The boundary-path space of a directed graph
56th AustMS AGM, Ballarat
S.B.G. Webster
University of Wollongong
25 September 2012
MODEL / ANALYSE / FORMULATE /
ILLUMINATE
CONNECT:IMIA
Definition
A directed graph E is a set E 0 of vertices and a set E 1 of directed
edges, with direction determined by range and source maps
r , s : E 1 → E 0.
Example
e
v
f
E 0 = {v , w }
s(e) = r (e) = r (f ) = v
w
E 1 = {e, f }
s(f ) = w
Paths
I
A sequence µ1 µ2 µ3 . . . of edges is a path if s(µi ) = r (µi+1 ) for
all i.
µ1
µn
r (µ)
s(µ)
Paths
I
A sequence µ1 µ2 µ3 . . . of edges is a path if s(µi ) = r (µi+1 ) for
all i.
µ1
µn
r (µ)
s(µ)
I
E n = {µ : µ is a path with n (possibly = ∞) edges}
I
E ∗ = {µ : µ has finitely many edges}.
Paths
I
A sequence µ1 µ2 µ3 . . . of edges is a path if s(µi ) = r (µi+1 ) for
all i.
µ1
µn
r (µ)
s(µ)
I
E n = {µ : µ is a path with n (possibly = ∞) edges}
I
E ∗ = {µ : µ has finitely many edges}.
I
For V ⊂ E 0 and F ⊂ E ∗ , define VF := F ∩ r −1 (V ).
I
In particular, for v ∈ E 0 , vF = F ∩ r −1 (v ).
Graph C ∗ -algebras
I
I
E ≤n := {µ ∈ E ∗ : |µ| = n, or |µ| < n and s(µ)E 1 = ∅}.
The graph C ∗ -algebra C ∗ (E ) is universal for C ∗ -algebras
containing a Cuntz-Krieger E -family: a family consisting of
mutually orthogonal projections {sv : v ∈ E 0 } and partial
isometries {sµ : µ ∈ E ∗ } such that {sµ : µ ∈ E ≤n } have mutually
orthogonal ranges for each n ∈ N, and such that
1. sµ∗ sµ = ss(µ) ;
2. sµ sν = sµν when s(µ) = r (ν);
3. sµ sµ∗ ≤ sr (µ) ; and
X
4. sv =
sµ sµ∗ for every v ∈ E 0 and n ∈ N such that
µ∈vE ≤n
|vE ≤n | < ∞.
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
I
For each n ∈ N, {sλ sλ∗ : λ ∈ E n } are mutually orthogonal
projections.
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
I
For each n ∈ N, {sλ sλ∗ : λ ∈ E n } are mutually orthogonal
projections.
I
Write µ λ ⇐⇒ λ = µµ0 . Then µ λ ⇐⇒ sλ sλ∗ ≤ sµ sµ∗ .
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
I
For each n ∈ N, {sλ sλ∗ : λ ∈ E n } are mutually orthogonal
projections.
I
Write µ λ ⇐⇒ λ = µµ0 . Then µ λ ⇐⇒ sλ sλ∗ ≤ sµ sµ∗ .
I
Denote the spectrum of D by ∆D .
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
I
For each n ∈ N, {sλ sλ∗ : λ ∈ E n } are mutually orthogonal
projections.
I
Write µ λ ⇐⇒ λ = µµ0 . Then µ λ ⇐⇒ sλ sλ∗ ≤ sµ sµ∗ .
I
Denote the spectrum of D by ∆D .Then for each φ ∈ ∆D and
µ λ, we have φ(sλ sλ∗ ) = 1 =⇒ φ(sµ sµ∗ ) = 1.
Diagonal sub-C ∗ -algebra
I
We call DE := C ∗ (sλ sλ∗ : λ ∈ E ∗ ) the diagonal C ∗ -subalgebra of
C ∗ (E ).
I
It can be deduced from the Cuntz-Krieger relations that
D = span{sλ sλ∗ : λ ∈ E ∗ }.
I
For each n ∈ N, {sλ sλ∗ : λ ∈ E n } are mutually orthogonal
projections.
I
Write µ λ ⇐⇒ λ = µµ0 . Then µ λ ⇐⇒ sλ sλ∗ ≤ sµ sµ∗ .
I
Denote the spectrum of D by ∆D .Then for each φ ∈ ∆D and
µ λ, we have φ(sλ sλ∗ ) = 1 =⇒ φ(sµ sµ∗ ) = 1.
I
Hence for each φ ∈ ∆D , the elements of {λ : φ(sλ sλ∗ ) = 1}
determine a path.
Boundary Paths
I
The paths we get turn out to be all infinite paths, and all finite
paths whose source is a singular vertex: elements v ∈ E 0
satisfying either
I
I
vE 1 = ∅, in which case we call v a source; or
|vE 1 | = ∞, in which case we call v an infinite receiver.
Boundary Paths
I
The paths we get turn out to be all infinite paths, and all finite
paths whose source is a singular vertex: elements v ∈ E 0
satisfying either
I
I
I
vE 1 = ∅, in which case we call v a source; or
|vE 1 | = ∞, in which case we call v an infinite receiver.
We define the boundary paths
∂E := E ∞ ∪ {µ ∈ E ∗ : s(µ) is singular}.
Boundary Paths
I
The paths we get turn out to be all infinite paths, and all finite
paths whose source is a singular vertex: elements v ∈ E 0
satisfying either
I
I
vE 1 = ∅, in which case we call v a source; or
|vE 1 | = ∞, in which case we call v an infinite receiver.
I
We define the boundary paths
∂E := E ∞ ∪ {µ ∈ E ∗ : s(µ) is singular}.
I
The formula
hE (x)(sµ sµ∗ )
(
1
=
0
if µ x
otherwise.
uniquely determines a bijection from ∂E onto ∆D [W].
Topology
I
Following the approach of [PW], define α : E ∗ ∪ E ∞ → {0, 1}E
by
(
1 if x = µµ0
α(x)(µ) =
0 otherwise.
I
Give E ∗ ∪ E ∞ the initial topology induced by α.
∗
Topology
I
Following the approach of [PW], define α : E ∗ ∪ E ∞ → {0, 1}E
by
(
1 if x = µµ0
α(x)(µ) =
0 otherwise.
I
Give E ∗ ∪ E ∞ the initial topology induced by α.
I
For µ ∈ E ∗ , define Z(µ) := {µµ0 ∈ E ∗ ∪ E ∞ }.
S
For G ⊂ E ∗ , we write Z(µ \ G ) := Z(µ) \ ν∈G Z(ν).
I
I
∗
The cylinder sets {Z(µ \ G ) : µ ∈ E ∗ , G ⊂ s(µ)E 1 is finite} are a
basis for our topology. [W].
Topology
I
Following the approach of [PW], define α : E ∗ ∪ E ∞ → {0, 1}E
by
(
1 if x = µµ0
α(x)(µ) =
0 otherwise.
I
Give E ∗ ∪ E ∞ the initial topology induced by α.
I
For µ ∈ E ∗ , define Z(µ) := {µµ0 ∈ E ∗ ∪ E ∞ }.
S
For G ⊂ E ∗ , we write Z(µ \ G ) := Z(µ) \ ν∈G Z(ν).
I
∗
I
The cylinder sets {Z(µ \ G ) : µ ∈ E ∗ , G ⊂ s(µ)E 1 is finite} are a
basis for our topology. [W].
I
With this topology, E ∗ ∪ E ∞ is locally compact and Hausdorff
[W].
Boundary Paths
I
Fix a path µ ∈ E ∗ with 0 < |s(µ)E 1 | < ∞.
Then {µ} = Z µ \ {s(µ)E 1 } an open set.
I
Then
I
U :=
is open.
[
{µ} : µ ∈ E ∗ such that 0 < |s(µ)E 1 | < ∞
Boundary Paths
I
Fix a path µ ∈ E ∗ with 0 < |s(µ)E 1 | < ∞.
Then {µ} = Z µ \ {s(µ)E 1 } an open set.
I
Then
I
U :=
[
{µ} : µ ∈ E ∗ such that 0 < |s(µ)E 1 | < ∞
is open.
I
So ∂E = U c is closed in E ∗ ∪ E ∞ , and hence locally compact
and Hausdorff.
Boundary Paths
I
Fix a path µ ∈ E ∗ with 0 < |s(µ)E 1 | < ∞.
Then {µ} = Z µ \ {s(µ)E 1 } an open set.
I
Then
I
U :=
[
{µ} : µ ∈ E ∗ such that 0 < |s(µ)E 1 | < ∞
is open.
I
So ∂E = U c is closed in E ∗ ∪ E ∞ , and hence locally compact
and Hausdorff.
I
The map hE : ∂E → ∆D is a homeomorphism [W].
Desingularisation
Drinen and Tomforde developed a construction they called
desingularisation [DT]:
I
Suppose E has some singular vertices. Fix µ ∈ ∂E ∩ E ∗ .
I
If |s(µ)E 1 | = 0, then append on an infinite path:
µ
•
•
Desingularisation
Drinen and Tomforde developed a construction they called
desingularisation [DT]:
I
Suppose E has some singular vertices. Fix µ ∈ ∂E ∩ E ∗ .
I
If |s(µ)E 1 | = 0, then append on an infinite path:
µ
µ
•
•
•
•
becomes
...
Desingularisation
Drinen and Tomforde developed a construction they called
desingularisation [DT]:
I
Suppose E has some singular vertices. Fix µ ∈ ∂E ∩ E ∗ .
I
If |s(µ)E 1 | = 0, then append on an infinite path:
µ
µ
•
•
•
•
becomes
I
...
If |s(µ)| = ∞, then append an infinite path, and distribute the
incoming edges along it:
•
...
s(µ)
Desingularisation
Drinen and Tomforde developed a construction they called
desingularisation [DT]:
I
Suppose E has some singular vertices. Fix µ ∈ ∂E ∩ E ∗ .
I
If |s(µ)E 1 | = 0, then append on an infinite path:
µ
µ
•
•
•
•
becomes
I
...
If |s(µ)| = ∞, then append an infinite path, and distribute the
incoming edges along it:
•
•
...
s(µ)
...
becomes
s(µ)
•
•
•
...
Desingularisation
I
Let E be a directed graph, and F be a Drinen-Tomforde
desingularisation of E .
I
This gives a homeomorphism φ∞ : E 0 F ∞ → ∂E [DT,W].
I
Then there exists a full projection p and an isomorphism
π : C ∗ (E ) → pC ∗ (F )p [DT].
All together now
I
For each directed graph E , we have hE : ∂E ∼
= ∆DE . [W]
All together now
I
I
For each directed graph E , we have hE : ∂E ∼
= ∆DE . [W]
Given a desingularisation of E , we have φ∞ : E 0 F ∞ ∼
= ∂E .
[DT,W].
All together now
I
I
I
For each directed graph E , we have hE : ∂E ∼
= ∆DE . [W]
Given a desingularisation of E , we have φ∞ : E 0 F ∞ ∼
= ∂E .
[DT,W].
π induces a homeomorphism π ∗ : ∆pDF p → ∆DE [W].
All together now
I
I
I
I
For each directed graph E , we have hE : ∂E ∼
= ∆DE . [W]
Given a desingularisation of E , we have φ∞ : E 0 F ∞ ∼
= ∂E .
[DT,W].
π induces a homeomorphism π ∗ : ∆pDF p → ∆DE [W].
These maps commute [W]:
E 0F ∞
φ∞
∂E
η
∆pDF p
hE
π∗
∆DE
Where η is essentially the restriction of hF
to paths with ranges in E 0 .
References
[DT] D. Drinen and M. Tomforde, The C ∗ -algebras of arbitrary
graphs, Rocky Mountain J. Math. 35 (2005), 105–135.
[PW] A.L.T. Paterson and A.E. Welch, Tychonoff’s theorem for locally
compact spaces and an elementary approach to the topology of
path spaces, Proc. Amer. Math. Soc. 133 (2005), 2761–2770.
[W] S.B.G. Webster, The path space of a directed graph, Proc.
Amer. Math. Soc., to appear.