General recipe for building C*-algebras out of combinatorial object

Question

I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out of those objects; but I want to understand what philosophy lies under the recipe. Say for discrete groups I know group C*-algebras are made of as universal object of unitary elements of groups elements with relations coming from group operation. Now what I understand is they take unitary because if we fix an element $g$ in the group $G$, the map $T_g:G\rightarrow G$ defined by $T_g(h)=gh$ is one-one onto morphism. For Ring C*-algebras also this works. In that case where the map is not onty they take isometry. But my confusion starts with graph C*-algebra. I don't understand why they take projections for vertices and partial isometrys for edges (and the given relations).

Answer 1

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.

Answer 2

Benjamin's answer nicely describes where the relations for graph $C^\ast$-algebras come from. Here is an attempt to answer the question suggested by the title: what is the general recipe for constructing $C^\ast$-algebras from other mathematical objects. This may not work for every type of $C^\ast$-construction, but it seems to be the general formula behind at least group $C^\ast$-algebras, crossed products, graph $C^\ast$-algebras (and more generally, $C^\ast$-algebras of inverse semigroups), semigroup $C^\ast$-algebras, and ring $C^\ast$-algebras.

To the mathematical object in question is associated a natural and concrete Hilbert space and a natural collection of operators on that Hilbert space (often called the regular representation). Take the obvious $*$-algebraic relations that hold between these operators; then these are the relations used to define the universal $C^\ast$-algebra.

Here is this formula worked out for a few examples:

Group $C^\ast$-algebras: For a discrete group $G$, the natural concrete Hilbert space is $\ell^2(G)$ (with canonical ONB $(\xi_g)_{g \in G}$), and to each group element $g \in G$ is associated the operator $\lambda_g$ defined by $$ \lambda_g(\xi_h) = \xi_{gh}. $$ The obvious relations here are, that $\lambda_g$ is a unitary and $\lambda_g\lambda_h = \lambda_{gh}$; that is to say, that $\lambda$ is a group homomorphism between $G$ and the unitary group of $\mathcal{B}(\ell^2(G))$.

Of course, the $C^*$-algebra generated by $\{\lambda_g\}$ is called the reduced $C^\ast$-algebra of $G$, while the universal one (generated by $\{u_g: g \in G\}$ satisfying the relations that they are unitary and $u_gu_h = u_{gh}$) is called the (full) group $C^*$-algebra of $G$.

Dynamical systems: Though this can be done more generally, let's stick to a discrete group $G$ acting by homeomorphisms $\alpha_g$ on a compact metric space $X$. The natural concrete Hilbert space is $\ell^2(G \times X)$ (with ONB $(\xi_{g,x})_{g \in G, x \in X}$). For each $g \in G$, we may associate the operator $\lambda_g$ defined by $$ \lambda_g(\xi_{h,x}) = \xi_{gh,x}. $$ For each $f \in C(X)$, we may also associate the operator $D_f$ defined by $$ D_f(\xi_{g,x}) = f(g^{-1}x)\xi_{g,x}. $$ The obvious relations are: that $g \mapsto \lambda_g$ is a group homomorphism from $G$ to the unitary group; that $f \mapsto D_f$ is a $*$-homomorphism; and that $$ \lambda_g D_f \lambda_g^* = f \circ \alpha_g. $$

Graph $C^\ast$-algebras: The details of this construction are more or less explained in Benjamin's answer. The concrete Hilbert space has an orthonormal basis $(\xi_{\pi})$ indexed by the nonempty paths in the graph. An edge $e$ gives rise to the operator $v_e$ given by $$ v_e(\xi_{\pi}) = \begin{cases} \xi_{e\pi},\ &\text{if $e\pi$ is a path,} \\ 0,\ &\text{otherwise.} \end{cases} $$ A vertex $v$ gives rise to the operator $p_v$ which is the projection onto the span of $$ \{\xi_{\pi}: \pi \text{ is a non-empty path beginning at $v$}\}. $$