Graphs, Groupoids, and Cuntz–Krieger Algebras

Diagonal-preserving gauge-invariant isomorphisms of graph C⁎-algebras

“…Matsumoto has since then used this aproach to study gauge-invariant isomorphisms of Cuntz-Krieger algebras [10,11], and has, among other results, proved a converse to [6 In [9] Kumjian, Pask, Raeburn, and Renault extended the definition of Cuntz-Krieger algebras when they used groupoids to construct C * -algebras from directed graphs that are not assumed to be finite. These graph C * -algebras have since attracted a lot of interest, and by using that graph C * -algebras can be constructed from groupoids, [12,Theorem 2.3] has recently been transfered to the setting of graph C * -algebras [1,2].…”

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

“…Let E = (E 0 , E 1 , r E , s E ) be the graph constructed from G as in Definition 3.12. We relate the path structure of E to that of G. In particular, we show that G satisfies Condition (K) as in [11] if and only if E satisfies Condition (K) as in [13]. Condition (K) was introduced in [13] to characterise those graphs in whose C * -algebras every ideal is gauge-invariant.…”

“…We relate the path structure of E to that of G. In particular, we show that G satisfies Condition (K) as in [11] if and only if E satisfies Condition (K) as in [13]. Condition (K) was introduced in [13] to characterise those graphs in whose C * -algebras every ideal is gauge-invariant. In Section 6, we will combine our results in this section with our main result Theorem 5.22 to deduce from Kumjian, Pask, Raeburn and Renault's result the corresponding theorem for ultragraph C * -algebras.…”

ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γOF A GAUGE ACTION ON A GRAPH C*-ALGEBRA

“…Let E be a row finite directed graph and C * (E) be the graph C * -algebra of E generated by a universal Cuntz-Krieger E family {p v , s e } (for example, see [1,3,18,19,22]). Then by the universal property, the gauge action γ of T, γ z (p v ) = p v , γ z (s e ) = zs e , is well defined and the fixed point algebra C * (E) γ turns out to be an AF algebra.…”

“…Then by the universal property, the gauge action γ of T, γ z (p v ) = p v , γ z (s e ) = zs e , is well defined and the fixed point algebra C * (E) γ turns out to be an AF algebra. In fact, it is known in [17] using results of [25] and [19] on groupoid C * -algebras that C * (E) γ is strong Morita equivalent (hence stably isomorphic by [7]) to the graph C * -algebra C * (E Z × E) of the Cartesian product graph E Z × E (E Z × E is the graph Z × E in [17]). Since E Z × E has no loops, its graph C * -algebra C * (E Z × E) is an AF algebra ( [18]).…”