tends to zero and the left one to $\int_{{\mathcal{SC}_*}}\langle A\sigma,\theta\rangle d\mu$.

We compute the trace of the identity operator ${\operatorname{Id}}_p$ on $C_p^{(2)}({\mathcal{SC}_*},\mu)$: $$\begin{aligned}
{\operatorname{tr}}_p^\mu({\operatorname{Id}}_p)&=\int_{{\mathcal{SC}_*}}\sum_{\tau\in\mathcal{T}_p}\frac{\langle {\operatorname{Id}}_p\tau,\tau\rangle}{p+1}d\mu([K,x])\\
&=\int_{{\mathcal{SC}_*}}\frac{\deg_p(x)}{p+1}d\mu([K,x])=\mathbb{E}(\deg_p)/(p+1).\end{aligned}$$ If $\mu$ is the unimodular [*rrsc*]{} associated to a finite simplicial complex $K$ this is equal to $\frac{|K(p)|}{|V(K)|}$.

Let $\mu$ be an unimodular random rooted simplicial complex such that the $p$-th Laplace operator is a bounded operator.

\[bettinumbers\] We define the *p-th $\ell^2$-Betti number of $\mu$* as the trace of the projection onto the kernel of the Laplace operator, i.e. $$\begin{aligned}
\beta_p^{(2)}(\mu)\coloneqq{\operatorname{tr}}_p^\mu(\mathrm{proj}_{\ker(\Delta_p)}).\end{aligned}$$

\[example-betti-numbers\] We compute the $\ell^2$-Betti numbers of the [*rrsc*]{}s defined in Example \[example-measures\]. We denote by $b_p(K)$ the $p$-th ordinary Betti number of the simplicial complex $K$ and by $\beta
_p^{(2)}(K,\Gamma)$ the $\ell^2$-Betti numbers of a $\Gamma$-simplicial complex with respect to the group von Neumann algebra. Gaboriau gives a definition of $\ell^2$-Betti numbers of $\mathcal{R}$-simplicial complexes which we denote by $\beta_p(\Sigma,\mathcal{R},\nu)$. If the $\mathcal{R}$-simplicial complex is $p$-connected, the definition is independent of the $\mathcal{R}$-simplicial complex $\Sigma$ and hence it is called the $p$-th $\ell^2$-Betti number $\beta_p(\mathcal{R},\nu)$ of the equivalence realtion $\mathcal{R}$ [@gaboriau].

1.  Given a finite simplicial complex $L$ with its associated unimodular [*rrsc*]{} $\mu_L$ and let $P$ denote the projection onto 