The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries (but I don't yet understand how to do it). I am aware of the existence of the well-known $C^*\!$-algebras derived from directed graphs, called graph $C^*\!$-algebras...but these seem to be different. My questions are:

- does anyone recognize this (as an inverse semigroup or as a $C^*\!$-algebra)?
- any hints as to how to derive the $C^*\!$-algebra from the inverse semigroup? (for instance, what is the norm?)

Here is the definition. I use $*$ to denote the inverse. The inverse semigroup has 0, and also a partially defined sum: if $x$ and $y$ are orthogonal ($xy^*\!=x^*\!y=0$) then we define $x+y$. Multiplication uses the distributive law, and $*$ distributes over sums. The generators are the edges of the graph. Suppose that $a_m\colon w_m\to v$ and $b_n\colon v\to x_n$ are the incoming and outgoing edges at vertex $v$. Then we have these relations:

- if $i\ne j$ then $a_i a_j^*=0$
- if $i\ne j$ then $b_i^* b_j=0$
- $\sum_{i=1}^m a_i^* a_i=\sum_{j=1}^n b_j b_j^*$

That's all the relations if you think of this as an inverse category rather than an inverse semigroup. For each edge $e\colon v\to w$ add an edge $e^*\colon w\to v$ to the directed graph. The allowed multiplications are those that follow directed paths. If you want an inverse semigroup, make all the nonallowed multiplications equal to zero.