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

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