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:

1. does anyone recognize this (as an inverse semigroup or as a $C^*\!$-algebra)?
2. 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:

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

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:

1. does anyone recognize this (as an inverse semigroup or as a $C^*\!$-algebra)?
2. 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:

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

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.

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:

1. does anyone recognize this (as an inverse semigroup or as a $C^*\!$-algebra)?
2. 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. 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:

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

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:

1. does anyone recognize this (as an inverse semigroup or as a $C^*\!$-algebra)?
2. 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:

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