Algebras Embed Into the Cuntz Algebra

108 Pages·2016·0.79 MB·English

Separable Exact C∗-Algebras Embed Into the Cuntz Algebra
Paul Skoufranis
July 8, 2016
Abstract
Doyouwanttoreadthesenotes? Yousureyouwanttoknow? Theproofscontainedinthisdocument
are not for the faint of heart. If somebody said it was a happy little tale, if somebody told you it was just
your average straightforward proof not a technicality in sight... somebody lied.
The purpose of these notes is to prove the embedding theorem of Kirchberg and Phillips, Theorem
11.11,thatstateseveryunital,separable,exactC∗-algebrahasaunitalembeddingintotheCuntzalgebra
O . These notes are based are based on several references that will be acknowledged at the beginning
2
of each chapter. These notes are meant to be as self-contained as possible except for some well-known
results in theory of C∗-algebras. In particular, the author assumes the reader has a basic knowledge of
the following ideas:
1. Basic C∗-Algebra Theory (including C∗-norms, invertible elements, normal operators, self-adjoint
operators,positiveoperators,continuousfunctionalcalculus,abelianC∗-algebras,finitedimensional
C∗-algebras, polar decomposition, ideals, quotients, pure states, representations, irreducible repre-
sentations, GNS, continuity of ∗-homomorphisms, compact operators, C∗-bounded approximate
identities, quasicentral C∗-bounded approximate identities)
2. BasicvonNeumannAlgebraTheory(WOT-convergence,SOT-convergence,vonNeumann’sDouble
CommutantTheory,Borelfunctionalcalculus,partialisometries,MurrayvonNeumannequivalence
of projections, polar decomposition, commutants, the Strong Kadison Transitivity Theorem)
3. CompletelyPositiveMaps(definitions,operatorsystems,completelyboundednorms,Stinespring’s
Theorem,Arveson’sExtensionTheorem,injectivity,conditionalexpectations,point-normtopology,
bounded-weak topology)
4. Tensor Products of C∗-Algebras (minimal and maximal tensor products, theory of states and rep-
resentations on tensor products)
5. Nuclear C∗-Algebras (tensor product and completely positive map definition, examples of nuclear
C∗-algebras)
6. Exact C∗-Algebras (completely positive map and tensor product definition, examples of exact C∗-
algebras)
7. Inductive Limits of C∗-Algebras (including AF C∗-algebras)
8. Cross Products of C∗-Algebras (definitions of reduced and full cross products, cross product of a
nuclear C∗-algebra by Z is nuclear, reduced and full cross products by Z are the same)
9. Quasidiagonal C∗-Algebras (definition given, cones of C∗-algebras are quasidiagonal)
Two excellent references that cover most of these topics are [Da] and [BO].
This document is for educational purposes and should not be referenced. Please contact the author
of this document if you need aid in finding the correct reference. Comments, corrections, and recom-
mendations on these notes are always appreciated and may be e-mailed to the author (see his website
for contact info).
Contents
1 Basic Properties of the Cuntz Algebras 3
2 Purely Infinite C∗-Algebras 13
3 Tensor Products of Purely Infinite C∗-Algebras 19
4 K-Theory for Purely Infinite C∗-Algebras 27
5 Approximation Properties of Purely Infinite C∗-Algebras 39
6 ∗-Homomorphisms From O 48
2
7 On O ⊗ O 58
2 min 2
8 States on Purely Infinite C∗-Algebras 62
9 Non-Standard Results on Completely Positive Maps 66
10 Completely Positive Maps on Purely Infinite C∗-Algebras 76
11 Embedding into O 87
2
12 O ⊗ A(cid:39)O 95
2 min 2
13 O ⊗ A(cid:39)A 103
∞ min
2
1 Basic Properties of the Cuntz Algebras
InthischapterwewilldevelopsomebasicpropertiesoftheCuntzalgebras. Tobemorespecific,wewillshow
that the Cuntz algebras are simple and nuclear. In fact, in our proof that the Cuntz algebras are simple we
willproveastrongerresultwhich,inthenextchapter,willimplythattheCuntzalgebrasarepurelyinfinite.
The results for this chapter were developed from the excellent book [Da] (if you are reading these notes,
youshoulddefinitelyinvestinthisbook)andfromtheoriginalpaper[Cu2]. NotethatLemmaV.4.5in[Da]
has a small problem at the end as Lemma V.4.4 does not apply directly. In these notes, we modify Lemma
V.4.4 to correct this mistake.
We begin with the definition of the Cuntz algebras.
Definition 1.1. For a natural number n ≥ 2, the Cuntz algebra O is the universal C∗-algebra generated
n
by n isometries S ,S ,...,S such that (cid:80)n S S∗ =I. The Cuntz algebra O is the universal C∗-algebra
1 2 n i=1 i i ∞
generated by an infinite collection of isometries {S }∞ such that (cid:80)n S S∗ ≤I for all n∈N.
i i=1 i=1 i i
Remarks 1.2. The statement “the Cuntz algebra O is the universal C∗-algebra generated by n isometries
n
S ,S ,...,S such that (cid:80)n S S∗ =I” means that if A is any C∗-algebra with n isometries {TA}n ⊆A
1 2 n i=1 i i i i=1
such that (cid:80)n TA(TA)∗ =I (such C∗-algebra exists by considering the specific isometries in B(H)) then
i=1 i i A
there exists a C∗-homomorphism π : O → A such that π(S ) = TA. We note that such a universal C∗-
n i i (cid:13) (cid:13)
algebra exists by taking a direct sum of all such C∗-algebras and the fact that (cid:13)⊕ATiA(cid:13)=1 for all i so the
norm of any element in ∗-alg{⊕ TA,...,⊕ TA} is finite. The same remarks apply for O .
A 1 A n ∞
Remarks 1.3. Clearly O and each O are separable being the closure of a ∗-algebra generated by a
∞ n
countable number of operators. Using the fact that the S ’s in O are isometries and (cid:80)n S S∗ =I imply
i n i=1 i i
that S S∗ are projections and thus S∗S =0 if i(cid:54)=j (and the same for O ). Thus S∗S =δ I.
i i i j ∞ i j i,j
To discuss the Cuntz algebras, it is useful to develops some notation.
Notation 1.4. For a word µ=(i ,i ,...,i ) with i ∈{1,2,...,n} (or N for O ), we define
1 2 m j ∞
S :=S S ···S .
µ i1 i2 im
Let |µ| denote the length of the word µ.
With this notation in hand, we make the following observations using Remarks 1.3.
Lemma 1.5. Let µ and ν be words in {1,...,n} (or N) such that S∗S (cid:54)=0. Then
µ ν
1. If |µ|=|ν| then µ=ν and S∗S =I.
µ ν
2. If |µ|>|ν| then there exists a word µ(cid:48) such that µ=νµ(cid:48) (as words) and S∗S =S∗ .
µ ν µ(cid:48)
3. If |µ|<|ν| then there exists a word ν(cid:48) such that ν =µν(cid:48) (as words) and S∗S =S .
µ ν ν(cid:48)
As a simple corollary, we have the following.
Corollary 1.6. For n≥2 or n=∞, every element in ∗-alg{S }n can be written as a linear combination
i i=1
of elements of the form S S∗ where µ and ν are words with letters in {1,...,n}.
µ ν
To prove the desired properties of the Cuntz algebras, we will need a specific C∗-subalgebra which will
be of vital importance.
Notation 1.7. For each n≥2 or n=∞ and for each k ∈N, let
Fn :=span{S S∗ | |µ|=|ν|=k,µ and ν are words with letters in {1,...,n}}
k µ ν
Let Fn =(cid:83) Fn. Notice that Fn,Fn ⊆O for all m≥n.
k≥1 k k m
3
Lemma 1.8. For n≥2, Fn (cid:39)M (C) and Fn is the UHF algebra with supernatural number n∞. Moreover
k nk
F∞ (cid:39)K and F∞ is an AF C∗-algebra.
k
Proof. To see that Fn (cid:39)M (C), we simply note that the set
k nk
{S S∗ | |µ|=|ν|=k,µ and ν are words with letters in {1,...,n}}
µ ν
is a set of matrix units for Fn by Lemma 1.5 with precisely nk elements. To see that Fn is the UHF algebra
k
with supernatural number n∞, we need to analyze the embeddings of Fn into Fn .
k k+1
To see that Fn embeds into Fn with the ‘correct’ embedding, we notice for any word µ and ν with
k k+1
letters in {1,...,n} and |µ|=|ν|=k that
(cid:32) n (cid:33) n
(cid:88) (cid:88)
S S∗ =S S S∗ S∗ = S S∗
µ ν µ i i ν µi νi
i=1 i=1
Therefore, by grouping the matrix units of Fn in the appropriate way, we obtain that Fn is the UHF
k+1
algebra with supernatural number n∞.
The proof that F∞ (cid:39) K is identical. To see that F∞ is AF, we note since Fn ⊆ Fn ⊆ Fn+1 for all n
k k k+1 k+1
that F∞ =(cid:83) Fn and thus the result follows.
n≥1 n
Next we note that there exists a very important map from O to Fn for all n≥2 and n=∞.
n
Theorem 1.9. There exists a faithful conditional expectation Φ :O →Fn for all n≥2 or n=∞. That
n n
is, Φ :O →Fn is a unital, (completely) positive map such that Φ (T)=T for all T ∈Fn.
n n n
Proof. Fix n≥2 or n=∞. For each λ∈T, we notice that {λS }n are also a set of isometries that satisfy
i i=1
the universal property of the Cuntz algebras. Therefore there must exists an ∗-automorphism ρ of O such
λ n
that ρ (S ) = λS . Hence ρ (S∗) = λ−1S and ρ (S S∗) = λ|µ|−|ν|S S∗. Thus the map from T to O
λ i i λ i i λ µ ν µ ν n
defined by λ (cid:55)→ ρ (T) is continuous for all T ∈ ∗-alg{S }n . Therefore, since ∗-alg{S }n is dense in O
λ i i=1 i i=1 n
and (cid:107)ρ (cid:107)=1 for all λ∈T, the map T to O defined by λ(cid:55)→ρ (T) is continuous for all T ∈O .
λ n λ n
Define Φ :O →O by
n n n
(cid:90)
Φ (T)= ρ (T)dλ
n λ
T
which exists by continuity. We notice for all words µ and ν with letters in {1,2,...,n} that
(cid:90) (cid:26)
0 if |µ|=(cid:54) |ν|
Φ (S S∗)= λ|µ|−|ν|S S∗dλ=
n µ ν µ ν S S∗ if |µ|=|ν|
T µ ν
Hence it is easy to see that Φ maps into Fn. Moreover, if T ∈Fn then Φ (T)=T. Hence, by extending by
n k n
continuity, Φ | = Id . In addition, since each ρ is a ∗-homomorphism and the integration of positive
n Fn Fn λ
(or matrices of positive) operators is positive, Φ is a conditional expectation onto Fn.
n
To see that Φ is faithful, let T ∈ O be positive with T (cid:54)= 0. Therefore there exists a state ϕ on O
n n n
such that ϕ(T) > 0. Since ρ (T) = T, ρ (T) ≥ 0 for all λ, and λ (cid:55)→ ρ (T) is continuous, the function
1 λ λ
λ (cid:55)→ ϕ(ρ (T)) is a continuous function from T to [0,∞) that is strictly positive at 1. Hence standard
λ
integration theory implies
(cid:90)
φ(Φ (T))= φ(ρ (T))dλ>0
n λ
T
so Φ (T)(cid:54)=0. Hence Φ is faithful.
n n
To prove that O is simple, the above conditional expectation will need to be examined further. To
n
begin, we need a technical lemma.
Lemma 1.10. Let n ≥ 2 or n = ∞. Let µ and ν be words in {1,2,...,n} such that |µ| (cid:54)= |ν|. Let
m≥max{|µ|,|ν|} and let S =SmS . Then S∗(S∗S )S =0.
γ 1 2 γ µ ν γ
4
Proof. Since |µ| (cid:54)= |ν|, Lemma 1.5 implies that if S∗S (cid:54)= 0, then either S∗S = S∗ where µ(cid:48) is a word of
µ ν µ ν µ(cid:48)
length at least one and at most m or S∗S = S where ν(cid:48) is a word of length at least one and at most m.
µ ν ν(cid:48)
In the first case, (S∗S )S =S∗ S is non-zero only if S =S|µ(cid:48)| as |µ(cid:48)|≤m. However, if S =S|µ(cid:48)| then
µ ν γ µ(cid:48) γ µ(cid:48) 1 µ(cid:48) 1
S∗(S∗S )S =S∗(S∗)|µ(cid:48)|S =S∗(S∗)mSm−|µ(cid:48)|S =0
γ µ ν γ γ 1 γ 2 1 1 2
as S∗S =0.
1 2
In the second case, S∗(S∗S )=S∗S is non-zero only if S =S|ν(cid:48)| as |ν(cid:48)|≤m. However, if S =S|ν(cid:48)|
γ µ ν γ ν(cid:48) ν(cid:48) 1 ν(cid:48) 1
then
S∗(S∗S )S =S∗(S )|ν(cid:48)|S =S∗(S∗)m−|ν(cid:48)|SmS =0
γ µ ν γ γ 1 γ 2 1 1 2
as S∗S =0.
2 1
Theorem 1.11. Let n ≥ 2. For each m ∈ N there exists an isometry W ∈ O that commutes with Fn
n,m n m
such that Φ (T)=W∗ TW ∈Fn for all
n n,m n,m m
T ∈span{S S∗ | |µ|,|ν|≤m,µ and ν are words with letters in {1,...,n}}.
µ ν
Proof. Let S = SmS and let W = (cid:80) S S S∗. We claim that W is an isometry. To see this,
γ 1 2 n,m |δ|=m δ γ δ n,m
we notice that
(cid:88) (cid:88) (cid:88)
W∗ W = S S∗S∗S S S∗ = S S∗S S∗ = S S∗ =I
n,m n,m (cid:15) γ (cid:15) δ γ δ δ γ γ δ δ δ
|(cid:15)|=|δ|=m |δ|=m |δ|=m
where (cid:80) S S∗ =I comes from the fact that (cid:80)n S S∗ =I, by dividing the sum into all S that start
|δ|=m δ δ i=1 i i δ
with the same m−1 letters, using the identity to decrease the length of the words, and repeating.
To see that W commutes with Fn (and to begin to obtain the other equality), we notice that if µ is
n,m m
a word of length m then
W S =S S and S∗W =S S∗
n,m µ µ γ µ n,m γ µ
Therefore, if S S∗ is one of the matrix units for Fn (so |µ|=|ν|=m) then
µ ν m
W S S∗ =S S S∗ =S S∗W
n,m µ ν µ γ ν µ ν n,m
Hence W must commute with Fn. Moreover, from the above computation, the fact that W is an
n,m m n,m
isometry,andourknowledgeofΦ fromTheorem1.8,weobtainthatW∗ S S W =S S =Φ (S S∗).
n n,m µ ν n,m µ ν n µ ν
Next notice that if µ and ν are words with letters in {1,...,n} of length at most m with |µ|=(cid:54) |ν|, then
(cid:88)
W∗ S S∗W = S S∗S∗S S∗S S S∗ =0=Φ (S S∗)
n,m µ ν n,m δ γ δ µ ν (cid:15) γ (cid:15) n µ ν
|(cid:15)|=|δ|=m
as if S∗S S∗S is non-zero, it can be written as S∗ S with |µ(cid:48)| = m − |µ| (cid:54)= m − |ν| = |ν(cid:48)| and so
δ µ ν (cid:15) µ(cid:48) ν
S∗S∗S S∗S S =S∗S∗ S S =0 by Lemma 1.9. Hence the result follows.
γ δ µ ν (cid:15) γ γ µ(cid:48) ν(cid:48) γ
Using the above proof, it is easy to prove the following for O .
∞
Theorem 1.12. Let n ≥ 2. For each m ∈ N there exists an isometry W(cid:48) ∈ O such that Φ (T) =
n,m ∞ ∞
(W(cid:48) )∗TW(cid:48) ∈Fn ⊆O for all
n,m n,m m ∞
T ∈span{S S∗ | |µ|,|ν|≤m,µ and ν are words with letters in {1,...,n}}
µ ν
Using the above isometries and some clever tricks, we are finally able to prove the following.
Theorem 1.13. Let n≥2. If X ∈O is non-zero then there exists A,B ∈O such that AXB =I.
n n
5
Proof. Since X (cid:54)= 0, X∗X (cid:54)= 0 and thus Φ (X∗X) (cid:54)= 0 as Φ is faithful. Hence we may assume without
n n
loss of generality that (cid:107)Φ (X∗X)(cid:107) = 1. By density, we can choose Y in the algebraic span of elements of
n
the form S S∗ such that (cid:107)X∗X−Y(cid:107) < 1. By considering the real part of Y, we may assume that Y is
µ ν 4
self-adjoint. Thus (cid:107)Φ (X∗X)−Φ (Y)(cid:107)≤ 1 so (cid:107)Φ (Y)(cid:107)≥ 3.
n n 4 n 4
Since Y is in the algebraic span of elements of the form S S∗, there exists an m ∈ N such that Y is a
µ ν
linearcombinationofelementsoftheformS S∗ where|µ|,|ν|≤m. Therefore,byTheorem1.10,thereexists
µ ν
an isometry W such that Φ (Y) = W∗ YW ∈ Fn. Since (cid:107)Φ (Y)(cid:107) ≥ 3 and Φ (Y) is a self-adjoint
n,m n n,m n,m m n 4 n
element of a matrix algebra, there exists a rank one projection P ∈Fn such that
m
3
PΦ (Y)=Φ (Y)P =(cid:107)Φ (Y)(cid:107)P ≥ P
n n n 4
Moreover, sinceP andSm(S∗)m arebothrankoneprojectionsinFn, thereexistsanisometryU ∈Fn such
1 1 m m
that UPU∗ =Sm(S∗)m.
1 1
Finally, let
1
Z := (S∗)mUPW∗ ∈O .
1 1 n,m n
(cid:107)Φn(Y)(cid:107)2
Then
(cid:107)Z(cid:107)≤ 1 (cid:107)S∗(cid:107)m(cid:107)U(cid:107)(cid:107)P(cid:107)(cid:13)(cid:13)W∗ (cid:13)(cid:13)≤ √2
(cid:107)Φn(Y)(cid:107)12 1 n,m 3
(as S , U, and W are isometries and P is a projection) and
1 n,m
1
ZYZ∗ = (S∗)mUPW∗ YW PU∗Sm =(S∗)mUPU∗Sm =(S∗)mSm(S∗)mSm =I.
(cid:107)Φ (Y)(cid:107) 1 n,m n,m 1 1 1 1 1 1 1
n
Hence
41 1
(cid:107)I−ZX∗XZ∗(cid:107)=(cid:107)Z(Y −X∗X)Z∗(cid:107)≤(cid:107)Z(cid:107)2(cid:107)Y −X∗X(cid:107)≤ =
34 3
so ZX∗XZ∗ is a self-adjoint, invertible operator.
Let B =Z∗(ZX∗XZ∗)−21. Then
(B∗X∗)XB =(ZX∗XZ∗)−12ZX∗XZ∗(ZX∗XZ∗)−12 =I
as desired.
If we follow the above proof with n = ∞, we notice at the step where Y is chosen that we can bound
the number of letters used in the words in the algebraic expression for Y as Y is a finite sum of operators of
the form S S∗. Therefore, by applying Theorem 1.11, we see that the remainder of the proof follows (with
µ ν
W replaced with W(cid:48) ). Hence we obtain the following.
n,m n,m
Theorem 1.14. If X ∈O is non-zero, then there exists A,B ∈O such that AXB =I.
∞ ∞
Using the above theorems, we easily obtain the following result.
Theorem 1.15. O and O are simple for all n≥2. Moreover, if T ,...,T ∈B(H) are isometries such
∞ n 1 n
that (cid:80)n T T∗ = I, then C∗(T ,...,T ) (cid:39) O . In addition, if {T }∞ ∈ B(H) are isometries such that
i=1 i i 1 n n i i=1
(cid:80)n T T∗ ≤I for all n∈N, then C∗({T }∞ )(cid:39)O .
i=1 i i i i=1 ∞
Proof. The proof that the C∗-algebras are simple is trivial.
If T ,...,T ∈ B(H) are isometries such that (cid:80)n T T∗ = I, then, by the universal property of the
1 n i=1 i i
Cuntz algebra, there exists a ∗-homomorphism π :O →C∗(T ,...,T ) such that π(S )=T . Clearly this
n 1 n i i
implies that π is surjective. Since O is simple and π is not the zero map, π must be injective.
n
The O proof is similar.
∞
6
With the above result in hand, we can prove that if A is a C∗-algebra generated by n isometries, then A
is either O or a quotient of A is isomorphic to O .
n n
Lemma 1.16. LetAbeaC∗-algebrageneratedbynisometriesS ,S ,...,S suchthat(cid:80)n S S∗ =P <I.
1 2 n i=1 i i
Then the ideal (cid:104)I−P(cid:105) generated by I−P is isomorphic to the compact operators and A/K(cid:39)O .
n
Proof. Since P contains the range of each S , (I −P)S = 0 = S∗(I −P) for all i. Therefore, since it is
i i i
trivial to see that Lemma 1.5 applies to A, we obtain that (cid:104)I−P(cid:105) has
span{S (I−P)S∗ | |µ|<∞,|ν|<∞}
µ ν
as a dense subset. Moreover, it is trivial to verify that
(cid:26) 0 if ν (cid:54)=µ(cid:48)
(S (I−P)S∗)(S (I−P)S∗)=
µ ν µ(cid:48) ν(cid:48) S (I−P)S∗ if ν =µ(cid:48)
µ ν(cid:48)
and thus {S (I −P)S∗} forms an infinite collection of matrix units whose span is dense in (cid:104)I −P(cid:105). Hence
µ ν
(cid:104)I−P(cid:105)(cid:39)K as claimed.
To see that A/(cid:104)I −P(cid:105) (cid:39) O , we notice that if π : A → A/K is the canonical quotient map, then π(S )
n i
are isometries in A/K such that
n
(cid:88)
π(S )π(S )∗ =π(P)=π(P)+π(I−P)=π(I)
i i
i=1
which is the unit of A/K. Hence, as A/K is generated by π(S ), we obtain that A/K(cid:39)O as claimed.
i n
Remarks 1.17. Notice that the above result implies that O contains a C∗-subalgebra A such that A/K(cid:39)
m
O for all m>n≥2 . Similarly, for all n≥2, O contains a C∗-subalgebra A such that A/K(cid:39)O for all
n ∞ n
n≥2.
Our next goal is to show that each O and O are nuclear C∗-algebras. The idea behind the proof is to
n ∞
construct a C∗-algebra B that is the reduced cross product of a nuclear C∗-algebra A by the integers and
show that O is isomorphic to a compression of this cross product C∗-algebra. We remark that the reduced
n
crossproductofanuclearC∗-algebrabytheintegersisnuclear(seeChapter4of[BO]forthisproofandthe
constructionofthereducedcrossproduct. TheideaoftheproofofnuclearityistocompressBbyprojections
corresponding to finite subsets of Z. This operation is a completely positive map into A⊗ M (C) where
min n
n is the number of elements of the finite subset of Z. Then a completely positive map is constructed from
A⊗ M (C) to B that asymptotically does the right thing as long as Følner sets are taken for the finite
min n
subsets of Z. Then B is nuclear as each A⊗ M (C) is nuclear. This also can be used to show that the
min n
reduced cross product is the same as the full cross product) and the compression of a nuclear C∗-algebra
is nuclear (as if C ⊆ D are nuclear and there is a conditional expectation of D onto C, then C must be
nuclear by elementary arguments). To begin this proof, we start with a fixed n≥2 as we will deal with O
∞
separately.
Notation 1.18. For all j ∈ Z let A = ⊗∞ M (C) (where this means the closure of all operators of the
j i=j n
form A ⊗···⊗A ⊗I⊗I⊗··· with respect to the infinite tensor norm). Then A (cid:39)Fn for all j.
j m j
Construction 1.19. With the notation as above, we have a canonical sequence of embeddings
···(cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→···
3 2 1 0 −1 −2
where the inclusion A (cid:44)→A is given by X (cid:55)→E ⊗X ∈M (C)⊗ A (cid:39)A (where {E } are the
j j−1 1,1 n min j j−1 i,j
canonical matrix units of M (C)). Let B be the C∗-algebra that is the direct limit of this chain. Hence
n
B is an inductive limit of AF C∗-algebras and thus B is AF. In fact B (cid:39) K ⊗ Fn (to see this, we
min
notice that the embeddings do not change the A (cid:39) Fn term and K = lim M (C) with the embeddings
0 → nk
M (C)(cid:44)→M (C) by T (cid:55)→T ⊕0 ⊕···⊕0 ). Therefore, since B is AF, B is nuclear.
nk nk+1 n n
7
Since each A is isomorphic, there is a canonical automorphism of B, which we will denote Ψ, given by
j
shifting the sequence to the left. Notice if T ∈A then Ψ(T)∈A is the operator T ∈A which is the
j j+1 j+1
operator E ⊗T in A .
1,1 j
Let C = B(cid:111) Z. Thus C is a nuclear C∗-algebra by the above discussion. Let U ∈ C be the unitary
Ψ
implementing Ψ (that is Ψ(X) = UXU∗ for all X ∈ B). Notice that C is the closure of all operator of the
form
N
(cid:88)
A= T Ui
i
i=−N
whereT ∈BandN ∈N. BylettingT˜ =U−iT Ui (fori<0),weobtainthatCistheclosureofalloperator
i i i
of the form
A=(cid:88)UiT˜ +T +(cid:88)T Ui
i 0 i
i<0 i>0
where T˜ ∈B.
i
Let P ∈A be the unit. Therefore P ∈C is a projection. Notice that
0
UPU∗ =Ψ(P)=E ⊗P ∈A .
1,1 0
Hence UPU∗ = P(UPU∗) (as P is the unit for A ) and thus UP = PUP as U∗ is invertible. Therefore it
0
is easy to see that
PT UiP =(PT P)(UP)i for i>0 and PUiT˜P =(UP)∗PT˜P for i<0.
i i i i
Let V =UP. Thus
PAP =(cid:88)ViPT˜P +PT P +(cid:88)PT PVi.
i 0 i
i<0 i>0
Let E = PCP. Thus the above computations show that E is generated by PBP = A (think about it!)
0
with V. Moreover E is nuclear being the compression of a nuclear C∗-algebra. Our goal is to show that
E (cid:39) O . To show this, it suffices by Theorem 1.14 to construct n isometries in E that generate E with the
n
desired properties.
Theorem 1.20. With n and E as above, E(cid:39)O so O is nuclear when n≥2.
n n
Proof. Let S =(E ⊗P)V (where E ⊗P ∈A ) for i∈{1,...,n}. It suffices to show that each S is an
i i,1 i,1 0 i
isometry, (cid:80)n S S∗ =P, and E=C∗(S ,...,S ). To begin, we notice that
i=1 i i 1 n
S∗S =PU∗(E ⊗P)UP =PΨ−1(E ⊗P)P =PPP =P
i i 1,1 1,1
(where any elements and tensors are viewed in A ). Hence each S is an isometry. Moreover
0 i
S S∗ =(E ⊗P)UPPU∗(E ⊗P)=(E ⊗P)Ψ(P)(E ⊗P)=(E ⊗P)(E ⊗P)(E ⊗P)=E ⊗P.
j i j,1 1,i j,1 1,i j,1 1,1 1,i i,j
Thus
n n
(cid:88) (cid:88)
S S∗ = E ⊗P =I⊗P =P.
i i i,i
i=1 i=1
Thus it remains only to show that C∗(S ,...,S )=E. Since A and V generate E, it suffices to show that
1 n 0
A ∪{V}⊆C∗(S ,...,S ).
0 1 n
To see that A ⊆ C∗(S ,...,S ), we notice that A = ⊗∞ M (C) = M (C)⊗k ⊗A . Thus, a little
0 1 n 0 i=0 n n k
thought shows that
(cid:40) (cid:41)
(cid:91)
span {E ⊗···⊗E ⊗P | P the unit of A }
j1,i1 jk,ik k
k>0
8
is dense in A . To show that the above span is in C∗(S ,...,S ), we recall that S S∗ =E ⊗P and
0 1 n i j i,j
S (E ⊗P)S∗ = (E ⊗P)(UP(E ⊗P)PU∗)(E ⊗P)
k i,j (cid:96) k,1 i,j 1,(cid:96)
= (E ⊗P)(U(E ⊗P)U∗)(E ⊗P)
k,1 i,j 1,(cid:96)
= (E ⊗P)(E ⊗(E ⊗P))(E ⊗P)
k,1 1,1 i,j 1,(cid:96)
= E ⊗(E ⊗P)=E ⊗E ⊗P
k,(cid:96) i,j k,(cid:96) i,j
Thus, by repeating the above arguments, we see that if µ=(j ,...,j ) and ν =(i ,...,i ) then
1 k 1 k
S S∗ =E ⊗···⊗E ⊗P
µ ν j1,i1 jk,ik
and thus A ∈C∗(S ,...,S ).
0 1 n
Finally, to see that V ∈C∗(S ,...,S ), we notice that
1 n
VV∗ =UPU∗ ∈A .
0
Thus
V =UP =UP(P)P =UPU∗(E ⊗P)UP =VV∗(S )∈A ·S ∈C∗(S ,...,S )
1,1 1 0 1 1 n
as desired.
Thus E=C∗(S ,...,S )(cid:39)O so O is nuclear.
1 n n n
To prove that O is also nuclear, we will only sketch the differences that need to be taken and the proof
∞
will follow by similar arguments to those shown above.
Theorem 1.21. O is nuclear.
∞
Proof. Foreachj ∈N∪{0}letA =SjF∞(S∗)j ⊆O . ThenitisclearthatA (cid:39)A =F∞ forallj ≥0(by
j 1 1 ∞ j 0
the ∗-homomorphism T (cid:55)→ (S∗)jT(S )j). Moreover it is not difficult to see (but perhaps slightly annoying
1 1
to write down) that A (cid:39)CI+(K⊗ A ) where the CI comes from Sj−1I(S∗)j−1 ∈A and
j−1 min j 1 1 j−1
Sj−1(S S ···S S∗ ···S∗ S∗ )(S∗)j−1
1 i1 i2 ik jk j2 j1 1
corresponds to the operator
E ⊗(Sj(S ···S S∗ ···S∗ )(S∗)j)
i1,j1 1 i2 ik jk j2 1
in K⊗ A .
min j
Next we extend our notation by letting A = CI +(K⊗ A ) for all j ∈ Z. Then we can consider
j−1 min j
the sequence of C∗-algebras
···(cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→A (cid:44)→···
3 2 1 0 −1 −2
wheretheinclusionA (cid:44)→A isgivenbyX (cid:55)→E ⊗X ∈K⊗ A ⊆A (where{E }∞ arematrix
j j−1 1,1 min j j−1 i,j i,j=1
units for K). Let B be the C∗-algebra that is the direct limit of this chain. Since each A is AF, it is clear
j
that B is AF and thus nuclear. Since each A is isomorphic, let Ψ be the automorphism of B given by
j
shifting to the left. The remainder of the proof follows as in the O case.
n
To conclude this section, we desire to draw a relation between the various Cuntz algebras and show that
the matrix algebras of certain Cuntz algebras are Cuntz algebras. We will show that certain Cuntz algebras
embed into others and that O embeds into each O .
∞ n
Theorem 1.22. O can be unitarily embedded into O for all k ≥1. Moreover O can be embedded
k(n−1)+1 n ∞
in O for all n≥2.
n
9
Proof. Fix n≥2 and k ≥1. If k =1 then k(n−1)+1=n so O sits inside O . Otherwise suppose
k(n−1)+1 n
k ≥2. Let {S ,...,S } be the generators for O . Let
1 n n
X :={S(cid:96)S } ∪{Sk}.
n m 1≤m≤n−1,0≤(cid:96)≤k−1 n
Thus|X|=k(n−1)+1. Notice(S(cid:96)S )∗(S(cid:96)S )=I and(Sk)∗(Sk)=I forall(cid:96),minourranges. Moreover
n m n m n n
n−1k−1 n−1 n−2k−1
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
Sk(S∗)k+ S(cid:96)S S∗(S∗)(cid:96) = Sk(S∗)k+ Sk−1S S∗(S∗)k−1+ S(cid:96)S S∗(S∗)(cid:96)
n n n m m n n n n m m n n m m n
m=1(cid:96)=0 m=1 m=1(cid:96)=0
(cid:32) n (cid:33) n−1k−2
(cid:88) (cid:88) (cid:88)
= Sk−1 S S∗ (S∗)k−1+ S(cid:96)S S∗(S∗)(cid:96)
n m m n n m m n
m=1 m=1(cid:96)=0
n−1k−2
(cid:88) (cid:88)
= Sk−1(S∗)k−1+ S(cid:96)S S∗(S∗)(cid:96)
n n n m m n
m=1(cid:96)=0
.
.
.
n−1 1
(cid:88) (cid:88)
= S2(S∗)2+ S(cid:96)S S∗(S∗)(cid:96)
n n n m m n
m=1(cid:96)=0
n−1 n−1
(cid:88) (cid:88)
= S2(S∗)2+ S S S∗S∗+ S S∗
n n n m m n m m
m=1 m=1
(cid:32) n (cid:33) n−1
(cid:88) (cid:88)
= S S S∗ S∗+ S S∗
n m m n m m
m=1 m=1
n−1
(cid:88)
= S S∗+ S S∗ =I.
n n m m
m=1
Whence X generates a copy of O inside O as desired.
k(n−1)+1 n
Let S and S be two of the generators for O . Let X = {S(cid:96)S } . Notice (S(cid:96)S )∗(S(cid:96)S ) = I for all
1 2 n 1 2 (cid:96)≥0 1 2 1 2
(cid:96)≥0. Moreover (S(cid:96)S )∗(SkS )=0 if (cid:96)(cid:54)=k. Therefore {(S(cid:96)S )(S(cid:96)S )∗} are projections with orthogonal
1 2 1 2 1 2 1 2 (cid:96)≥0
ranges (as S(cid:96)S is an isometry) so (cid:80)n (S(cid:96)S )(S(cid:96)S )∗ ≤ I for all n ≥ 0. Whence X generates a copy of
1 2 (cid:96)=0 1 2 1 2
O inside O as desired.
∞ n
The following result is our first result that shows the matrix algebras of some Cuntz algebra is a Cuntz
algebra.
Proposition 1.23. If k divides n then M (O ) is isomorphic to O .
k n n
Proof. Suppose k divides n (n ≥ 2) and that O is generated by S ,...,S . Let {E } be the canonical
n 1 n i,j
matrix units for M (C) ⊆ M (O ). For 0 ≤ j < n and 1 ≤ i ≤ k, consider the operator T =
k k n k i,j
(cid:80)k(cid:96)=1Skj+(cid:96)Ei,(cid:96). We notice {Ti,j}0≤j<nk,1≤i≤k has k(cid:0)nk(cid:1)=n elements such that
(cid:32) k (cid:33)∗(cid:32) k (cid:33)
(cid:88) (cid:88)
T∗ T = S E S E
i,j i,j kj+m i,m kj+(cid:96) i,(cid:96)
m=1 (cid:96)=1
k
(cid:88)
= S∗ S E
kj+m kj+(cid:96) m,(cid:96)
m,(cid:96)=1
k
(cid:88)
= E =I
(cid:96),(cid:96) k
(cid:96)=1
10