Graph C*-algebra

In mathematics, particularly the theory of C*-algebras, a graph C*-algebra is a universal C*-algebra associated to a directed graph. They form a rich class of C*-algebras encompassing Cuntz algebras, Cuntz-Krieger algebras, the Toeplitz algebra, etc. Also every AF-algebra is Morita equivalent^[1] to a graph C*-algebra. As the structure of graph C*-algebras is fairly tractable with computable invariants, they play an important part in the classification theory of C*-algebras.

Definition

Let $E=(E^0, E^1, r, s)$ be a directed graph with a countable set of vertices $E^0$, a countable set of edges $E^1$, and maps $r, s : E^1 \rightarrow E^0$ identifying the range and source of each edge, respectively. The graph C*-algebra corresponding to $E$, denoted by $C^*(E)$, is the universal C*-algebra generated by mutually orthogonal projections $\{ p_v : v \in E^0 \}$ and partial isometries $\{ s_e : e \in E^1 \}$ with mutually orthogonal ranges such that :

(i) $s_e^*s_e = p_{r(e)}$ for all $e \in E^1$

(ii) $p_v = \sum_{s(e)=v} s_e s_e^*$ whenever $0 < |s^{-1}(v)| < \infty$

(iii) $s_e s_e^* \le p_{s(e)}$ for all $e \in E^1$.

Examples of graph C*-algebras

Directed graph (E)	Graph C*-algebra (C*(E))
	$\mathbb{C}$ - the set of complex numbers
	$C(S^1)$ - the set of complex-valued continuous functions on the circle
	$M_n(\mathbb{C})$ - the set of n x n matrices over $\mathbb{C}$
	$M_n(C(S^1))$ - the set of n x n matrices over $C(S^1)$
	$\mathcal{K}$ - the set of compact operators over a separable Hilbert space
	$\mathcal{T}$ - Toeplitz algebra
	$\mathcal{O}_n$ - Cuntz algebra

Notes

1. ↑ D. Drinen,Viewing AF-algebras as graph algebras, Proc. Amer. Math. Soc., 128 (2000), pp. 1991–2000.

References