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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

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

1/30

Pullback diagrams from quotients of graph C ∗-algebras

Samantha Brooker (joint work with Jack Spielberg)

Arizona State University C ∗-Algebras Seminar

September 30, 2020

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

2/30

Introduction

In a recent paper, Hajac, Reznikoff, and Tobolski ([2]) provide conditions they call admissibility on a pair of subgraphs of a row-finite directed graph, which imply that the C ∗-algebras of the three graphs fit into a pullback diagram that is dual to the pushout diagram of the graphs:

C ∗(F1 ∩ F2) C ∗(F1) C ∗(F2) C ∗(E) F1 ∩ F2 F1 F2 E

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

2/30

Introduction

In a recent paper, Hajac, Reznikoff, and Tobolski ([2]) provide conditions they call admissibility on a pair of subgraphs of a row-finite directed graph, which imply that the C ∗-algebras of the three graphs fit into a pullback diagram that is dual to the pushout diagram of the graphs:

C ∗(F1 ∩ F2) C ∗(F1) C ∗(F2) C ∗(E) F1 ∩ F2 F1 F2 E

Our goal was to generalize this result to include graphs that are not necessarily row-finite.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

3/30

Introduction

The authors of [2] use the strong relationship between subgraphs

  • f row-finite graphs and the gauge-invariant ideals of their graph
  • algebras. That relationship is not entirely preserved when you

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].

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pullbacks

Definition 1 (Pedersen) [3]

A commutative diagram of C ∗-algebras C A B X Y !σ ψ φ α β γ δ is called a pullback if ker(γ) ∩ ker(δ) = 0 and if every other pair

  • f 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.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pullbacks of quotients

Theorem 2 (Pedersen)

Let B be a C ∗-algebra and let αI : B → A/I and αJ : B → A/J be ∗-homomorphisms such that qI ◦ αI = qJ ◦ αJ. If IJ = {0} then there exists a unique ∗-homomorphism φ : B → A such that the following diagram commutes. A/(I + J) A/I A/J A B !φ αJ αI qI qJ πJ πI Moreover, if IJ = {0} then such a map φ need not exist.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

6/30

Graphs

Definition 3

A directed graph is a quadruple E = (E 0, E 1, r, s), where E 0 is the set of vertices, E 1 is the set of directed edges, and r, s : E 1 → E 0 are the range and source maps, respectively, so that if e ∈ E 1 is an edge from v to w, then r(e) = w and s(e) = v. We then say that v emits the edge e, and that w 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.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

7/30

Graph Algebras

Definition 4

Let A be a C ∗-algebra. A Cuntz-Krieger E-family in A consists

  • f a set of mutually orthogonal projections P := {Pv : v ∈ E 0}

and a set of partial isometries S := {Se : e ∈ E 1} satisfying the Cuntz-Krieger relations: (CK1) for all e ∈ E 1, S∗

e Se = Ps(e)

(CK2) for all e, f ∈ E 1, if e = f then S∗

e Sf = 0

(CK3) for all e ∈ E 1, Pr(e)Se = Se (CK4) for all regular v ∈ E 0, Pv =

  • e∈E 1:r(e)=v

SeS∗

e

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).

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Paths

Definition 5

  • Let E be a graph. A path of length n in E is a sequence

µ = µ1µ2 · · · µn of edges µi ∈ E 1 such that s(µi) = r(µi+1) for 1 ≤ i ≤ n − 1. We write |µ| := n for the length of µ, and we regard vertices as paths of length 0.

  • 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(µ1) and

s(µ) = s(µn).

  • For paths µ, ν we write µE nν 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 Sµ = Sµ1Sµ2 · · · Sµn, and for

v ∈ E 0 we define Sv = Pv.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

9/30

Ideal structure

Definition 6

Let E be a graph.

  • If v, w ∈ E 0, then we write v ≤ w and say that v is in the

range of w if there exists a path µ ∈ E ∗ such that r(µ) = v and s(µ) = w. A set H ⊆ E 0 is

  • 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.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

9/30

Ideal structure

Definition 6

Let E be a graph.

  • If v, w ∈ E 0, then we write v ≤ w and say that v is in the

range of w if there exists a path µ ∈ E ∗ such that r(µ) = v and s(µ) = w. A set H ⊆ E 0 is

  • 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 IH := Pv : v ∈ H for distinct saturated hereditary sets H ⊆ E 0.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Breaking vertices

If E is not row-finite, however, we encounter a problem:

Definition 7

If H is saturated and hereditary, then a vertex v ∈ E 0 \ H is called breaking for H if v is an infinite receiver, and all but a finite nonzero number of edges in r−1

E (v) have sources in H. We

denote the set of breaking vertices for H by BH. For such vertices, “the image of the projection Pv,H :=

  • r(e)=v,s(e)∈H

SeS∗

e

will be strictly smaller in C ∗(E)/IH than the image of Pv” ([1] p.4).

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Breaking vertices

Even if E is not row-finite, we can still characterize all the distinct gauge-invariant ideals: they are precisely those of the form JH,S = {Pv : v ∈ H} ∪ {Pv − Pv,H : v ∈ S}, where H ⊆ E 0 is saturated and hereditary, and S ⊆ BH. 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.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

12/30

Quotients and (sub)graphs

The quotient C ∗(E)/I by a gauge-invariant ideal I = JH,S can be realized as the graph algebra of a certain graph F, which can be constructed from E, H, and S. In the row-finite case, where JH,S = IH, the graph F is given by F 0 = E 0 \ H and F 1 = s−1(F 0), so that F is a subgraph of E.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

12/30

Quotients and (sub)graphs

The quotient C ∗(E)/I by a gauge-invariant ideal I = JH,S can be realized as the graph algebra of a certain graph F, which can be constructed from E, H, and S. In the row-finite case, where JH,S = IH, the graph F is given by F 0 = E 0 \ H and F 1 = s−1(F 0), so that F is a subgraph of E. Conversely, if you start with a subgraph G ⊆ E whose vertex complement H := E 0 \ G 0 is saturated and hereditary, and whose edges are given by G 1 = s−1(G 0), you recover C ∗(G) through this quotient: C ∗(E)/IH ∼ = C ∗(G). This fact was used in [2] to get pullback diagrams in correspondence with the pushout diagrams of the underlying graphs.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

13/30

Quotients and (sub?)graphs

The quotient C ∗(E)/I by a gauge-invariant ideal I = JH,S can be realized as the graph algebra of a certain graph F, which can be constructed from E, H, and S. If E is not row-finite, F is not always a subgraph of E. But it is when S = BH, ie, when I is the ideal given by H and the gap projections of all its breaking vertices [1]. In this case, like when E was row-finite, F = (E 0 \ H, s−1(E 0 \ H)).

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

13/30

Quotients and (sub?)graphs

The quotient C ∗(E)/I by a gauge-invariant ideal I = JH,S can be realized as the graph algebra of a certain graph F, which can be constructed from E, H, and S. If E is not row-finite, F is not always a subgraph of E. But it is when S = BH, ie, when I is the ideal given by H and the gap projections of all its breaking vertices [1]. In this case, like when E was row-finite, F = (E 0 \ H, s−1(E 0 \ H)). Even though this doesn’t capture every gauge-invariant ideal, we have chosen to phrase our discussion in terms of subgraphs in

  • rder to better illustrate how our result extends the

pushout/pullback duality of [2].

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

14/30

The journey

subgraphs pullback diagrams

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

14/30

The journey

subgraphs ideals/ quotients pullback diagrams

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

14/30

The journey

subgraphs saturated hereditary vertex sets ideals/ quotients pullback diagrams

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

14/30

The journey

subgraphs saturated hereditary vertex sets ideals/ quotients pullback diagrams

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Admissible decompositions

Our aim now is to describe a decomposition of a graph E into subgraphs F1 and F2 so that

  • the vertex complements Hi := E 0 \ F 0

i are saturated and

hereditary for i = 1, 2, and

  • if Ii := JHi,Si is an ideal coming from Hi and some Si ⊆ BHi

(i = 1, 2), we have I1I2 = {0}. This will give us a pullback diagram: C ∗(E)/(I1 + I2) C ∗(E)/I1 C ∗(E)/I2 C ∗(E)

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

16/30

Admissible decompositions

If we want that pullback diagram to be dual to the pushout diagram of the underlying graphs, as in [2]...

C ∗(E)/(I1 + I2) C ∗(E)/I1 C ∗(E)/I2 C ∗(E) F1 ∩ F2 F1 F2 E

then we must impose further conditions so that C ∗(E)/Ii ∼ = C ∗(Fi) (i = 1, 2) and C ∗(E)/(I1 + I2) ∼ = C ∗(F1 ∩ F2).

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

17/30

Row-finite admissible decompositions

First, let’s see how it works in the row-finite case:

Definition 8 (modified from [2])

If E is a row-finite graph, then a pair of subgraphs {F1, F2} of E is an admissible decomposition of E if: (1) E = F1 ∪ F2 (2) if v is a source in F1 ∩ F2, then v is a source in Fi, i = 1, 2 (3) F 1

1 ∩ F 1 2 = s−1 Fi (F 0 1 ∩ F 0 2 ), i = 1, 2.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

17/30

Row-finite admissible decompositions

First, let’s see how it works in the row-finite case:

Definition 8 (modified from [2])

If E is a row-finite graph, then a pair of subgraphs {F1, F2} of E is an admissible decomposition of E if: (1) E = F1 ∪ F2 (2) if v is a source in F1 ∩ F2, then v is a source in Fi, i = 1, 2 (3) F 1

1 ∩ F 1 2 = s−1 Fi (F 0 1 ∩ F 0 2 ), i = 1, 2.

Condition (2) implies that the sets Hi := E 0 \ F 0

i , i = 1, 2, and

H0 := E 0 \ (F 0

1 ∪ F 0 2 ) are saturated.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

17/30

Row-finite admissible decompositions

First, let’s see how it works in the row-finite case:

Definition 8 (modified from [2])

If E is a row-finite graph, then a pair of subgraphs {F1, F2} of E is an admissible decomposition of E if: (1) E = F1 ∪ F2 (2) if v is a source in F1 ∩ F2, then v is a source in Fi, i = 1, 2 (3) F 1

1 ∩ F 1 2 = s−1 Fi (F 0 1 ∩ F 0 2 ), i = 1, 2.

Condition (2) implies that the sets Hi := E 0 \ F 0

i , i = 1, 2, and

H0 := E 0 \ (F 0

1 ∪ F 0 2 ) are saturated.

Condition (3) implies that H0, H1, and H2 are hereditary.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

18/30

Admissible decompositions

Based on that model, we make the following definition:

Definition 9

Let E be a directed graph with a pair of subgraphs {F1, F2}. Let D be the subgraph defined by D0 = F 0

1 ∩ F 0 2 and D1 = D0E 1D0.

Then {F1, F2} is an admissible decomposition of E if (1) E = F1 ∪ F2 (2) For v ∈ D0, if v is regular in Fi then v is regular in D ∩ Fi, i = 1, 2 (3) r(s−1(D0)) ⊆ D0 (4) For v ∈ D0, if v is regular in D and is not a source either in F1 or in F2, then v is regular in F1 or in F2.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Condition 9(2) (Non-examples)

For v ∈ D0, if v is regular in Fi then v is regular in D ∩ Fi, i = 1, 2. This rules out decompositions such as:

v F1 H2 H1 F2 (∞) g1 . . . gm v F1 H2 H1 F2 f1 · · · fn g1 . . . gm

The first decomposition fails because v ∈ D0 is regular in F2, but not in D ∩ F2 (where it is a source). The second fails because v is regular in both F1 and in F2, but is a source in D.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Condition 9(4) (Non-example)

For v ∈ D0, if v is regular in D and is not a source either in F1

  • r in F2, then v is regular in F1 or in F2. This rules out

decompositions such as:

v F1 H2 H1 F2 (∞) (∞) e1 · · · ek

This fails to be admissible because v is regular in D, a non-source in F1 and in F2, but is not regular in either F1 or F2.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Lemmas

Lemma 10 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of a graph E. Let Hi := E 0 \ F 0

i , i = 1, 2. Then:

  • 1. H1, H2, and H := H1 ∪ H2 are hereditary subsets of E 0
  • 2. Hi is saturated in Fj (δi,j = 0) and H1, H2, and H are

saturated in E

  • 3. BH1, BH2 ⊆ D0(= F 0

1 ∩ F 0 2 )

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Lemmas

Lemma 10 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of a graph E. Let Hi := E 0 \ F 0

i , i = 1, 2. Then:

  • 1. H1, H2, and H := H1 ∪ H2 are hereditary subsets of E 0
  • 2. Hi is saturated in Fj (δi,j = 0) and H1, H2, and H are

saturated in E

  • 3. BH1, BH2 ⊆ D0(= F 0

1 ∩ F 0 2 )

Lemma 11 (B-Spielberg 2020)

For any pair {S1, S2 : Si ⊆ BHi}, let Ii := JHi,Si = {Pv : v ∈ Hi} ∪ {Pv − Pv,H : v ∈ Si}, i = 1, 2. Then I1I2 = {0}.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Ideals

Lemma 11 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of E, Si ⊆ BHi, i = 1, 2. Let Ii := JHi,Si, i = 1, 2. Then I1I2 = {0}. Combining this with Theorem (2), we get a pullback diagram: C ∗(E)/(I1 + I2) C ∗(E)/I1 C ∗(E)/I2 C ∗(E)

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

22/30

Ideals

Lemma 11 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of E, Si ⊆ BHi, i = 1, 2. Let Ii := JHi,Si, i = 1, 2. Then I1I2 = {0}. Combining this with Theorem (2), we get a pullback diagram: C ∗(E)/(I1 + I2) C ∗(E)/I1 C ∗(E)/I2 C ∗(E) But we would like to know what I1 + I2 is...

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  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Theorem 12 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of E, let Hi := E 0 \ F 0

i , i = 1, 2, and H := H1 ∪ H2. If

Ii := JHi,BHi , i = 1, 2, then I1 + I2 = JH,BH.

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  • f graph

C∗- algebras

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& J. Spielberg Background

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Admissibility

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Theorem 12 (B-Spielberg 2020)

Let {F1, F2} be an admissible decomposition of E, let Hi := E 0 \ F 0

i , i = 1, 2, and H := H1 ∪ H2. If

Ii := JHi,BHi , i = 1, 2, then I1 + I2 = JH,BH. This might seem too good to be true (it did to me, anyway). It feels unreasonable to expect that if v ∈ BH, then the gap projection Pv − Pv,H ∈ I1 + I2. Although we might expect that BH = BH1 ∪ BH2 (indeed it is), the subprojection Pv,H is dependent on H. We will now prove this happy surprise.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pv − Pv,H ∈ I1 + I2

Suppose v ∈ BH. WLOG v ∈ BH1, ie, |vE 1H1| = ∞ and 0 < |vE 1(HC

1 )| < ∞. Put another way, 0 < |vE 1F 0 1 | < ∞,

meaning v is regular in F1. Then by definition 9(4), v is regular in D ∩ F1.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pv − Pv,H ∈ I1 + I2

Suppose v ∈ BH. WLOG v ∈ BH1, ie, |vE 1H1| = ∞ and 0 < |vE 1(HC

1 )| < ∞. Put another way, 0 < |vE 1F 0 1 | < ∞,

meaning v is regular in F1. Then by definition 9(4), v is regular in D ∩ F1. v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek {ej}k

1 = vD1, 0 < k < ∞

{fj}n

1 = vE 1H2, 0 ≤ n < ∞

Pv,H =

  • e∈vE 1(HC )

SeS∗

e

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pv − Pv,H ∈ I1 + I2

Suppose v ∈ BH. WLOG v ∈ BH1, ie, |vE 1H1| = ∞ and 0 < |vE 1(HC

1 )| < ∞. Put another way, 0 < |vE 1F 0 1 | < ∞,

meaning v is regular in F1. Then by definition 9(4), v is regular in D ∩ F1. v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek {ej}k

1 = vD1, 0 < k < ∞

{fj}n

1 = vE 1H2, 0 ≤ n < ∞

Pv,H =

  • e∈vE 1(HC )

SeS∗

e

If n = 0,

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

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Pv − Pv,H ∈ I1 + I2

Suppose v ∈ BH. WLOG v ∈ BH1, ie, |vE 1H1| = ∞ and 0 < |vE 1(HC

1 )| < ∞. Put another way, 0 < |vE 1F 0 1 | < ∞,

meaning v is regular in F1. Then by definition 9(4), v is regular in D ∩ F1. v F1 H2 H1 F2 (∞) e1 · · · ek {ej}k

1 = vD1, 0 < k < ∞

{fj}n

1 = vE 1H2, 0 ≤ n < ∞

Pv,H =

  • e∈vE 1(HC )

SeS∗

e

If n = 0, then vE 1(HC) = vE 1(HC

1 ), so Pv,H = Pv,H1,

and the gap projection Pv − Pv,H1 ∈ I1 so we’re done.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

25/30

Pv − Pv,H ∈ I1 + I2

However, if n > 0, then Pv,H1 =

  • e∈vE 1(HC

1 )

SeS∗

e

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek

slide-41
SLIDE 41

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

25/30

Pv − Pv,H ∈ I1 + I2

However, if n > 0, then Pv,H1 =

  • e∈vE 1(HC

1 )

SeS∗

e

=

k

  • j=1

SejS∗

ej + n

  • j=1

SfjS∗

fj

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek

slide-42
SLIDE 42

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

25/30

Pv − Pv,H ∈ I1 + I2

However, if n > 0, then Pv,H1 =

  • e∈vE 1(HC

1 )

SeS∗

e

=

k

  • j=1

SejS∗

ej + n

  • j=1

SfjS∗

fj

= Pv,H +

n

  • j=1

SfjS∗

fj.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek

slide-43
SLIDE 43

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

25/30

Pv − Pv,H ∈ I1 + I2

However, if n > 0, then Pv,H1 =

  • e∈vE 1(HC

1 )

SeS∗

e

=

k

  • j=1

SejS∗

ej + n

  • j=1

SfjS∗

fj

= Pv,H +

n

  • j=1

SfjS∗

fj.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek Thus Pv − Pv,H = Pv − Pv,H1 +

n

  • j=1

SfjS∗

fj.

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SLIDE 44

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

26/30

Pv − Pv,H = Pv − Pv,H1 + n

j=1 SfjS∗ fj

So, you wonder, where is n

j=1 SfjS∗ fj? Well, I’m glad you asked.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek

slide-45
SLIDE 45

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

26/30

Pv − Pv,H = Pv − Pv,H1 + n

j=1 SfjS∗ fj

So, you wonder, where is n

j=1 SfjS∗ fj? Well, I’m glad you asked.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek For j ∈ {1, . . . , n}, s(fj) ∈ H2, so Ps(fj) ∈ I2.

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SLIDE 46

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

26/30

Pv − Pv,H = Pv − Pv,H1 + n

j=1 SfjS∗ fj

So, you wonder, where is n

j=1 SfjS∗ fj? Well, I’m glad you asked.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek For j ∈ {1, . . . , n}, s(fj) ∈ H2, so Ps(fj) ∈ I2. By (CK1) and the fact that Sfj is a partial isometry, SfjPs(fj) =

  • Sfj. So, Sfj is a multiple of an

element of I2, thus Sfj ∈ I2.

slide-47
SLIDE 47

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

26/30

Pv − Pv,H = Pv − Pv,H1 + n

j=1 SfjS∗ fj

So, you wonder, where is n

j=1 SfjS∗ fj? Well, I’m glad you asked.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek For j ∈ {1, . . . , n}, s(fj) ∈ H2, so Ps(fj) ∈ I2. By (CK1) and the fact that Sfj is a partial isometry, SfjPs(fj) =

  • Sfj. So, Sfj is a multiple of an

element of I2, thus Sfj ∈ I2. Therefore, (since ideals

  • f

C ∗-algebras are ∗-closed subalgebras), we have n

j=1 SfjS∗ fj ∈ I2.

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SLIDE 48

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

26/30

Pv − Pv,H = Pv − Pv,H1 + n

j=1 SfjS∗ fj

So, you wonder, where is n

j=1 SfjS∗ fj? Well, I’m glad you asked.

v F1 H2 H1 F2 f1 · · · fn (∞) e1 · · · ek For j ∈ {1, . . . , n}, s(fj) ∈ H2, so Ps(fj) ∈ I2. By (CK1) and the fact that Sfj is a partial isometry, SfjPs(fj) =

  • Sfj. So, Sfj is a multiple of an

element of I2, thus Sfj ∈ I2. Therefore, (since ideals

  • f

C ∗-algebras are ∗-closed subalgebras), we have n

j=1 SfjS∗ fj ∈ I2.

So, Pv − Pv,H = (Pv − Pv,H1) + n

j=1 SfjS∗ fj ∈ I1 + I2, as

promised.

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SLIDE 49

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

27/30

When do pullbacks correspond to graph pushouts?

Now we have a pullback diagram

C ∗(E)/I C ∗(E)/I1 C ∗(E)/I2 C ∗(E)

so all we need to do is say when C ∗(E)/Ii ∼ = C ∗(Fi), i = 1, 2 and C ∗(E)/I ∼ = C ∗(F1 ∩ F2).

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SLIDE 50

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

27/30

When do pullbacks correspond to graph pushouts?

Now we have a pullback diagram

C ∗(E)/I C ∗(E)/I1 C ∗(E)/I2 C ∗(E)

so all we need to do is say when C ∗(E)/Ii ∼ = C ∗(Fi), i = 1, 2 and C ∗(E)/I ∼ = C ∗(F1 ∩ F2). We know that for any saturated hereditary set K ⊆ E 0, C ∗(E)/JK,BK ∼ = C ∗(E 0 \ K, s−1(E 0 \ K)).

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SLIDE 51

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

27/30

When do pullbacks correspond to graph pushouts?

Now we have a pullback diagram

C ∗(E)/I C ∗(E)/I1 C ∗(E)/I2 C ∗(E)

so all we need to do is say when C ∗(E)/Ii ∼ = C ∗(Fi), i = 1, 2 and C ∗(E)/I ∼ = C ∗(F1 ∩ F2). We know that for any saturated hereditary set K ⊆ E 0, C ∗(E)/JK,BK ∼ = C ∗(E 0 \ K, s−1(E 0 \ K)). That means we need Fi = (F 0

i , s−1(F 0 i )), i = 1, 2, and

F1 ∩ F2 = (F 0

1 ∩ F 0 2 , s−1(F 0 1 ∩ F 0 2 )) = D.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

28/30

Strong admissibility

Definition 13

Let E be a directed graph with a pair of subgraphs {F1, F2}. Let D be the subgraph defined by D0 = F 0

1 ∩ F 0 2 and D1 = D0E 1D0.

Then {F1, F2} is a strongly admissible decomposition of E if (1) E = F1 ∪ F2 (2) For v ∈ D0, if v is regular in Fi then v is regular in D ∩ Fi, i = 1, 2 (3) r(s−1(D0)) ⊆ D0 (4) For v ∈ D0, if v is regular in D and is not a source either in F1 or in F2, then v is regular in F1 or in F2.

slide-53
SLIDE 53

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

28/30

Strong admissibility

Definition 13

Let E be a directed graph with a pair of subgraphs {F1, F2}. Let D be the subgraph defined by D0 = F 0

1 ∩ F 0 2 and D1 = D0E 1D0.

Then {F1, F2} is a strongly admissible decomposition of E if (1) E = F1 ∪ F2 (2) If v is singular in F1 ∩ F2 then v is singular in Fi, i = 1, 2 (3) r(s−1(D0)) ⊆ D0 (4) For v ∈ D0, if v is regular in D and is not a source either in F1 or in F2, then v is regular in F1 or in F2.

slide-54
SLIDE 54

Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

28/30

Strong admissibility

Definition 13

Let E be a directed graph with a pair of subgraphs {F1, F2}. Let D be the subgraph defined by D0 = F 0

1 ∩ F 0 2 and D1 = D0E 1D0.

Then {F1, F2} is a strongly admissible decomposition of E if (1) E = F1 ∪ F2 (2) If v is singular in F1 ∩ F2 then v is singular in Fi, i = 1, 2 (3) F 1

1 ∩ F 1 2 = s−1 Fi (F 0 1 ∩ F 0 2 ), i = 1, 2

(4) For v ∈ D0, if v is regular in D and is not a source either in F1 or in F2, then v is regular in F1 or in F2.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

28/30

Strong admissibility

Definition 13

Let E be a directed graph with a pair of subgraphs {F1, F2}. Then {F1, F2} is a strongly admissible decomposition of E if (1) E = F1 ∪ F2 (2) If v is singular in F1 ∩ F2 then v is singular in Fi, i = 1, 2 (3) F 1

1 ∩ F 1 2 = s−1 Fi (F 0 1 ∩ F 0 2 ), i = 1, 2

(4) If v is regular in F1 ∩ F2 and not a source in F1 or in F2, then v is regular in F1 or in F2.

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Pullback diagrams from quotients

  • f graph

C∗- algebras

  • S. Brooker

& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

29/30

Pushouts and Pullbacks

Theorem 14 (B-Spielberg 2020)

If {F1, F2} is a strongly admissible decomposition of a directed graph E, then

C ∗(F1 ∩ F2) C ∗(F1) C ∗(F2) C ∗(E)

is a pullback diagram of graph C ∗-algebras via quotient maps, dual to the corresponding graph pushout diagram:

F1 ∩ F2 F1 F2. E

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Pullback diagrams from quotients

  • f graph

C∗- algebras

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& J. Spielberg Background

Pullbacks Graph algebras

Admissibility

30/30

Thank You!

  • T. Bates, J. Hong, I. Raeburn, W. Szyma´

nski, The ideal structure of the C ∗-algebras of infinite graphs, Illinois J. Math., vol. 46 (2002), no. 4, 1159-1176.

  • P. Hajac, S. Reznikoff and M. Tobolski, Pullbacks of graph

C ∗-algebras from admissible pushouts of graphs, arXiv:1811.00100v4[math.OA] 2 June 2019.

  • G. Pedersen, Pullback and pushout construtions in

C ∗-algebra theory, J. Functional Analysis 167 no. 2 (1999), 243-344.

  • I. Raeburn, Graph Algebras, CBMS Regional Conference

Series in Mathematics, vol. 103, published for the Conference Board of the Matematical Sciences, Washington, D.C. 2005.