be $D-(p-1)$ of them. Because $\mu$ is only supported on simplicial complexes with vertex degree $\le D$, the $\mu$-essential supremum $\|\partial_p^*\|\le \sqrt{D-(p-1)}$. Putting the two estimates together, we get $$\|\Delta_p\|\le 2\sqrt{(p+2)D}.$$

Trace and spectral measure
==========================

We fix a $p\in{{\mathbb N}}$ and an unimodular [*rrsc*]{}$\mu$ for this section.

\[def-vonneumannalgebra\] The von Neumann algebra $\operatorname{Alg}_p(\mu)$ is given by bounded decomposable operators $T$ on $C_p^{(2)}({\mathcal{SC}_*},\mu)$ such that for almost all $[K,x]\in{\mathcal{SC}_*}$ and all non-rooted and not necessarily order preserving simplicial isomorphisms $(K,x)_{{\mathbb N}}\to (L,y)_{{\mathbb N}}$ of subcomplexes of ${\Delta(\mathbb{N}_0)}$ and for all simplicial chains $z,z'\in C_p^{(2)}((K,x)_{{\mathbb N}})$ $$\label{equation-vonneumann algebra on hilbert field}
\langle T_{[K,x]}z,z'\rangle=\langle T_{[L,y]}\phi(z),\phi(z')\rangle.$$ holds.

It is obvious that this condition is closed under the weak topology. Further, Equation \[equation-vonneumann algebra on hilbert field\] implies that the operators in $\operatorname{Alg}_p(\mu)$ are independent of the root. Therefore we will sometimes write $T_K$ instead of $T_{[K,x]}$.\
We already introduced the fundamental sequence $\{\delta_\sigma\;|\; \sigma \in {\Delta(\mathbb{N}_0)}(p)\}$. This time, we pick only the characteristic functions of oriented $p$-simplices which contain $0$ as a vertex, i.e. $$\mathcal{T}_p=\{\delta_\sigma\;|\; \sigma\in {\Delta(\mathbb{N}_0)}(p)\text{ and }0\in\sigma\},$$ and call it the *p-th trace family*. We define a *trace* on the von Neumann algebra $\operatorname{Alg}_p(\mu)$ by $$\label{equation-trace}
{\operatorname{tr}}_p^\mu(T)\coloneqq\sum_{\tau\in\mathcal{T}_p}\frac{\langle T\tau,\tau\rangle_\mu}{p+1}\in [0,\infty].$$ Note that 