Masters Thesis Defense

Introduction Graph Algebras Two Uniqueness Theorems Tensor Products of Graph Algebras

Graph C ∗ -algebras and Subalgebras of
C ∗ (E1 ) ⊗ C ∗ (E2 )

Dale Hobbs

Tennessee Technological University

July 10, 2012

1 / 53 Dale Hobbs Graph C ∗ -algebras and Subalgebras of C ∗ (E1 ) ⊗ C ∗ (E2 )


Introduction

Definition
An algebra is defined as a vector space V paired with a mapping
ϕ : V × V → V such that ϕ : a, b → ab. That is to say, V is an
algebra if it is a vector space and has an associative multiplication
operation defined on the vector space.

Definition
A norm defined on a vector space is a mapping · : X → [0, ∞]
for which the following three conditions are met:
(i.) αx = |α| x for all α ∈ R and all x ∈ X . (Homogeneity)
(ii.) x + y ≤ x + y for all x, y ∈ X (Triangle Inequality)
(iii.) x = 0 if and only if x = 0 (Positivity)



Introduction

Deﬁnition
A Banach space is a vector space, either real or complex, equipped
with a norm · in which the space is complete with respect to the
metric deﬁned by the norm.

Example
1. R equipped with absolute value |·| is a Banach space
2. C [a, b] equipped with norm f = sup{|f (x)| : x ∈ [a, b]}



Introduction

Definition
If A is an algebra, then the pair (A, · ) is called a normed algebra
if the norm is submultiplicative, which means

xy ≤ x y for all x, y ∈ A.

If A also has an identity element 1 such that 1 = 1 and
a · 1 = 1 · a = a for all a ∈ A, the we call A a unital normed algebra.

Definition
An involution mapping on an algebra A is a conjugate-linear map
∗ : A → A defined by x → x ∗ such that
1 x ∗∗ = x and (xy )∗ = y ∗ x ∗ for all x, y ∈ A
2 (λa + µb)∗ = λa∗ + µb ∗ for λ, µ ∈ C.
¯ ¯ (conjugate linearity)
The element x ∗ is termed the adjoint of x.



Introduction

Definition
If A is an algebra, then the pair (A, ∗) is called a ∗-algebra, and this
represents the algebra A equipped with the involution mapping ∗.

Definition
A Banach *-algebra is a ∗-algebra A that is complete with respect
to a submultiplicative norm · , and a∗ = a for all a ∈ A. If A
has an identity element, then A is called a unital Banach ∗-algebra.

Definition
An algebra A is called a C*-algebra if it is a Banach *-algebra such
that the norm on A satisfies
2
a∗ a = a for all a ∈ A.



Introduction

Example
Here are some examples of well-known C ∗ -algebras
1. C is a unital C ∗ -algebra where the involution ∗ is complex
conjugation.
2. Mn (C) the algebra of n-by-n matrices over C if the matrices
are considered operators, the norm · is the operator norm
on matrices, and the involution is the conjugate transpose.
3. B (H) the space of bounded linear operators on a Hilbert
space H which will be discussed shortly.



Introduction

Deﬁnition
If X and Y are vector spaces then a mapping T : X → Y is called
a linear mapping provided that for all x, y ∈ X and scalars
α, β ∈ C we have that

T (αx + βy ) = αT (x) + βT (y ) .

Deﬁnition
A linear mapping T : X → Y where (X , · x ) and Y , · y are
normed vector spaces is called a bounded linear map if there exists
an M ≥ 0 such that

T (x) y ≤M x x for all x ∈ X .



Introduction

Deﬁnition
If (X , · x ) is a normed vector space then the collection of all
bounded linear maps T : X → X is denoted by B (X ), and we call
these bounded linear maps from X to X bounded linear operators.

We can deﬁne a norm on B (X ) as

T B(X ) = inf{C ≥ 0 : T (x) X ≤C· x X}

though there are several other equivalent representations of this
norm on B (X ).



Introduction

Definition
An inner product space is a vector space X over a field F
(generally either R or C) paired with a mapping ·, · : X × X → F
satisfying the following conditions for all x, y , z ∈ X and α ∈ F:
(i.) x, y = y , x (Conjugate Symmetry)
(ii.) αx, y = α x, y and x + y , z = x, z + y , z (Linearity
in First Argument)
(iii.) x, x ≥ 0 with equality when x = 0. (Positive-definiteness)
This mapping is referred to as the inner product.

Definition
A Hilbert space is an inner product space (real or complex) with an
associated norm and metric that is complete with respect to the
norm.



Introduction

From here we can now talk about B (H) which is the algebra of
bounded operators on a Hilbert space H. Moreover, it can be
shown that a Hilbert space is indeed a Banach space with respect
to the norm induced by the inner product. A norm induced based
on the inner product is defined to be

x = x, x .

A Hilbert space has many nice properties and becomes even more
valuable when used as the space where we apply bounded linear
operators. We want to eventually see that B (H) is a C ∗ -algebra.

In order to do this though we need to first define the adjoint-∗
mapping in B (H).



Introduction

According to the Riesz representation theorem, the adjoint T ∗ of a
bounded operator T on a Hilbert space does exists and is deﬁned
by the property

x, T (y ) = T ∗ (x) , y for all x, y ∈ H.

Theorem
The space of bounded linear operators on a Hilbert space, B (H),
is a C*-algebra.

This C ∗ -algebra of bounded linear operators is a very concrete
space since linear operators are very familiar and relatively easy to
utilize. They are also important because of the following theorem.



Introduction

Theorem (Gelfand-Naimark)
If A is a C ∗ -algebra, then it has a faithful representation (H, ϕ)
where H is a Hilbert space and ϕ : A → B (H) is a
∗-homomorphism.

This is a critical theorem that shows that any C ∗ -algebra can be
viewed as some C ∗ -subalgebra of B (H). This is most helpful as it
sets up the stage for how we deﬁne elements in a general
C ∗ -algebra: we can deﬁne them by how their representations work
in B (H).



Introduction

Deﬁnition
A partial isometry is a mapping f from H into H such that f is an
isometry on M, a subspace of H, and is 0 on M ⊥ , the orthogonal
complement of M.

Deﬁnition
An orthogonal projection, or just projection, P on a Hilbert space
H is a linear map P : H → H such that P (x) = P ∗ (x) = (P (x))2
for all x ∈ H.
Projections and partial isometries are critical for talking about a
certain class of C ∗ -algebras called graph algebras introduced in the
next section.



Graph Algebras

Definition
A directed graph E is a grouping consisting of the sets E 0 and E 1
along with functions r , s : E 1 → E 0 called the range and source
functions respectively. The set E 0 has elements referred to as
vertices and the elements in the set E 1 are called edges. The
graph E is denoted as E = E 0 , E 1 , r , s .

Any vertex v ∈ E 0 that has no edges feeding into it is called a
source. Any vertex w ∈ E 0 from which no edges are emitted is
called a sink. Moreover, all the graphs that will be considered here
are row-finite which means that each vertex receives at most a
finite number of edges.



Graph Algebras

Example
In the ﬁgure below is the depiction of a directed graph with
vertices v and w and three edges f , e, and g . This graph would be
represented by E 0 = {v , w }, E 1 = {e, f , g }, r (f ) = v , r (e) = w ,
s (f ) = w , s (e) = v , and r (g ) = s (g ) = v .

f
v
g <v 6w
e

Figure : Example of a Directed graph with 3 edges and 2 vertices



Graph Algebras

We now represent our directed graph as bounded operators on a
Hilbert space.
Deﬁnition
A Cuntz-Krieger E-family {S, P} on a Hilbert space H is a set
{Pv : v ∈ E 0 } of mutually orthogonal projections on H and a set
{Se : e ∈ E 1 } of partial isometries on H such that the following 2
relations hold:
∗
(CK1) Se Se = Ps(e) for all e ∈ E 1 ; and
∗
(CK2) Pv = Se Se if v is not a source.
{e∈E 1 :r (e)=v }

Note: if v is a source, then r −1 (v ) is an empty set meaning there
are no edges e ∈ E 1 with r (e) = v



Graph Algebras

It may seem strange to represent a directed graph in this way, but
it will be seen later that this gives a very interesting C ∗ -algebra
generated by these partial isometries and projections.

The projections Pv being mutually orthogonal implies that the
closed subspaces Pv H are mutually orthogonal subspaces of H.
This allows for the decomposition of H into a direct sum of these
closed subspaces which is helpful when trying to ﬁnd partial
isometries Se and projections Pv satisfying CK1 and CK2 to make
a Cuntz-Krieger family for a directed graph.



Graph Algebras

Theorem
Let {S, P} be a Cuntz-Krieger E -family acting on a Hilbert space
H. Then
Se Ps(e) = Pr (e) Se = Se
for all e ∈ E 1 .

Proof.
∗
The Cuntz-Krieger (CK) relations say that Se Se = Ps(e) . Applying
∗
Se to both sides of this equality gives that Se Se Se = Se Ps(e) . In
∗
B (H), it is true that if Se is a partial isometry then Se Se Se = Se .
Thus, Se = Se Ps(e) .



Graph Algebras

proof continued.
∗
The CK relations also give that Pr (e) = Pv = {e∈E 1 :r (e)=v } Se Se .
This means for a particular e ∈ E 1 such that r (e) = v that
∗
Pv − Se Se = {f ∈E 1 :r (f )=v and f =e} Sf Sf∗ . Since Sf Sf∗ is a
projection and projections are positive operators, this means that

Sf Sf∗ (h) , h
{f ∈E 1 :r (f )=v and f =e}
∗
= Se Se (h) , h ≥ 0 for all h ∈ H
{f ∈E 1 :r (f )=v and f =e}

∗
showing that Pv − Se Se is a positive operator. From this it is true
that
∗ ∗
0 ≤ (Pv − Se Se ) (h) , h = Pv (h) , h − Se Se (h) , h
∗
⇒ Pv (h) , h ≥ Se Se (h) , h



Graph Algebras

proof continued.
∗
which means that Pv H ⊇ Se Se H. This implies that
∗ S H = S H. The reverse inclusion is found by
Pv Se H ⊇ Se Se e e
simply observing that Pv Se H ⊆ Se H.
Therefore, Se = Pv Se .
Since Pv Se and Se Ps(e) are both equal to Se , it is also true that
Pv Se = Se Ps(e) thus completing the proof.

The algebraic relationship

Se Ps(e) = Pr (e) Se = Se . (1)

is extremely helpful when making manipulations of Cuntz-Krieger
E -families.



Graph Algebras

It is important to note that a Cuntz-Krieger E -family can be
spoken of in relation to an arbitrary C ∗ -algebra B. These
projections p will satisfy p = p ∗ = p 2 and the partial isometries
will satisfy s = ss ∗ s. There is a useful theorem regarding s in
B (H) that says
Theorem
Let S ∈ B (H). Then
S is a partial isometry ⇐⇒ S ∗ S is a projection
⇐⇒ SS ∗ S = S ⇐⇒ S ∗ SS ∗ = S ∗ ⇐⇒ SS ∗ is a projection.

Then, the reason that this terminology carries over from B (H) to
classify partial isometries, and even projections, in an arbitrary
C ∗ -algebra B is thanks to the Gelfand-Naimark representation
theorem which states there is an injective ∗-homomorphism ϕ from
B to B (H).


Graph Algebras

Example (Directed graph of the Cuntz-Algebra O2 )

We shall ﬁnd a Cuntz-Krieger family can be found satisfying CK1
and CK2 for the directed graph pictured below. That means we
must ﬁnd partial isometries Sg and Sh and a projection Pv
∗ ∗ ∗ ∗
satisfying Sg Sg = Pv = Sh Sh and Pv = Sg Sg + Sh Sh on some
Hilbert space.

v ]b

g

h



Graph Algebras

Let H to be the space 2. For the partial isometries Sg and Sh ,
deﬁne these to be

Sg (x0 , x1 , x2 , x3 , . . .) = (x0 , 0, x1 , 0, x2 , 0, x3 , . . .)
Sh (x0 , x1 , x2 , x3 , . . .) = (0, x0 , 0, x1 , 0, x2 , 0, x3 , . . .) .

Let the adjoint of these just act in the reverse. It will need to be
checked that Sg and Sh are both partial isometries. The best way
is to use the following theorem.



Graph Algebras

∗ ∗ 2 ∗ ∗
Observe that Sg Sg = Sg Sg = Sg Sg with the way Sg is
deﬁned showing that Sg Sg ∗ is a projection. Therefore, S is a
g
partial isometry according to the proposition. Similarly, it is
possible to see that Sh is a partial isometry. Lastly, check that the
relations hold and discover the operator Pv . It can be easily
∗ ∗ ∗ ∗
checked to see that Sg Sg = 1 = Sh Sh and Sg Sg + Sh Sh = 1
where 1 is the identity operator in B (H). Thus, Pv = 1 making
{S, P} is a Cuntz-Krieger E -family for this particular graph.



Graph Algebras

Theorem
Let E be a row-finite graph and {S, P} be a Cuntz-Krieger
E -family in a C ∗ -algebra B. Then
∗
(a) the projections {Se Se | e ∈ E 1 } are mutually orthogonal;
∗
(b) If e = f , then Se Sf = 0;
(c) If s (e) = r (f ), then Se Sf = 0;
(d) If s (e) = s (f ), then Se Sf∗ = 0.

Part (c) in this proposition allows for talking about paths in a
directed graph. Notice, part (c) is saying that Se Sf = 0 unless ef
is a path. A path of length n is a sequence of edges µ1 , µ2 , . . . , µn
such that s (µi ) = r (µi+1 ). Moreover, the set E n is defined to be
the set of all paths of length n. Now define E ∗ := n≥0 E n . Then
the set E 0 clarifies our definition earlier: this is the set of paths of
length 0 which is the set of vertices.


Graph Algebras

Since the multiplication in B (H) is composition, which is
performed from right to left, it is important to note that a path of
length 2, say µ1 µ2 , will be represented as partial isometries and
that µ2 is traversed prior to µ1 . Because of this, the range and
source maps extend to E ∗ by

r (µ) = r µ1 µ2 · · · µ|µ| = r (µ1 )

and
s (µ) = s µ1 µ2 · · · µ|µ| = s µ|µ| .
Moreover, if ν and µ are paths such that s (ν) = r (µ), then νµ
can be expanded to ν1 · · · ν|ν| µ1 · · · µ|µ| a path of length |µ| + |ν|.



Graph Algebras

Similar algebraic relationships to those found in equation 1 can be
shown for any µ ∈ E ∗ . Simply deﬁne Sµ := Sµ1 Sµ2 · · · Sµn . From
proposition on slide 25 it can be seen that Sµ = 0 unless µ is a
path.
Then using the algebraic relations on single edges µi it can be seen
that
Sµ Ps(µ) = Pr (µ) Sµ = Sµ for all µ ∈ E ∗ .
The next two propositions will allow us to characterize the
C ∗ -algebra formed from the Cuntz-Krieger E -family {S, P}.



Graph Algebras

Theorem
Let E be a row-ﬁnite graph and {S, P} be a Cuntz-Krieger
E -family in a C ∗ -algebra B and µ and ν be paths in E . Then
a. if |µ| = |ν| but µ and ν are distinct paths, then
∗ ∗
Sµ Sµ (Sν Sν ) = 0;

Sµ ∗ if µ = νµ for some µ ∈ E ∗

∗
b. Sµ Sν = Sν if ν = µν for some ν ∈ E ∗

0 when paths µ and ν don’t overlap;


c. If Sµ Sν = 0, then µν is a path in E and Sµ Sν = Sµν ;
∗
d. If Sµ Sν = 0, then s (µ) = s (ν).



Graph Algebras

Theorem
Suppose that E is a row-ﬁnite graph and {S, P} is a Cuntz-Krieger
E -family in a C ∗ -algebra B. For any µ, ν, α, β ∈ E ∗ then

∗
Sµα Sβ if α = να

∗ ∗ ∗
(Sµ Sν ) Sα Sβ = Sµ Sβν if ν = αν

0 otherwise.




Graph Algebras

From the previous corollaries we can take any nonzero product of a
∗
ﬁnite number of partial isometries of the forms Sα and Sβ for
α, β ∈ E ∗ and reduce it to a form Sµ Sν for some µ, ν ∈ E ∗ where
∗

s (µ) = s (ν). To see this clearer assume that W is the nonzero
product of a ﬁnite number of Se and Sf∗ terms. If there are
adjacent Se ’s, then these can be lumped together into a single
term Sµ . Moreover, adjacent Sf∗ ’s can be combined into a single
∗ ∗
Sν . The product Sµ Sν from the corollary on slide 28 this will either
become Sα or Sβ ∗.

∗
A single Sµ term can be rewritten as
∗
(Sµ )∗ = Sµ Ps(µ) = Ps(µ) Sµ = Ps(µ) Sµ and since every
∗ ∗ ∗

projection is also a partial isometry then Ps(µ) can be written as
∗ ∗
Ss(µ) giving that Sµ = Ss(µ) Sµ . This means a single Sµ term will
∗
be Sµ = Sµ Ss(µ) by applying the ∗ to Sµ .∗



Graph Algebras

There are several cases that can be inspected now. Take the
∗ ∗
sequence Sµ Sν Sα Sβ for instance. The middle part Sν Sα will be
∗ from slide 28. For the former the sequence boils
either Sλ or Sλ
∗
down to Sµ Sλ Sβ = Sµλβ and the later case gives Sµ Sλ Sβ for which
∗ S S ∗ . This can now be
which Sβ can be changed to give Sµ Sλ β s(β)
collapsed into the desired form according to the corollary on slide
29.
All of these results ﬁnally lead to the amazing fact that all graph
C ∗ -algebras are linearly generated by terms like Sµ Sν which is the
∗

next corollary.
Theorem
Let E be a row-ﬁnite graph and {S, P} be a CK E -family. Then,
the C ∗ -algebra generated by the {S, P} is exactly

span{Sµ Sν | µ, ν ∈ E ∗ , s (µ) = s (ν)}.
∗



Graph Algebras

Proof.
It is clear that C ∗ (S, P) ⊆ span{Sµ Sν | µ, ν ∈ E ∗ , s (µ) = s (ν)}
∗

from the above statement. To explain, notice that product of any
∗
finite number of Sµ and Sν terms will reduce to the form Sα Sβ . ∗

Also notice that projections Pv can be rewritten as
Pv = f ∈E 1 :r (f )=v Sf Sf∗ . This means if there are projections in
these finite products of terms from C ∗ (S, P) then that product
∗
will still come down to the form Sα Sβ or a finite sum of terms
looking like Sα Sβ ∗ after distributing and simplifying. Moreover, the

finite sum of any product of elements from C ∗ (S, P) will boil
∗ ∗
down to the form Sµ Sν + Sα Sβ + · · · which is included in the
span. Finally, closure of the span will include infinite products and
sums of elements from C ∗ (S, P).
The reverse inclusion is obtained by showing the closed linear span
of Sµ Sν is a C ∗ -subalgebra of C ∗ (S, P).
∗



Graph Algebras

Proof.
First, the closure under addition and multiplication by scalars is
obvious since the span is a vector space. Moreover, if two elements
∗ ∗
Sµ Sν and Sα Sβ are multiplied the result will be something of the
form Sγ Sλ∗ according to the corollary on slide 29. Thus, the closed
∗
linear span is an algebra. Now take Sµ Sν in the span and apply the
∗ ∗ ∗
adjoint to this to see (Sµ Sν ) = Sν Sµ which is in the closed span.
This means that the closed linear span is closed under adjoints
making it a ∗-algebra. Applying the closure to the span gives
completeness making the closed linear span a Banach ∗-algebra.
Furthermore, the elements in the closed span are bounded
operators meaning that the C ∗ -norm property holds giving that the
closed linear span is a C ∗ -algebra. Since Sµ Sν generates the closed
∗

span and Sµ Sν ∗ ∈ C ∗ (S, P), this means the closed linear span of

Sµ Sν terms is a C ∗ -subalgebra of C ∗ (S, P). That is
∗

span{Sµ Sν | µ, ν ∈ E ∗ , s (µ) = s (ν)} ⊆ C ∗ (S, P) .
∗



Graph Algebras

It is possible for two Cuntz-Krieger E -families to generate
isomorphic C ∗ -algebras. To generalize this, a universal C ∗ -algebra
can be found for any directed graph E by mimicking the spanning
∗
set of Sµ Sν . This is termed the graph algebra of E denoted
C ∗ (E ). This universality property of C ∗ (E ) gives rise to the
following theorem.
Theorem (universality of the C ∗ -algebra of the graph E )
Let E be any row-ﬁnite directed graph. Then there exists a
C ∗ -algebra C ∗ (E ) generated by a Cuntz-Krieger E -family {S, P}
such that for every Cuntz-Krieger E -family {T , Q} in a C ∗ -algebra
B, there is a homomorphism πT ,Q : C ∗ (E ) → B such that
πT ,Q (Se ) = Se for every e ∈ E 1 and πT ,Q (Pv ) = Qv for every
v ∈ E 0.



Tensor Products

Theorem (Gauge-invariant uniqueness theorem)
Let E be a row-ﬁnite directed graph, and suppose that {T , Q} is a
Cuntz-Krieger E -family in a C ∗ -algebra B with each Qv = 0. If
there is a continuous action β : T → Aut B such that
βz (Te ) = zTe for every e ∈ E 1 and βz (Qv ) = Qv for every
v ∈ E 0 , then the universal homomorphism πT ,Q is an isomorphism
of C ∗ (E ) onto C ∗ (T , Q).

Theorem (Cuntz-Krieger uniqueness theorem)
Let E be a row-ﬁnite directed graph where every cycle has an entry
and let {T , Q} be a Cuntz-Krieger E -family in a C ∗ -algebra B
such that Qv = 0 for every v ∈ E 0 . Then the homomorphism
πT ,Q : C ∗ (E ) → B is an isomorphism of C ∗ (E ) onto C ∗ (T , Q).



Tensor Products

Definition
A group action of a group G on a set A is a homomorphism
γ : G → Aut (A) from a group G to the group of automorphisms
of A such that g → αg (a) for αg : A → A an automorphism of A
and such that for all a ∈ A
1 γg1 g2 (a) = γg1 (γg2 (a)) for all g1 , g2 ∈ G and
2 γ1 (a) = 1 (a) = a where 1 is the identity element of G
hold. If the underlying group G is compact, then γ is called a
compact group action.

For our specific group action, which will be defined later, the group
we use is T which is the topologically compact group (under
multiplication) formed from the unit circle.



Tensor Products

Theorem
Let E be a row-finite directed graph. Then there is an action γ of
T on C ∗ (E ) such that γz (se ) = zse for every e ∈ E 1 and
γz (pv ) = pv for every v ∈ E 0 .

This theorem guarantees an action exists on any C ∗ -graph algebra
which can often then be used to fulfill the conditions of the
gauge-invariant uniqueness theorem.
With any group action γ of the group G on a C ∗ -algebra A, we
can talk about the fixed-point algebra, denoted Aγ , which is

Aγ = {a ∈ A | γg (a) = a for all g ∈ G }

and is in fact a C ∗ -algebra itself.



Tensor Products

The power behind the gauge action γ found on slide 37 is that it
guarantees us a projection mapping onto the fixed point
subalgebra. This mapping is defined to be

Φ (a) = γg (a) dz for all a ∈ A (2)
T

for our specific case, and can be shown to be a non-zero projection
onto Aγ . In relation to a C ∗ -graph algebra C ∗ (E ) with a gauge
action γ, we are then able to observe another characterization for
the fixed-point subalgebra, C ∗ (E )γ , which is

C ∗ (E )γ = span{Sµ Sν : s (µ) = s (ν) , |µ| = |ν|}
∗

which I then was able to use in my thesis to prove the
gauge-invariant uniqueness theorem.



Tensor Products

Definition
Consider the bilinear mapping ⊗ : V × W → V ⊗ W defined by
ϕ (v , w ) → v ⊗ w for vector spaces V and W with v ∈ V and
w ∈ W defined over a field C. Then the tensor product of two
vector spaces V and W is the vector space created by the linear
span of the tensors v ⊗ w , that is
V ⊗ W = { n λi (vi ⊗ wi ) | λ ∈ C, vi ∈ V and wi ∈ W }. The
i=1
function ⊗ is bilinear satisfying
(i.) α (v ⊗ w ) = αv ⊗ w = v ⊗ αw ,
(ii.) v ⊗ (w + w ) = v ⊗ w + v ⊗ w ,
(iii.) (v + v ) ⊗ w = v ⊗ w + v ⊗ w
for all v , v ∈ V , w , w ∈ W and α ∈ F. When, in addition, V
and W are both ∗-algebras, then these conditions also hold:
(iv.) (v ⊗ w ) (v ⊗ w ) = vv ⊗ ww ;
(v.) (v ⊗ w )∗ = v ∗ ⊗ w ∗ .


Tensor Products

We will now turn our focus to the tensor product of the graph
algebras C ∗ (E1 ) and C ∗ (E2 ). It can be shown that
∗ ∗
C ∗ (E1 )⊗C ∗ (E2 ) = span{Sµ Sν ⊗Tα Tβ | s (µ) = s (ν) , s (α) = s (β)}

based on the fact that C ∗ (E1 ) and C ∗ (E2 ) are linearly generated
∗ ∗
by elements of the form Sµ Sν and Tα Tβ respectively. Moreover, if
γ1 and γ2 are group actions on C ∗ (E1 ) and C ∗ (E2 ), then it can be
shown that a mapping defined by β = γ1 ⊗ γ2 is a group action on
the tensor product C ∗ -algebra. When the actions γ1 and γ2 are
gauge actions then this gives rise to a conditional expectation onto
the fixed point subalgebra of C ∗ (E1 ) ⊗ C ∗ (E2 ). This fixed point
subalgebra of the action β is denoted (C ∗ (E1 ) ⊗ C ∗ (E2 ))β .



Tensor Products

Deﬁnition
A conditional expectation from a C ∗ -algebra A onto a
C ∗ -subalgebra B is a linear mapping Φ which is a projection of
norm one.
This conditional expectation can be found onto
(C ∗ (E1 ) ⊗ C ∗ (E2 ))β by making Φ deﬁned very similar to equation
2. It can then be shown that
∗ ∗
span Sµ Sν ⊗ Tα Tβ | s (µ) = s (ν) , s (α) = s (β) ,
|µ| − |ν| = |α| − |β|}

is the exactly (C ∗ (E1 ) ⊗ C ∗ (E2 ))β .



Tensor Products

Deﬁnition
The Cartesian Product E of the directed graphs E1 , E2 , . . . , En is
deﬁned to be the directed graph E = E 0 , E 1 , r , s with the set of
vertices E 0 = {(v1 , v2 , . . . , vn ) | vi ∈ Ei0 } and with the set of edges
E 1 = {(e1 , e2 , . . . , en ) | ei ∈ Ei1 }. The range and source maps act
on E 1 such that s (e1 , e2 , . . . , en ) = (s (e1 ) , s (e2 ) , . . . , s (en )) and
r (e1 , e2 , . . . , en ) = (r (e1 ) , r (e2 ) , . . . , r (en )) respectively.

These graphs are very complex so I have provided an example on
the next slides to show what these look like when doing the
Cartesian product graph of two directed graphs.



Tensor Products

Example

E1 E2
f1
w1
G
v2 7 w2
f2

v1 g1
g2
- u2

Figure : Two directed graphs E1 and E2



Tensor Products

Example (continued)
The Cartesian product graph E can be formed from these two
graphs and will consist of edges E 1 = {f1 f2 , f1 g2 , g1 f2 , g1 g2 } and
vertices E 0 = {v1 v2 , v1 w2 , v1 u2 , w1 v2 , w1 w2 , w1 u2 }. In order to get
where the edges should be placed, use the range and source
mapping deﬁnitions. Take the edge f1 g2 for instance. The source
map gives s (f1 g2 ) = s (f1 ) s (g2 ) = v1 v2 which is the vertex acting
as the source of the edge f1 g2 . Then applying the range map to
this edge gives r (f1 g2 ) = r (f1 ) r (g2 ) = v1 u2 which is the vertex
that receives the edge f1 g2 . Drawing in all the edges will give the
graph shown in slide 3.



Tensor Products

Example (continued)

E = E1 × E2
v1 v2 5
w1 u2
g1 g2
f1 g2
g1 f2
f1 f2 v1 u2 -w w
1 2

v1 w 2 w1 v2

Figure : The cartesian product graph of E1 and E2



Tensor Products

We desire to investigate if C ∗ (E) is isomorphic to any subalgebra
of C ∗ (E1 ) ⊗ C ∗ (E2 ). It can indeed be shown that C ∗ (E) is
isomorphic to
∗ ∗
span Sµ Sν ⊗ Tα Tβ | s (µ) = s (ν) , s (α) = s (β) , (3)
|µ| − |ν| = |α| − |β|} (4)

when E doesn’t contain any sources. From here denote the set in
equation 3 as B, which was earlier found to be the ﬁxed point
subalgebra of the action β composed of the gauge actions on the
graph algebras C ∗ (E1 ) and C ∗ (E2 ).



Tensor Products

Theorem
There is a Cuntz-Krieger E-family in the C ∗ -algebra
C ∗ (E1 ) ⊗ C ∗ (E2 ).

Proof.
Let A = C ∗ (E1 ) ⊗ C ∗ (E2 ). The graph algebra C ∗ (E) is generated
1 1
by partial isometries S(e,f ) where e ∈ E1 and f ∈ E2 and
projections P(v ,w ) where v ∈ E10 and w ∈ E 0 . The universality of
2
graph algebras guarantees there is a homomorphism π1 such that
1
π1 S(e,f ) = Se for every edge e ∈ E1 . Likewise, there exist a
homomorphism π2 such that π2 S(e,f ) = Tf for all f ∈ E2 . 1

Deﬁne a mapping π = π1 ⊗ π2 which is a homomorphism from
C ∗ (E) to A. On the generators of C ∗ (E) it gives
π S(e,f ) = π1 S(e,f ) ⊗ π2 S(e,f ) = Se ⊗ Tf and
π P(v ,w ) = π1 P(v ,w ) ⊗ π2 P(v ,w ) = Pv ⊗ Qw . I claim that
{π S(e,f ) , π P(v ,w ) } is a CK E-family in A.


Tensor Products

continued.
Observe ﬁrst that
∗
π S(e,f ) π S(e,f ) = [Se ⊗ Tf ]∗ [Se ⊗ Tf ]
= [Se ⊗ Tf∗ ] [Se ⊗ Tf ] = Se Se ⊗ Tf∗ Tf
∗ ∗

= Ps(e) ⊗ Ps(f )
= π1 Ps(e),s(f ) ⊗ π2 Ps(e),s(f )
= π Ps(e),s(f ) = π Ps(e,f ) .

Moreover, it can be shown that
∗
π S(e,f ) π S(e,f )
{(e,f ):r (e,f )=(v ,w )}
∗
= Se Se ⊗ Tf Tf∗ .
{e:r (e)=v } {f :r (f )=w }



Tensor Products

continued.
Now, since {S, P} and {T , Q} are Cuntz-Krieger families for the
graph algebras C ∗ (E1 ) and C ∗ (E2 ) respectively, then it is true
∗
that {e:r (e)=v } Se Se = Pv and {f :r (f )=w } Tf Tf∗ = Qw provided
that both v and w are not sources. Making these replacements in
the above gives
∗
π S(e,f ) π S(e,f )
{(e,f ):r (e,f )=(v ,w )}

= Pv ⊗ Qw = π P(v ,w )

where (v , w ) is not a source. Now to show is that the projections
π P(v ,w ) are mutually orthogonal. When (v1 , w1 ) = (v2 , w2 ),
then either v1 = v2 or w1 = w2 .



Tensor Products

continued.
Without loss of generality suppose that v1 = v2 and notice that

π P(v1 ,w1 ) π P(v2 ,w2 )
= (Pv1 ⊗ Qw1 ) (Pv2 ⊗ Qw2 ) = Pv1 Pv2 ⊗ Qw1 Qw2 .

Notice that the projections Pv are mutually orthogonal making the
left component of the tensor product 0. This gives that

π P(v1 ,w1 ) π P(v2 ,w2 ) = Pv1 Pv2 ⊗ Qw1 Qw2 = 0 ⊗ Qw1 Qw2 = 0.

Hence, the projections π P(v ,w ) are mutually orthogonal.
Therefore, it is true that {π S(e,f ) , π P(v ,w ) } forms a
Cuntz-Krieger family in the graph E for every (e, f ) ∈ E 1 and every
(v , w ) ∈ E 0 .



Tensor Products

Theorem
There is a subalgebra of C ∗ (E1 ) ⊗ C ∗ (E2 ) isomorphic to C ∗ (E).

Proof.
It was shown in slide 47 that there is a CK E-family inside the
C ∗ -algebra C ∗ (E1 ) ⊗ C ∗ (E2 ) := A. This family is speciﬁcally
0
π P(v ,w ) , π S(e,f ) . Now, if Pv = 0 for all v ∈ E1 and
0
Qw = 0 for all w ∈ E2 are chosen in the original CK E -families for
the graphs E1 and E2 , respectively, then each π P(v ,w ) = 0.
According to the universality of graph algebras, the mapping
π : C ∗ (E) → A detailed slide 47 is a homomorphism such that
π P(v ,w ) = Pv ⊗ Qw and π S(e,f ) = Se ⊗ Tf . Deﬁne an action
β to be β = γ ⊗ γ restricted to the compact group
{(z, z) | z ∈ T} where γ is the gauge action on C ∗ (E1 ) and γ is a
mapping to the identity automorphism of C ∗ (E2 ). This acts on an
element se ⊗ tf from A in the following way


Tensor Products

continued.

β(z,z) (se ⊗ tf ) = γz (se ) ⊗ γz (tf ) = zse ⊗ tf = z (se ⊗ tf )

and acts on the elements pv ⊗ qw ∈ A as

β(z,z) (pv ⊗ qw ) = γz (pv ) ⊗ γz (qw ) = pv ⊗ qw .

Then the β defined in this way is indeed a group action and can be
shown to be continuous. Since the action β : T → Aut A defined
above is a continuous action such that
βz π P(v ,w ) = βz (Pv ⊗ Qw ) = Pv ⊗ Qw for every (v , w ) ∈ E 0
and βz π S(e,f ) = βz (Se ⊗ Tf ) = z (Se ⊗ Tf ) for every
(e, f ) ∈ E 1 , then the conditions for the gauge-invariant uniqueness
theorem have been satisfied. Therefore, π is an isomorphism of
C ∗ (E) onto C ∗ π P(v ,w ) , π S(e,f ) .



Tensor Products

Lastly, in the case where the Cartesian product graph E has no
sources, it can be shown that

C ∗ (E)
∼ ∗ ∗
= span Sµ Sν ⊗ Tα Tβ |
s (µ) = s (ν) , s (α) = s (β) , |µ| − |ν| = |α| − |β|}

which is the ﬁxed-point algebra of the action β made from the
gauge actions on the graph algebras C ∗ (E1 ) and C ∗ (E2 ). Earlier
it was observed that there is a conditional expectation Φ onto B.
Since C ∗ (E) ∼ B, then we know there exist a conditional
=
expectation onto C ∗ (E) when the directed graph E has no sources.


Masters Thesis Defense

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