Let $A$ be a separable C*-algebra and ${pi_n}_{ninmathbb N}$ be a sequence of finite dimensional representations of $A$ such that for every $aneq 0$ there is $n$ such that $pi_n(a)neq 0$. In other words, $bigoplus_n pi_n $ is faithful. (such $A$ is said to be residually finite dimensional.)

Let $B$ be unital separable and ${pi_n}_{ninmathbb N}$ be such sequence. Assume every $pi_n$ is irreducible and has dimension $k(n)$. Also assume each representation in ${pi_n}_{ninmathbb N}$ repeats infinitely many times.

Suppose $l:mathbb Nto mathbb N$ is an increasing sequence.

For each $n$, define $phi_n:M_{k(n)}to M_{k(n)}(B)$ by viewing $M_{k(n)}$ as a subalgebra of $M_{k(n)}(B)$, since $B$ is unital. Then define $psi_n^1=phi_ncirc pi_n:Bto M_{k(n)}(B)$.

By defining $h_1:bmapsto text{diag}(b,psi_1^1(b),…,psi_{l(1)}^1(b))$

, $h_1$ is a homomorphism from $B=M_1(B)=M_{I(1)}(B)$ to $M_{I(2)}(B)$, where $I(1)=1$ and $I(2)=1+k(1)+…+k(l(1))$.

Suppose $h_m:M_{I(1)}(B)to M_{I(2)}(B)$ and $psi_n^m:M_{I(m)}(B)to M_{I(m)k(n)}(B)$ are defined. Define $psi_n^{m+1}=psi_n^1otimes 1_{I(m+1)}:M_{I(m+1)}(B)to M_{k(n)I(m+1)}(B)$ and define $h_{m+1}:bmapsto text{diag}(b,psi_1^{m+1}(b),…,psi_{l(m+1)}^{m+1}(b))$ as homomorphism from $M_{I(m+1)}(B)$ to $M_{I(m+2)}(B)$, where $I(m+2)=I(m+1)(1+k(1)+…+k(l(m+1)))$.

Then we have unital homomorphisms $h_m:M_{I(m)}(B)to M_{I(m+1)}(B)$ and therefore we can define the inductive limit C*-algebra $A=lim_{ntoinfty}M_{I(m)}(B)$.

Proposition. $A$ is simple.

This is from *An introduction to the classification of amenable C*-algebras*, p158. This definition is too complicated for me to understand.

Since $h_m$ maps $b$ to the matrix with $b$ on its diagonal, $h_m$ is a norm preserving injection. Therefore there are $M_{k(n)}(B)simeq B_nleq A$ with $mathbf{closure}(bigcup B_n)=A$. It only needs to show there is simple $M_{I(n)}(B)$ for arbitrarily large $n$. However, the definition is so complicated that I don’t even know where to start.

As in the proof the book gives, it says,

For $0neq yin M_{l(n)}$, there is $pi_m$ such that $pi_motimes text{id}_{M_{I(n)}}(y)neq 0$. Let $n+lgeq m$, by considering $h_{n+l}circ h_{n+l-1}circ…circ h_{n+1}$ we may write $h_{n,n+l}(y)=H(y)oplus (pi_motimes text{id}_{M_{I(n)}}otimes text{id}_{M_J})$ for some positive integer $J$ and some homomorphism $H$. Since $psi_m^1otimes text{id}_{M_{I(n)}}otimes text{id}_{M_J}(y)$ is a nonzero element in $M_{I(n+l)}$, the ideal generated by $psi_m^1otimes text{id}_{M_{I(n)}}otimes text{id}_{M_J}(y)$ contains $M_{I(n+l)}$.

I do not understand it at all.