On the Primitive Ideal spaces of the $C^*$-algebras of graphs

Abstract

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz-Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as Prim C * (E) are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph E from E such that C * ( E) is an AF-algebra and Prim C * (E) and Prim C * ( E) are homeomorphic.

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

“…By Proposition 7.2 we obtain H J = θ(H I ) and (H J ) fin ∞ \ B J = (H I ) fin ∞ \ V I . By [1,Corollary 3.5] there is an isomorphism ψ : C * (E)/J → C * (E I ). Let Q I ∈ M(C * (E I )) be the image of Q under ψ.…”

“…Since I ∩ C * (E) • ⊂ J ⊂ I, we have J ∩ C * (E) • = I ∩ C * (E) • . Theorem 3.6 of [1] implies that each gauge-invariant ideal of C * (E) is uniquely determined by its intersection with C * (E) • . Since both I and J are gauge invariant, it follows that I = J.…”

“…To do this we use the classifications of gauge-invariant ideals in graph algebras [1,Theorem 3.6] and in ultragraph algebras [11,Theorem 6.12]. We also describe quotients of ultragraph algebras by gauge-invariant ideals as full corners in graph algebras.…”

“…If H ⊂ E 0 is saturated hereditary, then we define H fin ∞ := {v ∈ E 0 : |s −1 E (v)| = ∞ and 0 < |s −1 E (v) ∩ r −1 E (E 0 \ H)| < ∞}. Theorem 3.6 of [1] shows that there is a bijection J → (H J , B J ) between gauge-invariant ideals of C * (E) and pairs (H, B) such that H ⊂ E 0 is saturated hereditary, and B ⊂ H fin ∞ . Specifically, H J = {x ∈ E 0 : q x ∈ J} and B J = {v ∈ (H J ) fin ∞ : q v − α∈s −1…”

Operator Algebras and C*-correspondences: A Survey

“…with |r −1 (p)| = 0, ∞ The relations ( †) have been refined in a series of papers by the Australian school and reached the above form in [5,49]. All refinements involved condition (5) and as it stands now, condition (5) gives the equality requirement for projections L p such that p is not a source and receives finitely many edges.…”