I^* \}.\end{aligned}$$ The set $B_0$ is closed under multiplication and involution. The algebra $B$ is the closure of $B_0$. Note that all the partial isometries $v(g_l)$ correspond to partial bijections of $X$ and that $p(h_l) \in \ell_\infty X$ is a characteristic function of a subset of $X$. It turns out that the image of the conditional expectation $E(v(g_1) p(h_1) \cdots v(g_n) p(h_n))$ has to be a projection. It follows that $A$ is generated by these countably many projections.

Define $g \in I^*$ by $(g_1(1), g_1(2), \cdots, g_2(1), g_2(2), \cdots, \cdots, g_n(1), g_n(2), \cdots)$. Define $v$ by $v(g_1) p(h_1) v(g_2) p(h_2) \cdots v(g_n) p(h_n) \in B_0$. The matrix coefficients of $v$ are $1$ only on a subset of $\phi(g) \subseteq X \times X$. It follows that $v^* v \le v(g)^* v(g)$. Since $v$ corresponds to a partial bijection on $X$, the projection $v^* v$ is in $\ell_\infty X$. By the equality $v = v(g) v^* v$, we have $E(v) = p(g) v^* v$. It follows that $E(B_0)$ is a subset of $B$. We conclude that $A$ is a subalgebra of $B$ and equal to $E(B)$.

Groupoid associated to label {#SubsectionGroupoidLabel}
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Let $\Omega$ be a Gelfand spectrum of the unital commutative 