Here is Cuntz's paper "Simple $C^*$-algebras Generated by Isometries". I'm having trouble understanding section 1.7. First, let me set up some notation:
With $n =2,3,...,\infty$ fixed, given $k \in \Bbb{N}$,
$$W^n_k := \{(j_1,...,j_k) \mid j_i \in \{1,...,n\}\},$$
$W_0^n := \{0\}$, and $W^n_{\infty} := \bigcup_{k=0}^{\infty} W_{k}^{n}$. Let $S_0 = 1$, and given $\alpha = (j_1,...,j_k) \in W_{k}^{n}$, define the isometry $S_{\alpha}$ as $S_{\alpha} = S_{j_i} S_{j_2} ... S_{j_k}$. It turns that any word in $\{S_i\} \cup \{S_i^*\}$ can be written as $S_{\mu} S_{v}^*$ for unique $\mu,v \in W_{\infty}^{n}$
Let $\mathcal{F}_0^{n} = \Bbb{C}1$, let $\mathcal{F}_{k}^{n}$ be the $C^*$-algebra generated by the set $\{S_{\mu} S_{v}^* \mid \mu, v \in W_{k}^{n} \}$, and let $\mathcal{F}^n$ be the $C^*$-algebra generated by $\bigcup_{k=0}^{\infty} \mathcal{F}_{k}^{n}$.
I think I've given all the notation needed to describe my issue (if not, I can try to clarify or you can look at the linked paper). In section 1.6, Cuntz sets out to describe the algebra $\mathcal{P}$ generated algebraically by $\{S_i\} \cup \{S_i^*\}$. Let $V$ denote $S_1$ and $V^{-1}$ denote $S_1^*$. He claims that
since any $A \in \mathcal{P}$ is a linear combination of words, $A$ can be written in the form $$A = \sum_{i=-N}^{-1} V^i A_i + A_0 + \sum_{i=1}^{N} A_i V^i,$$ where the $A_i \in \mathcal{F}^n$.
I'm having trouble verifying this decomposition holds. First, it doesn't quite make sense to me. The algebra $\mathcal{P}$ is generated algebraically from words in those isometries, whereas $\mathcal{F}^n$ is a $C^*$-algebra. Why do we need elements from a big $C^*$-algebra to describe elements in $\mathcal{P}$. I guess if someone could help me prove that $A$ decomposes in that way I would understand, but I don't see how to prove it.