Let me begin with inverse semigroup $C^{\ast}$-algebras. Inverse semigroups are semigroups $S$ with the property that for all $s\in S$, there is a unique element $s^*$ with $ss^\ast s=s$ and $s^\ast ss^\ast=s^\ast$. The key example of an inverse semigroup is the symmetric inverses semigroup of all partial bijections of a set $X$ (also called the rook monoid when $X$ is finite). The notion was invented independently by Preston and Wagner to abstract the structure of Lie pseudogroups of transformations.

Inverse semigroups are to partial symmetry as groups are to symmetry. So for example, the symmetry group of the Sierpinski gasket is the Dihedral group of order 6 but the inverse semigroup of partial symmetries is infinite and more interesting. I recommend Mark Lawson's book.

Now if you look at the inverse semigroup of all partial bijections of $X$, then the idempotents are the partial identity maps $1_Y$ with $Y\subset X$. These commute with each other. Ok, now let us try to lift this to operator algebras. Inverse semigroups have lots of idemptotents because $ss^\ast$ and $s^\ast s$ are idempotents. It is a nice fact, proved by Roger Penrose and Douglas Munn, that the idempotents commute.

A partial isometry of a Hilbert space is a bounded operator $a$ such that $a=aa^*a$. In this case, $a^\ast a$ and $aa^\ast$ are projections and $a$ induces an isometry between the images of these projections. So many have argued the natural way to represent inverse semigroups on a Hilbert space is via partial isometries. Idempotents of the inverse semigroup then become projections.

A nice example is the unilateral shift and its adjoint generate the bicyclic inverse semigroup $\langle x\mid x^\ast x=1\rangle$.

Now if you have a quiver $Q$, then we have a partial bijection of the set of non-empty paths associated to each possibly empty path $p$ which operates on paths by concatenation when it makes sense and otherwise is undefined. Empty paths are partial identities acting on the paths beginning from the corresponding vertex. These generate an inverse semigroup and under mild hypotheses give the graph inverse semigroup given by the presentation you might have seen. More generally, one can extend this to a representation on the Hilbert space with basis the non-empty paths and generate a $C^\ast$-algebra. The empty paths act now as projections and that is why vertices give projections in graph $C^\ast$-algebras. There are also some additive relations you pick up in this way. Under mild hypotheses this is the graph $C^\ast$-algebra.

For example, if your quiver has two loops edges $a,b$. Then the graph inverse semigroup has generators $a,b$ with relations $a^\ast a=1=b^\ast b$ and $ab^\ast=0=ba^\ast$. When you take the above representation and generate an algebra you pick up the relation $aa^\ast+bb^\ast=1$ because each non-empty word begins with either $a$ or $b$. So you get the Cunz algebra.

The papers of Ruy Exel go into more detail how to get operator algebras from inverse semigroups.

Let me begin with inverse semigroup $C^{\ast}$-algebras. Inverse semigroups are semigroups $S$ with the property that for all $s\in S$, there is a unique element $s^*$ with $ss^\ast s=s$ and $s^\ast ss^\ast=s^\ast$. The key example of an inverse semigroup is the symmetric inverses semigroup of all partial bijections of a set $X$ (also called the rook monoid when $X$ is finite). The notion was invented independently by Preston and Wagner to abstract the structure of Lie pseudogroups of transformations.

Inverse semigroups are to partial symmetry as groups are to symmetry. So for example, the symmetry group of the Sierpinski gasket is the Dihedral group of order 6 but the inverse semigroup of partial symmetries is infinite and more interesting. I recommend Mark Lawson's book.

Now if you look at the inverse semigroup of all partial bijections of $X$, then the idempotents are the partial identity maps $1_Y$ with $Y\subset X$. These commute with each other. Ok, now let us try to lift this to operator algebras. Inverse semigroups have lots of idemptotents because $ss^\ast$ and $s^\ast s$ are idempotents. It is a nice fact, proved by Roger Penrose and Douglas Munn, that the idempotents commute.

A partial isometry of a Hilbert space is a bounded operator $a$ such that $a=aa^*a$. In this case, $a^\ast a$ and $aa^\ast$ are projections and $a$ induces an isometry between the images of these projections. So many have argued the natural way to represent inverse semigroups on a Hilbert space is via partial isometries. Idempotents of the inverse semigroup then become projections.

A nice example is the unilateral shift and its adjoint generate the bicyclic inverse semigroup $\langle x\mid x^\ast x=1\rangle$.

Now if you have a quiver (=directed graph) $Q$, then we have a partial bijection of the set of non-empty paths associated to each possibly empty path $p$ which operates on paths by concatenation when it makes sense and otherwise is undefined. Empty paths are partial identities acting on the paths beginning from the corresponding vertex. These generate an inverse semigroup and under mild hypotheses give the graph inverse semigroup given by the presentation you might have seen. More generally, one can extend this to a representation on the Hilbert space with basis the non-empty paths and generate a $C^\ast$-algebra. The empty paths act now as projections and that is why vertices give projections in graph $C^\ast$-algebras. There are also some additive relations you pick up in this way. Under mild hypotheses this is the graph $C^\ast$-algebra.

For example, if your quiver has two loops edges $a,b$. Then the graph inverse semigroup has generators $a,b$ with relations $a^\ast a=1=b^\ast b$ and $ab^\ast=0=ba^\ast$. When you take the above representation and generate an algebra you pick up the relation $aa^\ast+bb^\ast=1$ because each non-empty word begins with either $a$ or $b$. So you get the Cunz algebra.

The papers of Ruy Exel go into more detail how to get operator algebras from inverse semigroups.

Let me begin with inverse semigroup $C^{\ast}$-algebras. Inverse semigroups are semigroups $S$ with the property that for all $s\in S$, there is a unique element $s^*$ with $ss^\ast s=s$ and $s^\ast ss^\ast=s^\ast$. The key example of an inverse semigroup is the symmetric inverses semigroup of all partial bijections of a set $X$ (also called the rook monoid when $X$ is finite). The notion was invented independently by Preston and Wagner to abstract the structure of Lie pseudogroups of transformations.

Inverse semigroups are to partial symmetry as groups are to symmetry. So for example, the symmetry group of the Sierpinski gasket is the Dihedral group of order 6 but the inverse semigroup of partial symmetries is infinite and more interesting. I recommend Mark Lawon's book.

Now if you look at the inverse semigroup of all partial bijections of $X$, then the idempotents are the partial identity maps $1_Y$ with $Y\subset X$. These commute with each other. Ok, now let us try to lift this to operator algebras. Inverse semigroups have lots of idemptotents because $ss^\ast$ and $s^\ast s$ are idempotents. It is a nice fact, proved by Roger Penrose and Douglas Munn, that the idempotents commute.

A partial isometry of a Hilbert space is a bounded operator $a$ such that $a=aa^*a$. In this case, $a^*a$ and $aa^*$ are projections and $a$ induces an isometry between the images of these projections. So many have argued the natural way to represent inverse semigroups on a Hilbert space is via partial isometries. Idempotents of the inverse semigroup then become projections.

A nice example is the unilateral shift and its adjoint generate the bicyclic inverse semigroups.

Now if you have a quiver $Q$, then we have a partial bijection of the set of non-empty paths associated to each path $p$ (including the empty path at each vertex) which operators on paths by concatenation when it makes sense and otherwise is undefined. Empty paths are partial identities acting on the paths beginning from those vertices. These generate an inverse semigroup and under mild hypotheses give the graph inverse semigroup given by the presentation you might have seen. More generally, one can extend this to a representation on the Hilbert space with basis the non-empty paths and generate an algebra. The empty paths act now as projections and that is why vertices give projections in graph $C^\ast$-algebras. There are also some additive relations you pick up in this way. Under mild hypotheses this is the graph $C^\ast$-algebra.

For example, if your graph has two loops edges $a,b$. Then the graph inverse semigroup has generators $a,b$ with relations $a^\ast a=1=b^\ast b$ and $aa^\ast=0=bb^\ast$. When you take the above representation and generate an algebra you pick up the relation $aa^\ast+bb^\ast=1$ because each non-empty word begins with either $a$ or $b$. So you get the Cunz algebra.

The papers of Ruy Exel go into more detail how to get operator algebras from inverse semigroups.

Let me begin with inverse semigroup $C^{\ast}$-algebras. Inverse semigroups are semigroups $S$ with the property that for all $s\in S$, there is a unique element $s^*$ with $ss^\ast s=s$ and $s^\ast ss^\ast=s^\ast$. The key example of an inverse semigroup is the symmetric inverses semigroup of all partial bijections of a set $X$ (also called the rook monoid when $X$ is finite). The notion was invented independently by Preston and Wagner to abstract the structure of Lie pseudogroups of transformations.

Inverse semigroups are to partial symmetry as groups are to symmetry. So for example, the symmetry group of the Sierpinski gasket is the Dihedral group of order 6 but the inverse semigroup of partial symmetries is infinite and more interesting. I recommend Mark Lawson's book.

Now if you look at the inverse semigroup of all partial bijections of $X$, then the idempotents are the partial identity maps $1_Y$ with $Y\subset X$. These commute with each other. Ok, now let us try to lift this to operator algebras. Inverse semigroups have lots of idemptotents because $ss^\ast$ and $s^\ast s$ are idempotents. It is a nice fact, proved by Roger Penrose and Douglas Munn, that the idempotents commute.

A partial isometry of a Hilbert space is a bounded operator $a$ such that $a=aa^*a$. In this case, $a^\ast a$ and $aa^\ast$ are projections and $a$ induces an isometry between the images of these projections. So many have argued the natural way to represent inverse semigroups on a Hilbert space is via partial isometries. Idempotents of the inverse semigroup then become projections.

A nice example is the unilateral shift and its adjoint generate the bicyclic inverse semigroup $\langle x\mid x^\ast x=1\rangle$.

Now if you have a quiver $Q$, then we have a partial bijection of the set of non-empty paths associated to each possibly empty path $p$ which operates on paths by concatenation when it makes sense and otherwise is undefined. Empty paths are partial identities acting on the paths beginning from the corresponding vertex. These generate an inverse semigroup and under mild hypotheses give the graph inverse semigroup given by the presentation you might have seen. More generally, one can extend this to a representation on the Hilbert space with basis the non-empty paths and generate a $C^\ast$-algebra. The empty paths act now as projections and that is why vertices give projections in graph $C^\ast$-algebras. There are also some additive relations you pick up in this way. Under mild hypotheses this is the graph $C^\ast$-algebra.

For example, if your quiver has two loops edges $a,b$. Then the graph inverse semigroup has generators $a,b$ with relations $a^\ast a=1=b^\ast b$ and $ab^\ast=0=ba^\ast$. When you take the above representation and generate an algebra you pick up the relation $aa^\ast+bb^\ast=1$ because each non-empty word begins with either $a$ or $b$. So you get the Cunz algebra.

The papers of Ruy Exel go into more detail how to get operator algebras from inverse semigroups.