# Graph C*-algebra

In mathematics, particularly the theory of C*-algebras, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. The graph C*-algebras form a rich subclass of C*-algebras that include Cuntz algebras, Cuntz-Krieger algebras, the Toeplitz algebra, and all stable AF-algebras. In addition, every AF-algebra is Morita equivalent to a graph C*-algebra and every Kirchberg algebra with free K1-group is Morita equivalent to a graph C*-algebras. 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) $s_{e}s_{e}^{*}\leq p_{s(e)}$ for all $e\in E^{1}$ .

When a graph is row-finite (i.e., each vertex emits at most a finite number of edges) condition (iii) is implied by condition (ii).

## 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