Rohlin flows on the Cuntz algebra $O_\infty$

0.41 MB·English

by  Ola Bratteli

Rohlin flows on the Cuntz algebra O
∞
5
0
0
Ola Bratteli
2

n Department of Mathematics, University of Oslo
a
J Blindern, P.O.Box 1053, N-0316, Norway

8
Akitaka Kishimoto
1

] Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
A
and
O
Derek W. Robinson
.
h
t Centre for Mathematics and its Applications, Australian National University
a
m
Canberra, ACT 0200, Australia
[

1 January, 2005
v
4
6
2 Abstract
1
0 Itisshownthatcertainquasi-freeflowsontheCuntzalgebraO havetheRohlin
∞
5
property and therefore are cocycle-conjugate with each other. This, in particular,
0
/ shows that any unital separable nuclear purely infinite simple C∗-algebra has a
h
Rohlin flow.
t
a
m
: 1 Introduction
v
i
X
We are concerned here with Rohlin flows; a flow α on a unital C∗-algebra A is said to have
r
a
the Rohlin property (or to be a Rohlin flow) if for any p ∈ R there is a central sequence
(u ) in U(A), the unitary group of A, such that max kα (u )−eiptu k→0 as n→∞.
n |t|≤1 t n n
A major consequence of this property can be paraphrased as any α-cocycle is almost a
coboundary. This consequence, combined with enough information on U(A), may lead us
to a classification theory of Rohlin flows up to cocycle conjugacy. This is a goal we have
in mind (see [14, 17, 18, 20, 19]).
Since the property is rather stringent, it is not easy to present a Rohlin flow in general.
But we managed to give Rohlin flows on the Cuntz algebra O with n finite; moreover we
n
can identify the quasi-free flows which have the Rohlin property. In this paper we show
that certain quasi-free flows on O have the Rohlin property. Hence it follows that any
∞
unital C∗-algebra A with A ∼= A⊗O has Rohlin flows; the class of such A includes all
∞
unital separable nuclear purely infinite simple C∗-algebras, due to Kirchberg.
1
We have left some quasi-free flows O undecided whether they have the Rohlin prop-
∞
erty or not. But, as in [17], we show that all the Rohlin flows on O are cocycle conjugate
∞
with each other in the class of quasi-free flows. This is true for a wider class of flows. Not-
ing that there is a certain maximal abelian C∗-subalgebra C of O whose elements the
∞ ∞
quasi-free flows fix, we show that Rohlin flows are cocycle-conjugate in the class of flows
which are C1+ǫ on C (see below for details and note that our terminology of quasi-free
∞
flows is restrictive).
We will now describe the contents more precisely.
For each n = 2,3,... the Cuntz algebra O is generated by n isometries s ,s ,...,s
n 1 2 n
such that n s s∗ = 1. For n = ∞ the Cuntz algebra O is generated by a sequence
k=1 k k ∞
(s ,s ,...)Pof isometries such that n s s∗ ≤ 1 for all n. It is shown in [6] that O with
1 2 k=1 k k n
n = 2,3,... or n = ∞ is a simple pPurely infinite nuclear C∗-algebra.
For a finite (resp. infinite) sequence (p ,p ,...,p ) in R we define a flow α, called a
1 2 n
quasi-free flow, on O (resp. O ) by
n ∞
α (s ) = eipkts .
t k k
(In the case n = ∞, a more general flow can be induced by a unitary flow U on the
closed linear subspace H spanned by s ,s ,..., where the inner product h·,·i is given by
1 2
y∗x = hx,yi1, x,y ∈ H, if the generator of U is not diagonal. But we will exclude them
from the quasi-free flows in this paper.) It is known in [11, 21] that if p ,p ,...generate R
1 2
as a closed subsemigroup, then the crossed product O × R is simple and purely infinite
n α
(whether n is finite or infinite). It is also known in [17, 19] that if n is finite, the flow α
has the Rohlin property if and only if O × R is simple and purely infinite. For n = ∞,
∞ α
it is known in [11, 14] that if α has the Rohlin property then O × R is simple and
∞ α
purely infinite. In this paper we shall give a partial converse to this fact:
Theorem 1.1 Let (p ) be an infinite sequence in R such that p ,p ,...,p generate R
k 1 2 n
as a closed subsemigroup for some n. Then the quasi-free flow α on O defined by
∞
α (s ) = eipkts has the Rohlin property.
t k k
We shall prove that each α is α-invariantly approximately inner, i.e., for each t ∈
t
R there is a sequence (u ) in U(O ) such that α (x) = limAdu (x), x ∈ O and
n ∞ t n ∞
max kα (u )−u k→0. Then we would get the above theorem, by [18, 20], from the
s∈[0,1] s n n
fact that O × R is simple and purely infinite.
∞ α
Let E be the C∗-subalgebra of O = C∗(s ,s ,...) generated by s ,s ,...,s . Then
n ∞ 1 2 1 2 n
E is left invariant under α and the union E is dense in O . Hence, to prove the asser-
n n n ∞
tionintheprevious paragraph, it suffices tSoshow thatα|E isα-invariantlyapproximately
n
inner for all large n. Let us state formally:
Proposition 1.2 Let s ,s ,...,s be isometries such that
1 2 n
n
s s∗ (cid:8) 1
k k
X
k=1
2
and let E be the C∗-algebra generated by these s ,...,s . Let (p ,p ,...,p ) be a finite
n 1 n 1 2 n
sequence in R such that p ,...,p generate R as a closed subsemigroup and define a
1 n
quasi-free flow α on E by α (s ) = eipkts . Then each α is α-invariantly approximately
n t k k t
inner.
To prove this we use the following facts. Let J be the ideal of E generated by
n n
e0 = 1 − n s s∗. Then J is isomorphic to the C∗-algebra K of compact operators
n k=1 k k n
(on a sepaPrable infinite-dimensional Hilbert space) and is left invariant under α. The
quotient E /J is isomorphic to O by mapping s +J into s (the latter s ’s satisfy the
n n n k n k k
equality n s s∗ = 1 and generate O ). By the assumption on (p ) the induced flow
k=1 k k n k
α˙ on O Phas the Rohlin property [19], from which follows that each α˙ is α-invariantly
n t
approximately inner. We will translate this property to α on E by using the fact that
t n
∼
J = K. See Section 3 for details.
n
Before embarking on the proof of the above proposition, we will have to prove that if
α is a Rohlin flow on O , then each α is not only α-invariantly approximately inner but
n t
also α-invariantly asymptotically inner, i.e., there is a continuous map u : [0,∞)→U(O )
n
such that α (x) = lim Adu(s)(x) for x ∈ O and max kα (u(s)) − u(s)k→0.
t s→∞ n t1∈[0,1] t1
This will be proved for a wider class of C∗-algebras (see 2.2 for details). (As a matter of
fact we do not know of a single example of α without the above property of α-invariant
asymptotical innerness if it has covariant irreducible representations; we expect that this
property holds fairly in general whether it has the Rohlin property or not.)
As a corollary to the above theorem we get that any purely infinite simple separable
nuclear C∗-algebra has a Rohlin flow; because such a C∗-algebra A satisfies that A ∼=
A⊗O due to Kirchberg (see [9]) and a flow α on O induces a flow on A via id⊗α on
∞ ∞
A⊗O which has the Rohlin property if α has.
∞
Let C denote the C∗-subalgebra of O generated by s s ···s s∗ ···s∗ with all
∞ ∞ i1 i2 ik ik i1
finite sequences (i ,i ,...,i ) in N. Then C is a weakly regular maximal abelian
1 2 k ∞
C∗-subalgebra of O (weakly regular in the sense that {u ∈ PI(O ) | uu∗,u∗u ∈
∞ ∞
C , uC u∗ = C uu∗} generates O , where PI(O ) is the set of partial isometries
∞ ∞ ∞ ∞ ∞
of O ). Moreover there is a projection of norm one of O onto C and there is a charac-
∞ ∞ ∞
ter of C which extends uniquely to a state of O . (When a weakly regular masa satisfies
∞ ∞
these two additional conditions, we will say that it is a weak Cartan masa.) We note that
if α is a quasi-free flow (in our sense) then α is the identity on C ; in other words, if δ
t ∞ α
denotes the generator of α, then D(δ ) ⊃ C and δ |C = 0. We consider the following
α ∞ α ∞
condition for a flow γ on O : D(δ ) ⊃ C and sup k(γ −id)δ (x)k converges
∞ γ ∞ x∈C∞,kxk≤1 t γ
to zero as t→0; which we express by saying that γ is C1+ǫ on C below. This is obviously
∞
satisfied if γ is C2 on C or D(δ2) ⊃ C (because then δ2|C is bounded).
∞ γ ∞ α ∞
We will also show:
Corollary 1.3 Any two Rohlin flows on O are cocycle conjugate with each other if they
∞
are C1+ǫ on C .
∞
The proof consists of two parts. In the first part we show that if the flow γ is C1+ǫ on
C then δ |C is inner, i.e., there is an h = h∗ ∈ O such that δ (x) = adih(x), x ∈ C
∞ γ ∞ ∞ γ ∞
3
(see 5.6). Thus we can assume, by inner perturbation, that δ |C = 0. In the second part
γ ∞
we show that any two Rohlin flows are cocycle-conjugate with each other if they fix each
element of C (see 5.11).
∞
Acknowledgement. One of the authors (A.K.) visited at Australian National University in
March, 2004 and at University of Oslo in August-September, 2004 duringthis collaboration. He
acknowledges partial financial supports from these institutions.
2 Rohlin property
In this section we consider the class of purely infinite simple nuclear separable C∗-algebra
satisfying the universal coefficient theorem, which is classified by Kirchberg and Phillips
[9, 10] in terms of K-theory.
Let A be a unital C∗-algebra of the above class. Let ℓ∞(A) be the C∗-algebra of
bounded sequences in A and let, for a free ultrafilter ω on N, c (A) be the ideal of ℓ∞(A)
ω
consisting of x = (x ) with lim kx k = 0. If α is a flow on A, i.e., a strongly continuous
n ω n
one-parameter automorphism group of A, we can define an action of R on ℓ∞(A) by
t 7→ (α (x )) for x = (x ). Let ℓ∞(A) be the maximal C∗-subalgebra of ℓ∞(A) on which
t n n α
this action is continuous; we will denote this flow by α. We set
Aω = ℓ∞(A)/c (A), Aω = ℓω(A)/c (A).
ω α α ω
We embed A into ℓ∞(A) by constant sequences. Since A∩c (A) = {0}, we regard A as
α ω
a C∗-subalgebra of Aω ⊂ Aω.
α
We recall the following result [18, 20]:
Theorem 2.1 Let A be a unital separable nuclear purely infinite simple C∗-algebra sat-
isfying the universal coefficient theorem and let α be a flow on A. Then the following
conditions are equivalent.
1. α has the Rohlin property.
2. (A′ ∩Aω)α is purely infinite and simple, K ((A′ ∩Aω)α) ∼= K (A′ ∩Aω) induced by
α 0 α 0
the embedding, and Spec(α|A′ ∩Aω) = R.
α
3. The crossed product A× R is purely infinite and simple and the dual action αˆ has
α
the Rohlin property.
4. The crossed product A× R is purely infinite and simple and α is α-invariantly
α t0
approximately inner for every t ∈ R.
0
If the above conditions are satisfied, it also follows that K ((A′ ∩Aω)α) ∼= K (A′ ∩Aω),
1 α 1
which is induced by the embedding.
4
In the last condition of the above theorem, α (for a fixed t ) is α-invariantly ap-
t0 0
proximately inner if there is a sequence (u ) in U(A) such that α = limAdu and
n t0 n
max kα (u )−u k→0. We will strengthen this condition as follows.
t∈[0,1] t n n
Lemma 2.2 Let α be a Rohlin flow on a unital C∗-algebra A of the above class (or
in particular O ). Then each α is α-invariantly asymptotically inner, i.e., there is a
n t0
continuous map u : [0,∞)→U(A) such that α = lim Adu(s) and
t0 s→∞
lim max kα (u(s))−u(s)k = 0.
t
s→∞t∈[0,1]
Proof. Since KK(α ) = KK(id), α is asymptotically inner [25], i.e., there is a continu-
t0 t0
ous map v : [0,∞)→U(A) such that
α = lim Adv(s).
t0 s→∞
Let w(s,t) = v(s)α (v(s)∗), s ∈ [0,∞), t ∈ R. Then for each s ∈ [0,∞), the map
t
t 7→ w(s,t) is an α-cocycle, i.e., t 7→ w(s,t) is a continuous function into U(A) such that
w(s,t +t ) = w(s,t )α (w(s,t )), t ,t ∈ R. Since for each x ∈ A,
1 2 1 t1 2 1 2
k[w(s,t),x]k ≤ kAdv(s)(α (x))−α (x)k+kAdv(s)(α (x))−xk.
−t0−t −t −t0
we get, for any T ≫ 0 and for any x ∈ A, that
sup k[w(s,t),x]k→0
0≤t≤T
as s→∞.
More specifically let F be a finite subset of A and ǫ > 0. Then there exists an a > 0
such that if s ≥ a, then kAdv(s)(α (x))−α (x)k < ǫ/22 for x ∈ F and t ∈ [0,T],
−t0−t −t
which entails that k[w(s,t),x]k < ǫ/11 for x ∈ F and t ∈ [0,T].
Furthermore, for any bounded interval I of [0,∞), there is a continuous map z :
I ×[0,T]→U(A) such that
z(s,0) = 1,
z(s,T) = w(s,T),
kz(s,t )−z(s,t )k ≤ (16π/3+ǫ)|t −t |/T,
1 2 1 2
k[z(s,t),x]k < 10ǫ/11, x ∈ F,
for s ∈ I and t,t ,t ∈ [0,T]. (Here we used the estimate for a particular construction of
1 2
z(s,t) that
k[z(s,t),x]k < 9 max k[w(s,t ),x]k+ǫ′
1
0≤t1≤T
foranyǫ′ > 0; see[24]or2.7of[18].) Byusingthisz,wegetacontinuousmapU : I→U(A)
such that
kw(s,t)−U(s)α (U(s)∗)k ≤ 6π|t|/T +ǫ,
t
k[U(s),x]k ≤ ǫ, x ∈ F,
5
where we have assumed that 16π/3+ǫ < 6π.
We recall how U (s) = U(s) is defined [14]. We define a unitary U˜ in C(R/Z)⊗A
T T
by
U˜ (t) = w(s,Tt)α (z (s,Tt)∗),
T T(t−1) T
where R/Z is identified with [0,1]/{0,1} and z (s,t) = z(s,t) is defined above, and we
T
embed C(R/Z)⊗Ainto A approximatelyby using the Rohlinproperty. (Ifτ is the flow on
C(R/Z) induced by translations on R/Z, then k1⊗w(s,t)−U˜(τ ⊗α )(U˜∗)k ≤ 6π|t|/T.
t/T t
Wefindanapproximatehomomorphism φ ofC(R/Z)⊗Ainto Asuch thatφ◦(τ ⊗α ) ≈
t/T t
α φ and φ(1 ⊗ x) ≈ x, x ∈ A.) Since z is defined in terms of w(s,t), t ∈ [0,T] and
t T
other elements which almost commute with them, we may assume that S ∈ [0,T]→z is
S
continuous; hence that S ∈ [0,T]→U˜ ∈ U(C(R/Z)⊗ A) is continuous. Note also that
S
U˜ commutes with any element to the same degree as U˜ does with it. Since U˜ = 1, we
S T 0
may thus assume that there is a continuous path (U , t ∈ [0,1]) in the space of continuous
t
maps of I into U(A) such that U (s) = 1, U (s) = U(s), and k[U (s),x]k < ǫ, x ∈ F.
0 1 t
Then we set v (s) = U(s)∗v(s) for s ∈ I, which satisfies that
1
kAdv (s)(x)−α (x)k ≤ 2ǫ, x ∈ α (F),
1 t0 −t0
max kα (v (s))−v (s)k ≤ 6π/T +ǫ.
t 1 1
0≤t≤1
Thus we have shown the following assertion: For any finite subset F of A and ǫ > 0,
there exists an a ∈ [0,∞) such that for any compact interval I of [a,∞) we find a
continuous v : I ×[0,1]→U(A) such that
I
kAdv (s,t)(x)−α (x)k < ǫ, x ∈ F, (s,t) ∈ I ×[0,1],
I t0
and
v (s,0) = v(s), s ∈ I,
I
max kα (v (s,1))−v (s,1)k < ǫ, s ∈ I,
t I I
0≤t≤1
where v : [0,∞)→U(A) has been chosen so that α = lim Adv(s).
t0 s→∞
Let (F ) be an increasing sequence of finite subsets of A such that F is dense in A
n n n
and (ǫ ) a decreasing sequence in (0,∞) such that lim ǫ = 0. We chSoose an increasing
k k k
sequence (a ) in (0,∞) such that if I is a compact interval of [a ,∞) then there is a
k k
continuous v : I × [0,1]→U(A) such that the above conditions are satisfied for F = F
k
and ǫ = ǫ .
k
Let a = 0 and I = [a ,a ] for k = 0,1,2,.... For each k = 1,2,... we choose
0 k k k+1
v : I ×[0,1]→U(A) for F and ǫ as above and define v : I ×[0,1]→U(A) by v (s,t) =
k k k k 0 0 0
v(s), s ∈ I . If v (a ) = v (a ) for k = 1,2,..., we would be finished by defining a
0 k−1 k k k
continuous function v : [0,∞)→U(A) with the desired properties in an obvious way. But
note that v (a )v (a )∗ is connected to 1 by a continuous path (w (s), s ∈ [0,1]) such
k k k−1 k k
that w (0) = 1, w (1) = v (a )v (a )∗, and
k k k k k−1 k
k[w (s),x]k < 2ǫ , x ∈ α (F )
k k−1 t0 k−1
6
for s ∈ [0,1]. By modifying the path w (s), s ∈ [0,1], we have to impose the condition
k
that max kα (w (s)) − w (s)k is small; then the path s 7→ w (s)v (s) connects
0≤t≤1 t k k k k−1
v (a ) with v (a ) and has the desired property with respect to α. Thus it suffices to
k−1 k k k
prove the following lemma by assuming that (α (F )) is sufficiently rapidly increasing
t0 k k
and (ǫ ) is sufficiently rapidly decreasing. (cid:3)
k
Lemma 2.3 For any finite subset F of A and ǫ > 0 there exist a finite subset G of A and
δ > 0 satisfying the following condition: If a continuous v : [0,1]→U(A) satisfies that
v(0) = 1,
k[v(s),x]k < δ, s ∈ [0,1], x ∈ G,
kα (v(1))−v(1)k < δ, t ∈ [0,1],
t
then there exists a continuous u : [0,1]→U(A) such that
u(0) = 1,
u(1) = v(1),
k[u(s),x]k < ǫ, s ∈ [0,1], x ∈ F,
kα (u(s))−u(s)k < ǫ, t ∈ [0,1], s ∈ [0,1].
t
Proof. Suppose that v satisfies that v(0) = 1, and
k[v(s),α (x)]k < δ, x ∈ G, t ∈ [0,1],
−t
max kα (v(1))−v(1)k < δ.
t
0≤t≤1
We define w(s,t) = v(s)α (v(s)∗). Then t 7→ w(s,t) is an α-cocycle for each s ∈ [0,1] and
t
satisfies that w(0,t) = 1, and
max k[w(s,t),x]k < 2δ,
0≤t≤1
max kw(1,t)−1k < δ.
0≤t≤1
From the latter condition there are b,h ∈ A such that b ≈ 0, h ≈ 0, and w(1,t) =
sa
eibz(h)α (e−ib), where z(h) is a differentiable α-cocycle such that dz /dt| = ih [15]. By
t t t t=0
connecting the α-cocycle t 7→ w(1,t) with the trivial α-cocycle 1 by the path of α-cocycles
s 7→ (t 7→ eisbz(sh)α (e−isb)) and squeezing it around0 ∈ T = R/Z, we get anα-cocycle W
t t
in C(T)⊗A with respect to the flow id⊗α such that W(0,t) = 1 and W(s,t) ≈ w(s,t).
Henceitsufficestoshowthefollowinglemma,becausethenwefindaunitaryZ ∈ C(T)⊗A
with appropriate commutativity such that Z(0) = 1 and W(·,t) ≈ Zα (Z)∗, and replace
t
v by the path s 7→ Z(s)∗v(s) which are almost α-invariant and moves from v(0) = 1 to
v(1). (cid:3)
7
Lemma 2.4 For any finite subset F of A and ǫ > 0 there exists a finite subset G of A
and δ > 0 satisfying the following condition: Let α = id⊗α be the flow on C(T)⊗A and
let t 7→ W be an α-cocycle such that W (0) = 1 at 0 ∈ T = [0,1]/{0,1} and
t t
max k[W ,1⊗x]k < δ, x ∈ G.
t
0≤t≤1
Then there exists a unitary Z in C(T)⊗A such that Z(0) = 1 and
k[Z,1⊗x]k < ǫ, x ∈ F,
max kW −Zα (Z∗)k < ǫ.
t t
0≤t≤1
Proof. We just sketch the proof; see [14] or the first part of the proof of 2.2 for details.
To meet the last condition we choose T ∈ N such that T−1 < ǫ/6π. Then we impose
the condition that max k[W ,1 ⊗ x]k < δ/T for x ∈ α (F), which can be
0≤t≤1 t −T≤t≤0 t
replaced by a finite subset because it is compact. Since mSax k[W ,1 ⊗x]k < δ, we
0≤t≤T t
find a continuous path (U , t ∈ [0,T]) in U(C(T) ⊗ A) such that U = 1, U = W ,
t 0 T T
U (0) = 1, kU − U k ≤ 6π|t − t |/T, and k[U ,x]k < 9δ. By using W and U and
t t1 t2 1 2 t
the Rohlin property for α, we define a unitary Z ∈ C(T) ⊗ A such that Z(0) = 1,
max kW −Zα (Z∗)k < ǫ, and k[Z,1⊗x]k < 10δ, x ∈ F. (cid:3)
0≤t≤1 t t
We also give the following technical results which will be used in the next section. We
assume that α is a Rohlin flow on A as before.
Lemma 2.5 For any finite subset F of A and ǫ > 0 there exists a finite subset G of A
and δ > 0 satisfying the following condition: If (u(s), s ∈ [0,1]) is a continuous path in
U(A) such that
k[u(s),x]k < δ, x ∈ G,
max kα (u(s))−u(s)k < δ, s ∈ [0,1],
t
0≤t≤1
then there exists a rectifiable path v(s), s ∈ [0,1] such that v(0) = u(0), v(1) = u(1),
k[v(s),x]k < ǫ, x ∈ F,
max kα (v(s))−v(s)k < ǫ, s ∈ [0,1],
t
0≤t≤1
and the length of the path v is less than 17π/3. If F = ∅, then G = ∅ is possible.
Proof. Without the conditions with respect to α, this is shown in [24].
To define v we use certain elements of A which almost commute with u(s), s ∈ [0,1].
They are a certain compact subset of O , which is then embedded centrally in A in [24],
∞
by using a result due to Kirchberg and Phillips. In the present case, to meet the condition
of almost α-invariance, those elements embedded in A should be almost invariant under
α. For this we use the fact that (A′ ∩Aω)α is purely infinite and simple [18].
α
8
Explicitly we assume that those elements of O = C∗(s ,s ,...) (before the embed-
∞ 1 2
dingintoA)isinthelinearsubspace spannedbyafinitenumber ofmonomialsins ,...,s
1 k
and their adjoints for some k. We find a finite sequence (T ,...,T ) of isometries in
1 k
(A′∩Aω)α such that k T T∗ (cid:8) 1. Each T is represented by a central sequence (t (m))
α i=1 i i i i
of isometries in A suchPthat k t (m)t (m)∗ (cid:8) 1 and max kα (t (m))−t (m)k→0 as
i=1 i i t∈[0,1] t i i
m→∞. We then express thoPse elements in terms of t (m),...,t (m) in place of s ,...,s
1 k 1 k
respectively for a sufficiently large m. Thus we get the required condition involving α. (cid:3)
We will denote by δ the generator of α, which is a closed derivation from a dense
α
∗-subalgebra D(δ ) into A. See [5, 2, 27] for the theory of generators and derivations.
α
Lemma 2.6 Let (u(s), s ∈ [0,∞)) be a continuous path in U(A) such that
max kα (u(s))−u(s)k
t
0≤t≤1
converges to zero as s→∞. Then there is a continuous path (v(s), s ∈ [0,∞)) of unitaries
such that v(s) ∈ D(δ ) and δ (v(s)) and u(s)−v(s) converge to zero as s→∞.
α α
Proof. Let f be a non-negative C∞-function on R of compact support such that the
integral is 1. We set
z(s) = b(s) f(b(s)t)α (u(s))dt,
Z t
where b : [0,∞)→(0,∞) is a continuous decreasing function such that lim b(s) = 0,
s
kz(s)−u(s)k < 1, and kz(s)−u(s)k→0. Then it follows that z(s) ∈ D(δ ) and
α
kδ (z(s))k ≤ b(s) |f′(t)|dt,
α Z
which converges to zero as s→∞. We set v(s) = z(s)|z(s)|−1, which satisfies the required
(cid:3)
conditions.
Lemma 2.7 For a finite subset F of A and ǫ > 0 there exists a finite subset G of A
and δ > 0 satisfying the following condition. Let u be a unitary in C[0,1]⊗A such that
u(0) = 1, u(t) ∈ D(δ ), kδ (u(t))k < δ, and k[u(t),x]k < δ, x ∈ G. Then there exist an
α α
h ∈ D(δ )∩A for i = 1,2,...,10 such that
i α sa
u(1) = eih1eih2···eih10,
kh k < π,
i
kδ (h )k < ǫ,
α i
k[h ,x]k < ǫ, x ∈ F.
i
If F = ∅, then G = ∅ is possible.
9
Proof. We may assume, by 2.5, that the length of the path u is smaller than 17π/3 < 18.
Then we choose 0 < s < s < ··· < s < 1 such that ku(s ) − u(s )k < 9/5 < 2 for
1 2 9 i i−1
i = 1,2,...,10, where s = 0 and s = 1. Note that
0 10
u(1) = u(s )∗u(s )·u(s )∗u(s )···u(s )∗u(s ).
0 1 1 2 9 10
Since ku(s )∗u(s )−1k < 9/5, the spectrum of u(s )∗u(s ) is contained in
i−1 i i−1 i
S = {eiθ | |θ| < θ },
0
where θ = π − 2cos−1(9/10) < π. Let Arg denote the function eiθ 7→ θ from S onto
0
the interval (−θ ,θ ) and set h = Arg(u(s )∗u(s )). Then we have that kh k < π and
0 0 i i−1 i i
u(1) = eih1eih2 ···eih10. We shall show that these h satisfy the other conditions for a
i
sufficiently small δ > 0.
In general if v is a unitary with Spec(v) ⊂ S, then h = Arg(v) can be obtained as
1
h = (logz)(z −v)−1dz,
2πi I
C
where logz is the logarithmic function on C\(−∞,0] with values in {z | |ℑz| < π} and
C is a simple rectifiable path surrounding S in the domain of log. We fix C and let r be
the distance between C and S. Since
1
δ (h) = logz(z −v)−1δ (v)(z −v)−1dz,
α 2πi I α
C
we have the estimate
kδ (h)k ≤ (2π)−1M|C|r−2kδ (v)k,
α α
where M is the maximum of |logz|, z ∈ C and |C| is the length of C. Similarly we have
the estimate k[h,x]k ≤ (2π)−1M|C|r−2k[v,x]k for any x ∈ A. (See [5, 27] for details.)
(cid:3)
Thus we get the conclusion.
3 Proof of Proposition 1.2
We recall that E = C∗(s ,...,s ), where s ,...,s are isometries such that e0 = 1 −
n 1 n 1 n n
n s s∗ is a non-zero projection, and that J is the ideal of E generated by e0. Let
k=1 k k n n n
SP= {1,2,...,n}∗ denote the set of all finite sequences including an empty sequence,
denoted by ∅. ForI = (i ,i ,...,i ) ∈ S with m = |I|, we set s = s s ···s , where |I|
1 2 m I i1 i2 im
is the length of I; if |I| = 0 or I = ∅, then s = 1. It then follows that {s e s∗ | I,J ∈ S}
I I n J
forms a family of matrix units and spans J . Thus, in particular, J is isomorphic
n n
to the C∗-algebra K of compact operators (on an infinite-dimensional separable Hilbert
space). Hence there is a unique (up to unitary equivalence) irreducible representation π
0
of E such that π |J is non-zero or π (e0) is a one-dimensional projection. We call this
n 0 n 0 n
10