Graph C*-algebra

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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[1] 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.


Let be a directed graph with a countable set of vertices , a countable set of edges , and maps identifying the range and source of each edge, respectively. The graph C*-algebra corresponding to , denoted by , is the universal C*-algebra generated by mutually orthogonal projections and partial isometries with mutually orthogonal ranges such that :

(i) for all

(ii) whenever

(iii) for all .

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[edit]

Directed graph (E) Graph C*-algebra (C*(E))
Graph complex numbers.png - the set of complex numbers
Graph circle algebra.png - the set of complex-valued continuous functions on the circle
Directed graph matrix.png - the set of n x n matrices over
Graph circle matrix.png - the set of n x n matrices over
Graph compact.png - the set of compact operators over a separable Hilbert space
Graph Toeplitz algebra.png - Toeplitz algebra
Graph cuntz.png - Cuntz algebra


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