}$, then we obtain an $L\times K$ matrix $\Phi \in M_{L\times K}({\mathbb{N}}_0\times {\mathbb{N}}_0)$ given by
\[
\Phi := \begin{pmatrix}
(a_{1,1}, b_{1,1}) & (a_{1,2}, b_{1,2}) & \ldots & (a_{1,K}, b_{1,K}) \\
(a_{2,1}, b_{2,1}) & (a_{2,2}, b_{2,2}) & \ldots & (a_{2,K}, b_{2,K}) \\
\vdots & \vdots & \vdots & \vdots \\
(a_{L,1}, b_{L,1}) & (a_{L,2}, b_{L,2}) & \ldots & (a_{L,K}, b_{L,K})
\end{pmatrix}
\]
This is called the \emph{signature} matrix of $\varphi$. \cref{prop_iso_fm} and \cref{prop_fm_circle} together yield the following theorem.

\begin{theorem}\label{thm_homomorphism_circle}
Let $A = \bigoplus_{j=1}^K C(\mathbb{T})\otimes M_{n_j}$ and $B = \bigoplus_{i=1}^L C(\mathbb{T})\otimes M_{\ell_i}$, and suppose $\varphi : A\to B$ is a diagonal $\ast$-homomorphism with signature matrix $\Phi$ as above. Then, for any $m\geq 1$,
\[
F_m(\varphi) : \bigoplus_{j=1}^K V_{n_j}^m \to \bigoplus_{i=1}^L V_{\ell_i}^m
\]
is given by multiplication by $\Phi$, in the sense that
\[
F_m(\varphi)((x_j,y_j)_{1\leq j\leq K}) = \left(\sum_{j=1}^K a_{i,j}x_j, \sum_{j=1}^K b_{i,j}y_j\right)_{1\leq i\leq L}
\]
\end{theorem}

Finally, with the results of \cref{sec_hom_circle}, we arrive at the first main result of the paper. A few words of caution before we describe the result though. Given an A$\mathbb{T}$-algebra $A$, we may write $A$ as an inductive limit $A = \lim_{n\to\infty} (A_n,\psi_n)$ where each $A_n$ is a circle algebra, and $\psi_n : A_n\to A_{n+1}$ is a $\ast$-homomorphism. It is not, in general, possible to replace $\psi_n$ by a suitable diagonal $\ast$-homomorphism. However, when computing $F_m(\cdot)$ (or any other continuous homology theory), one may use the results of \cref{sec_hom_circle} to replace $\psi_n$ by maps that are diagonal. Now, the resulting inductive limit algebra is \emph{not necessarily} isomorphic to the algebra $A$. However, the functor $F_m(\cdot)$ is blind to the difference, which is what allows this theorem to work.

\begin{theorem}\label{thm_at_alg_rat_hom}
Let $A$ be an A$\mathbb{T}$-algebra that is expressed as an inductive limit of circle algebras
\[
A_1\xrightarrow{\psi_1} A_2\xrightarrow{\psi_2} A_3\to \ldots \to A.
\]
Then, for each $m\geq 1$, the group $F_m(A)$ may be computed as the inductive limit of a sequence
\[
F_m(A_1) \xrightarrow{\Phi_1} F_m(A_2)\xrightarrow{\Phi_2} F_m(A_3)\to \ldots \to F_m(A),
\]
where the connecting maps are given by multiplication by the signature matrices of certain diagonal $\ast$-homomorphism $\varphi_n : A_n\to A_{n+1}$.
\end{theorem}
\begin{proof}
Fix $m\geq 1$. Given an inductive sequence $A_1\xrightarrow{\psi_1} A_2 \xrightarrow{\psi_2} A_3 \to \ldots \to A$, we may assume by Thomsen's theorem (\cref{thomsen_1}) that each $\psi_i$ is a map of Type A. For each $i \in {\mathbb{N}}$, \cref{thm_homotopy_type_ab} tells us that there is a $\ast$-homomorphism $\eta_i : A_i \to A_{i+1}$ of Type B such that $\psi_i \sim_h \eta_i$. Suppose $B$ is the inductive limit of the sequence
\[
A_1\xrightarrow{\eta_1} A_2 \xrightarrow{\eta_2} A_3 \to \ldots \to B.
\]
Then, it follows that $F_m(A)$ is naturally isomorphic to $F_m(B)$ (even though $A$ and $B$ may not be isomorphic). Once again, by \cref{prop_type_bc}, we may replace $\eta_i$ by another map $\delta_i : A_i \to A_{i+1}$ that is of Type C and is unitarily equivalent to $\eta_i$, so that $B \cong \lim (A_n, \delta_n)$. Finally, by \cref{prop_type_cd}, there are diagonal $\ast$-homomorphism $\varphi_i : A_i\to A_{i+1}$ such that $\varphi_i \sim_h \delta_i$. If $A'$ denotes the inductive limit of the sequence
\[
A_1\xrightarrow{\varphi_1} A_2 \xrightarrow{\varphi_2} A_3 \to \ldots \to A',
\]
then $F_m(A) \cong F_m(A')$. The result now follows by appealing to \cref{thm_homomorphism_circle}.
\end{proof}

The next two examples illustrate this theorem.

\begin{ex}\label{ex_bunce_deddens}
Let $B$ be the Bunce-Deddens algebra given as the inductive limit of the sequence
\[
C(\mathbb{T})\xrightarrow{\varphi_0} C(\mathbb{T})\otimes M_2\xrightarrow{\varphi_1} C(\mathbb{T})\otimes M_4\xrightarrow{\varphi_2} \to \ldots\to B
\]
where $\varphi_n(f):C(\mathbb{T})\otimes M_{2^n}\to C(\mathbb{T})\otimes M_{2^{n+1}}$ is given by
\[
\varphi_n(f)(t) = (u(t)\otimes I_{2^n})\begin{pmatrix}
f(e^{\pi i t}) & 0\\
0 & f(e^{\pi i(1+t)})
\end{pmatrix}(u(t)\otimes I_{2^n})^{\ast}
\]
and $u:[0,1]\to \u_2$ is a path of unitaries in $M_2$ satisfying $u(0)=I$ and $u(1)=\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}$. In order to calculate $F_m(B)$, we may replacing $\varphi_n$ by the recipe laid out in \cref{thm_homotopy_type_ab}, and consider the inductive sequence
\[
C(\mathbb{T})\xrightarrow{\psi_0} C(\mathbb{T})\otimes M_2\xrightarrow{\psi_1} C(\mathbb{T})\otimes M_4\to \ldots
\]
where each $\psi_n$ is a diagonal $\ast$-homomorphism given by
\[
\psi_n(f)(z) =\begin{pmatrix}
f(1) & 0\\
0 & f(z)
\end{pmatrix}
\]
Here, the identity map on $\mathbb{T}$ has winding number $1$ and $\psi_n$ has multiplicity $2$. Hence for $m\geq 1$, $\psi_n$ has signature $(2,1)$. In other words, $F_m(\psi_n) : V_{2^n}^m \to V_{2^{n+1}}^m$ is given by
\[
F_m(\psi_n)(x,y)= (2x,y).
\]
Now,
\[
F_m(C(\mathbb{T})\otimes M_{2^n}) = \begin{cases}
{\mathbb{Q}}\oplus 0 &: \text{ if } 1 \leq m\leq 2^{n+1}-1, m \text{ odd}\\
0\oplus {\mathbb{Q}} &: \text{ if } 1\leq m\leq 2^{n+1}-1, m \text{ even}\\
0 &: \text{ otherwise}.
\end{cases}
\]
Therefore, $F_m(B) \cong {\mathbb{Q}}$ for all $m \in {\mathbb{N}}$.
\end{ex}

\begin{ex}\label{ex_goodearl}
Let $(r_n)$ and $(p_n)$ be sequences of positive integers such that $r_n$ divides $r_{n+1}$ and $p_n< r_{n+1}/r_n$ for each $n \in {\mathbb{N}}$. Choose a finite set $F_n = \{z_{n,1}, z_{n,2}, \ldots, z_{n,p_n}\} \subset \mathbb{T}$, and define $\psi_n : C(\mathbb{T})\otimes M_{r_n} \to C(\mathbb{T})\otimes M_{r_{n+1}}$ by
\[
\psi_n(f)(z)=\diag(f(z_{n,1}), f(z_{n,2}),\ldots,f(z_{n,p_n}), f(z),f(z),\ldots, f(z)).
\]
Consider the algebra $G$ given as an inductive limit of the sequence
\[
C(\mathbb{T})\otimes M_{r_1}\xrightarrow{\psi_1} C(\mathbb{T})\otimes M_{r_2}\xrightarrow{\psi_2} C(\mathbb{T})\otimes M_{r_3}\to \ldots\to G.
\]
If $\varphi_n : C(\mathbb{T})\otimes M_{r_n} \to C(\mathbb{T})\otimes M_{r_{n+1}}$ denotes the map
\[
\varphi_n(f)(z) = \diag(f(1), f(1), \ldots, f(1), f(z),f(z),\ldots, f(z)),
\]
then $\psi_n\sim_h \varphi_n$. Observe that for $m\geq 1$, the signature of $\varphi_n$ is $(r_{n+1}/r_n, r_{n+1}/r_n-p_n)$. Therefore, for each $m\geq 1$, $F_m(\psi_n) : V_{r_n}^m \to V_{r_{n+1}}^m$ is thus given by
\[
F_m(\psi_n)(x,y)= ((r_{n+1}/r_n)x,(r_{n+1}/r_n-p_n)y)).
\]
Hence, $F_m(G)={\mathbb{Q}}$ for all $m\geq 1$ (since $r_{n+1}/r_n > p_n \geq 1$). Note that if $\bigcup_{n=k}^{\infty} F_n$ is dense in $\mathbb{T}$ for each $k \in {\mathbb{N}}$, then $G$ is simple and thus a Goodearl algebra.
\end{ex}

\section{K-Stability}\label{sec_k_stability_at}

We now turn to a proof of \cref{mainthm_k_stable} and begin with the definition of K-stability. As mentioned before, this was first studied by Thomsen \cite{thomsen}. Its rational counterpart was also discussed by Farjoun and Schochet \cite{farjoun}.

\begin{defn}\label{defn:k_stable}
Let $A$ be a C*-algebra and $j\geq 2$. Define $\iota^A_j: M_{j-1}(A)\to M_j(A)$ to be the natural inclusion map
\[
a\mapsto\begin{pmatrix}
a&0\\
0&0
\end{pmatrix}.
\]
$A$ is said to be \emph{$K$-stable} if $G_k(\iota^A_j): G_k(M_{j-1}(A))\to G_k(M_j(A))$ is an isomorphism for all $k\geq 0$ and all $j\geq 2$. $A$ is said to be \emph{rationally $K$-stable} if $F_m(\iota^A_j):F_m(M_{j-1}(A))\to F_m(M_j(A))$ is an isomorphism for all $m\geq 1$ and all $j\geq 2$.
\end{defn}

Note that, for a $K$-stable C*-algebra, $G_k(A) \cong K_{k+1}(A)$ and for a rationally $K$-stable $C^{\ast}$-algebra, $F_m(A) \cong K_{m+1}(A)\otimes {\mathbb{Q}}$. A variety of interesting C*-algebras are known to be $K$-stable (see \cite[Remark 1.5]{apurva_pv_cx}). Clearly, $K$-stability implies rational $K$-stability. By \cite[Theorem B]{apurva_pv_af}, the converse is true for AF-algebras. However, the converse is not true in general (see \cite[Example 2.1]{apurva_pv_cx_rat}). The aim of this section is to show that, for the class of A$\mathbb{T}$-algebras, both these notions are equivalent. We begin with the following observation.

\begin{rem}\label{rem_generic_hom_limit}
Let $A$ be an A$\mathbb{T}$-algebra, given as an inductive limit $A = \lim (A_n, \varphi_n)$. By \cref{thomsen_1}, we may assume that each $\varphi_n$ is of Type A. In \cref{thm_homotopy_type_ab}, we proved that there is another inductive sequence $(A_n, \psi_n)$, where each $\psi_n$ is a $\ast$-homomorphism of Type B such that $\varphi_n \sim_h \psi_n$ for each $n\in {\mathbb{N}}$. Hence, the inductive limit $A' = \lim (A_n, \psi_n)$ has the property that $G_k(A) \cong G_k(A')$ for each $k\geq 0$. \\

Now suppose $A$ is $K$-stable, then we prove that $A'$ is also $K$-stable (the same argument applies for rational $K$-stability as well). To do this, write $\alpha_n : A_n\to A$ and $\beta_n : A_n\to A'$ be the $\ast$-homomorphisms defining $A$ and $A'$ respectively. In other words, $\alpha_{n+1}\circ \varphi_n = \alpha_n$ and $\beta_{n+1}\circ \psi_n = \beta_n$ for all $n\in {\mathbb{N}}$. Fix $k\geq 0$. For simplicity, we will show that $\eta := G_k(\iota^{A'}_2) : G_k(A')\to G_k(M_2(A'))$ is an isomorphism, under the assumption that $\rho := G_k(\iota^A_2) : G_k(A) \to G_k(M_2(A))$ is an isomorphism. Now observe that the following diagrams commute
\[
\xymatrix{
& G_k(A_i)\ar[rr]^{(\iota^{A_i}_2)_{\ast}}\ar[ld]_{(\alpha_i)_{\ast}}\ar[dd]_{(\varphi_i)_{\ast}} & & G_k(M_2(A_i))\ar[ld]_{(\alpha^{(2)}_i)_{\ast}}\ar[dd]_{(\varphi^{(2)}_i)_{\ast}} \\
G_k(A)\ar@{..>}[rr]^{\rho} && G_k(M_2(A)) \\
& G_k(A_{i+1})\ar[rr]_{(\iota^{A_{i+1}}_2)_{\ast}}\ar[ul]^{{(\alpha_{i+1})_{\ast}}} & & G_k(M_2(A_{i+1}))\ar[ul]^{{(\alpha^{(2)}_{i+1})_{\ast}}} \\
& G_k(A_i)\ar[rr]^{(\iota^{A_i}_2)_{\ast}}\ar[ld]_{(\beta_i)_{\ast}}\ar[dd]_{(\psi_i)_{\ast}} & & G_k(M_2(A_i))\ar[ld]_{(\beta^{(2)}_i)_{\ast}}\ar[dd]_{(\psi^{(2)}_i)_{\ast}} \\
G_k(A')\ar@{..>}[rr]^{\eta} && G_k(M_2(A')) \\
& G_k(A_{i+1})\ar[rr]_{(\iota^{A_{i+1}}_2)_{\ast}}\ar[ul]^{{(\beta_{i+1})_{\ast}}} & & G_k(M_2(A_{i+1}))\ar[ul]^{{(\beta^{(2)}_{i+1})_{\ast}}}
}
\]
(Note that if $\theta : C\to D$ is a $\ast$-homomorphism, we write $\theta^{(2)} : M_2(C)\to M_2(D)$ for the induced $\ast$-homomorphism). Since $(\varphi_i)_{\ast} = (\psi_i)_{\ast}$ and $(\varphi_i^{(2)})_{\ast} = (\psi_i^{(2)})_{\ast}$, the universal property of the inductive limit also tells us that there are isomorphisms $\lambda : G_k(A)\to G_k(A')$ and $\mu : G_k(M_2(A))\to G_k(M_2(A'))$ such that the following diagrams commute
\[
\xymatrix{
& G_k(A_i)\ar[dl]_{(\alpha_i)_{\ast}}\ar[rd]^{(\beta_i)_{\ast}} & & & G_k(M_2(A_i))\ar[dl]_{(\alpha^{(2)}_i)_{\ast}}\ar[rd]^{(\beta^{(2)}_i)_{\ast}} \\
G_k(A)\ar[rr]^{\lambda} & & G_k(A') & G_k(M_2(A))\ar[rr]^{\mu} && G_k(M_2(A'))
}
\]
A short argument shows that
\[
\eta \circ \lambda \circ (\alpha_i)_{\ast} = (\beta^{(2)}_i)_{\ast}\circ (\iota^{A_i}_2)_{\ast} = \mu \circ \rho\circ (\alpha_i)_{\ast}.
\]
Hence, $\eta\circ \lambda = \mu\circ \rho$. In this expression, both $\lambda$ and $\mu$ are isomorphisms. Therefore, if $\rho$ is an isomorphism, so is $\eta$. Now, the same argument applies for each such map $G_k(M_{n-1}(A'))\to G_k(M_n(A'))$ and we conclude that $A'$ is $K$-stable. Since the relationship between $A$ and $A'$ is symmetric, we see that $A$ is $K$-stable if and only if $A'$ is $K$-stable.\\

Therefore, when considering questions of $K$-stability (or rational $K$-stability), we need only consider the case where the connection homomorphisms in the inductive limit are of Type B. In fact, we may repeat this argument using \cref{prop_type_bc} and \cref{prop_type_cd} and assume that the connecting maps are diagonal. This is what we now do.
\end{rem}

Suppose $A$ is an A$\mathbb{T}$-algebra, and $(A_n,\varphi_n)$ is an inductive sequence such that $A = \lim (A_n,\varphi_n)$, and each $\varphi_n$ is a diagonal $\ast$-homomorphism. For each $n \in {\mathbb{N}}$, let $B_n$ be the quotient of $A_n$ obtained by evaluating each summand of $A_n$ at $1\in \mathbb{T}$. Since the connecting maps are diagonal, we get an inductive sequence $(B_n, \psi_n)$ of finite dimensional C*-algebras and maps $\pi_n : A_n\to B_n$ such that $\psi_n\circ \pi_n = \pi_{n+1}\circ \varphi_n$ for all $n\in {\mathbb{N}}$. If $B = \lim (B_n, \psi_n)$, then

\begin{lem}\label{lem_rat_k_stable_af}
If $A$ is rationally K-stable, so is $B$.
\end{lem}
\begin{proof}
It suffices to show that the map $F_m(\iota^B_2) : F_m(B)\to F_m(M_2(B))$ is an isomorphism for each $m\geq 1$. If $m$ is even, $F_m(B) = F_m(M_2(B)) = 0$ by \cite[Lemma 3.2]{apurva_pv_af}, so let $m \in {\mathbb{N}}$ be odd. For each $n\in {\mathbb{N}}$, consider the short exact sequence
\[
0 \to C_{\ast}(\mathbb{T}, B_n) \to A_n \xrightarrow{\pi_n} B_n \to 0
\]
Since the connecting maps $\varphi_n : A_n\to A_{n+1}$ are diagonal, the restriction gives a $\ast$-homomorphism $\widetilde{\varphi_n} : C_{\ast}(\mathbb{T}, B_n) \to C_{\ast}(\mathbb{T}, B_{n+1})$, and we have a commuting diagram of extensions
\[
\xymatrix{
0\ar[r] & C_{\ast}(\mathbb{T}, B_n)\ar[r]\ar[d]_{\widetilde{\varphi_n}} & A_n\ar[r]^{\pi_n}\ar[d]^{\varphi_n} & B_n\ar[r]\ar[d]^{\psi_n} & 0 \\
0\ar[r] & C_{\ast}(\mathbb{T}, B_{n+1})\ar[r] & A_{n+1}\ar[r]^{\pi_{n+1}} & B_{n+1}\ar[r] & 0 \\
}
\]
Now, $F_m(C_{\ast}(\mathbb{T}, B_j)) \cong F_{m+1}(B_j) = 0$ for all $j\in {\mathbb{N}}$ by \cite[Lemma 3.2]{apurva_pv_af}, and thus $F_m(\pi_n) : F_m(A_n) \to F_m(B_n)$ is an isomorphism. This induces an isomorphism $\theta : F_m(A) \to F_m(B)$. \\

Furthermore, the same argument applies with $M_2(A_n)$ instead of $A_n$, and we obtain an isomorphism $F_m(\pi_n^{(2)}) : F_m(M_2(A_n))\to F_m(M_2(B_n))$ such that the following diagram commutes
\[
\xymatrixcolsep{3pc}\xymatrix{
F_m(A_n)\ar[r]^{F_m(\pi_n)}\ar[d]_{F_m(\iota^{A_n}_2)} & F_m(B_n)\ar[d]^{F_m(\iota^{B_n}_2)} \\
F_m(M_2(A_n))\ar[r]^{F_m(\pi_n^{(2)})} & F_m(M_2(B_n))
}
\]
Once again, there is an induced isomorphism $\rho : F_m(M_2(A))\to F_m(M_2(B))$. Since each horizontal map is an isomorphism, we obtain a commuting diagram at the level of inductive limits
\[
\xymatrix{
F_m(A)\ar[r]^{\theta}\ar[d]_{F_m(\iota^{A}_2)} & F_m(B)\ar[d]^{F_m(\iota^{B}_2)} \\
F_m(M_2(A))\ar[r]^{\rho} & F_m(M_2(B))
}
\]
Since $F_m(\iota^A_2)$ is an isomorphism, so is $F_m(\iota^B_2)$.
\end{proof}

We now need to revisit some notation from \cite[Section 3]{apurva_pv_af} and adapt it to our setting here. If $C$ is a finite dimensional C*-algebra, we write
\[
\min\dim(C) = \min\{\text{square root of the dimension of a simple summand of } C\}.
\]
In other words, if $C = \bigoplus_{j=1}^K M_{n_j}$, then $\min\dim(C) = \min\{n_j : 1\leq j\leq K\}$. Also, we write $C^{(j)}$ to be the direct sum of all simple summands of $C$ of dimension equal to $j^2$ (adopting the convention that the direct sum over an empty index set is the zero C*-algebra). Similarly, $C^{(>j)}$ denotes the direct sum of all simple summands of $C$ whose dimension is $>j^2$, and $C^{(<j)}$ is the direct sum of all simple summands of $C$ whose dimension is $<j^2$.  Hence,
\[
C = C^{(<j)}\oplus C^{(j)}\oplus C^{(>j)}.
\]
Now if $A = C(\mathbb{T})\otimes C$ is a circle algebra and $C$ is finite dimensional, we write $\min\dim(A) = \min\dim(C)$, $A^{(j)} := C(\mathbb{T})\otimes C^{(j)}$, and $A^{(>j)}$ and $A^{(<j)}$ are also defined analogously. \\

We are now ready to prove \cref{mainthm_k_stable}, and we do so using the following lemmas. The first of these lemmas is an analogue (and indeed a consequence) of \cite[Lemma 3.7]{apurva_pv_af}.

\begin{lem}\label{lem_rat_k_stable_min_dim}
Let $A$ be a rationally $K$-stable A$\mathbb{T}$-algebra, and suppose $A = \lim (A_p,\varphi_p)$ where each $A_p$ is a circle algebra and each $\varphi_p$ is a diagonal $\ast$-homomorphism. Then, for each $m \in {\mathbb{N}}$, there is a sequence $(A_{m,p}, \varphi_p^{m})$ of circle algebras with diagonal connecting maps such that $A = \lim_{p\to \infty} (A_{m,p}, \varphi_p^{m})$ and
\[
\min\dim(A_{m,p}) \geq m
\]
for all $p \in {\mathbb{N}}$.
\end{lem}
\begin{proof}
The result is true if $m=1$, so we may assume that the result is true for $0 \leq i\leq m$, and now construct the sequence $(A_{m+1,p}, \varphi_p^{m+1})$. Let $(A_{m,p}, \varphi_p^m)$ be an inductive sequence such that $A = \lim_{p\to \infty} (A_{m,p}, \varphi_p^m)$ and $\min\dim(A_{m,p}) \geq m$ for all $p \in {\mathbb{N}}$. Taking quotients as before, we let $(B_{m,p}, \psi_p^{m})$ be the associated sequence of finite dimensional algebras obtained from $(A_{m,p},\varphi_p^{m})$, and let $\pi_p^{m} : A_{m,p} \to B_{m,p}$ to be the natural quotient maps (evaluation at $1\in \mathbb{T}$). If $B_m := \lim (B_{m,p}, \psi_p^{m})$, then $B_m$ is rationally $K$-stable by \cref{lem_rat_k_stable_af}. \\

We now adopt the approach of \cite[Lemma 3.7]{apurva_pv_af}. By hypothesis, $\min\dim(A_{m,p})\geq m$ for each $p \in {\mathbb{N}}$. If $\min\dim(A_{m,p})\geq m+1$ for all but finitely many $p$, then we may ignore these finitely many terms and write $A_{m+1,p} = A_{m,p}$ and $\psi_p^{m+1} = \psi_p^{m}$ for all $n\in {\mathbb{N}}$. Therefore, we may assume that $\min\dim(A_{m,p}) = m$ for infinitely many $p\in {\mathbb{N}}$, and also that $\min\dim(A_{m,1}) = m$. \\

Since the connecting maps $\varphi_p^{m} : A_{m,p}\to A_{m,p+1}$ are diagonal maps, it follows that the connecting maps $\psi_p^{m} : B_{m,p} \to B_{m,p+1}$ are injective. By the argument of \cite[Lemma 3.7]{apurva_pv_af}, there is a subsequence $(B_{m,n_j}, \psi_{n_j}^{m})$ such that $B_{m,n_j}^{(m)}$ is an orphan for each $j \in {\mathbb{N}}$ (in the sense that $B_{m,n_j}^{(m)}$ is not the target of any arrow emanating from $B_{m,n_j-1}$). In the sequence $(A_{m,p}, \varphi^{m}_p)$, this means that $A_{m,n_j}^{(m)}$ must also be an orphan. Let $\iota_j : A_{m,n_j}^{(>m)} \to A_{m,n_j}$ and $\pi_j : A_{m,n_j}\to A_{m,n_j}^{(>m)}$ be the natural inclusion and quotient maps respectively. Then, it follows that
\[
\iota_j \circ \pi_j \circ \varphi_{n_j,n_{j-1}}^{m} = \varphi_{n_j,n_{j-1}}^{m}
\]
for all $j\geq 1$ with the convention that $n_0 = 1$. Hence, the following diagram commutes
\[
\xymatrixcolsep{4pc}\xymatrix{
\ldots\ar[r] & A_{m,n_{j-1}}\ar[rr]^{\varphi^{m}_{n_j,n_{j-1}}}\ar[d]_{\pi_j\circ\varphi^{m}_{n_j,n_{j-1}}} && A_{m,n_j}\ar[d]^{\pi_{j+1}\circ\varphi^{m}_{n_{j+1},n_j}}\ar[r]^{\varphi^{m}_{n_{j+1},n_j}} & \ldots \\
\ldots\ar[r] & A_{m,n_j}^{(>m)}\ar[rr]_{\pi_{j+1}\circ\varphi^{m}_{n_{j+1},n_j}\circ \iota_j}\ar[rru]^{\iota_j} && A_{m,n_{j+1}}^{(>m)}\ar[r] & \ldots
}
\]
We set $(A_{m+1,j}, \varphi^{m+1}_j)$ to be the terms in the lower row. Then, it follows from \cite[Exercise 6.8]{rordam}, that $\lim (A_{m+1,j}, \varphi^{m+1}_j) \cong A$. Furthermore, by construction, we have $\min\dim(A_{m+1,j})\geq m+1$ for all $j\in {\mathbb{N}}$. Finally, since each $\varphi_j^{m}$ is a diagonal $\ast$-homomorphism, so is $\varphi_j^{m+1}$.
\end{proof}

We are now in a position to prove the first part of \cref{mainthm_k_stable}. The argument follows the same line of reasoning as that of \cite[Theorem 3.8]{apurva_pv_af}.

\begin{theorem}\label{thm_at_alg_k_stable}
An A$\mathbb{T}$-algebra $A$ is $K$-stable if and only if it is rationally $K$-stable.
\end{theorem}
\begin{proof}
It suffices to prove that rational $K$-stability implies $K$-stability. So suppose $A$ is a rationally $K$-stable A$\mathbb{T}$-algebra. By \cref{rem_generic_hom_limit}, we may assume that $A$ is described as an inductive limit $A = \lim (A_p, \varphi_p)$, where each connecting map $\varphi_p : A_p\to A_{p+1}$ is a diagonal $\ast$-homomorphism. \\

Now fix $k \in {\mathbb{N}}$, and we wish to prove that $G_k(\iota^A_j) : G_k(M_{j-1}(A)) \to G_k(M_j(A))$ is an isomorphism for each $j \geq 2$. Since $A$ is rationally $K$-stable, we may assume by \cref{lem_rat_k_stable_min_dim} that
\[
\min\dim(A_p) \geq \left\lceil \frac{k}{2}\right\rceil + 1.
\]
for each $p \in {\mathbb{N}}$. Now, $M_{j-1}(A)$ is an inductive limit of the algebras $\{M_{j-1}(A_p)\}$ and
\[
\min\dim(M_{j-1}(A_p)) \geq \left\lceil \frac{k}{2}\right\rceil + 1.
\]
for each $p\in {\mathbb{N}}$. Therefore, replacing $M_{j-1}(A)$ by $A$, it suffices to prove that the inclusion map $\iota_2^A : A\to M_2(A)$ induces an isomorphism $G_k(\iota_2^A) : G_k(A)\to G_k(M_2(A))$. \\

Now write $A_p = C(\mathbb{T})\otimes B_p$ where $B_p = M_{\ell_1^p}({\mathbb{C}})\oplus M_{\ell_2^p}({\mathbb{C}}) \oplus \ldots \oplus M_{\ell_{k_p}^p}({\mathbb{C}})$. By construction,
\[
k+1 < 2\ell_i^p
\]
for all $p$ and all $1\leq i\leq k_p$. Hence, the inclusion maps $\iota_2^{B_p} : B_p\to M_2(B_p)$ induce isomorphisms $G_k(B_p) \to G_k(M_2(B_p))$ for all $p \in {\mathbb{N}}$ by \cite[Chapter 2, Corollary 3.17]{mimura}. Also, by the natural isomorphisms $G_k(C_{\ast}(\mathbb{T}, B_p)) \cong G_{k+1}(B_p)$ and $G_k(M_2(C_{\ast}(\mathbb{T}, B_p))) \cong G_k(C_{\ast}(\mathbb{T}, M_2(B_p))) \cong G_{k+1}(M_2(B_p))$, the map
\[
G_k(\iota_2^{C_{\ast}(\mathbb{T},B_p)}) : G_k(C_{\ast}(\mathbb{T}, B_p)) \to G_k(M_2(C_{\ast}(\mathbb{T}, B_p))
\]
is also an isomorphism. Furthermore, the following diagram commutes
\[
\xymatrix{
0\ar[r] & C_{\ast}(\mathbb{T}, B_p)\ar[r]\ar[d]_{\iota_2^{C_{\ast}(\mathbb{T},B_p)}} & A_p\ar[r]^{\pi_p}\ar[d]^{\iota_2^{A_p}} & B_p\ar[r]\ar[d]^{\iota_2^{B_p}} & 0 \\
0\ar[r] & M_2(C_{\ast}(\mathbb{T}, B_p))\ar[r] & M_2(A_p)\ar[r]^{\pi_p^{(2)}} & M_2(B_p)\ar[r] & 0 \\
}
\]
Since $(G_k)$ is a homology theory, this induces a diagram of long exact sequences. By the Five Lemma, $G_k(\iota^{A_p}_2) : G_k(A_p) \to G_k(M_2(A_p))$ is an isomorphism for each $p \in {\mathbb{N}}$. Once again, the following diagram commutes
\[
\xymatrix{
0\ar[r] & C_{\ast}(\mathbb{T}, B_p)\ar[r]\ar[d]_{\widetilde{\varphi_p}} & A_p\ar[r]^{\pi_p}\ar[d]^{\varphi_p} & B_p\ar[r]\ar[d]^{\psi_p} & 0 \\
0\ar[r] & C_{\ast}(\mathbb{T}, B_{p+1})\ar[r] & A_{p+1}\ar[r]^{\pi_{p+1}} & B_{p+1}\ar[r] & 0 \\
}
\]
Therefore, $G_k(\iota_2^A) : G_k(A) \to G_k(M_2(A))$ is an isomorphism, and $A$ is $K$-stable.
\end{proof}

Our proof of the next part of \cref{mainthm_k_stable} is now a refinement of these arguments given above. We now recall an important definition (see, for instance, \cite[Definition 3.1.1]{rordam_stormer}).

\begin{defn}\label{defn_slow_dim_growth}
Let $(A_p,\varphi_p)$ be a sequence of circle algebras, then the sequence is said to have \emph{slow dimension growth} if $\lim_{p\to\infty} \min\dim(A_p) = +\infty$. An A$\mathbb{T}$-algebra $A$ is said to have slow dimension growth if it is the inductive limit of some sequence that has slow dimension growth.
\end{defn}

\begin{theorem}\label{thm_k_stable_sdg}
An A$\mathbb{T}$-algebra is $K$-stable if and only if it has slow dimension growth.
\end{theorem}
\begin{proof}
Let $A$ be an A$\mathbb{T}$-algebra. If $A$ has slow dimension growth, then for each $m \in {\mathbb{N}}$, there is a sequence $(A_{m,p}, \varphi^{m}_p)$ of circle algebras such that $A = \lim_{p\to \infty} (A_{m,p}, \varphi^{m}_p)$ and $\min\dim(A_{m,p}) \geq m$ for each $p \in {\mathbb{N}}$. By the results of \cref{sec_hom_circle}, we may assume that $\varphi^{m}_p$ are all diagonal, therefore proof of \cref{thm_at_alg_k_stable} applies and  shows that $A$ must be $K$-stable. \\

Conversely, suppose $A$ is $K$-stable, then $A$ is rationally $K$-stable. Write $A = \lim_{p \to \infty} (A_{1,p}, \varphi^{1}_p)$ where each $A_{1,p}$ is a circle algebra and each $\varphi^{1}_p$ is of type A . By \cref{rem_generic_hom_limit}, there is a sequence $(\widetilde{A}_{1,p}, \psi^{1}_p)$ such that $\widetilde{A}_{1,p} = A_{1,p}$ for all $p \in {\mathbb{N}}$, whose connecting maps are diagonal maps and whose limit $\widetilde{A} = \lim_{p\to \infty} (\widetilde{A}_{1,p},\psi^{1}_p)$ is also rationally K-stable.  \\

By the proof of \cref{lem_rat_k_stable_min_dim}, there is a subsequence $(p(1,j))_{j=1}^{\infty} \subset {\mathbb{N}}$ such that $\widetilde{A}_{1,p(1,j)}^{(1)}$ is an orphan for each $j \in {\mathbb{N}}$. By the construction in \cref{thm_homotopy_type_ab}, it is then clear that $A_{1,p(1,j)}^{(1)}$ must also be an orphan in the original sequence $(A_{1,p}, \varphi_p^1)$ for each $j \in {\mathbb{N}}$. Let $A_{2,j} = A_{1,p(1,j)}^{(>1)}$. Then, as in \cref{lem_rat_k_stable_min_dim}, we get an inductive sequence $(A_{2,p}, \varphi^2_p)$ such that $A = \lim_{p\to \infty} (A_{2,p}, \varphi^2_p)$ with $\min\dim(A_{2,p}) \geq 2$ for each $p \in {\mathbb{N}}$. \\

Thus proceeding, for each $m \in {\mathbb{N}}$, there is a sequence $(A_{m,p}, \varphi_p^m)$ of circle algebras such that $A = \lim_{p\to \infty}(A_{m,p}, \varphi_p^m)$ and $\min\dim(A_{m,p}) \geq m$ for each $p \in {\mathbb{N}}$. Moreover, for each $m \in {\mathbb{N}}$, there is a subsequence $(p(m,j))_{j=1}^{\infty} \subset {\mathbb{N}}$ and quotient maps $\pi^m_j : A_{m,p(m,j)} \to A_{m+1,j}$ such that the following diagram commutes
\[
\xymatrixcolsep{5pc}\xymatrix{
A_{m,p(m,j)}\ar[d]^{\pi^m_j}\ar[r]^{\varphi^{m}_{p(m,j+1),p(m,j)}} & A_{m,p(m,j+1)}\ar[d]^{\pi^m_{j+1}} \\
A_{m+1,j}\ar[r]_{\varphi^{m+1}_j} & A_{m+1,j+1}
}
\]
where $\varphi^{m}_{k,\ell} = \varphi_{k-1}\circ \varphi_{k-2}\circ \ldots \circ \varphi_{\ell} : A_{m,\ell} \to A_{m,k}$ whenever $k>\ell$. Furthermore, we may assume without loss of generality that $p(m,j+1)\geq j$ for each $m,j\in {\mathbb{N}}$. Now define $\psi_m : A_{m,1} \to A_{m+1,1}$ by
\[
\psi_m := \pi^m_1\circ \varphi^m_{p(m,1),1}.
\]
Then, $(A_{m,1},\psi_m)$ is an inductive sequence with $\min\dim(A_{m,1})\geq m$ for each $m \in {\mathbb{N}}$. We claim that $A = \lim_{m\to \infty} (A_{m,1},\psi_m)$. To see this, let $B := \lim_{m\to \infty} (A_{m,1},\psi_m)$, and let $\beta_m : A_{m,1}\to B$ be $\ast$-homomorphisms defining $B$ such that $\beta_{m+1}\circ \psi_m = \beta_m$. Also, let $\alpha^m_p : A_{m,p} \to A$ be the $\ast$-homomorphisms such that $\alpha^m_{p+1}\circ \varphi^m_p = \alpha^m_p$ for each $m,p \in {\mathbb{N}}$. Then, it follows that the following diagram commutes
\[
\xymatrix{
A_{m,1}\ar[rd]_{\alpha^m_1}\ar[rr]^{\psi_m} && A_{m+1,1}\ar[ld]^{\alpha^{m+1}_1} \\
& A
}
\]
By the universal property of the inductive limit, there is a $\ast$-homomorphism $\lambda : B\to A$ such that $\lambda\circ \beta_m = \alpha^m_1$ for each $m \in {\mathbb{N}}$. We claim that $\lambda$ is an isomorphism. \\

For surjectivity, it suffices to show that
\[
\bigcup_{k=1}^{\infty} \alpha^1_k(A_{1,k}) = \bigcup_{m=1}^{\infty} \alpha^m_1(A_{m,1})
\]
since $A = \lim_{k\to \infty} (A_{1,k}, \varphi^1_k)$. So fix $a \in \bigcup_{k=1}^{\infty} \alpha^1_k(A_{1,k})$, and choose $k_1\in {\mathbb{N}}$ such that $a \in \alpha^1_{k_1}(A_{1,k_1})$ and assume $k_1 > 1$. Since $p(1,k_1-1) \geq k_1$ by hypothesis, it follows that
\[
a \in \alpha^1_{p(1,k_1-1)}(A_{1,p(1,k_1-1)})
\]
However, for each $m,j\in {\mathbb{N}}$, the following diagram commutes
\[
\xymatrix{
A_{m,p(m,j)}\ar[rr]^{\pi^m_j}\ar[rd]_{\alpha^m_{p(m,j)}} && A_{m+1,j}\ar[ld]^{\alpha^{m+1}_j} \\
& A
}
\]
Therefore, $a\in \alpha^2_{k_2}(A_{2,k_2})$ where $k_2 = k_1-1 < k_1$. If $k_2 = 1$, then we may stop this process. Else, we may repeat this until we obtain $N \in {\mathbb{N}}$ such that $a \in \alpha^N_1(A_{N,1})$. Thus, $a \in \bigcup_{m=1}^{\infty} \alpha^m_1(A_{m,1})$ and we have proved that $\lambda$ is surjective. \\

For injectivity of $\lambda$, it suffices to prove that $\ker(\alpha^m_1) \subset \ker(\beta_m)$ for each $m \in {\mathbb{N}}$ (by \cite[Proposition 6.2.4]{rordam}). So fix $m \in {\mathbb{N}}$ and let $a \in \ker(\alpha^m_1)$. Then, $\lim_{n\to \infty}\|\varphi^m_{n,1}(a)\| = 0$. Therefore, if $\epsilon > 0$, then there exists $k_1 \in {\mathbb{N}}$ such that $\|\varphi^m_{k_1,1}(a)\| < \epsilon$. Assume first that $k_1 > 1$. Once again, since $p(m,k_1-1) \geq k_1$, it follows that
\[
\|\varphi^m_{p(m,k_1-1),1}(a)\| < \epsilon,
\]
and thus $\|\pi^m_{k_1-1}\circ \varphi^m_{p(m,k_1-1),1}(a)\| < \epsilon$. However, the following diagram commutes
\[
\xymatrixcolsep{5pc}\xymatrix{
A_{m,1}\ar[d]^{\psi_m}\ar[r]^{\varphi^m_{p(m,k_1-1),1}} & A_{m,p(m,k_1-1)}\ar[d]^{\pi^m_{k_1-1}} \\
A_{m+1,1}\ar[r]^{\varphi^{m+1}_{k_1-1}} & A_{m+1,k_1-1}
}
\]
Therefore, $\|\varphi^{m+1}_{k_2}\circ \psi_m(a)\| < \epsilon$ where $k_2 = k_1-1 < k_1$. Thus proceeding, there exists $N\in {\mathbb{N}}$ such that $\|\varphi^N_1\circ \psi_{N,m}(a)\| < \epsilon$ (where $\psi_{N,m} = \psi_{N-1}\circ \psi_{N-2}\circ \ldots \circ \psi_m$). This implies that $\|\varphi^N_{p(N,1),1}\circ \psi_{N,m}(a)\| < \epsilon$, which in turn proves that
\[
\|\psi_{N+1,m}(a)\| < \epsilon.
\]
This is true for any $\epsilon > 0$, so $\lim_{n\to \infty} \|\psi_{n,m}(a)\| = 0$. In other words, $a\in \ker(\beta_m)$. We conclude that $\ker(\alpha^m_1)\subset \ker(\beta_m)$, so $\lambda$ is an isomorphism. \\

Thus, $A = \lim_{m\to \infty} (A_{m,1},\psi_m)$. Since $\min\dim(A_{m,1}) \geq m$ for each $m\in {\mathbb{N}}$, we conclude that $A$ has slow dimension growth.
\end{proof}

\begin{ex}
\cref{mainthm_k_stable} has a number of interesting consequences, some of which we describe below.
\begin{enumerate}
\item If $A$ is a simple, infinite dimensional A$\mathbb{T}$-algebra, then $A$ has slow dimension growth (see \cite{dnnp}). Therefore, every such algebra is $K$-stable.
\item In particular, the simple noncommutative tori are all $K$-stable. However, this class of C*-algebras is known to be $K$-stable from the results of Rieffel \cite{rieffel2}.
\item Similarly, the Bunce-Deddens algebra from \cref{ex_bunce_deddens} is $K$-stable because it is simple \cite{bunce_deddens}. Once again, this may be deduced from another result due to Zhang which states that any non-elementary, simple C*-algebra with real rank zero and stable rank one is $K$-stable \cite{zhang}.
\item Finally, the algebra $G$ from \cref{ex_goodearl} is also $K$-stable because it has slow dimension growth. Notice that $G$ need not be simple in general.
\end{enumerate}
\end{ex}

\textbf{Acknowedgements:} The first named author is supported by UGC Junior Research Fellowship No. 1229, and the second named author was supported by the SERB Grant No. MTR/2020/000385.

redpajama_set_name: RedPajamaArXiv


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We're excited to present to you yet another cool Kickstarter, for a locally-developed educational deck-building card game! It sounds really neat, and you can play it at Modern Myths this week if you want to get a feel for it before you actually contribute to the Kickstarter. But also it's a card-game where the goal is to get enough fat layers to take a long nap and that's worth supporting sight unseen in our opinion. Not nearly enough games officially end with naps.
Also it's coming up in a couple weeks still but everyone get excited for the next Nerd Nite! It's going to be a special Third Anniversary Show with three presenters! Such excitement and wonder you would expect to cost an arm and a leg but no it's only $5!
Also we're really excited to hear ourselves on The Chairmen of the Boards, a lovely local podcast! We talk about our work here and with Nerd Nite and it's a good conversation we think you'll like it. It should be up in a bit over a week!
Create a simple multi-page booklet that's easy to copy and distribute and is most importantly very very cool! Ages 11-16, register here because space is limited!
Learn how to use 3D design software, use a 3D printer, use an engraver, and more things in order to create cool stuff! A neat trick is to 3D print a 3D printer so you can keep working at home.
Nancy Schön is famous for excellent sculpture works inspired by children's literature like the Make Way for Ducklings sculpture in the Boston Public Garden! Meet Nancy and learn about public art projects and her work on a skate park! Radical.
Heroes! Clix! Put them together and what do you get? Clicky heroes, that's right. This week try out a team of only people who can turn into animals. That's Beast Boy, Snowbird, I guess we can count Animal Man too!
Modified is about GMOs, which is an issue that bears a lot of reading about on account of how the way that US public policy feels about food determines what life is like for a lot of people around the world, and this is a great opportunity to learn from a journalist who did a lot of work finding out about GMOs! BUT ALSO this is just great copy so we're just going to quote it "Modified is a rare breed of book that will at once make you weep at the majestic beauty of our Great Plains and force you to harvest deep seeds of doubt about the invisible monsters currently infiltrating our food and our land and threatening our future." It will make your dog mow the lawn! It will make your house unfold into nth-dimensional space! It will active the X-Gene lying dormant within you!
This book explores the lives of 36 inspiring women who struggle to overcome their mental health and relationship challenges. It was written as a tribute and gift to women everywhere. Helene DeMontreux Houston, MS, APRN, one of the co-authors, will discuss how the book came to be and how these stories were chosen.
An exciting panel from the authors of Finding Wonders, Local Girl Swept Away, and Still a Work In Progress! They'll read and talk about their works! One's about groundbreaking female scientists, one's about a mysterious death in Provincetown, and a younger sibling trying to maintain while his older sister is having a crisis.
A party for a book about pets of course features giveaways and fun surprises, but also a chance to bring in donations of pet food, toys, bedding, etc for the Dakin Humane Society! You get a poster if you do, which is nice, but you also just made puppies and kittens more comfortable which is the best gift of all.
Potluck + Kin-dza-dza! Soviet Anti-Utopian Comedy!
Two hapless Russians accidentally teleport to the telepathic planet Pluke! Hijinks ensue, as they are wont to. Bring something to eat or drink! Join the meetup to see where to go, or just bring food and drink to some general location and lay it there as a respectful tribute to this meetup.
Fiction and nonfiction celebrating the relationships between women and their dogs. Dogs are great. Author Michelle A. Gabow is a playwright honored to be be named the Roxbury Repertory Theatre's playwright laureate in 2011!
Local creator and Modern Myther Jesse Daniels is publishing a new card game called Hibernation, which is about being a bear! It's kind of very cute. It's a deck-building game where you try to make sure you've got enough food to develop layers of fat. First to five fat layers gets to take a long nap! It's educational and also a game where the winner takes a nap which needs to happen more often. Support it on Kickstarter here. You can learn to play the game at this event with the creator!
What's Benford's law? Does 1 appear more often than 7 in everyday life? Can math detect tax fraud? What do the statistics of numerals inform us about corporate sales? WELL? THE TEST IS NEXT WEEK AND YOU STILL DON'T KNOW? Better attend this SciTech cafe with Steven J. Miller, associate professor of Mathematics at Williams College. He researches number theory and he'll help explain things to you.
Keep track of the epic history of your family line in order to get straight exactly which sword glows with power when you wield it and what thrones of which nations are your right by birth! Theresa Chamberland and Jan Sadler will present on how to gather, choose, and organize the material we want to preserve!
The election of 1912. Based on our very good American history understanding and entirely without using the internet we know it was between… Grover Cleveland and … another? WELP nope it's the one that was between three people who all were or became President at some point we're awful. Woodrow Wilson, William Taft, and Teddy "The Internet's Favorite Tough Prez" Roosevelt! Learn more about this complicated three-way race!
The 11th annual gala and art auction honoring artist Allen Say, angel Lee and Low books, bridge Steven Heller and mentor Regina Hayes. Awards presented by author Gregory Maguire! Get some real cool artwork and support the Carle!
Put your drinking to work helping the library! That's efficient use of your time. Music by Mark Ricker and Jimmy Burgoff! Tickets are $20 each and available at the library!
Drop-in to craft a Star Wars thing before seeing the movie in Pulaski Park! Maybe craft a drop-ship to keep with the theme!
There's a bunch of fun stuff at this! A children's farmer's market where kids can have fun at the bounce house, getting their face painted, playing the distressingly-named "Cow Plop Bingo", going to the bake sale, and just generally food and fun!
Make things! Use a 3D printer, a CNC router, and lots of materials! Also, they'll demonstrate balloon sculptures, the 3D printers of the circus world. Ages 8-12 welcome!
Nerve tells of the global fight against the incineration of chemical weapons and other hazardous substances, due to the fact that it can be an unsafe disposal method. The film in particular highlights the Chemical Weapons Working Group's important work to illustrate how grassroots activism can change everything!
It was a pretty fun movie, right? See it in a park! See it in a park with other friendly people, it'll be great!
Come view Emily Dickinson's landscape through the lens of an archaeologist during this presentation and walking tour at the Museum. Faculty and members of the University of Massachusetts Amherst archaeological field school will present the findings from their work on the grounds of the Emily Dickinson Museum in the spring of 2016. See the mask that Emily wore when she fought the Wayward Face! See the ray gun she stole back from Barsoom! It's all exciting stuff.


DataKit is a Swift package for loading, manipulating and analysing datasets in memory. Functionality is built around DataSet and Column classes with a DataSet containing a collection of Column instances, together representing tabular data.
The DataSet class takes inspiration from DataFrame classes in R, Python Pandas and Julia DataFrames.jl libraries.
DataKit is developed to explore machine learning algorithms in Swift. See TreeKit for decision tree classification built on DataKit.
Explicitly deal with data types throughout. Don't make any assumptions about what type is intended.
Handle missing data explicitly as Swift Optionals.
Use Swift Generics for common functionality, for example through Generic column methods.
// module or a test suite.
// packages which this package depends on.
See the docs folder for interactive examples in playgrounds that can be run using Xcode or see the Contents.swift file in the playground folder.
Date column type added, using string column as underlying data type, but enabling date functions on a column.


4/22/09 Drew Chambers Paper #9- Compare and Contrast Purcellville Effective Writing Classs Frozen Cow Juice v. Your George Washingtons Finding the Best Vanilla Ice Creams for Your Money From its birth in the 4th century, conceived in the mind of the Roman Emperor Nero, and King Tang of China in the 6th century, ice cream has reined a coveted and lusted after delicacy, with vanilla its king. However, with this current economic position, money is one thing Americans cannot afford to waste. Is it possible to find a delicious-tasting ice cream without taking out a small loan? A blind study was conducted to answer just that question- which brand of vanilla ice cream is best for your money? The contenders: Giant foods, Bloom foods, Breyer's, Edy's, and Haagen ~ Dazs. Among the standards are fat grams, cost, and creaminess. The Raw Data Starting with the number of calories per ounce we see that Bloom has an early lead with just 30 calories per ounce. ...read more.
The other tier to this component is how well the flavor would do when introduced to other flavors, or, in other words, its compatibility. Essentially, will the ice cream dominate the flavor arena? Now, the scores. Once again Bloom takes the lead with a score of 5.25, but in a tie, Giant also scores a 5.25 for "vanilla-ness". In a solo-second, Edy's scored a 3.5. And in last place we have Breyer's and Haagen ~ Dazs both with 1.75. In the category of creaminess, Giant, Edy's, and Haagen ~ Dazs, with its first win, all take first place with a score of an even 6 each. Coming in second we have Breyer's with 4 points and in its first last place, Bloom with 2 points. Moving on the fifth and final criterion sweetness, we find for the third time in a row, Giant wins with s score of 6. In a close second, Haagen ~ Dazs and Edy's score an even 4. ...read more.
As expected all of the ice cream samples were not the exact same amount, causing the smaller portions the melt faster than the bigger portions. This melting could, and possibly did, create a false sense of creaminess. Flavor shouldn't be only for those with a civilization of greenbacks in their wallets, but should be for everyone, a mutual bond. However, I say, why buy vanilla? Go out and buy a carton of chocolate ice cream from your local Giant! Criteria: Rank Weight Edy's Giant Breyer's Hg ~ Dz Bloom Vanillaness 2 1.75 2/3.5 3/5.25 1/1.75 1/1.75 3/5.25 Creaminess 1 2 3/6 3/6 2/4 3/6 1/2 Sweetness 1 2 2/4 3/6 2/4 3/6 1/2 # Calories 3 1 2/2 2/2 3/3 1/1 3/3 Grams Fat 3 1 3/3 3/3 3/3 1/1 3/3 Cost 2 1.75 2/3.5 2/3.5 2/3.5 1/1.75 3/5.25 Total 3rd- 21 1st-25.75 4th-17.25 5th-15.5 2nd-21.5 Table 1- Goal: Best Vanilla Ice Cream for Your Money Rank: 1 Best- 3 Worst Weight: 2 Best- 1 Worst Table 2- Raw Numbers Numbers: Edy's Giant Breyer's Hg ~ Dz Bloom Cost/Ounce $0.09/Oz. $0.07/Oz. $0.08/Oz. $0.29/Oz. $0.05/Oz. Fat Gram/Ounce 2g/Oz. 2g/Oz. 2g/Oz. 4g/Oz. 2g/Oz. Calories/Ounce 35/Oz. 38/Oz. 32/Oz. 63/Oz. 30/Oz. ...read more.
Problems of Structural Violence * Structural violence: violence exerted by institutions, situations, and social, political, and economic spheres. Example - the economic collapse of East Asian countries in 1997, when these countries had to make drastic cuts in social services in order to survive, with a calamitous effect on the citizens (Haviland).
The Greek culture involves a lot of olive oil, honey, yoghurt, fresh fruits and vegetables, fish and lamb. The Italians seems to involve more carbohydrates in their meals especially bread. I don't see honey, yoghurt or fish present in many of the meals my family consumes.
We are told how to experience nature, and this takes away from the personal interaction. Nature exists only when "filtered through out optical nerves, our 'senses', our fiercely romantic expectations." (Oates, p.230) We expect what we have been told to expect, and get just that, nothing more.
According to Reporters Without Borders, China ranked 174/179 in the index of free press, which is a damning statistic for China that boasts around 2,000 newspapers and 374 television networks, none of which broadcast external media. As with the internet, there are many laws in place which limit freedom of the press.


text:
Vengeance: Night of Champions foi um evento em pay-per-view de luta livre profissional produzido pela World Wrestling Entertainment (WWE) e patrocinado pelo RAW Attitude Energy Drink, que aconteceu em 24 de junho de 2007, no Toyota Center, em Houston, Texas. Esta foi a sétima edição na cronologia do Vengeance e a primeira na do Night of Champions. Contou com a participação dos lutadores dos programas Raw, SmackDown e ECW. Todos os títulos ativos na empresa foram defendidos.

Antes do evento

Vengeance: Night of Champions teve combates de luta profissional de diferentes lutadores com rivalidades e histórias pré-determinadas, que se desenvolveram no Raw,  SmackDown e ECW — programas de televisão da WWE. Os lutadores interpretaram um vilão ou um mocinho seguindo uma série de eventos para gerar tensão, culminando em várias lutas.

Ver também
WWE Night of Champions

Ligações externas
Site oficial do Vengeance: Night of Champions

2007 na luta profissional
2007

redpajama_set_name: RedPajamaWikipedia


CEDAR CITY – A semi truck hauling granite rolled Wednesday night leaving a Cedar City man with minor injuries.
At approximately 10 p.m., a 55-year-old man was traveling north on Interstate 15 when the semi truck he was driving rolled at mile marker 44 near New Harmony.
The driver, who chose not to give his name, said his semi truck was carrying a full load of granite.
It was then, Allred said, the truck veered to the right and rolled into a culvert. The driver said he woke up just as his semi truck started drifting off the freeway.
Paramedics on scene evaluated the driver and released him to his wife, who also declined to provide her name. She told Cedar City News she was taking her husband to the hospital for further examination. Allred said the man had complained of slight neck pain.
The driver's wife said her husband had been driving semi trucks for decades but Wednesday's accident was his first.
It was too dark for authorities to assess the full damage of the semi truck or the load it was carrying, Allred said.


This madeleine WordPress GPL template comes with a 100% GPL license giving you the freedom to run, study, share, and modify everything in it. This madeleine wordpress theme is also known as template 59014. It has had a total of 165 downloads since it was created by Jetimpex.
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Cone Crusher - 911 MetallurgistOct 19, 2016 . Protect Cone Crusher from Tramp Iron. Protection against tramp iron or other uncrushable materials is afforded by an accumulator in the hydraulic system. This consists of a neoprene rubber oil-resistant bladder inside a steel shell. This bladder is inflated with nitrogen to a predetermined pressure higher.accumulator function in cone crusher,DEM Modelling and Simulation of Cone Crushers and High .of compressive breakage machines such as Cone Crushers and HPGRs is presented. .. Prediction function. Φ. Vandermonde matrix j. ∆. Error deviation constraint ε. Error estimate x. Optimization design variable vector g. Inequality constraints h ... and the volume of the accumulator determines the spring properties.MP® Series Cone Crushers - MP cone crushers perform consistently, no matter what the application. A track record of success. From the first day they were ... tor to be stationed near the crusher during operation. A nitrogen/hydraulic accumulator system with patented dual function hydraulic cylinders provides overload protection and a safe, quick, and.
Jun 19, 2014 . Hydro-pneumatic accumulators are pressure vessels designed and built to store fluids under pressure. They are charged with nitrogen which is separated from the fluid section by a piston, bladder or diaphragm. HYDAC has over 50 years of experience in the research, development, design and production.
TRIO's design and manufacturing teams have many years of experience building the TC cone crusher series. These crushers combine . Accumulators maintain an even and constant pressure during clamping. . TRIO utilizes an integrated PLC system to monitor and control vital crusher functions as standard equipment.
Better Overload Protection. 's exclusive, patented design limits the amount of force a cone crusher needs to absorb when it encounters overloads or tramp metal – providing the best protection for internal parts. A limited force is maintained as the crusher relieves, unlike spring or accumulator systems that experience.
CS420 Cone Crusher For Achieving High Performance .
CS420 cone crusher has a hydraulically supported main shaft that is supported at both ends. . Hydroset™ system provides safety and setting-adjustment functions; ASRi™ automatically adapts crusher to feed conditions; Unibody mainframe ensures optimal strength and less maintenance; Lifting from above.
Feb 23, 2017 . Superior Industries says the simple design of its Patriot Cone Crusher's inverted tramp relief system ensures the hydraulic cylinder rod seal is not exposed during operation. This greatly reduces chances of contamination and failure. The tramp relief system has fewer accumulators to minimize maintenance.
All Trimax cone crushers are high-speed cone crushers of heavy duty construction, with high reliability and durability that give exceptional production capacity relative to their size. As a standard safety feature within the Trimax NH series cone crusher, the accumulator & hydraulic system functions as an Automated Overload.
Oct 19, 2016 . Protect Cone Crusher from Tramp Iron. Protection against tramp iron or other uncrushable materials is afforded by an accumulator in the hydraulic system. This consists of a neoprene rubber oil-resistant bladder inside a steel shell. This bladder is inflated with nitrogen to a predetermined pressure higher.
of compressive breakage machines such as Cone Crushers and HPGRs is presented. .. Prediction function. Φ. Vandermonde matrix j. ∆. Error deviation constraint ε. Error estimate x. Optimization design variable vector g. Inequality constraints h ... and the volume of the accumulator determines the spring properties.
Taken the theory of laminating crushing combined with high power, with optimized crushing chamber, GPY Series Hydraulic Cone Crusher has the general characteristics in smart weight, large production capacity, good fineness shape and wearable wearing parts, etc. The machine can be widely used in metallurgy,.
Extra-coarse, coarse and medium type crushers are equipped with a feed hopper which has material level detection function; while fine type is equipped with a material distributor. 2. The displacement sensor can quickly, and accurately detect and set up the size of the discharge opening. 3. Accumulator and throttle valve.
Feb 6, 2010 . Hydrocone Crusher. A new fast robust design of the hydraulic system. Johan Vestman. Patrik Sjöberg. Fluid and Mechanical Engineering Systems .. In the bigger models of Hydrocone crusher there is a pilot operated pressure relief valve instead of accumulators. Figure 3, schematics and assembly.
Jaw crushers. ®. C Series™. Uncompromised reliability and productivity. The reliability of C Series jaw crushers is based on a revolution- ary pinned and bolted, non-welded frame construction. This ... rock breaker solutions are complete function- ... accumulator combination provides better reactivity.
A gyrating cone-shaped head cooperates with a stationary bowl support to crush rock or the like admitted into the crusher. The gyrating action of the head is . Motor 204 drives a hydraulic pump 205 connected on its input side to a pair of accumulators 206 by a conduit 207. Accumulators 206 are in communication with each.
All Hydraulic and Rotary functions are synchronized and controlled by the hydraulic and electrical control panels. A very compact Hydraulic power pack controls all the Hydraulic and Lubrication systems. SMAN SH/LH Gyratory Hydraulic Cone Crushers are equipped with the most advanced crushing techniques. The outer.


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Q: Loading html image description div at specified height before and after image loads Ok, so I have a page that loads an image onto a div, but it happens after one sec. The image has some description and story related to it that also loads but much faster. In fact the description loads at top of screen, and then "shifts" when the image completes loading. Obviously this is bad. I want such that even if the image hasn't loaded, the description loads exactly where it would be expected to be after the image has loaded.
The structure is:
<div image />
<div description />

Please help so I can go home for the weekend ;)

A: You can show the description when image is loaded - detect it in this way with jQuery:
$("#image").load(function (){
   alert("Image loaded!");
});

see this answer for deails

A: There are several ways to solve your problem. I suggest you read this article. There you will find a solution that exactly suits your needs. At least it will reduce image jumping noticeably

A: You have two possibilities (as far as I see):

*

*Add/display the description as soon as the image is loaded (using JS to watch, when the image is loaded)

*Pre-define the width + height (or aspect ratio) of the image and reserve that space, so the image loads into it and even the text is below it, it will be in the correct position from the beginning

A: One way is to add style information onto the description tag:
style="position: absolute; top: 1000px;"

Here the 1000px is the height of the actual image. This will ensure that the description loads at 1000 pixels below the top margin, leaving space for the image to then load after 1 second, without shifting the description.

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You mean authoritarianism and monopoly rather than as in history, an extended family of various traits relying on various archetypes.
It is very easy to convince people that what they do is good. It is very hard to show them that what they deem good is bad. Principally because their intuitions have been trained by deception and ritual, because there was no means of training them by reason and practice.
No cult survives falsification by science, and no good of cults cannot be produced by training. There are only two religions that are not evil: stoicism and shintoism, with original buddhism simply a pre-scientific attempt (as in philosophy vs science) to produce an insular stoicism; where stoicism sought to produce a civic discipline.