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 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) $s_{e}s_{e}^{*}\leq p_{s(e)}$ 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