Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Aperiodicity Conditions in Topological k-Graphs
Sarah E. Wright
The College of the Holy Cross
2011 Fall Central Sectional Meeting of the AMS
University of Nebraska, Lincoln
Special Session ∼ Recent Progress in Operator Algebras
October 15, 2011
Aperiodicity Conditions in Topological k-Graphs
1 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s Going On Here?
1
Graph Algebras
2
Topological k-Graphs
3
Aperiodicity Conditions
4
Proof of Equivalence
5
Example(s)
Aperiodicity Conditions in Topological k-Graphs
2 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
Define projections
{Pv | v ∈ E 0 }
en
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
and partial
isometries
{Se | e ∈ E 1 }
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
and partial
isometries
{Se | e ∈ E 1 }
Use Cuntz-Krieger
Realtions
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
and partial
isometries
{Se | e ∈ E 1 }
Use Cuntz-Krieger
Realtions
Se∗ Se = Ps(e) and
X
Pv =
Se Se∗
r(e)=v
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
Build a C∗ -algebra
and partial
isometries
{Se | e ∈ E 1 }
Use Cuntz-Krieger
Realtions
Se∗ Se = Ps(e) and
X
Pv =
Se Se∗
r(e)=v
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
Build a C∗ -algebra
and partial
isometries
{Se | e ∈ E 1 }
C ∗ (En ) = On
Use Cuntz-Krieger
Realtions
C ∗ (E) = M4 (C)
Se∗ Se = Ps(e) and
X
Pv =
Se Se∗
r(e)=v
C ∗ (F ) = C(Sq7 )
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
What’s a Graph Algebra?
Begin with a
directed graph
v
e1
···
en
Define projections
{Pv | v ∈ E 0 }
Build a C∗ -algebra
and partial
isometries
{Se | e ∈ E 1 }
C ∗ (En ) = On
Use Cuntz-Krieger
Realtions
C ∗ (E) = M4 (C)
Se∗ Se = Ps(e) and
X
Pv =
Se Se∗
r(e)=v
C ∗ (F ) = C(Sq7 )
What do the combinatorics of the graph tell us about the C∗ -algebra?
Aperiodicity Conditions in Topological k-Graphs
3 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
All AF-algebras (up to ME)
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
All AF-algebras (up to ME)
All SPIN algebras with free K1 -group (up to ME)
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
All AF-algebras (up to ME)
All SPIN algebras with free K1 -group (up to ME)
•
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
All AF-algebras (up to ME)
All SPIN algebras with free K1 -group (up to ME)
•
•
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Why do I Care About Graph Algebras?
Graph algebras provide a rich class of accessible examples. They
include:
The Toeplitz and Cuntz-Krieger algebras
The compact operators and C(T)
All AF-algebras (up to ME)
All SPIN algebras with free K1 -group (up to ME)
•
•
•
Aperiodicity Conditions in Topological k-Graphs
4 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
y
µ3
µ2
µ4
µ
µ1
e
µ5
w
ν1
µ8
ν2
µ6
ν
µ7
ν4
ν3
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
y
µ3
µ2
µ4
µ
µ1
e
µ5
w
ν1
µ8
ν2
µ6
ν
µ7
ν4
ν3
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
y
µ3
µ2
µ4
µ
µ1
e
µ5
w
ν1
µ8
ν2
µ6
ν
µ7
ν4
ν3
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
y
µ3
µ2
µ4
µ
µ1
e
µ5
w
ν1
µ8
ν2
µ6
ν
µ7
ν4
ν3
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
So... Aperiodicity is Important?
Let E be a row finite such that every
cycle has an entry. Then C ∗ (E) is
simple if and only if E is cofinal.
y
µ3
µ2
µ4
µ
µ1
e
µ5
w
ν1
µ8
ν2
µ6
ν
µ7
ν4
ν3
In the groupoid
model of the
C∗ -algebra, the
elements are paths
which differ only by
an initial segment.
Aperiodicity and
cofinality cause
more infinite paths
to be “related”.
Aperiodicity Conditions in Topological k-Graphs
5 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Topological k-Graphs
For k ∈ N, a topological k-graph is a pair (Λ, d) consisting of a
category Λ = (Obj(Λ), Mor(Λ), r, s) and a functor d : Λ → Nk ,
called the degree map, which satisfy:
1
Obj(Λ) and Mor(Λ) are second countable, locally compact
Hausdorff spaces;
2
r, s : Mor(Λ) → Obj(Λ) are continuous and s is a local
homeomorphism;
3
Composition ◦ : Λ ×c Λ → Λ is continuous and open, where Λ ×c Λ
has the relative topology inherited from the product topology on
Λ × Λ;
4
d is continuous, where Nk is given the discrete topology;
5
The unique factorization property: For all λ ∈ Λ and m, n ∈ Nk
with d(λ) = m + n, there exists unique (µ, ν) ∈ Λ ×c Λ such that
λ = µν, d(µ) = m and d(ν) = n.
Aperiodicity Conditions in Topological k-Graphs
6 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Topological k-Graphs
For k ∈ N, a topological k-graph is a pair (Λ, d) consisting of a
category Λ = (Obj(Λ), Mor(Λ), r, s) and a functor d : Λ → Nk ,
called the degree map, which satisfy:
1
Obj(Λ) and Mor(Λ) are second countable, locally compact
Hausdorff spaces;
2
r, s : Mor(Λ) → Obj(Λ) are continuous and s is a local
homeomorphism;
3
Composition ◦ : Λ ×c Λ → Λ is continuous and open, where Λ ×c Λ
has the relative topology inherited from the product topology on
Λ × Λ;
4
d is continuous, where Nk is given the discrete topology;
5
The unique factorization property : For all λ ∈ Λ and m, n ∈ Nk
with d(λ) = m + n, there exists unique (µ, ν) ∈ Λ ×c Λ such that
λ = µν, d(µ) = m and d(ν) = n.
Aperiodicity Conditions in Topological k-Graphs
6 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Topological k-Graphs
For k ∈ N, a topological k-graph is a pair (Λ, d) consisting of a
category Λ = (Obj(Λ), Mor(Λ), r, s) and a functor d : Λ → Nk ,
called the degree map, which satisfy:
1
Obj(Λ) and Mor(Λ) are second countable, locally compact
Hausdorff spaces;
2
r, s : Mor(Λ) → Obj(Λ) are continuous and s is a local
homeomorphism;
3
Composition ◦ : Λ ×c Λ → Λ is continuous and open, where
Λ ×c Λ has the relative topology inherited from the product
topology on Λ × Λ;
4
d is continuous, where Nk is given the discrete topology;
5
The unique factorization property: For all λ ∈ Λ and m, n ∈ Nk
with d(λ) = m + n, there exists unique (µ, ν) ∈ Λ ×c Λ such that
λ = µν, d(µ) = m and d(ν) = n.
Aperiodicity Conditions in Topological k-Graphs
6 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices
v
w
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices and edges of shape ei
v
w
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices and edges of shape ei
e
v
w
f
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices and edges of shape ei
a0
a
e
v
w
f
b
b0
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices and edges of shape ei
a0
a
e
c
v
w
d
f
b
b0
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing Higher Rank Graphs
We represent k graphs by drawing their 1-skeletons, which consist
of the vertices and edges of shape ei , and giving the appropriate
factorization rules if necessary.
a0
a
e
c
v
w
d
f
b
b0
da = a0 c, f a = a0 e, bd = cb0 , bf = eb0
Aperiodicity Conditions in Topological k-Graphs
7 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
a0
a
e
c
v
w
d
f
b
b0
da = a0 c, f a = a0 e,
bd = cb0 , bf = eb0
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram
a0
a
e
c
v
w
d
f
b
b0
da = a0 c, f a = a0 e,
bd = cb0 , bf = eb0
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram
.. . .
.. . .
.. . .
.. . .
.. . . .. . . .. . . .. . .. . .
.
.
.
....
.
.
.
.
..
..
..
..
...
...
.
.
.
.
.
.
.
.
.
.
..
.. .
...
a0
a
e
c
v
w
d
f
b
b0
da = a0 c, f a = a0 e,
bd = cb0 , bf = eb0
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
0
.
.
.
.
b
..
..
..
..
0
da = a c, f a = a0 e,
.. .
bd = cb0 , bf = eb0
...
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
0
.
.
.
.
b
..
..
..
..
0
da = a c, f a = a0 e,
.. .
bd = cb0 , bf = eb0
...
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
0
.
.
.
.
b
..
..
..
..
0
da = a c, f a = a0 e,
.. .
c
e
bd = cb0 , bf = eb0
...
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
da = a c, f a = a0 e,
.. .
c
e
bd = cb0 , bf = eb0
...
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
w
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
da = a c, f a = a0 e,
.. .
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
b
...
w
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
e
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
d w
b
...
w
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
e
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
..
..
..
..
...
d w
b
...
w b0
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
e
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
.. f
..
..
..
...
w
d w
b
f
...
w b0
w
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
e
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
.. f
..
..
..
...
d w
d w
b
f
...
w b0
w
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
b
e
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . .. . . .. . . .. . .. . .
a
.
.
.
....
v
w
.
.
.
.
e c
d
.. f
..
..
..
...
d w
d w
b
f
...
w b0
w b
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
0
b
e b
da = a c, f a = a0 e,
.. .
c v
c v
e
bd = cb0 , bf = eb0
...
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Infinite Paths
An infinite path in a k-graph is an infinite k-dimensional
commutative diagram which is labeled with edges from the
1-skeleton in a way which respects the unique factorization
property, and the range and source maps.
.. . .
.. . .
.. . .
.. . .
a0
.. . . e .. . . e .. . . e .. . .. . .
c v
c v
c v
c v
a
. e . e . e ....
v
v
v
0
0
a . v
a
v
w
a .
a .
.
.
..
.
e c
d
.
.
.
.
0
0
f a
f a
f a
a
...
d w
d w
d w
d w
b
f
f
f
...
w b0
w b
w b0
w b
0
.
.
.
.
b
.
.
.
.
.
.
.
.
0
0
0
b
e b
e b
e b
da = a c, f a = a0 e,
.. .
c v
c v
c v
c v
e
e
e
bd = cb0 , bf = eb0
...
v
v
v
v
f
Aperiodicity Conditions in Topological k-Graphs
8 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
σ n (x) is the infinite path obtained by removing the initial segment
of degree n from x.
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
σ n (x) is the infinite path obtained by removing the initial segment
of degree n from x.
For x below,
x
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
.. . .
. .
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
σ n (x) is the infinite path obtained by removing the initial segment
of degree n from x.
For x below, this is σ (2,0) (x).
x
..
.
.
..
.
.
σ (2,0) (x)
..
..
.
.
.
.
..
.
.
.. . .
. .
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
σ n (x) is the infinite path obtained by removing the initial segment
of degree n from x.
For x below, this is σ (1,1) (x).
x
..
.
.
σ (1,1) (x)
..
..
.
.
.
.
..
.
.
..
.
.
.. . .
. .
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Shift Map
We can shift an infinite path of a k graph, x, using the shift map σ.
σ n (x) is the infinite path obtained by removing the initial segment
of degree n from x.
For x below,
x
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
.. . .
. .
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
.
.
.
.
.
. ...
We say that a path x is aperiodic if σ m x = σ n x only when m = n.
Aperiodicity Conditions in Topological k-Graphs
9 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Yeend’s Condition (A)
Definition
We say a topological k-graph (Λ, d) is aperiodic, or satisfies
Condition (A), if for every open set V ⊆ Λ0 there exists an infinite
aperiodic path x ∈ V Λ.
Proposition (Yeend, 2007)
Suppose Λ is a compactly aligned topological k-graph that satisfies
Condition (A). Then, GΛ is topologically principal.
Aperiodicity Conditions in Topological k-Graphs
10 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Theorem
Let (Λ, d) be a compactly aligned topological k-graph and V be
any nonempty open subset of Λ0 . The following conditions are
equivalent.
Aperiodicity Conditions in Topological k-Graphs
11 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Theorem
Let (Λ, d) be a compactly aligned topological k-graph and V be
any nonempty open subset of Λ0 . The following conditions are
equivalent.
(A) There exists an aperiodic path x ∈ V ∂Λ.
Aperiodicity Conditions in Topological k-Graphs
11 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Theorem
Let (Λ, d) be a compactly aligned topological k-graph and V be
any nonempty open subset of Λ0 . The following conditions are
equivalent.
(A) There exists an aperiodic path x ∈ V ∂Λ.
(B) For any pair m 6= n ∈ Nk there exists a path λV,m,n ∈ V Λ such
that d(λ) ≥ m ∨ n and
λ(m, m + d(λ) − (m ∨ n)) 6= λ(n, n + d(λ) − (m ∨ n)).
(?)
Aperiodicity Conditions in Topological k-Graphs
11 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Theorem
Let (Λ, d) be a compactly aligned topological k-graph and V be
any nonempty open subset of Λ0 . The following conditions are
equivalent.
(A) There exists an aperiodic path x ∈ V ∂Λ.
(B) For any pair m 6= n ∈ Nk there exists a path λV,m,n ∈ V Λ such
that d(λ) ≥ m ∨ n and
λ(m, m + d(λ) − (m ∨ n)) 6= λ(n, n + d(λ) − (m ∨ n)).
(?)
(C) There is a vertex v ∈ V and paths α, β ∈ Λ with s(α) = s(β) = v
such that there exists a path τ ∈ s(α)Λ with MCE(ατ, βτ ) = ∅.
Aperiodicity Conditions in Topological k-Graphs
11 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
n
m
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
n
λ
m
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
n
λ
m
v
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
d(λ)
n
m∨n
m
v
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
d(λ)
n
m∨n
m
v
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Visualizing the Conditions
d(λ)
n+l
n
m∨n
m+l
m
v
V
Aperiodicity Conditions in Topological k-Graphs
12 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Ever Important Lemma
Tube Lemma
Let V ⊂ Λ0 be open, m 6= n ∈ Nk , and λ ∈ V Λ satisfy (?). Then
there exists a compact neighborhood E ⊂ V Λd(λ) of λ such that
every µ ∈ E satisfies (?).
Aperiodicity Conditions in Topological k-Graphs
13 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Ever Important Lemma
Tube Lemma
Let V ⊂ Λ0 be open, m 6= n ∈ Nk , and λ ∈ V Λ satisfy (?). Then
there exists a compact neighborhood E ⊂ V Λd(λ) of λ such that
every µ ∈ E satisfies (?).
“Proof”:
Aperiodicity Conditions in Topological k-Graphs
13 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Ever Important Lemma
Tube Lemma
Let V ⊂ Λ0 be open, m 6= n ∈ Nk , and λ ∈ V Λ satisfy (?). Then
there exists a compact neighborhood E ⊂ V Λd(λ) of λ such that
every µ ∈ E satisfies (?).
“Proof”:
λ = λ(0, m)λ(m, m + d(λ) − (m ∨ n))λ(m + d(λ) − (m ∨ n), d(λ))
and
λ = λ(0, n)λ(n, n + d(λ) − (m ∨ n))λ(n + d(λ) − (m ∨ n)), d(λ))
Aperiodicity Conditions in Topological k-Graphs
13 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Ever Important Lemma
Tube Lemma
Let V ⊂ Λ0 be open, m 6= n ∈ Nk , and λ ∈ V Λ satisfy (?). Then
there exists a compact neighborhood E ⊂ V Λd(λ) of λ such that
every µ ∈ E satisfies (?).
“Proof”:
λ = λ(0, m) λ(m, m + d(λ) − (m ∨ n)) λ(m + d(λ) − (m ∨ n), d(λ))
and
λ = λ(0, n) λ(n, n + d(λ) − (m ∨ n)) λ(n + d(λ) − (m ∨ n)), d(λ))
Aperiodicity Conditions in Topological k-Graphs
13 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The Ever Important Lemma
Tube Lemma
Let V ⊂ Λ0 be open, m 6= n ∈ Nk , and λ ∈ V Λ satisfy (?). Then
there exists a compact neighborhood E ⊂ V Λd(λ) of λ such that
every µ ∈ E satisfies (?).
“Proof”:
λ(m, m + d(λ) − (m ∨ n))
λ(m + d(λ) − (m ∨ n), d(λ))
λ = λ(0, m)
Em
λ(n, n + d(λ) − (m ∨ n))
λ(n + d(λ) − (m ∨ n)), d(λ))
λ = λ(0, n)
En
Aperiodicity Conditions in Topological k-Graphs
13 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
λ2
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
Ei := F1 . . . Fi ∂Λ.
λ2
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
Ei := F1 . . . Fi ∂Λ.
·
λ2
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
Ei := F1 . . . Fi ∂Λ.
··
λ2
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Now... Use That Lemma!
(B) =⇒ (A):
Fix an open set V1 ⊂ Λ0 and let{(mi , ni )}∞
i=1 be a listing of the
set (m, n) ∈ Nk × Nk : m 6= n .
Vi := interior of s(Fi−1 ),
λi := λVi ,mi ,ni satisfy (?) for mi and ni ,
Fi := a compact neighborhood of λi given by the Tube Lemma, and
Ei := F1 . . . Fi ∂Λ.
·
··
λ2
V2
λ1
V1
Aperiodicity Conditions in Topological k-Graphs
14 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources,
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor. Then the pair Λ ×τ X, d˜
is a topological k-graph.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor. Then the pair Λ ×τ X, d˜ , with object and
morphism sets
Obj(Λ ×τ X) := Obj(Λ) × X and Mor(Λ ×τ X) := Mor(Λ) × X,
is a topological k-graph.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor. Then the pair Λ ×τ X, d˜ , with object and
morphism sets
Obj(Λ ×τ X) := Obj(Λ) × X and Mor(Λ ×τ X) := Mor(Λ) × X,
range and source maps
r(λ, x) := (r(λ), τλ (x)) and s(λ, x) := (s(λ), x) ,
is a topological k-graph.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor. Then the pair Λ ×τ X, d˜ , with object and
morphism sets
Obj(Λ ×τ X) := Obj(Λ) × X and Mor(Λ ×τ X) := Mor(Λ) × X,
range and source maps
r(λ, x) := (r(λ), τλ (x)) and s(λ, x) := (s(λ), x) ,
and composition
(λ, τµ (x)) ◦ (µ, x) = (λµ, x),
whenever s(λ) = r(µ) in (Λ, d)
is a topological k-graph.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Twisted Product Graphs
Let (Λ, d) be a finitely aligned k-graph with no sources, X be a
second countable, locally compact, Hausdorff space, and
τ : Λ → {τλ : X → X | τλ is a local homeomorphism}
a continuous functor. Then the pair Λ ×τ X, d˜ , with object and
morphism sets
Obj(Λ ×τ X) := Obj(Λ) × X and Mor(Λ ×τ X) := Mor(Λ) × X,
range and source maps
r(λ, x) := (r(λ), τλ (x)) and s(λ, x) := (s(λ), x) ,
and composition
(λ, τµ (x)) ◦ (µ, x) = (λµ, x),
whenever s(λ) = r(µ) in (Λ, d), and degree functor
˜ x) = d(λ)
d(λ,
is a topological k-graph.
Aperiodicity Conditions in Topological k-Graphs
15 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The 1-Skeleton
n
αn-1
α12
α01
v0
β01
α02
v1
β02
β12
..
.
α23
α13
α03
v2
β03
β13
β23
α1n
α0n
v3
···
vn
···
β0n
β1n
vn+1
..
.
n
βn-1
Aperiodicity Conditions in Topological k-Graphs
16 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The 1-Skeleton
n
αn-1
α12
α01
v0
β01
α02
v1
β02
β12
..
.
α23
α13
α03
v2
β03
β13
β23
α1n
α0n
v3
···
vn
···
β0n
β1n
vn+1
..
.
n
βn-1
n αn+1
αin βjn+1 = βi+1
j+1
Aperiodicity Conditions in Topological k-Graphs
16 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The 1-Skeleton
n
αn-1
α12
α01
v0
β01
α02
v1
β02
β12
..
.
α23
α13
α03
v2
β03
β13
β23
α1n
α0n
v3
···
vn
···
β0n
β1n
vn+1
..
.
n
βn-1
n αn+1
αin βjn+1 = βi+1
j+1
X := T
Aperiodicity Conditions in Topological k-Graphs
16 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
The 1-Skeleton
n
αn-1
α12
α01
v0
β01
α02
v1
β02
β12
..
.
α23
α13
α03
v2
β03
β13
β23
α1n
α0n
v3
···
vn
···
β0n
β1n
vn+1
..
.
n
βn-1
n αn+1
αin βjn+1 = βi+1
j+1
X := T
ταni (z) = τβjn (z) := z n
Aperiodicity Conditions in Topological k-Graphs
16 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
1/m+n+1
λ = βjm+n+1
,
z
0 −n
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
1/m+n+1
λ = βjm+n+1
,
z
0 −n
m+n+1 1/m+n+1
m+1 k/m+n
m k
,
z
β
,
z
α i1 , z
. . . αim+n
µλ = αi0 , z
j
−n
n
0
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
1/m+n+1
λ = βjm+n+1
,
z
0 −n
m+n+1 1/m+n+1
m+1 k/m+n
m k
,
z
β
,
z
α i1 , z
. . . αim+n
µλ = αi0 , z
j
−n
n
0
m+n
m+n+1 1/m+n+1
k/m+n
= αim0 , z k αim+1
,
z
.
.
.
β
,
z
α
,
z
j0 −n+1
in +1
1
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
1/m+n+1
λ = βjm+n+1
,
z
0 −n
m+n+1 1/m+n+1
m+1 k/m+n
m k
,
z
β
,
z
α i1 , z
. . . αim+n
µλ = αi0 , z
j
−n
n
0
m+n
m+n+1 1/m+n+1
k/m+n
= αim0 , z k αim+1
,
z
.
.
.
β
,
z
α
,
z
j0 −n+1
in +1
1
m+n 1/m+n+1
k/m+n
= βjm0 +1 , z k αim+1
,
z
.
.
.
α
,
z
in +1
0 +1
Aperiodicity Conditions in Topological k-Graphs
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (C)
Fix and open set V ⊂ Λ0 , µ, ν ∈ V Λ, with r(µ) = r(ν),
s(µ) = s(ν), and d(µ) ∧ d(ν) = 0.
m+n−1 m+n
m+n
m+n m+n−1
µ = αim0 , z k . . . αim+n−2
,
z
α
,
z
α
,
z
in−1
in
n−2
m+n−1 m+n
m+n
m+n m+n−1
ν = βjm0 , z k . . . βjm+n−2
,
z
β
,
z
β
,
z
jn−1
jn
n−2
1/m+n+1
λ = βjm+n+1
,
z
0 −n
m+n+1 1/m+n+1
m+1 k/m+n
m k
,
z
β
,
z
α i1 , z
. . . αim+n
µλ = αi0 , z
j
−n
n
0
m+n
m+n+1 1/m+n+1
k/m+n
= αim0 , z k αim+1
,
z
.
.
.
β
,
z
α
,
z
j0 −n+1
in +1
1
m+n 1/m+n+1
k/m+n
= βjm0 +1 , z k αim+1
,
z
.
.
.
α
,
z
in +1
0 +1
Aperiodicity Conditions in Topological k-Graphs
,
17 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (A)
1
Consider a path x in “standard form”.
Aperiodicity Conditions in Topological k-Graphs
18 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (A)
Consider a path x in “standard form”.
x = α01 , z β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
1
Aperiodicity Conditions in Topological k-Graphs
18 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (A)
Consider a path x in “standard form”.
x = α01 , z β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
1
Calculate σ (1,0) x and σ (0,1) x.
σ (1,0) x = β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
2
Aperiodicity Conditions in Topological k-Graphs
18 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (A)
Consider a path x in “standard form”.
x = α01 , z β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
1
Calculate σ (1,0) x and σ (0,1) x.
σ (1,0) x = β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
2
3
Find a formula for σ m x.
4
Deduce range of σ m x, |m|, and m1 and m2 .
5
Conclude x is aperiodic.
6
Other paths and open sets?
Aperiodicity Conditions in Topological k-Graphs
18 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
Checking Condition (A)
Consider a path x in “standard form”.
x = α01 , z β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
1
Calculate σ (1,0) x and σ (0,1) x.
σ (1,0) x = β0i+1 , z 1/i+1 α0i+2 , z 1/(i+1)(i+2) β0i+3 , z 1/(i+1)(i+2)(i+3) . . .
2
3
Find a formula for σ m x.
4
Deduce range of σ m x, |m|, and m1 and m2 .
5
Conclude x is aperiodic.
6
Other paths and open sets?
7
More complicated 1-skeletons, factorizations, spaces, twistings...
EEEKK!
Aperiodicity Conditions in Topological k-Graphs
18 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
You’re The BEST!
, THANKS! ,
Aperiodicity Conditions in Topological k-Graphs
19 / 20
Graph Algebras
Topological k-Graphs
Aperiodicity Conditions
Proof of Equivalence
Example(s)
References
Iain Raeburn, Graph Algebras, CBMS Regional Conference Series in
Mathematics, vol. 103.
Takeshi Katsura, A class of C∗ -algebras generalizing both graph algebras
and homeomorphism C∗ -algebras I, II, III, and IV.
Alex Kumjian and David Pask, Higher rank graph C∗ -algebras, New York
J. Math.
David I. Robertson and Aidan Sims, Simplicity of C∗ -algebras associated
to higher-rank graphs, Bull. Lond. Math. Soc.
Trent Yeend, Topological higher-rank graphs and the C∗ -algebras of
topological 1-graphs, Operator theory, operator algebras, and applications
Sarah Wright, Aperiodicity Conditions in Topological k-Graphs, Thesis,
Dartmouth College, May 2010
Aperiodicity Conditions in Topological k-Graphs
20 / 20