\mathbb{B}(\ell_2 X)$. The operators satisfy the relation $v(-i) = v(i)^*$. Let $I^*$ be the index set $\bigsqcup_{n = 1}^\infty I^n$. For $$\begin{aligned}
g = (g(1), g(2), \cdots, g(n)) \in I^n \subseteq I^*,\end{aligned}$$ we define a partial bijection $\phi(g)$ by the composition $\phi(g(1)) \circ \phi(g(2)) \circ \cdots \circ \phi(g(n))$, restricting the domain in the obvious way. The composition $\circ$ is identical to the product as controlled sets.

Let $E \colon \mathbb{B}(\ell_2 X) \rightarrow \ell_\infty X$ be the conditional expectation onto $\ell_\infty X$. We define operators $v(g), p(g)$ in $C^*_\mathrm{u}(X)$ as follows:

-   a partial isometry $v(g) = v(g(1)) v(g(2)) \cdots v(g(n))$,

-   a projection $p(g) = E(v(g))$.

Let $B = B_L$ be the unital C$^*$-subalgebra of $C^*_\mathrm{u}(X)$ generated by $\{ v(g) \ |\ g \in I^* \}$ and $\{ p(g) \ |\ g \in I^* \}$. Let $A$ be the commutative C$^*$-algebra generated by $E(B) \subseteq \ell_\infty X$.

\[Lemma;First\] The algebra $A$ is generated by at most countably many projections. The algebra $B$ contains $A$. The algebra $A$ is equal to $E(B)$.

Let $B_0$ be the linear span of the set $$\begin{aligned}
\{ v(g_1) p(h_1) v(g_2) p(h_2) \cdots v(g_n) p(h_n) \ | \ g_1, h_1, g_2, h_2, \cdots, g_n, h_n \in 