On normalizers of $C^{*}$-subalgebras in the Cuntz algebra $\mathcal{O}_{n}$

0.14 MB·English

by  Tomohiro Hayashi

ON NORMALIZERS OF C∗-SUBALGEBRAS IN THE CUNTZ
ALGEBRA O
n
2
1
TOMOHIRO HAYASHI
0
2

n
a
J Abstract. In this paper we investigate the normalizer NOn(A) of a C∗-
5 subalgebra A ⊂ Fn where Fn is the canonical UHF-subalgebra of type n∞
2
in the Cuntz algebra On. Under the assumption that the relative commutant
] A′ ∩Fn is finite-dimensional, we show several facts for normalizers of A. In
A particular it is shown that the automorphism group {Adu|A | u ∈ NFn(A)}
O has a finite index in {AdU|A | U ∈NOn(A)}.
.
h
t
a 1. Introduction
m
[ The purpose of this paper is to investigate the normalizer of C∗-subalgebras

1 in the Cuntz algebra O [1]. Let F be the canonical UHFsubalgebra of O . In
n n n
v
5 the paper [4], it is shown that the normalizer group NOn(Fn) is a subset of Fn.
9
(In [4], more general results are shown.) More generally, if A is an irreducible
1
5 C∗-subalgebra of O , then the normilizers N (A) is a subset of F . In this
. n On n
1 paper we investigate the normalizer N (A) where A is a C∗-subalgebra of F
0 On n
2 witha finite-dimensional relative commutant inF . Inthissetting thenormalizer
n
1
N (A) is not a subset of F in general. However we can show that the inner
v: On n
i automorphism group induced by the elements in NF (A) has a finite index in the
X n
inner automorphism group induced by the elements in N (A). In order to show
r On
a this fact, we show that the relative commutant A′∩O is also finite-dimensional.
n
As a corollary of our investigation, it is shown that the irreducibility A′∩F = C
n
implies that A′ ∩O = C. Hence in this case the normalizer group N (A) is a
n On
subset of F .
n
We would like to explain the motivation of this paper. There is a one-to-one
correspondencebetweenallunitariesU(O )andallendomorphismsEnd(O )such
n n
that
U(O ) ∋ u 7→ λ ∈ End(O )
n u n
where λ is defined by λ (S ) = uS . The endomorphism λ is called local-
u u i i u
ized if the corresponding unitary is a matrix in the UHF-algebra F [6, 2]. In the
n
paper [9] Szymanski showed that the localized endomorphism λ is an inner auto-
u
morphism if and only if u can be written in some special form. As a consequence,
if the localized endomorphism λ is an inner automorphism, then there exists a
u
1
2 TOMOHIROHAYASHI
unitary U ∈ F such that λ = AdU. Keeping this in mind, we would like to
n u
consider the following problem. Let λ and λ be two localized endomorphisms.
u v
If they satisfy AdU ◦ λ = λ , what can we say about U? Can we determine
u v
such a unitary U? Unfortunately in this paper we cannot say nothing about this
problem. But we remark that a localized endomorphism has finite index [6, 8].
Therefore the C∗-algebras λ (F )′∩F and λ (O )′∩O are finite-dimensional.
u n n u n n
Soweexpect thatourinvestigationwouldbehelpful onthisprobleminthefuture.
The author wishes to express his hearty gratitude to Professor Wojciech Szy-
manski for valuable comments and discussion on this paper. The author is also
grateful to Professor Roberto Conti for valuable comments. The author would
like to thank Professor Takeshi Katsura for useful advice and comments.
2. Main Results
The Cuntz algebra O is the C∗-algebra generated by isometries S ,...,S
n 1 n
satisfying n S S ∗ = 1. The gauge action γ (z ∈ T) on O is defined by
i=1 i i z n
γ (S ) = zSP. Let F be the fixed point algebra of the gauge action. This algebra
z i i n
is isomorphic to the UHF-algebra of type n∞. So F has the unique tracial state
n
τ. We have a conditional expectation E : O → F defined by
n n
E(x) = γ (x)dz.
z
Z
T
The canonical shift ϕ is defined by ϕ(x) = n S xS∗. It is easy to see that
i=1 i i
S x = ϕ(x)S and xS ∗ = S ∗ϕ(x). For eaPch x ∈ O , we have the Fourier
i i i i n
expansion
∞ ∞
x = S∗kx +x + x Sk
1 −k 0 k 1
Xk=1 Xk=1
where x = E(xS∗k), x = E(Skx) and x = E(x). (The right-hand side
k 1 −k 1 0
converges in the Hilbert space generated by the GNS-representation with respect
to τ ◦E.) For example, if x = S S S∗S∗2S∗, then S2x ∈ F and x = S∗2(S2x) =
1 2 5 8 3 1 n 1 1
S∗2E(S2x).
1 1
For the inclusion of C∗-algebras A ⊂ B, the normalizer group is defined by
∗ ∗ ∗
N (A) = {u ∈ B | uAu = A, uu = u u = 1}.
B
For a unitary operator u, we define the inner automorphism by Adu(x) = uxu∗.
We denote by Adu| the restriction of Adu on A.
A
The following two theorems are the main results of this paper.
Theorem 2.1. Let A be a C∗-subalgebra of F . If the relative commutant A′∩F
n n
is finite-dimensional, then the algebra A′ ∩O is also finite-dimensional.
n
ON NORMALIZERS OF C∗-SUBALGEBRAS IN THE CUNTZ ALGEBRA O 3
n
Theorem 2.2. Let A be as above. We consider two subgroups of the automor-
phism group Aut(A) as follows.
G = {AdU| | U ∈ N (A)}, H = {Adu| | u ∈ N (A)}.
A On A Fn
Then H is a subgroup of G with finite index.
We need some preparations to show these theorems.
Lemma 2.3. For X ∈ A′∩O , we set x = E(XS∗k) and x = E(SkX). Then
n k 1 −k 1
for any a ∈ A we have ax = x ϕk(a), x a = ϕk(a)x and x x∗, x∗ x ∈
k k −k −k k k −k −k
A′ ∩F
n
Proof. For any a ∈ A, we see that
ax = aE(XS∗k) = E(aXS∗k) = E(XaS∗k)
k 1 1 1
= E(XS∗kϕk(a)) = E(XS∗k)ϕk(a) = x ϕk(a)
1 1 k
and therefore
x x∗a = x (a∗x )∗ = x (x ϕk(a)∗)∗ = x ϕk(a)x∗ = ax x∗.
k k k k k k k k k k
In the same way we also have x a = ϕk(a)x and x∗ x a = ax∗ x . (cid:3)
−k −k −k −k −k −k
Lemma 2.4. There is a positive integer N satisfying the following properties. For
any integer k ≥ N and any element X ∈ A′ ∩ O , we have x = E(XS∗k) = 0
n k 1
and x = E(SkX) = 0.
−k 1
Proof. We compute
x∗x = E(XS∗k)∗E(XS∗k) ≤ E(SkX∗XS∗k) ≤ ||X||2E(SkS∗k) = ||X||2SkS∗k.
k k 1 1 1 1 1 1 1 1
′′
Let R = F be the hyperfinite II -factor. We take the polar decomposition
n 1
x = v |x | in R. Then the above computation shows that v∗v ≤ SkS∗k. On the
k k k k k 1 1
other hand, since x x∗ is an element of the finite-dimensional C∗-algebra A′∩F ,
k k n
we have v v∗ ∈ A′∩F . Since the C∗-algebra A′∩F is finite- dimensional, there
k k n n
is a positive number c satisfying τ(p) ≥ c for any non-zero projection p ∈ A′∩F .
n
We can take a positive integer N satisfying τ(SkS∗k) = 1 < c for any k ≥ N.
1 1 nk
Then we see that τ(v v∗) = τ(v∗v ) ≤ τ(SkS∗k) < c and hence v v∗ = 0. So
k k k k 1 1 k k
we conclude that x = 0 for k ≥ N. In the same way we also have x = 0 for
k −k
k ≥ N. (cid:3)
Lemma 2.5. Let N be the positive integer in the previous lemma. For any
X ∈ A′ ∩O , we have
n
N N
X = S∗kx +x + x Sk
1 −k 0 k 1
Xk=1 Xk=1
4 TOMOHIROHAYASHI
where x = E(XS∗k), x = E(SkX) and x = E(X).
k 1 −k 1 0
Proof. We have the Fourier expansion
∞ ∞
X = S∗kx +x + x Sk.
1 −k 0 k 1
Xk=1 Xk=1
(cid:3)
Thus by the previous lemma, we are done.
Lemma 2.6. We define the isomorphism π on A by
k
x 0
π (x) = , x ∈ A.
k (cid:18)0 ϕk(x)(cid:19)
Then we have
0 x 0 x∗
k , −k ∈ π (A)′ ∩M (F )
(cid:18)x∗k 0(cid:19) (cid:18)x−k 0 (cid:19) k 2 n
where x = E(XS∗k), x = E(SkX) for X ∈ A′ ∩O .
k 1 −k 1 n
Proof. This is an immediate consequence of the relations ax = x ϕk(a) and
k k
x a = ϕk(a)x for a ∈ O . (cid:3)
−k −k n
Lemma 2.7. The C∗-algebra π (A)′ ∩M (F ) is finite-dimensional.
k 2 n
Proof. We set
1 0 0 0
′
B = π (A) ∩M (F ), e = ∈ B, f = = 1−e.
k 2 n (cid:18)0 0(cid:19) (cid:18)0 1(cid:19)
Then we see that eBe ≃ A′ ∩ F and fBf ≃ ϕk(A)′ ∩ F ≃ M (C) ⊗ (A′ ∩
n n nk
F ). So the both algebra eBe and fBf are finite-dimensional and hence B is
n
finite-dimensional. Indeed the von Neumann algebras eB′′e and fB′′f are finite-
dimensional. So the center of B′′ is finite-dimensional. Therefore we may assume
that B′′ is a factor. Then the finite-dimensionality of eB′′e and fB′′f ensures
that B′′ is finite-dimensional. Therefore B is finite-dimensional.
(cid:3)
Proof of Theorem 2.1. Consider the vector space
V = {x = E(XS∗k) | X ∈ A′ ∩O }.
k k 1 n
Since the map
0 x
′
V ∋ x 7→ ∈ π (A) ∩M (F )
k (cid:18)x∗ 0(cid:19) k 2 n
is injective and R-linear, the vector space V is finite-dimensional. In the same
k
way the vector space
V = {x = E(SkX) | X ∈ A′ ∩O }
−k k 1 n
ON NORMALIZERS OF C∗-SUBALGEBRAS IN THE CUNTZ ALGEBRA O 5
n
is also finite-dimensional. On the other hand, the element x = E(X) belongs to
0
the finite-dimensional C∗-algebra A′∩F . Combining these with Lemma 2.5, we
n
see that A′ ∩O is finite-dimensional. (cid:3)
n
Proposition2.8. Thereexistsan orthogonalfamilyof minimalprojectionse ,...,e ∈
1 l
A′ ∩O satisfying the following.
n
(i) l e = 1 and e ,...,e ∈ A′ ∩F .
i=1 i 1 l n
(ii) TPhere are integers k ,...,k such that Adu (x) = γ (x) for x ∈ A′ ∩O
1 l z z n
where uz = zk1e1 +···+zklel.
Proof. Since A′∩O is finite-dimensional and γ-invariant, there exists an orthog-
n
onal family of minimal projections e ,...,e ∈ A′ ∩ O and integers k ,...,k
1 l n 1 l
such that Aduz(x) = γz(x) where uz = zk1e1+···+zklel and x ∈ A′∩On. Then
e ∈ (A′ ∩O )γ = A′ ∩F . (cid:3)
i n n
Corollary 2.9. If A is an irreducible C∗-subalgebras of F , then we have A′ ∩
n
O = C.
n
Proof. By the previous proposition, we know that there are minimal projections
in A′ ∩O such that they belong to A′ ∩F . Thus we are done. (cid:3)
n n
In the rest of this paper we frequently use the projections e ,...,e ∈ A′ ∩F
1 l n
and the unitary uz = zk1e1 +···+zklel in the above proposition.
Remark 2.1. The Bratteli diagramof the inclusion A′∩F ⊂ A′∩O has a special
n n
form. They have a common family of minimal projections. So for each vertex
corresponding to a direct summand of A′∩F , there is only one edge which starts
n
onthis vertex. Forexample, if A′∩F = C, then A′∩O = C. If A′∩F = C⊕C,
n n n
then A′ ∩O is isomorphic to either C⊕C or M (C).
n 2
Lemma 2.10. Let U ∈ O be a unitary satisfying UAU∗ ⊂ F . Then we have
n n
(i) U∗γ (U) ∈ A′ ∩O .
z n
(ii) There exists a unitary w ∈ A′ ∩ O and integers m ,...,m such that
n 1 l
γ (Uwe ) = zmiUwe
z i i
Proof. For any a ∈ A, since UaU∗ ∈ F , we see that
n
∗ ∗ ∗
γ (U)aγ (U) = γ (UaU ) = UaU .
z z z
Thuswehave U∗γz(U) ∈ A′∩On. Itiseasytosee thatthefamily{U∗γz(U)uz}z∈T
is a unitary group. Indeed since U∗γ (U) ∈ A′ ∩O , we see that
z n
∗ ∗ ∗ ∗
U γ (U)u U γ (U)u = U γ (U)Adu (U γ (U))u
z1 z1 z2 z2 z1 z1 z2 z1z2
∗ ∗ ∗
= U γ (U)γ (U )γ (U)u = U γ (U)u .
z1 z1 z1z2 z1z2 z1z2 z1z2
6 TOMOHIROHAYASHI
Since {U∗γz(U)uz}z∈T is a unitary group in the finite-dimensional C∗-algebra
A′ ∩ O , we can take a unitary w ∈ A′ ∩ O and integers n ,...,n such that
n n 1 l
w∗U∗γz(U)uzw = zn1e1 +···+znlel. Then we see that
γz(Uwei) = γz(U)uzwu∗zei = {Uw(zn1e1 +···+znlel)w∗u∗z}uzwu∗zei
= Uw(zn1e1 +···+znlel)u∗zei = zni−kiUwei.
(cid:3)
Remark 2.2. Bythepreviouslemma, weknowthattheFourierexpansionofU can
bewrite down ina finite sum. Indeed if m > 0, then Uwe = (Uwe S∗mi)Smi and
i i i 1 1
Uwe S∗mi ∈ F . On the other hand if m < 0, then Uwe = S∗−mi(S−miUwe )
i 1 n i j 1 1 j
and S−miUwe ∈ F . Therefore the Fourier expansion of Uw is a finite sum.
1 j n
Combining this with Lemma 2.5, we can show that the Fourier expansion of U is
a finite sum.
Proposition 2.11. For any normalizer U ∈ N (A), there exist unitaries v ∈
O
n
A′ ∩F and w ∈ A′ ∩O satisfying
n n
vUw ∈ N (e F e ⊕···⊕e F e ).
On 1 n 1 l n l
Proof. By the previous lemma we have γ (Uwe ) = zmiUwe . Then we get
z i i
γ (Uwe w∗U∗) = Uwe w∗U∗ and hence Uwe w∗U∗ ∈ F . Since UA′ ∩ O U∗ =
z i i i n n
A′ ∩ O , we have Uwe w∗U∗ ∈ A′ ∩ F . Thus {Uwe w∗U∗} is a family of
n i n i i
minimal projections in the finite-dimensional C∗-algebra A′ ∩ F . So we can
n
find a unitary v ∈ A′ ∩ F satisfying vUwe w∗U∗v∗ = e . Since γ (vUwe ) =
n i j z i
vγ (Uwe ) = zmivUwe for any x ∈ F , we see that γ ◦ AdvUw(e xe ) =
z i i n z i i
AdvUw(e xe ). Therefore (vUw)e F e (vUw)∗ ⊂ e F e . On the other hand,
i i i n i j n j
since γ (w∗U∗v∗e ) = γ (vUwe )∗ = (zmiUwe )∗ = z−miw∗U∗v∗e , we also have
z j z i i j
(vUw)∗e F e (vUw) ⊂ e F e . Therefore we have
j n j i n i
AdvUw(e F e ⊕···⊕e F e ) ⊂ e F e ⊕···⊕e F e ,
1 n 1 l n l 1 n 1 l n l
∗ ∗ ∗
Adw U v (e F e ⊕···⊕e F e ) ⊂ e F e ⊕···⊕e F e
1 n 1 l n l 1 n 1 l n l
and hence
vUw ∈ N (e F e ⊕···⊕e F e ).
On 1 n 1 k n k
(cid:3)
Remark 2.3. The normalizer N (F ) is a subset of F . However the structure
On n n
of N (e F e ⊕···⊕e F e ) is not simple in general. See Examples 2.1 and 2.2.
On 1 n 1 l n l
Lemma 2.12. Let e ∈ F be a projection. If a partial isometry u ∈ O satisfies
n n
u∗u = uu∗ = e and ueF eu∗ = eF e, then we have u ∈ F .
n n n
ON NORMALIZERS OF C∗-SUBALGEBRAS IN THE CUNTZ ALGEBRA O 7
n
Proof. Since u∗γ (u) ∈ (eF e)′∩eO e = e(F′ ∩O )e = Ce, we have γ (u) = zmu
z n n n n z
for some integer m. We will show m = 0. Suppose that m > 0. Set v = uS∗m.
1
Then we have γ (v) = v and hence v ∈ F . Then we compute v∗v = SmeS∗m =
z n 1 1
ϕm(e)SmS∗m and vv∗ = uu∗ = e. So we see that τ(e) = τ(vv∗) = τ(v∗v) =
1 1
τ(ϕm(e)SmS∗m) = τ(e) × τ(SmS∗m) = 1 τ(e) < τ(e). This is a contradiction.
1 1 1 1 nm
On the other hand, if m < 0, we have γ (u∗) = z−mu∗. So by the same way we
z
get a contradiction.
(cid:3)
Lemma 2.13. Let B be the abelian C∗-algebra generated by e ,...,e . Then we
1 l
have
N (A)∩N (B) ⊂ N (e F e ⊕···⊕e F e ).
On On On 1 n 1 k n k
Proof. The proof is essentially same as that of Lemma 2.10 and Proposition 2.11.
For any U ∈ N (A)∩N (B), since U ∈ N (B), we have U∗γ (U) ∈ B′ and
On On On z
hence U∗γ (U)u ∈ B′. Therefore we can take w = 1 in the proof of Lemma 2.10.
z z
Then since U ∈ N (B), we have Uwe w∗U∗ = Ue U∗ = e and hence we can
On i i j
take v = 1 in the proof of Proposition 2.11. Thus by Proposition 2.11, we have
U ∈ N (e F e ⊕···⊕e F e ). (cid:3)
On 1 n 1 k n k
Proof of Theorem 2.2. We can choose a finite family of unitaries U ,...,U ∈
1 N
N (A) ∩ N (B) ⊂ N (e F e ⊕ ··· ⊕ e F e ) satisfying the following. For
On On On 1 n 1 k n k
any V ∈ N (A)∩N (B), there exists U such that AdV = AdU on B.
On On i i
For any U ∈ N (A), by Proposition 2.11 there exist unitaries v ∈ A′ ∩ F
On n
and w ∈ A′ ∩ O satisfying vUw ∈ N (e F e ⊕ ··· ⊕ e F e ). Then since
n On 1 n 1 k n k
vUw ∈ N (A) ∩ N (B), we can take U satisfying AdU∗vUw = id on B.
On On i i
Combining this with the fact that U ∈ N (e F e ⊕···⊕e F e ) we see that
i On 1 n 1 k n k
U∗vUwe ∈ N (e F e ) ⊂ F and hence U∗vUw ∈ N (A). Here we used
i j ejOnej j n j n i Fn
Lemma 2.12. Therefore we see that AdU| = AdvUw| ∈ (AdU | )H. This
A A i A
implies that the index [G : H] is finite.
(cid:3)
Example 2.1. Let e be a projection in F . Consider the C∗-algebra A = eF e⊕
n n
(1−e)F (1−e). Here we remark that A′∩F = Ce⊕C(1−e). We will show that
n n
N (A) ⊂ F and hence G = H. This can be shown by K-theoretic argument as
On n
follows.
For any U ∈ N (A), if UeU∗ = e, it follows from Lemma 2.12 that U ∈ F .
On n
So we consider the case UeU∗ = 1−e. Since U∗γ (U)e ∈ (eF e)′∩eO e = e(F′ ∩
z n n n
8 TOMOHIROHAYASHI
O )e = Ce, we have γ (U)e = zmUe for some integer m. We will show m = 0.
n z
Suppose that m > 0. Set v = UeS∗m. Then we have γ (v) = v and hence v ∈ F .
1 z n
Then we compute v∗v = SmeS∗m = ϕm(e)SmS∗m and vv∗ = UeU∗ = 1−e. So
1 1 1 1
we see that 1−τ(e) = τ(vv∗) = τ(v∗v) = τ(ϕm(e)SmS∗m) = τ(e)×τ(SmS∗m) =
1 1 1 1
1 τ(e). Since F is the UHF-algebra of type n∞, we can write τ(e) = q . So we
nm n np
get 1− q = 1 q and hence
np nmnp
nm+p = q(1+nm).
This is impossible. Indeed, consider the prime factorization n = pk1 ×···×pkn.
1 n
Then we have
(pk1 ×···×pkn)m+p = q(1+(pk1 ×···×pkn)m).
1 n 1 n
Therefore we must have
1+(pk1 ×···×pkn)m = pl1 ×···×pln.
1 n 1 n
However this cannot occur because the left hand side has the remainder 1 when
dividing by p .
1
Example 2.2. We can write
O ⊃ F = M (C)⊗M (C)⊗··· .
2 2 2 2
Consider two projections
1 0
e = ⊗1⊗1⊗···
(cid:18)0 0(cid:19)
and
0 0 1 0
f = ⊗ ⊗1⊗··· .
(cid:18)0 1(cid:19) (cid:18)0 0(cid:19)
Since ϕ(e)S S ∗ ∈ M (C)⊗M (C) and τ(ϕ(e)S S ∗) = 1, there exists a partial
1 1 2 2 1 1 4
isometry v ∈ M (C)⊗M (C) such that v∗v = ϕ(e)S S ∗ and vv∗ = f. We set
2 2 1 1
∗
U = vS +(vS ) +(1−e−f).
1 1
Then it is easy to see that
U ∈ N (eF e⊕fF f ⊕(1−e−f)F (1−e−f)).
On n n n
We let A = eF e⊕fF f ⊕(1−e−f)F (1−e−f). Since τ(UeU∗) = τ(f) =
n n n
1 6= 1 = τ(e), we have
4 2
AdU| 6∈ {Adu| | u ∈ N (A)}.
A A Fn
Therefore we see that G 6= H.
ON NORMALIZERS OF C∗-SUBALGEBRAS IN THE CUNTZ ALGEBRA O 9
n
Remark 2.4. If A is of the form A = e F e ⊕···⊕e F e , then we have A′∩F =
1 n 1 l n l n
A′ ∩ O = Ce ⊕ ···⊕ Ce . On the other hand, in Remark 2.1 we see that the
n 1 l
Bratteli diagram of the inclusion A′ ∩ F ⊂ A′ ∩ O has a special form. So we
n n
might expect that A′ ∩F = A′ ∩O . However this is wrong in general. Indeed
n n
thereexists aC∗-subalgebra A ⊂ F withfiniteindex suchthatA′∩F 6= A′∩O .
n n n
We can take A = λ (F ) where λ is a localized endomorphism. See [4, 6].
u n u
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(TomohiroHayashi)NagoyaInstituteofTechnology,Gokiso-cho,Showa-ku,Nagoya,
Aichi, 466-8555, Japan
E-mail address, Tomohiro Hayashi: [email protected]