Masters Thesis Defense

Introduction Graph Algebras Two Uniqueness Theorems Tensor Products of Graph Algebras

Graph C ∗-algebras and Subalgebras ofC ∗ (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 mappingϕ : V × V → V such that ϕ : 〈a, b〉 7→ ab. That is to say, V is analgebra if it is a vector space and has an associative multiplicationoperation 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, equippedwith a norm ‖·‖ in which the space is complete with respect to themetric 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 algebraif 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 anda · 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 7→ x∗ such that

1 x∗∗ = x and (xy)∗ = y∗x∗ for all x , y ∈ A

2 (λa + µb)∗ = λ̄a∗ + µ̄b∗ 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 thisrepresents the algebra A equipped with the involution mapping ∗.

Definition

A Banach *-algebra is a ∗-algebra A that is complete with respectto a submultiplicative norm ‖·‖, and ‖a∗‖ = ‖a‖ for all a ∈ A. If Ahas 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 suchthat the norm on A satisfies

‖a∗a‖ = ‖a‖2 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 complexconjugation.

2. Mn (C) the algebra of n-by-n matrices over C if the matricesare considered operators, the norm ‖·‖ is the operator normon matrices, and the involution is the conjugate transpose.

3. B (H) the space of bounded linear operators on a Hilbertspace H which will be discussed shortly.



Introduction

Definition

If X and Y are vector spaces then a mapping T : X → Y is calleda linear mapping provided that for all x , y ∈ X and scalarsα, 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 existsan 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 allbounded linear maps T : X → X is denoted by B (X ), and we callthese 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 thisnorm 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 → Fsatisfying 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〉 (Linearityin 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 anassociated norm and metric that is complete with respect to thenorm.



Introduction

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

‖x‖ =√〈x , x〉.

A Hilbert space has many nice properties and becomes even morevaluable when used as the space where we apply bounded linearoperators. 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 abounded operator T on a Hilbert space does exists and is definedby 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 concretespace since linear operators are very familiar and relatively easy toutilize. 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 beviewed as some C ∗-subalgebra of B (H). This is most helpful as itsets up the stage for how we define elements in a generalC ∗-algebra: we can define them by how their representations workin B (H).



Introduction

Definition

A partial isometry is a mapping f from H into H such that f is anisometry on M, a subspace of H, and is 0 on M⊥, the orthogonalcomplement of M.

Definition

An orthogonal projection, or just projection, P on a Hilbert spaceH 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 acertain class of C ∗-algebras called graph algebras introduced in thenext 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 sourcefunctions respectively. The set E 0 has elements referred to asvertices and the elements in the set E 1 are called edges. Thegraph 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 asource. Any vertex w ∈ E 0 from which no edges are emitted iscalled a sink. Moreover, all the graphs that will be considered hereare row-finite which means that each vertex receives at most afinite number of edges.



Graph Algebras

Example

In the figure below is the depiction of a directed graph withvertices v and w and three edges f , e, and g . This graph would berepresented 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 .

v we

66

f

vvg <<

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



Graph Algebras

We now represent our directed graph as bounded operators on aHilbert 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 2relations hold:

(CK1) S∗e Se = Ps(e) for all e ∈ E 1; and

(CK2) Pv =∑

{e∈E1:r(e)=v}

SeS∗e if v is not a source.

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



Graph Algebras

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

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



Graph Algebras

Theorem

Let {S ,P} be a Cuntz-Krieger E -family acting on a Hilbert spaceH. Then

SePs(e) = Pr(e)Se = Se

for all e ∈ E 1.

Proof.

The Cuntz-Krieger (CK) relations say that S∗e Se = Ps(e). ApplyingSe to both sides of this equality gives that SeS∗e Se = SePs(e). InB (H), it is true that if Se is a partial isometry then SeS∗e Se = Se .Thus, Se = SePs(e).



Graph Algebras

proof continued.

The CK relations also give that Pr(e) = Pv =∑{e∈E1:r(e)=v} SeS∗e .

This means for a particular e ∈ E 1 such that r (e) = v thatPv − SeS∗e =

∑{f ∈E1:r(f )=v and f 6=e} Sf S∗f . Since Sf S∗f is a

projection and projections are positive operators, this means that⟨ ∑{f ∈E1:r(f )=v and f 6=e}

Sf S∗f (h) , h

⟩

=∑

{f ∈E1:r(f )=v and f 6=e}

〈SeS∗e (h) , h〉 ≥ 0 for all h ∈ H

showing that Pv − SeS∗e is a positive operator. From this it is truethat

0 ≤ 〈(Pv − SeS∗e ) (h) , h〉 = 〈Pv (h) , h〉 − 〈SeS∗e (h) , h〉⇒ 〈Pv (h) , h〉 ≥ 〈SeS∗e (h) , h〉



Graph Algebras

proof continued.

which means that PvH ⊇ SeS∗eH. This implies thatPvSeH ⊇ SeS∗e SeH = SeH. The reverse inclusion is found bysimply observing that PvSeH ⊆ SeH.

Therefore, Se = PvSe .Since PvSe and SePs(e) are both equal to Se , it is also true thatPvSe = SePs(e) thus completing the proof.

The algebraic relationship

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

is extremely helpful when making manipulations of Cuntz-KriegerE -families.



Graph Algebras

It is important to note that a Cuntz-Krieger E -family can bespoken of in relation to an arbitrary C ∗-algebra B. Theseprojections p will satisfy p = p∗ = p2 and the partial isometrieswill satisfy s = ss∗s. There is a useful theorem regarding s inB (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) toclassify partial isometries, and even projections, in an arbitraryC ∗-algebra B is thanks to the Gelfand-Naimark representationtheorem which states there is an injective ∗-homomorphism ϕ fromB to B (H).



Graph Algebras

Example (Directed graph of the Cuntz-Algebra O2)

We shall find a Cuntz-Krieger family can be found satisfying CK1and CK2 for the directed graph pictured below. That means wemust find partial isometries Sg and Sh and a projection Pv

satisfying S∗gSg = Pv = S∗hSh and Pv = SgS∗g + ShS∗h on someHilbert space.

v

g

]]

h

bb



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 bechecked that Sg and Sh are both partial isometries. The best wayis to use the following theorem.



Graph Algebras


Observe that SgS∗g =(SgS∗g

)2=(SgS∗g

)∗with the way Sg is

defined showing that SgS∗g is a projection. Therefore, Sg is apartial isometry according to the proposition. Similarly, it ispossible to see that Sh is a partial isometry. Lastly, check that therelations hold and discover the operator Pv . It can be easilychecked to see that S∗gSg = 1 = S∗hSh and SgS∗g + ShS∗h = 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-KriegerE -family in a C ∗-algebra B. Then

(a) the projections {SeS∗e | e ∈ E 1} are mutually orthogonal;

(b) If e 6= f , then S∗e Sf = 0;

(c) If s (e) 6= r (f ), then SeSf = 0;

(d) If s (e) 6= s (f ), then SeS∗f = 0.

Part (c) in this proposition allows for talking about paths in adirected graph. Notice, part (c) is saying that SeSf = 0 unless efis a path. A path of length n is a sequence of edges µ1, µ2, . . . , µnsuch that s (µi ) = r (µi+1). Moreover, the set En is defined to bethe set of all paths of length n. Now define E ∗ :=

⋃n≥0 En. Then

the set E 0 clarifies our definition earlier: this is the set of paths oflength 0 which is the set of vertices.



Graph Algebras

Since the multiplication in B (H) is composition, which isperformed from right to left, it is important to note that a path oflength 2, say µ1µ2, will be represented as partial isometries andthat µ2 is traversed prior to µ1. Because of this, the range andsource maps extend to E ∗ by

r (µ) = r(µ1µ2 · · ·µ|µ|

)= r (µ1)

ands (µ) = s

(µ1µ2 · · ·µ|µ|

)= s

(µ|µ|).

Moreover, if ν and µ are paths such that s (ν) = r (µ), then νµcan 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 beshown for any µ ∈ E ∗. Simply define Sµ := Sµ1Sµ2 · · · Sµn . Fromproposition on slide 25 it can be seen that Sµ = 0 unless µ is apath.

Then using the algebraic relations on single edges µi it can be seenthat

SµPs(µ) = Pr(µ)Sµ = Sµ for all µ ∈ E ∗.

The next two propositions will allow us to characterize theC ∗-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-KriegerE -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(SµS∗µ

)(SνS∗ν ) = 0;

b. S∗µSν =

S∗µ ′ if µ = νµ ′ for some µ ′ ∈ E ∗

Sν ′ if ν = µν ′ for some ν ′ ∈ E ∗

0 when paths µ and ν don’t overlap;

c. If SµSν 6= 0, then µν is a path in E and SµSν = Sµν ;

d. If SµS∗ν 6= 0, then s (µ) = s (ν).



Graph Algebras

Theorem

Suppose that E is a row-finite graph and {S ,P} is a Cuntz-KriegerE -family in a C ∗-algebra B. For any µ, ν, α, β ∈ E ∗ then

(SµS∗ν )(SαS∗β

)=

Sµα ′S

∗β if α = να ′

SµS∗βν ′ if ν = αν ′

0 otherwise.



Graph Algebras

From the previous corollaries we can take any nonzero product of afinite number of partial isometries of the forms Sα and S∗β forα, β ∈ E ∗ and reduce it to a form SµS∗ν for some µ, ν ∈ E ∗ wheres (µ) = s (ν). To see this clearer assume that W is the nonzeroproduct of a finite number of Se and S∗f terms. If there areadjacent Se ’s, then these can be lumped together into a singleterm Sµ. Moreover, adjacent S∗f ’s can be combined into a singleS∗ν . The product S∗µSν from the corollary on slide 28 this will eitherbecome Sα or S∗β .

A single S∗µ term can be rewritten as

(Sµ)∗ =(SµPs(µ)

)∗= P∗s(µ)S

∗µ = Ps(µ)S

∗µ and since every

projection is also a partial isometry then Ps(µ) can be written asSs(µ) giving that S∗µ = Ss(µ)S

∗µ. This means a single Sµ term will

be Sµ = SµS∗s(µ) by applying the ∗ to S∗µ.



Graph Algebras

There are several cases that can be inspected now. Take thesequence SµS∗νSαSβ for instance. The middle part S∗νSα will beeither Sλ or S∗λ from slide 28. For the former the sequence boilsdown to SµSλSβ = Sµλβ and the later case gives SµS∗λSβ for whichwhich Sβ can be changed to give SµS∗λSβS∗s(β). This can now becollapsed into the desired form according to the corollary on slide29.

All of these results finally lead to the amazing fact that all graphC ∗-algebras are linearly generated by terms like SµS∗ν which is thenext 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{SµS∗ν | µ, ν ∈ E ∗, s (µ) = s (ν)}.



Graph Algebras

Proof.

It is clear that C ∗ (S ,P) ⊆ span{SµS∗ν | µ, ν ∈ E ∗, s (µ) = s (ν)}from the above statement. To explain, notice that product of anyfinite number of Sµ and S∗ν terms will reduce to the form SαS∗β .Also notice that projections Pv can be rewritten asPv =

∑f ∈E1:r(f )=v Sf S∗f . This means if there are projections in

these finite products of terms from C ∗ (S ,P) then that productwill still come down to the form SαS∗β or a finite sum of termslooking like SαS∗β after distributing and simplifying. Moreover, thefinite sum of any product of elements from C ∗ (S ,P) will boildown to the form SµS∗ν + SαS∗β + · · · which is included in thespan. Finally, closure of the span will include infinite products andsums of elements from C ∗ (S ,P).

The reverse inclusion is obtained by showing the closed linear spanof SµS∗ν is a C ∗-subalgebra of C ∗ (S ,P).



Graph Algebras

Proof.

First, the closure under addition and multiplication by scalars isobvious since the span is a vector space. Moreover, if two elementsSµS∗ν and SαS∗β are multiplied the result will be something of theform SγS∗λ according to the corollary on slide 29. Thus, the closedlinear span is an algebra. Now take SµS∗ν in the span and apply theadjoint to this to see (SµS∗ν )∗ = SνS∗µ which is in the closed span.This means that the closed linear span is closed under adjointsmaking it a ∗-algebra. Applying the closure to the span givescompleteness making the closed linear span a Banach ∗-algebra.Furthermore, the elements in the closed span are boundedoperators meaning that the C ∗-norm property holds giving that theclosed linear span is a C ∗-algebra. Since SµS∗ν generates the closedspan and SµS∗ν ∈ C ∗ (S ,P), this means the closed linear span ofSµS∗ν terms is a C ∗-subalgebra of C ∗ (S ,P). That is

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



Graph Algebras

It is possible for two Cuntz-Krieger E -families to generateisomorphic C ∗-algebras. To generalize this, a universal C ∗-algebracan be found for any directed graph E by mimicking the spanningset of SµS∗ν . This is termed the graph algebra of E denotedC ∗ (E ). This universality property of C ∗ (E ) gives rise to thefollowing theorem.

Theorem (universality of the C ∗-algebra of the graph E )

Let E be any row-finite directed graph. Then there exists aC ∗-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 ∗-algebraB, there is a homomorphism πT ,Q : C ∗ (E )→ B such thatπT ,Q (Se) = Se for every e ∈ E 1 and πT ,Q (Pv ) = Qv for everyv ∈ E 0.



Tensor Products

Theorem (Gauge-invariant uniqueness theorem)

Let E be a row-finite directed graph, and suppose that {T ,Q} is aCuntz-Krieger E -family in a C ∗-algebra B with each Qv 6= 0. Ifthere is a continuous action β : T→ AutB such thatβz (Te) = zTe for every e ∈ E 1 and βz (Qv ) = Qv for everyv ∈ E 0, then the universal homomorphism πT ,Q is an isomorphismof C ∗ (E ) onto C ∗ (T ,Q).

Theorem (Cuntz-Krieger uniqueness theorem)

Let E be a row-finite directed graph where every cycle has an entryand let {T ,Q} be a Cuntz-Krieger E -family in a C ∗-algebra Bsuch that Qv 6= 0 for every v ∈ E 0. Then the homomorphismπT ,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 homomorphismγ : G → Aut (A) from a group G to the group of automorphismsof A such that g 7→ αg (a) for αg : A → A an automorphism of Aand such that for all a ∈ A

1 γg1g2 (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 acompact group action.

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



Tensor Products

Theorem

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

This theorem guarantees an action exists on any C ∗-graph algebrawhich can often then be used to fulfill the conditions of thegauge-invariant uniqueness theorem.

With any group action γ of the group G on a C ∗-algebra A, wecan 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 itguarantees us a projection mapping onto the fixed pointsubalgebra. This mapping is defined to be

Φ (a) =

∫Tγg (a) dz for all a ∈ A (2)

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

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

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



Tensor Products

Definition

Consider the bilinear mapping ⊗ : V ×W → V ⊗W defined byϕ (v ,w) 7→ v ⊗ w for vector spaces V and W with v ∈ V andw ∈W defined over a field C. Then the tensor product of twovector spaces V and W is the vector space created by the linearspan of the tensors v ⊗ w , that isV ⊗W = {

∑ni=1 λi (vi ⊗ wi ) | λ ∈ C, vi ∈ V and wi ∈W }. The

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, Vand 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 graphalgebras C ∗ (E1) and C ∗ (E2). It can be shown that

C ∗ (E1)⊗C ∗ (E2) = span{SµS∗ν⊗TαT ∗β | s (µ) = s (ν) , s (α) = s (β)}

based on the fact that C ∗ (E1) and C ∗ (E2) are linearly generatedby elements of the form SµS∗ν and TαT ∗β respectively. Moreover, if

γ1 and γ2 are group actions on C ∗ (E1) and C ∗ (E2), then it can beshown that a mapping defined by β = γ1 ⊗ γ2 is a group action onthe tensor product C ∗-algebra. When the actions γ1 and γ2 aregauge actions then this gives rise to a conditional expectation ontothe fixed point subalgebra of C ∗ (E1)⊗ C ∗ (E2). This fixed pointsubalgebra of the action β is denoted (C ∗ (E1)⊗ C ∗ (E2))β.



Tensor Products

Definition

A conditional expectation from a C ∗-algebra A onto aC ∗-subalgebra B is a linear mapping Φ which is a projection ofnorm one.

This conditional expectation can be found onto(C ∗ (E1)⊗ C ∗ (E2))β by making Φ defined very similar to equation2. It can then be shown that

span{

SµS∗ν ⊗ TαT ∗β | 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 isdefined to be the directed graph E =

(E0, E1, r , s

)with the set of

vertices E0 = {(v1, v2, . . . , vn) | vi ∈ E 0i } and with the set of edges

E1 = {(e1, e2, . . . , en) | ei ∈ E 1i }. The range and source maps act

on E1 such that s (e1, e2, . . . , en) = (s (e1) , s (e2) , . . . , s (en)) andr (e1, e2, . . . , en) = (r (e1) , r (e2) , . . . , r (en)) respectively.

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



Tensor Products

Example

E1 E2

v1

w1 v2 w2

u2

g1

GG

f1

��f2

77

g2--

Figure : Two directed graphs E1 and E2



Tensor Products

Example (continued)

The Cartesian product graph E can be formed from these twographs and will consist of edges E1 = {f1f2, f1g2, g1f2, g1g2} andvertices E0 = {v1v2, v1w2, v1u2,w1v2,w1w2,w1u2}. In order to getwhere the edges should be placed, use the range and sourcemapping definitions. Take the edge f1g2 for instance. The sourcemap gives s (f1g2) = s (f1) s (g2) = v1v2 which is the vertex actingas the source of the edge f1g2. Then applying the range map tothis edge gives r (f1g2) = r (f1) r (g2) = v1u2 which is the vertexthat receives the edge f1g2. Drawing in all the edges will give thegraph shown in slide 3.



Tensor Products

Example (continued)

E = E1 × E2

v1v2

v1u2

v1w2

w1u2

w1w2

w1v2

f1f2

&&

f1g2

�� g1f2 --

g1g2

55

Figure : The cartesian product graph of E1 and E2



Tensor Products

We desire to investigate if C ∗ (E) is isomorphic to any subalgebraof C ∗ (E1)⊗ C ∗ (E2). It can indeed be shown that C ∗ (E) isisomorphic to

span{

SµS∗ν ⊗ TαT ∗β | 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 inequation 3 as B, which was earlier found to be the fixed pointsubalgebra of the action β composed of the gauge actions on thegraph algebras C ∗ (E1) and C ∗ (E2).



Tensor Products

Theorem

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

Proof.

Let A = C ∗ (E1)⊗ C ∗ (E2). The graph algebra C ∗ (E) is generatedby partial isometries S(e,f ) where e ∈ E 1

1 and f ∈ E 12 and

projections P(v ,w) where v ∈ E 01 and w ∈ E 0

2 . The universality ofgraph algebras guarantees there is a homomorphism π1 such thatπ1(S(e,f )

)= Se for every edge e ∈ E 1

1 . Likewise, there exist ahomomorphism π2 such that π2

(S(e,f )

)= Tf for all f ∈ E 1

2 .Define a mapping π = π1 ⊗ π2 which is a homomorphism fromC ∗ (E) to A. On the generators of C ∗ (E) it givesπ(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 ]

= [S∗e ⊗ T ∗f ] [Se ⊗ Tf ] = S∗e Se ⊗ T ∗f 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∑{(e,f ):r(e,f )=(v ,w)}

[π(S(e,f )

)] [π(S(e,f )

)]∗=

∑{e:r(e)=v}

SeS∗e ⊗∑

{f :r(f )=w}

Tf T ∗f .



Tensor Products

continued.

Now, since {S ,P} and {T ,Q} are Cuntz-Krieger families for thegraph algebras C ∗ (E1) and C ∗ (E2) respectively, then it is truethat

∑{e:r(e)=v} SeS∗e = Pv and

∑{f :r(f )=w} Tf T ∗f = Qw provided

that both v and w are not sources. Making these replacements inthe above gives ∑

{(e,f ):r(e,f )=(v ,w)}

[π(S(e,f )

)] [π(S(e,f )

)]∗= Pv ⊗ Qw = π

(P(v ,w)

)where (v ,w) is not a source. Now to show is that the projectionsπ(P(v ,w)

)are mutually orthogonal. When (v1,w1) 6= (v2,w2),

then either v1 6= v2 or w1 6= w2.



Tensor Products

continued.

Without loss of generality suppose that v1 6= v2 and notice that[π(P(v1,w1)

)] [π(P(v2,w2)

)]= (Pv1 ⊗ Qw1) (Pv2 ⊗ Qw2) = Pv1Pv2 ⊗ Qw1Qw2 .

Notice that the projections Pv are mutually orthogonal making theleft component of the tensor product 0. This gives that[π(P(v1,w1)

)] [π(P(v2,w2)

)]= Pv1Pv2⊗Qw1Qw2 = 0⊗Qw1Qw2 = 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 ) ∈ E1 and every(v ,w) ∈ E0.



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 theC ∗-algebra C ∗ (E1)⊗ C ∗ (E2) := A. This family is specifically{π(P(v ,w)

), π(S(e,f )

)}. Now, if Pv 6= 0 for all v ∈ E 0

1 andQw 6= 0 for all w ∈ E 0

2 are chosen in the original CK E -families forthe graphs E1 and E2, respectively, then each π

(P(v ,w)

)6= 0.

According to the universality of graph algebras, the mappingπ : C ∗ (E)→ A detailed slide 47 is a homomorphism such thatπ(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 amapping to the identity automorphism of C ∗ (E2). This acts on anelement se ⊗ tf from A in the following way



Tensor Products

continued.

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

and acts on the elements pv ⊗ qw ∈ A as

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

Then the β defined in this way is indeed a group action and can beshown to be continuous. Since the action β : T→ AutA definedabove is a continuous action such thatβz(π(P(v ,w)

))= βz (Pv ⊗ Qw ) = Pv ⊗ Qw for every (v ,w) ∈ E0

and βz(π(S(e,f )

))= βz (Se ⊗ Tf ) = z (Se ⊗ Tf ) for every

(e, f ) ∈ E1, then the conditions for the gauge-invariant uniquenesstheorem have been satisfied. Therefore, π is an isomorphism ofC ∗ (E) onto C ∗

(π(P(v ,w)

), π(S(e,f )

)).



Tensor Products

Lastly, in the case where the Cartesian product graph E has nosources, it can be shown that

C ∗ (E)∼= span

{SµS∗ν ⊗ TαT ∗β |

s (µ) = s (ν) , s (α) = s (β) , |µ| − |ν| = |α| − |β|}

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


Masters Thesis Defense

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