Graph C* algebra - Alchetron, The Free Social Encyclopedia

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 → 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 ∈ E 0 } and partial isometries { s e : e ∈ E 1 } with mutually orthogonal ranges such that :

(i) s e ∗ s e = p r ( e ) for all e ∈ E 1

(ii) p v = ∑ s ( e ) = v s e s e ∗ whenever 0 < | s − 1 ( v ) | < ∞

(iii) s e s e ∗ ≤ p s ( e ) for all e ∈ E 1 .

References

Graph C*-algebra Wikipedia

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