Let $\sigma:\Sigma^{\ast}\times\Sigma^{\ast}\rightarrow\Sigma^{\ast}$ be any (possibly nontotal, possibly many-to-one) 2-ary function. We say $\sigma$ is overstrong if and only if for no $f\in{\rm FP}$ with $f:\{1,2\}\times\Sigma^{\ast}\times\Sigma^{\ast}\rightarrow\Sigma^{\ast}\times\Sigma^{\ast}$ does it hold that for each $i\in\{1,2\}$ and for all strings $z,a\in\Sigma^{\ast}$ :

In what follows, we shall consider cut–downs of operators in $M$ of the form

where $r$ is a projection and $b$ is self-adjoint. Note that we may view $b^{\prime}$ as an element of $M$ or the von Neumann algebra $rMr$ . If $s$ is a real number then the cut down spectral interval projections determined by $b,r$ and $s$ are the spectral projections of $b^{\prime}$ determined by the intervals $(-\infty,s)$ and $(-\infty,s]$ . We compute the spectrum and the corresponding spectral projections in the algebra $M_{r}$ where the projection $r$ is the identity.

If $(s_{1},\mathbf{t}_{1}),(s_{2},\mathbf{t}_{2}),\dots,(s_{k},\mathbf{t}_{k})$ is a sequence of spectral pairs, then its associated facial complex is defined inductively as follows. Write

Let $q_{s_{2},\mathbf{t}_{2}}^{\pm}$ denote the associated cut down spectral interval projections determined by $b_{\mathbf{t}_{2}},r_{1}$ and $s_{2}$ and write

Now suppose that for some $1<i<k$ we have defined sequences

such that if $2\leq j\leq i$ , then

where $q_{s_{j},\mathbf{t}_{j}}^{\pm}$ are the cut down spectral interval projections determined by $b_{\mathbf{t}_{j}},r_{j-1}$ and $s_{j}$ .

We may then set

let $q_{s_{i+1},\mathbf{t}_{i+1}}^{\pm}$ denote the spectral interval projections determined by $b_{\mathbf{t}_{i+1}},r_{i}$ and $s_{i+1}$ and write

This completes the induction.

Thus the facial complex determined by the spectral pairs $(s_{1},\mathbf{t}_{1}),\dots,(s_{k},\mathbf{t}_{k})$ consists of the projections $p_{s_{1},\mathbf{t}_{1}}^{\pm}$ together with the sequences

as defined above.