Pullback diagrams from quotients of graph C ∗ - algebras Pullback diagrams from quotients of graph S. Brooker & J. C ∗ -algebras Spielberg Background Pullbacks Graph algebras Samantha Brooker Admissibility (joint work with Jack Spielberg) Arizona State University C ∗ -Algebras Seminar September 30, 2020 1/30
Pullback Introduction diagrams from quotients of graph C ∗ - In a recent paper, Hajac, Reznikoff, and Tobolski ([2]) provide algebras conditions they call admissibility on a pair of subgraphs of a S. Brooker row-finite directed graph, which imply that the C ∗ -algebras of & J. Spielberg the three graphs fit into a pullback diagram that is dual to the Background pushout diagram of the graphs: Pullbacks Graph algebras C ∗ ( E ) E Admissibility C ∗ ( F 1 ) C ∗ ( F 2 ) F 1 F 2 C ∗ ( F 1 ∩ F 2 ) F 1 ∩ F 2 2/30
Pullback Introduction diagrams from quotients of graph C ∗ - In a recent paper, Hajac, Reznikoff, and Tobolski ([2]) provide algebras conditions they call admissibility on a pair of subgraphs of a S. Brooker row-finite directed graph, which imply that the C ∗ -algebras of & J. Spielberg the three graphs fit into a pullback diagram that is dual to the Background pushout diagram of the graphs: Pullbacks Graph algebras C ∗ ( E ) E Admissibility C ∗ ( F 1 ) C ∗ ( F 2 ) F 1 F 2 C ∗ ( F 1 ∩ F 2 ) F 1 ∩ F 2 Our goal was to generalize this result to include graphs that are not necessarily row-finite. 2/30
Pullback Introduction diagrams from quotients of graph C ∗ - algebras S. Brooker & J. Spielberg The authors of [2] use the strong relationship between subgraphs Background of row-finite graphs and the gauge-invariant ideals of their graph Pullbacks algebras. That relationship is not entirely preserved when you Graph algebras Admissibility drop the row-finite assumption; while subgraphs can be used to describe many gauge-invariant ideals, they are not enough on their own to capture all such ideals. In spite of this, our results today are phrased in terms of subgraphs in order to better parallel those in [2]. 3/30
Pullback Pullbacks diagrams from quotients of graph Definition 1 (Pedersen) [3] C ∗ - algebras A commutative diagram of C ∗ -algebras S. Brooker & J. Spielberg Y φ ψ Background ! σ Pullbacks Graph algebras γ δ X Admissibility A B α β C is called a pullback if ker( γ ) ∩ ker( δ ) = 0 and if every other pair of morphisms φ : Y → A and ψ : Y → B from a C ∗ -algebra Y that are coherent (meaning α ◦ φ = β ◦ ψ ) factors through X , ie, φ = δ ◦ σ and ψ = γ ◦ σ for a (necessarily unique) morphism σ : Y → X . 4/30
Pullback Pullbacks of quotients diagrams from quotients of graph Theorem 2 (Pedersen) C ∗ - algebras S. Brooker Let B be a C ∗ -algebra and let α I : B → A / I and α J : B → A / J & J. Spielberg be ∗ -homomorphisms such that q I ◦ α I = q J ◦ α J . If IJ = { 0 } then there exists a unique ∗ -homomorphism φ : B → A such that Background Pullbacks the following diagram commutes. Graph algebras Admissibility B α I α J ! φ π J π I A A / I A / J q I q J A / ( I + J ) Moreover, if IJ � = { 0 } then such a map φ need not exist. 5/30
Pullback Graphs diagrams from quotients of graph C ∗ - Definition 3 algebras A directed graph is a quadruple E = ( E 0 , E 1 , r , s ), where E 0 is S. Brooker & J. the set of vertices, E 1 is the set of directed edges, and Spielberg r , s : E 1 → E 0 are the range and source maps, respectively, so Background that if e ∈ E 1 is an edge from v to w , then r ( e ) = w and Pullbacks Graph algebras s ( e ) = v . We then say that v emits the edge e , and that w Admissibility receives the edge e . A vertex v ∈ E 0 is called... • a source if it receives no edges, that is, r − 1 ( v ) = ∅ . • an infinite receiver if | r − 1 ( v ) | = ∞ . • singular if v is either a source or an infinite receiver. • regular if v is not singular. E is called row-finite if it has no infinite receivers. 6/30
Pullback Graph Algebras diagrams from quotients Definition 4 of graph C ∗ - Let A be a C ∗ -algebra. A Cuntz-Krieger E-family in A consists algebras of a set of mutually orthogonal projections P := { P v : v ∈ E 0 } S. Brooker & J. and a set of partial isometries S := { S e : e ∈ E 1 } satisfying the Spielberg Cuntz-Krieger relations: Background Pullbacks (CK1) for all e ∈ E 1 , S ∗ e S e = P s ( e ) Graph algebras (CK2) for all e , f ∈ E 1 , if e � = f then S ∗ Admissibility e S f = 0 (CK3) for all e ∈ E 1 , P r ( e ) S e = S e (CK4) for all regular v ∈ E 0 , � S e S ∗ P v = e e ∈ E 1 : r ( e )= v The Cuntz-Krieger algebra of the graph E is the C ∗ -algebra generated by a universal Cuntz-Krieger E -family, and is denoted C ∗ ( E ). 7/30
Pullback Paths diagrams from quotients of graph C ∗ - algebras Definition 5 S. Brooker & J. • Let E be a graph. A path of length n in E is a sequence Spielberg µ = µ 1 µ 2 · · · µ n of edges µ i ∈ E 1 such that s ( µ i ) = r ( µ i +1 ) Background for 1 ≤ i ≤ n − 1. We write | µ | := n for the length of µ , Pullbacks Graph algebras and we regard vertices as paths of length 0. Admissibility • We write E n for the set of paths in E of length n , and E ∗ = � n ≥ 0 E n . For µ ∈ E n we define r ( µ ) = r ( µ 1 ) and s ( µ ) = s ( µ n ). • For paths µ, ν we write µ E n ν for all paths of the form µαν where α ∈ E n , r ( α ) = s ( µ ) , and s ( α ) = r ( ν ). • For µ ∈ � n i =1 E 1 , we define S µ = S µ 1 S µ 2 · · · S µ n , and for v ∈ E 0 we define S v = P v . 8/30
Pullback Ideal structure diagrams from quotients of graph Definition 6 C ∗ - algebras Let E be a graph. S. Brooker & J. • If v , w ∈ E 0 , then we write v ≤ w and say that v is in the Spielberg range of w if there exists a path µ ∈ E ∗ such that r ( µ ) = v Background and s ( µ ) = w . Pullbacks A set H ⊆ E 0 is Graph algebras Admissibility • hereditary if whenever v ∈ H and w ∈ E 0 with v ≤ w , it follows that w ∈ H . • saturated if for every regular vertex v ∈ E 0 , if s ( r − 1 E ( v )) ⊆ H , then v ∈ H ; that is, if every vertex that sends an edge to v is in H , then v must be in H . 9/30
Pullback Ideal structure diagrams from quotients of graph Definition 6 C ∗ - algebras Let E be a graph. S. Brooker & J. • If v , w ∈ E 0 , then we write v ≤ w and say that v is in the Spielberg range of w if there exists a path µ ∈ E ∗ such that r ( µ ) = v Background and s ( µ ) = w . Pullbacks A set H ⊆ E 0 is Graph algebras Admissibility • hereditary if whenever v ∈ H and w ∈ E 0 with v ≤ w , it follows that w ∈ H . • saturated if for every regular vertex v ∈ E 0 , if s ( r − 1 E ( v )) ⊆ H , then v ∈ H ; that is, if every vertex that sends an edge to v is in H , then v must be in H . If E is row-finite, then the distinct gauge-invariant ideals of C ∗ ( E ) are precisely those of the form I H := � P v : v ∈ H � for distinct saturated hereditary sets H ⊆ E 0 . 9/30
Pullback Breaking vertices diagrams from quotients of graph C ∗ - If E is not row-finite, however, we encounter a problem: algebras S. Brooker & J. Definition 7 Spielberg If H is saturated and hereditary, then a vertex v ∈ E 0 \ H is Background called breaking for H if v is an infinite receiver, and all but a Pullbacks Graph algebras finite nonzero number of edges in r − 1 E ( v ) have sources in H . We Admissibility denote the set of breaking vertices for H by B H . For such vertices, “the image of the projection � S e S ∗ P v , H := e r ( e )= v , s ( e ) �∈ H will be strictly smaller in C ∗ ( E ) / I H than the image of P v ” ([1] p.4). 10/30
Pullback Breaking vertices diagrams from quotients of graph C ∗ - algebras S. Brooker & J. Spielberg Even if E is not row-finite, we can still characterize all the distinct gauge-invariant ideals: they are precisely those of the Background Pullbacks form Graph algebras J H , S = �{ P v : v ∈ H } ∪ { P v − P v , H : v ∈ S }� , Admissibility where H ⊆ E 0 is saturated and hereditary, and S ⊆ B H . Note that if E is row-finite, there are no breaking vertices for any saturated hereditary vertex set, so this description of the ideals captures that case, as well. 11/30
Pullback Quotients and (sub)graphs diagrams from quotients of graph C ∗ - algebras S. Brooker The quotient C ∗ ( E ) / I by a gauge-invariant ideal I = J H , S can be & J. Spielberg realized as the graph algebra of a certain graph F , which can be constructed from E , H , and S . Background Pullbacks In the row-finite case, where J H , S = I H , the graph F is given by Graph algebras F 0 = E 0 \ H and F 1 = s − 1 ( F 0 ), so that F is a subgraph of E . Admissibility 12/30
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