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 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 .
Examples of graph C*-algebras
Directed graph (E) | Graph C*-algebra (C*(E)) |
---|---|
- the set of complex numbers | |
- the set of complex-valued continuous functions on the circle | |
- the set of n x n matrices over | |
- the set of n x n matrices over | |
- the set of compact operators over a separable Hilbert space | |
- Toeplitz algebra | |
- Cuntz algebra |
Notes
- ↑ D. Drinen,Viewing AF-algebras as graph algebras, Proc. Amer. Math. Soc., 128 (2000), pp. 1991–2000.