Families of Type {\rm III KMS} States on a Class of $C^*$-Algebras containing $O_n$ and $\mathcal{Q}_\N$

0.43 MB·English

by  A. L. Carey

Families of Type III KMS States on a Class of C -Algebras containing O and
∗ n N
Q
A.L. Careya, J. Phillipsb,c, I.F. Putnamb, A. Renniea
a Mathematical Sciences Institute, Australian National University, Canberra, ACT, AUSTRALIA
b Department of Mathematics and Statistics, University of Victoria, Victoria, BC, CANADA
c Corresponding Author. email: [email protected]; telephone: [phone]; fax: [phone]
0
Abstract
1
0
We construct a family of purely infinite C -algebras, λ for λ (0,1) that are classified by their
2 ∗ Q ∈
K-groups. There is an action of the circle T with a unique KMS state ψ on each λ. For λ = 1/n,
n Q
a 1/n ∼= On, with its usual T action and KMS state. For λ = p/q, rational in lowest terms, λ ∼= On
Q Q
J (n = q p+1) with UHF fixed point algebra of type (pq) . For any n > 0, λ = O for infinitely
− ∞ Q ∼ n
4 many λ with distinct KMS states and UHF fixed-point algebras. For any λ (0,1), λ = O . For λ
irrational the fixed point algebras, are NOT AF and the λ are usually NO∈T CuntzQalg6ebra∞s. For λ
] Q
A transcendental, K1 ∼= K0 ∼= Z∞, so that λ is Cuntz’ N, [Cu1]. If λ±1 are both algebraic integers,
Q Q
O the only O which appear satisfy n 3(mod 4). For each λ, the representation of λ defined by the
n
≡ Q
h. KMS state ψ generates a type IIIλ factor. These algebras fit into the framework of modular index
(twisted cyclic) theory of [CPR2, CRT] and [CNNR].
t
a
m
Keywords: KMS state, III factor, modular index, twisted cyclic theory, K-Theory.
λ
[

1 AMS Classification codes: 46L80, 58J22, 58J30.
v
4
2
4 1. Introduction
0
.
1
In this paper we introduce some new examples of KMS states on a large class of purely infinite C -
0 ∗
algebras that were motivated by the ‘modular index theory’ of [CPR2, CNNR]. We were aiming to
0
1 find examples of algebras that were not Cuntz-Krieger algebras (or the CAR algebra) and were not
v: previously known in order to explore the possibilities opened by [CNNR]. These algebras, denoted by
i λ for 0 < λ < 1, are not constructed as graph algebras, but as “corner algebras” of certain crossed
X Q
product C -algebras. The λ have similar structural properties to the Cuntz algebras, however there
r ∗ Q
a are important new features, such as
1) when λ = p/q is rational in lowest terms, then λ = O as mentioned in the Abstract,
∼ q p+1
Q −
2) when λ is algebraic, the K-groups depend on the minimal polynomial (and its coefficients) of λ,
3) when λ is transcendental, λ ∼= N, Cuntz’ algebra, [Cu1].
Q Q
We prove in Section 3 that the λ are purely infinite, simple, separable, nuclear C -algebras, so there
∗
Q
is no nontrivial trace on them. Also in Section 3 we determine in many cases the K-groups of these
algebras and use classification theory to identify them when these algebras have the same K-groups
as others found previously (these facts are summarised in the Abstract). As each λ comes equipped
Q
with a gauge action of the circle, our results thus give an uncountable family of distinct circle actions
on , each with its own unique KMS state. Indeed, for all 0 < λ < 1, we find a unique KMS state,
N
Q
[BR2], for this gauge action, and we prove in Section 4 that the GNS representation of λ associated
Q
to our KMS state generates a type III von Neumann algebra. The result of [CPR2] that motivated
λ
this paper was the construction of a ‘modular spectral triple’ with which one may compute an index
pairing using the KMS state. In [CNNR] it was shown how modular spectral triples arise naturally for
1
2
KMS states of circle actions and lead to ‘twisted residue cocycles’ using a variation on the semifinite
residue cocycle of [CPRS2]. It is well known that such twisted cocycles can not pair with ordinary
K . In [CPR2, CRT] a substitute was introduced which is called ‘modular K ’. The correct definition
1 1
of modular K was found in [CNNR], and there is a general spectral flow formula which defines the
1
pairing of modular K with our ‘twisted residue cocycle’.
1
There is a strong analogy with the local index formula of noncommutative geometry in the 1, -
∞
L
summable case, however, there are important differences: the usual residue cocycle is replaced by
a twisted residue cocycle and the Dixmier trace arising in the standard situation is replaced by a
KMS-Dixmier functional. The common ground with [CPRS2] stems from the use of the spectral
flow formula of [CP2] to derive the twisted residue cocycle and this has the corollary that we have a
homotopy invariant. To illustrate the theory for these examples we compute, for particular modular
unitariesinmatrixalgebrasoverthealgebras λ, theprecisenumericalvaluesarisingfromthegeneral
Q
formalism.
2. The algebras λ for 0 < λ < 1.
Q
2.1. The C -algebras C (Γ ) = C(Γˆ ) and their K-theory. We will construct our algebras λ
∗ ∗ λ λ
Q
as “corner” algebras in certain crossed product C -algebras but first we need some preliminaries. For
∗
0 < λ < 1, let Γ be the countable additive abelian subgroup of R defined by:
λ
k=N
Γ = n λk N 0 and n Z .
λ k k
( (cid:12) ≥ ∈ )
k=X−N (cid:12)(cid:12)
Loosely speaking, Γ consists of Laurent polyno(cid:12)mials in λ and λ 1 with integer coefficients. It is not
λ (cid:12) −
only a dense subgroup of R, but is clearly a unital subring of R.
Proposition 2.1. Let 0 < λ < 1.
(1) If λ = p/q where 0 < p < q are integers in lowest terms, then Γ = Z[1/n], where n = pq.
λ
(2) If λ and λ 1 are both algebraic integers, then Γ = Z+Zλ+ +Zλd 1 is an internal direct
− λ −
···
sum where d 2 is the degree of the minimal (monic) polynomial in Z[x] satisfied by λ.
≥
(3) If λ is transcendental then, Γ = Zλk is an internal direct sum.
λ k Z
(4) If λ = 1/√n with n 2 a square-fr∈ee positive integer, then Γ = Z[1/n] + Z[1/n] √n is an
λ
≥ L ·
internal direct sum.
(5) In general, if λ is algebraic with minimal polynomial, nλd+ +m = 0 over Z, then
···
1 1 1
Z Zλ Zλd 1 Γ Z[ ] Z[ ]λ Z[ ]λd 1.
− λ −
⊕ ⊕···⊕ ⊆ ⊆ mn ⊕ mn ⊕···⊕ mn
Hence, rank(Γ ) := dim (Γ Q) = d.
λ Q λ Z
⊗
Proof. In case (1), since gcd(p,q) = 1, there exist a,b Z so that 1 = ap+bq. Therefore, 1 = ap+bq =
∈ q q
aλ+b Γ ; and similarly, 1 Γ . Since, Γ is a commutative ring, for any k,m Z with k 1 we
∈ λ p ∈ λ λ ∈ ≥
have: m = m is in Γ . That is, Z[1/n] Γ . On the other hand, for k 1 we have
nk (pq)k λ ⊆ λ ≥
pk 1 1 qk 1 1
λk = = p2k = p2k Z[1/n] and λ k = = q2k = q2k Z[1/n].
−
qk (pq)k nk ∈ pk (pq)k nk ∈
That is, Z[1/n] = Γ .
λ
In case (2), it is not hard to see the minimal polynomial of λ in Z[x] is not only monic, but also
has constant term = 1; say, p(λ) = λd +aλd 1 + 1 = 0. Clearly, λ Z+Zλ+ +Zλd 1.
− −
± ···± ∈ ···
Since λ 1p(λ) = 0, we also have λ 1 Z+Zλ+ +Zλd 1. By an easy induction, we have λk
− − −
∈ ··· ∈
3
Z+Zλ+ +Zλd 1, for all k Z. Hence, Γ = Z+Zλ+ +Zλd 1. The sum is direct by the
− λ −
··· ∈ ···
minimality of the degree of the minimal polynomial.
In case (3) the sum is direct because if λ satisfied a Laurent polynomial over Z, then by multipling by
a high power of λ it would also satisfy a genuine polynomial over Z.
Case (4) is an easy calculation which we leave to the reader. Case (5) is proved by similar methods
used in case (2). Again, the sum Z[ 1 ]+Z[ 1 ]λ+ +Z[ 1 ]λd 1 is direct by the minimality of the
degree of the minimal polynomial. mn mn ··· mn − (cid:3)
Proposition 2.2. Let 0 < λ < 1.
(1) If λ = p/q is rational in lowest terms so that Γ = Z[1/n], where n = pq, then
λ
K (C(Γˆ )) = Z[1 ] and K (C(Γˆ )) = Z[1/n].
0 λ Γˆλ) 1 λ
(2) If λ and λ 1 are both algebraic integers, so that Γ = Z+Zλ+ +Zλd 1 is an internal direct
− λ −
···
sum as above, then
even d k odd d k
K (C(Γˆ )) = (Γ ) = (Γ ) and K (C(Γˆ )) = (Γ ) = (Γ ).
0 λ λ λ 1 λ λ λ
k=0,keven k=1,kodd
^ M ^ ^ M ^
(3) If λ is transcendental then,
even k odd k
∞ ∞
K (C(Γˆ )) = (Γ ) = (Γ ) and K (C(Γˆ )) = (Γ ) = (Γ ).
0 λ λ λ 1 λ λ λ
k=0,keven k=1,kodd
^ M ^ ^ M ^
(4) If λ = 1/√n with n 2 a square-free positive integer, then
≥
K (C(Γˆ )) = Z Z[1/n] and K (C(Γˆ )) = Z[1/n] Z[1/n]
0 λ ∼ 1 λ ∼
⊕ ⊕
(5) In general, if λ is algebraic with nλd+ +m = 0 over Z then the composition of the inclusions
···
1 1 1
Z Zλ Zλd 1 Γ Z[ ] Z[ ]λ Z[ ]λd 1
− λ −
⊕ ⊕···⊕ ⊆ ⊆ mn ⊕ mn ⊕···⊕ mn
induces an inclusion on K-Theory, so that both of the following maps are one-to-one
even odd
(Zd) ∼= K0(C∗(Z Zλd−1)) ֒ K0(C(Γˆλ)) and (Zd) ∼= K1(C∗(Z Zλd−1)) ֒ K1(C(Γˆλ)).
⊕··· → ⊕··· →
^ ^
Proof. In case (1), Γ = limZ where each map is multiplication by n, so that Γˆ = limT. Since
λ λ
K (C(T)) = Z[1] is generat−e→d by multiples of the trivial rank one bundle, the maps in the←d−irect limit
0
K (C(Γˆ ) = limK (C(T)) are the identity map in each case, so that K (C(Γˆ )) = Z[1]. On the other
0 λ 0 0 λ
hand, K (C(T−→)) is generated by the maps on C(T) z zk, and each map in the direct limit is the
1
7→
same map induced by z zn. Thus, K (C(Γˆ )) = Z[1/n].
1 λ
7→
Cases (2) and (3) are well-known facts about the K-Theory of tori.
Case (4): first one uses item (4) of the previous Proposition, then the proof of case (1) above in order
to apply Proposition 2.11 of [Sc]. The proof is finished off with the easily proved observation that
Z[1/n] Z[1/n] = Z[1/n].
⊗
Case (5) the composed embedding is just containment:
Z Zλ Zλd 1 Z[ 1 ] Z[ 1 ]λ Z[ 1 ]λd 1. Since we know that K (C (Z))
⊕ ⊕ ··· ⊕ − ⊆ mn ⊕ mn ⊕ ··· ⊕ mn − ∗ ∗ →
4
K (C (Z[1/mn])) is one-to-one (even an isomorphism after tensoring with Q), an application of C.
∗
∗
Schochet’s Ku¨nneth Theorem, [Sc] shows that the induced map on K-Theory:
1 1 1
K (C (Z Zλ Zλd 1)) K (C (Z[ ] Z[ ]λ Z[ ]λd 1))
∗ − ∗ −
∗ ⊕ ⊕···⊕ −→ ∗ mn ⊕ mn ⊕···⊕ mn
is one-to-one (even an isomorphism after tensoring with Q). (cid:3)
Corollary 2.3. If λ is algebraic with minimal polynomial of degree d so that rank(Γ ) = d then
λ
even odd
rank(K (C(Γˆ ))) = rank( (Zd)) = 2d 1 = rank( (Zd)) = rank(K (C(Γˆ ))).
0 λ − 1 λ
^ ^
Proof. For each N d 1, let Γ = Zλ N + +ZλN Γ . Then each Γ is a finitely generated
N − λ N
≥ − ··· ⊆
torsion free (and hence free abelian) subgroup of Γ . Moreover,
λ
1 1 1
Z Zλ Zλd 1 Γ Γ Z[ ] Z[ ]λ Z[ ]λd 1,
− N λ −
⊕ ⊕···⊕ ⊆ ⊆ ⊆ mn ⊕ mn ⊕···⊕ mn
so that by tensoring with Q the induced inclusions are all equalities, and hence all are Q-vector spaces
of dimension d. Since Γ is free abelian, Γ = Zd. Now,
N N ∼
even odd
K0(C∗(ΓN)) ∼= K0(C(Td)) ∼= (Zd) ∼= Z2d−1 and K1(C∗(ΓN)) ∼= K1(C(Td)) ∼= (Zd) ∼= Z2d−1.
So, each K (C (Γ )) Q is a^Q-vector space of dimension 2d 1 and the map: ^
i ∗ N Z −
⊗
K (C (Z Zλ Zλd 1)) Q K (C (Γ )) Q
∗ − Z ∗ N Z
∗ ⊕ ⊕···⊕ ⊗ −→ ∗ ⊗
is one-to-one and hence an isomorphism of Q-vector spaces. Since the corresponding isomorphism
onto K (C (Γ )) Q factors through K (C (Γ )) Q the maps
∗ N+1 Z ∗ N Z
∗ ⊗ ∗ ⊗
K (C (Γ )) Q K (C (Γ )) Q
∗ N Z ∗ N+1 Z
∗ ⊗ → ∗ ⊗
are all isomorphisms. Now, C (Γ ) = lim C (Γ ) and so K (C (Γ )) = lim K (C (Γ )), and
∗ λ N ∗ N i ∗ λ N i ∗ N
therefore,
Ki(C∗(Γλ)) ZQ = limKi(C∗(ΓN)) ZQ ∼= Q2d−1
⊗ N ⊗
for each i = 1,2. (cid:3)
Now, let G G0 be the following countable discrete groups of matrices:
λ ⊃ λ
λn a 1 a
G = a Γ , n Z G0 = a Γ .
λ 0 1 ∈ λ ∈ ⊃ λ 0 1 ∈ λ
(cid:26)(cid:18) (cid:19) (cid:12) (cid:27) (cid:26)(cid:18) (cid:19) (cid:12) (cid:27)
(cid:12) (cid:12)
Of course, G0λ is isomorphic to the(cid:12)(cid:12)additive group Γλ, and Gλ is semidirec(cid:12)(cid:12)t product of Z acting on
G0 = Γ . We let G act on R as an “ax+b” group, noting that the action leaves Γ invariant. That
λ ∼ λ λ λ
is,
λn a
for t R and g = G define g t := λnt+a.
∈ 0 1 ∈ λ ·
(cid:18) (cid:19)
Notation. For such an element g G we will use the notation g := [λn : a] in place of the matrix
λ
for g and g := det(g) = λn for the∈determinant of g. Note: G0 = g G g = 1 ⊳G .
| | λ { ∈ λ | | | } λ
We use this action on R to define the transpose action α of G on (R) :
λ ∞
L
α (f)(t) = f(g 1t) for f (R) and t R.
g − ∞
∈ L ∈
Now let Cλ(R) be the separable C -subalgebra of (R) generated by the countable family of pro-
0 ∗ L∞
jections where a,b Γ . That is,
[a,b) λ
X ∈
5
n
Cλ(R) = closure c c C; a ,b Γ .
0 ( kX[ak,bk) (cid:12) k ∈ k k ∈ λ)!
Xk=1 (cid:12)
(cid:12)
We observe that C0λ(R) is a commutative AF-algebra. (cid:12)(cid:12)Clearly, C0(R) ⊂ C0λ(R) and since αg(X[a,b)) =
both are invariant under the action α of G . We define the separable C -algebras Aλ Aλ
X[g(a),g(b)) λ ∗ ⊃ 0
as the crossed products:
Aλ = G ⋊ Cλ(R) = Z⋊(G0 ⋊ Cλ(R)) Aλ = G0 ⋊ Cλ(R).
λ α 0 λ α 0 ⊃ 0 λ α 0
Since G and G0 are amenable these equal the reduced crossed products by [Ped, Theorem 7.7.7 ].
λ λ
Let Cλ (R) denote the dense -subalgebra of Cλ(R) consisting of finite linear combinations of the
00 ∗ 0
generating projections, , and let Aλ l1(G ,Cλ(R)) Aλ denote the dense -subalgebra of Aλ
X[a,b) c ⊂ α λ 0 ⊂ ∗
consisting of finitely supported functions x : G Cλ (R). Similarly we define Aλ Aλ.
λ → 00 0,c ⊂ 0
Proposition 2.4. For any λ (0,1) Aλ and Aλ are in the bootstrap class N .
∈ 0 nuc
Proof. Since Aλ = Z⋊Aλ, it suffices to see that Aλ is in N . By the proof of the previous Corollary,
0 0 nuc
we can write Γ as an increasing union of finitely generated torsion-free abelian groups Γ which are
λ N
free abelian group of finite rank so that Aλ is the direct limit of crossed products of the separable
0
commutative C -algebra Cλ(R) by Zmi and hence is in N . (cid:3)
∗ 0 nuc
Notation: We remind the reader of the crossed product operations in our setting (Definition 7.6.1 of
[Ped]) together with some particular notations we use. To this end, let x,y l1(G ,Cλ(R)) then we
∈ α λ 0
have the product and adjoint formulas:
(x y)(g) = x(h)α (y(h 1g)) for g G ;
h − λ
· ∈
hX∈Gλ
x (g) = α ((x(g 1)) ) for g G .
∗ g − ∗ λ
∈
If x l1(G ,Cλ(R)) is supported on the single element g G and x(g) = f Cλ(R), then we write
∈ α λ 0 ∈ λ ∈ 0
x = f δ . Since Aλ (respectively, Aλ ) is dense in Aλ (respectively, Aλ) we often do our calculations
· g c 0,c 0
with these elements and we have the following easily verified calculus for them.
Lemma 2.5. Let f δ ,f δ ,f δ Aλ, then:
1· g1 2· g2 · g ∈ c
(1) (f δ ) (f δ ) = f α (f ) δ
(2) (f1·δgg)1∗ =· α2g−·1(gf2¯) δg−11.g1 2 · g1g2
· ·
(3) f δ is self-adjoint if and only if f is self-adjoint and g = 1.
g
·
(4) f δ is a projection if and only if f is a projection and g = 1.
g
·
(5) f δ is a partial isometry if and only if f is a projection.
g
· | |
(6) The product of partial isometries of the form δ is a partial isometry of the same form.
[a,b) g
X ·
(7) Consider the partial isometry, v = δ . Any two of the following: vv , v v, g completely
[a,b) g ∗ ∗
X ·
determine the interval [a,b) and the element g.
Definition2.6. Lete Aλ betheprojectione = δ .WedefinetheseparableunitalC -algebras:
∈ 0,c X[0,1)· 1 ∗
λ := eAλe eAλe =: Fλ.
Q ⊃ 0
We will also have occasion to use the dense subalgebras λ := eAλe, and Fλ := eAλ e.
Qc c c 0,c
Proposition 2.7. The orthogonal family of projections e = δ Aλ for n Z are mutually
n X[n,n+1)· 1 ∈ 0 ∈
equivalent by partial isometries in Aλ of the form V := δ where g = [1 : (n k)].
0 n,k X[n,n+1)· gn−k n−k −
6
tMhaotreover, the finite sums EN := Nn=−−1N en = X[−N,N) ·δ1 form an approximate identity for Aλ so
Aλ = Pλ (l2(Z)) and Aλ = Fλ (l2(Z)).
∼ Q ⊗K 0 ∼ ⊗K
Proof. By Lemma 2.5, one easily calculates that:
for each pair n,k Z, V V = e and V V = e .
∈ n,k n∗,k n n∗,k n,k k
Now for each positive integer N if we have y Aλ that satisfies supp(y ) [ N,N) for all h, then
∈ c h ⊆ −
using Lemma 2.5 again we see that E y = y. Since the collection of all such elements y Aλ is
N · ∈ c
dense in Aλ, we see that the increasing sequence of projections E form an approximate identity
N
{ }
for Aλ. (cid:3)
Corollary 2.8. It follows from Proposition 2.4.7 of [RS] and Proposition 2.4 that for any λ (0,1),
∈
λ and Fλ are both in N .
nuc
Q
Lemma 2.9. (cf. [PhR, Proposition 3.1, Lemma 3.6]) The algebra Cλ(R) is a commutative separable
0
AF algebra consisting of all functions f : R C which vanish at and: are right continuous at
→ ∞
each x Γ ; have a finite left-hand limit at each x Γ ; and are continuous at each x (R Γ ).
λ λ λ
∈ ∈ ∈ \
\
Moreover, if φ Cλ(R), (the space of all nonzero -homomorphisms: Cλ(R) C) then there exists a
∈ 0 ∗ 0 →
unique x R such that:
0
∈
(1) if x (R Γ ) then φ(f) = f(x ) for all f Cλ(R),
0 ∈ \ λ 0 ∈ 0
φ(f) = f(x ) for all f Cλ(R), or
(2) if x Γ then either 0 ∈ 0
0 ∈ λ (cid:26) φ(f) = f−(x0) = limx→x−0 f(x) for all f ∈ C0λ(R).
Proof. Since generating functions for Cλ(R) satisfy each of the properties above which are clearly
0
preserved by passing to uniform limits, we see that any function in Cλ(R) satisfies these properties.
0
Conversely, it is easy to show that any function satisfying these properties can be uniformly approxi-
mated by a finite linear combination of the generators. The remainder of the proof is given in [PhR,
Lemma 3.6]. (cid:3)
\
Notation. We denote the dual space, Cλ(R) by R and endow it with the relative weak- topology,
0 λ ∗
that is the topology of pointwise convergence on Cλ(R). Of course, R is a locally compact Hausdorff
0 λ
space, and Cλ(R) = C (R ).
0 ∼ 0 λ
Proposition 2.10. The algebras Aλ and Aλ (and hence λ and Fλ) are simple C -algebras. More-
0 Q ∗
over, Aλ is purely infinite and hence so is λ.
Q
Proof. Now, both G and G0 act on Cλ(R) as countable discrete groups of outer automorphisms.
λ λ 0
Thus, we can apply Theorem 3.2 of [E] once we check that neither action has any nontrivial invariant
ideals in Cλ(R) and that the actions are properly outer in the sense of Definition 2.1 of [E].
0
To do this we look at the induced action of G and G0 on R . So, for g G we have g acting on R
λ λ λ ∈ λ λ
via g(φ) = φ α 1 so that for φ = φ given by evaluation at x R, we have as expected g(φ ) = φ .
◦ g− x ∈ x g(x)
Now, if x Γ we use the notation φ to denote the -homomorphism φ (f) = f (x) = f(x ) =
∈ λ x− ∗ x− − −
lim f(y). One easily checks that since g(x) Γ , we have g(φ ) = φ .
y→x− ∈ λ x− g(x)−
Next we claim that each of the sets φ m Γ and φ m Γ is dense in R in the relative
{ m | ∈ λ} { m− | ∈ λ} λ
weak- topology. For example, we show that the second set is dense. To approximate φ for some
x
∗
x R we let m be a sequence in Γ converging to x from the right in R. Let f Cλ(R) so that f
∈ { n} λ ∈ 0
is right continuous at x. One easily shows that φ (f) φ (f) 0; that is, the sequence φ
| mn− − x | → { mn−}
converges to φ in the relative weak- topology.
x
∗
7
It is easy to see that the action of G0 on R has dense orbits, and so, of course, the action of G has
λ λ λ
dense orbits also. This implies that the actions of G0 and G on Cλ(R) have no nontrivial invariant
λ λ 0
ideals since the induced action on R has no nontrivial invariant closed sets. We complete the proof
λ
by showing that the action is properly outer in the sense of Definition 2.1 of [E]. Since there are
no nontrivial α-invariant ideals and Cλ(R) is commutative this is the condition that for each g = 1
0 6
and each nonzero closed two sided ideal invariant under α we have (α Id) = 2. Since is
nonzero there is a nonempty open subset,I of R so that ˆ =g . But skincegg−= 1|aInkd is not finIite
λ
O I O 6 O
there exists y such that g(y) = y and g(y) . Let x = g(y) so that g 1(x) = y and
−
∈ O 6 ∈ O ∈ O ∈ O
x = g 1(x). So we can choose a continuous compactly supported real-valued function f on with
−
6 O
f(x) = 1, f(g 1(x)) = 1 and f = 1. But then f and
−
− k k ∈ I
2 (α Id) (α Id)(f) = α (f) f f(g 1(x)) f(x) = 2.
≥ k g − |Ik| ≥ k g − k k g − k ≥ | − − |
Now that we know Aλ is simple, we can easily apply Theorem 9 of [LS] to conclude that Aλ satisfies
hypothesis(v)ofProposition4.1.1(page66)of[RS]. ForsimpleC -algebras,thisisequivalenttobeing
∗
purely infinite by Definition 4.1.2 of [RS]: the authors of [LS] had used one of the earlier definitions of
purely infinite in their paper (namely, hypothesis (v)). By Proposition 4.1.8 of [RS] λ is also purely
infinite. Q (cid:3)
Corollary 2.11. It follows from Corollaries 8.2.2 and 8.4.1 (Kirchberg-Phillips) of [RS] and the fact
that Aλ is stable that for any λ (0,1), Aλ is classified up to isomorphism (among Kirchberg algebras
in N ) by its K-theory. ∈
nuc
Since we need to calculate with elements of λ and Fλ, we make the following observations.
Q
Lemma 2.12. Now, λ (respectively, Fλ) is the norm closure of finite linear combinations of the ele-
Q
ments of the form e( δ )e, where g G (respectively, g G0), henceforth called the generators.
X[a,b)· g ∈ λ ∈ λ
Thus, we calculate
(1) If f δ Aλ, (respectively, f δ Aλ) where f Cλ(R), then
· g ∈ · g ∈ 0 ∈ 0
e(f δ )e = f δ where [a,b) = [0,1) [g(0),g(1)).
g [a,b) g
· X · ∩
(2) Thus, for g G , (respectively, g G0) f δ is in λ (respectively, Fλ) iff supp(f) [0,1)
∈ λ ∈ λ · g Q ⊆ ∩
[g(0),g(1)). In particular, for g G , (respectively, g G0) δ is in λ (respectively, Fλ) iff
∈ λ ∈ λ X[a,b) · g Q
[a,b) [0,1) [g(0),g(1)).
⊆ ∩
Proof. The first item is an easy calculation using part (1) of Lemma 2.5 and the fact that α ( ) =
g [a,b)
. The second item follows easily from the first. X (cid:3)
[g(a),g(b))
X
Proposition 2.13. If λ is rational, then Aλ and Fλ are AF-algebras. In particular, if λ = p/q where
0
0 < p < q are in lowest terms, then Fλ is the UHF algebra n where n = pq. Moreover, the minimal
∞
projections in the finite-dimensional subalgebras can all be chosen from the canonical commutative
subalgebra Cλ(R) δ .
0 · I
Proof. We have shown in Proposition 2.1 that if λ = p/q where 0 < p < q are in lowest terms, then
Γ = Z[1/n], where n = pq. Now, any element in Z[1/n] has the form m/nk = m(1/nk) where k 1.
λ
≥
Therefore any of the generating partial isometries δ Aλ can (by bringing a,b and c to a
X[a,b) · [1:c] ∈ 0
common denominator) be written (assuming c > 0) as a finite linear combination of partial isometries
oftheform δ .ForpartialisometriesinFλ wewouldhavetorestrict0 l nk 1
X[l/nk,(l+1)/nk)· [1:1/nk] ≤ ≤ −
and such partial isometries generate an nk by nk matrix subalgebra of Fλ. It should now be clear that
Fλ is a UHF algebra of type n . (cid:3)
∞
8
At this point we define some special elements in λ which behave very much like the isometries
Q
S O , except for the fact that some of them are not isometries.
µ n
∈
Definition 2.14. Fix 0 < λ < 1 and let k be a positive integer. Define m to be the unique positive
k
integer satisfying: m λk < 1 (m +1)λk. For 0 m m define partial isometries S λ
k k k k,m
≤ ≤ ≤ ∈ Q
via:
S = δ where g = [λk : mλk].
k,m X[mλk,(m+1)λk)· gk,m k,m
Note: for m < m the S are actually isometries, and S is an isometry iff 1 = (m +1)λk.
k k,m k,mk k
Remarks. Thedefininginequalitiesm λk < 1 (m +1)λk forthepositiveintegerm areequivalent
k k k
≤
to: 0 < λ k m 1. In particular, these differences are positive and bounded above by 1. In the
− k
− ≤
case of 1/n we have m = nk 1. Generally we have mk m < 1 (m +1) (m +1)k.
Q k − 1 ≤ k ≤ k ≤ 1
Lemma 2.15. With the previously defined elements we have:
Sk∗,m = X[0,1)·δgk−,1m and Sk∗,mk = X[0,λ−k−mk)·δgk−,1mk where for all m, gk−,1m = [λ−k : −m].
Moreover, for 0 ≤ m < mk, Sk∗,mSk,m = X[0,1)·δ1 = e while Sk∗,mkSk,mk = X[0,λ−k−mk)·δ1.
Finally, for 0 m < m , S S = δ while S S = δ , so that
≤ k k,m k∗,m X[mλk,(m+1)λk)· 1 k,mk k∗,mk X[mkλk,1)· 1
mk
S S = δ = e.
k,m k∗,m X[0,1)· 1
m=0
X
Proof. These are just straightforward calculations based on Lemma 2.5 which we leave to the reader.
(cid:3)
Theorem 2.16. For each λ with 0 < λ < 1, consider the partial isometries S for m = 0,1,...,m
1,m 1
where m λ < 1 (m +1)λ. For m < m , S is an isometry and m1 S S = 1. For λ = 1/n,
1 ≤ 1 1 1,m m=0 1,m 1∗,m
m = n 1, S is also an isometry, and 1/n = O , the usual Cuntz algebra.
1 − 1,m1 Q ∼ n P
Proof. The first statement is clear. With λ = 1/n we have inside 1/n, n isometries one for each
Q
m = 0,1,...,(n 1) defined by:
−
Sm = X[mn,mn+1)·δgm where gm = [1/n : m/n] and so Sm∗ = X[0,1)·δgm−1 where gm−1 = [n : −m].
Using Lemma 2.12, we easily see that for each m, S 1/n.
m
∈ Q
Then, using item (1) of Lemma 2.5 we calculate:
n 1
−
S S = δ = e and S S = δ and so S S = δ = e.
m∗ m X[0,1)· 1 m m∗ X[mn,mn+1)· 1 m m∗ X[0,1)· 1
m=0
X
Since e is the identity of 1/n, we have constructed a unital copy of O inside 1/n. Now one shows by
n
Q Q
induction that for each k > 0 the product of exactly k of these n isometries has the form S where
k,m
S has the same defining equation as S above but with nk in place of n and m = 0,1,...,(nk 1).
k,m m
−
These new isometries have range projections S S = δ which therefore lie in this copy
k,m k∗,m X[nmk,mn+k1)· 1
of O . By adding up some of these projections, we can get any projection of the form δ where
n [a,b) 1
X ·
0 a < b 1 and both a,b have the form m/nk. But any element a Γ can be written as a = m
≤ ≤ ∈ 1/n nk
for a sufficiently large k 0 and some m Z depending on k, and any pair a,b can be brought to a
≥ ∈
common denominator nk. Hence any projection of the form δ in 1/n is in this copy of O .
[a,b) 1 n
X · Q
9
Now, a straightforward calculation gives us:
nk 1 nk 1
− −
(1) S S = δ = δ O .
k,m k∗,m−1 X[mn+k1,mn+k2)· [1:1/nk] X[1/nk,1)· [1:1/nk] ∈ n
m=1 m=1
X X
Finally, let δ 1/n be an arbitrary generator. By taking adjoints if necessary we can assume
[a,b) g
X · ∈ Q
that g has the form g = [nk : ] where k 0. Since S is an isometry in O it suffices to prove that
k,0 n
∗ ≥
S ( δ ) O . That is, we are reduced to the case g = [1 : c] and again by taking adjoints
k,0 [a,b) g n
X · ∈
if necessary we can assume that c 0. The case c = 0 is done and so we can assume that c > 0. So
≥
(with possibly new a,b) we have δ where 0 < c 1 and [a,b) [0,1) [c,c+1) = [c,1).
[a,b) [1:c]
X · ≤ ⊆ ∩
But, δ = δ = δ δ and we already know that δ O .
[a,b) [1:c] [a,b) [c,1) [1:c] [a,b) 1 [c,1) [1:c] [a,b) 1 n
X · X X · X · X · X · ∈
Therefore it suffices to see that δ O . However, c = l/nk for some 0 < l < nk and so:
[c,1) [1:c] n
X · ∈
l
δ = δ = δ
X[c,1)· [1:c] X[l/nk,1)· [1:l/nk] X[1/nk,1)· [1:1/nk]
(cid:16) (cid:17)
which is in O by Equation 1. Since all generators for 1/n are in O we’re done. (cid:3)
n n
Q
2.2. K-Theory of λ for λ rational. : Since Aλ is stable and stably isomorphic to the UHF
Q 0
algebra F , each of its projections is equivalent to one in some finite-dimensional subalgebra and
λ
hence to some projection in Cλ(R), and in this case the trace induces an isomorphism from K (Aλ)
0 0 0
onto Γ = Z[1/(pq)] R. This isomorphism carries the projection e = δ which is the identity
λ [0,1) 1
⊂ X ·
of λ and Fλ onto 1 Z[1/(pq)]. Now, since Aλ is AF, K (Aλ) = 0 , and since Aλ = Z⋊ Aλ we
Q ∈ 0 1 0 { } λ 0
can use the Pimsner-Voiculescu exact sequence to calculate K (Aλ) = K ( λ). When we do this we
∗ ∗ Q
get:
K ( λ) = 0 , and K ( λ) = Z[1/(pq)]/(1 λ)Z[1/(pq)].
1 0
Q { } Q −
Proposition 2.17. For λ rational with λ = p/q in lowest terms, we have
K ( λ) = 0 , and K ( λ) = Z[1/(pq)]/(1 λ)Z[1/(pq)] = Z .
1 0 ∼ ∼ (q p)
Q { } Q − −
Proof. By Proposition 2.1, Γ = Z[1/(pq)], so we must show that
λ
Z[1/(pq)]/(1 (1/(pq))Z[1/(pq)] = Z .
∼ (q p)
− −
Since (q p) = (1 p/q)q and every element of Z[1/(pq)] is of the form m/(pq)N, it is easy to see that
− −
(q p)Z[1/(pq)] = (1 p/q)Z[1/(pq)]. Now, (q p) and (pq)N are relatively prime for any N and so
− − −
there exist a,b Z so that 1 = a(q p)+b(pq)N and hence m/(pq)N = (q p)am/(pq)N +mb. That
∈ − −
is, m/(pq)N and mb represent the same element in the quotient. So, every element in the quotient has
an integer representative. Two integers c,d represent the same element in the quotient if and only if
c d = (p q)n/(pq)N, or (c d)(pq)N = n(q p). But then:
− − − −
(c d) = (c d)[a(q p)+b(pq)N] = (c d)a(q p)+b(c d)(pq)N = [(c d)a+bn](q p).
− − − − − − − −
That is, c,d represent the same element in Z/(q p)Z = Z . On the other hand if (c d) is in
(q p)
(q p)Z then clearly, [c] = [d] in Z[1/(pq)]/(1 (1−/(pq))Z[1/(p−q)] and we are done. − (cid:3)
− −
Corollary 2.18. If λ = p/q in lowest terms, then
Fλ = Fp/q ∼= UHF((pq)∞) and λ = p/q ∼= O(q p+1).
Q Q −
In particular, if λ = k then
k+1
Fλ ∼= UHF((k(k+1))∞) and λ ∼= O2.
Q
10
Proof. Since each λ is separable, nuclear, simple, purely infinite and in the bootstrap category
nuc
Q N
once we show that the class of the identity e λ is a generator for K ( λ) = Z/(q p)Z, the
0
∈ Q Q −
Kirchberg-Phillips Classification Theorem, Theorem 8.4.1 of [RS], shows that λ = O . To this
∼ (q p+1)
end we observe that since e is mapped to 1 in Z[1/pq], we must show thatQ[1] is a−generator for
K ( λ) = Z[1/pq]/(1 (p/q))Z[1/pq]. Now, by the proof of the previous proposition, k[1] = [k 1] =
0
Q − ·
0 Z[1/pq]/(1 (p/q))Z[1/pq] if and only if [k 1] = 0 Z/(q p)Z if and only if k 0 = m(q p)
∈ − · ∈ − − −
for some m Z if and only if k is a multiple of (q p). That is, [1],[2 1],...,[(q p 1) 1] are all
∈ − · − − ·
nonzero in K ( λ) = Z/(q p)Z and hence [1] is a generator. (cid:3)
0
Q −
2.3. The K-Theory of the Algebras Aλ for λ irrational. The case λ rational is much simpler,
0
and while it does fit into the following scheme, it does not need this deeper machinery. Initially, we
(and others) believed that the algebras Aλ were AF algebras when λ is irrational. In fact we will
0
show that Aλ is never AF when λ is irrational. We will set up our examples to fit the situation on
0
page 1487 of [Put2] so that we can apply the six-term exact sequence of Theorem 2.1 on page 1489 of
[Put2].
We let Γ = Γ = G0. Thus, Γ R is a countable dense subgroup of R which acts on R by translations.
λ ∼ λ ⊂
Before looking at the crossed product of Γ acting on Cλ(R) = C (R ) (which gives us Aλ) we first
0 0 λ 0
consider the crossed product of Γ acting on C (R). Since Γ acts on R by translation we can Fourier
0
transform to get an isomorphism:
Γ⋊C (R) = Rˆ ⋊C(Γˆ).
0 ∼
Then, by Connes’ Thom isomorphism we get for i = 0,1:
K (Γ⋊C (R)) = K (Rˆ ⋊C(Γˆ)) = K (C(Γˆ)).
i 0 ∼ i ∼ i+1
Proposition 2.19. The composition:
K (C (R)) i∗ K (Γ⋊C (R)) b K (Rˆ ⋊C(Γˆ)) ∼= K (C(Γˆ))
1 0 1 0 1 0
−→ −→ −→
takes the generator [u] K (C (R)) = Z [u]; where u is the Bott element in C (R)1 defined by
1 0 0
∈ ·
u(t) = 1+it; to [1 ] where 1 is the identity function in C(Γˆ).
1 it Γˆ Γˆ
−
Proof. We first work on the right hand side of this sequence of maps. Let u(t) = 1+ε(t), then by the
proof of Connes’ Thom isomorphism from
K (C(Γˆ)) K (C (R)) K (R⋊C(Γˆ))
0 Z 1 0 1
⊗ −→
we see that [1 ] [u] gets mapped to the class [1+(convolution by εˆ 1 )]. Now in this displayed
Γˆ ⊗ · Γˆ
equation, K0(C(Γˆ))⊗ZK1(C0(R)) = K0(C(Γˆ))⊗ZZ·[u] = K0(C(Γˆ))·[u] ∼= K0(C(Γˆ)). Thus, [1Γˆ] in
K (C(Γˆ)) gets mapped to the class [1+(convolution by εˆ 1 )] by the Thom isomorphism.
0 · Γˆ
On the other hand, the map K ((C (R)1) K ((Γ⋊C (R))1) takes [u] [δ ε+1] and by the
1 0 1 0 0
−→ 7−→ ·
Fourier transform this goes to [(convolution by εˆ 1 )+1] in K (R⋊C(Γˆ)). Combining these we get:
· Γˆ 1
1 Z [u] Z [u] = K ((C (R))1) = K (C (R)) [1 ] K ((C(Γˆ)).
∈ 7−→ ∈ · 1 0 1 0 7−→ Γˆ ∈ 0
(cid:3)
Now, by Proposition 2.1 we know Γ in many cases so that these last groups are quite computable. In
the notation of [Put2] we define the transformation groupoids:
G := R ⋊Γ, G := R⋊Γ, and H := Γ⋊Γ.
λ ′
Then, Aλ = C (G) is the reduced C -algebra of G; Γ⋊C (R) = C (G) is the reduced C -algebra of
0 r∗ ∗ 0 r∗ ′ ∗
G; and (l2(Γ))isthereducedC -algebraofH. BytheproofofProposition2.10thereisacontinuous
′ ∗
K