Decomposition Theory of a Certain Class of O N ( en )
Yuan-Ching Huang (黃苑青)
Department of International Business, Yu Da College of Business
No. 168, Hsueh-Fu Rd, Chaochiao Township, Miaoli County, 361, Taiwan, R.O.C..
Tel: 037-651188-6025 Fax: 037-651201
Email: [email protected]
Abstract
If O N (e n ) has a certain period, then O N (e n ) has a nice decomposition theory.
Furthermore, there
exists an interesting correspondence between the irreducible factors over Z[x] of x k  1 , where k is the
number of periodic points of O N (e n ) and the orthogonal invariant subspaces of O N (e n ) .
Key words: Cuntz algebras, permutative representations, irreducible subspaces, decomposition, periods
1. Introduction
Recall that the Cuntz algebra of order N ,
denoted by O N , is the unital C * -algebra generated
by
N
isometiries {si }iN1
with the following
relation:
si* s j   ij 1 ;
i1 si si*  1
N
to
B(H ) (bounded operators on a
Hilbert Space H ).
Let H be a separable Hilbert
space with an orthonormal basis   {en }n .
A
representation  of O N on H with respect to
 is a permutative if there exist  i :  ,
i  0,..., N  1 such that
 i ()   j ()   , for all i  j
such that
Eq. (1)
N 1
S *j (em )  0 .
j1 , , j k 1 are
When
defined, let j k be the unique j  Z N such that
S *jk  S *j1 (em )  0 .
A function system is called multiplicity-free if
([Cun])
A representation  on O N is a *-homomorphism
from O N
m  ,  (m)  ( j1 , j2 ,, jk ,)  Z N , where j1
is the unique element j in Z N  {0, , N  1}
the coding map is injective.
map σ
We say that the coding
is partially injective if
n 
and
i1 ,, ik  Z N and  (n)   ( i1  ik (n)) , imply
n   i1  ik (n) . In this case, the function system
is then called to be regular.
For a  H , denote ON (a)  the closure of
the
linear
{SI S (a)  Si1 Si2
*
J
J  ( j1 ,
span
*
j
Sik S S
, j ), i1 ,
, ik , j1 ,
of
S (a) | I  (i1 ,
*
j 1
*
j1
, ik ),
, j  Z N and k , } .
 i ()  
Eq. (2)
Define
 i is injective
Eq. (3)
R(m)  m1 , where m1 is the unique element in 
i 1
R : 
as follow:
Let
and
such that  j (m1 )  m for some j Z N .
S j (en )  e j ( n) , for all j  0, , N 1, where
ON (n)  { I R (n)   i1  i2
S j   (s j )
k
Eq. (4)
The collection of such maps  0 ,,  N 1
satisfying Eq. (1) to Eq. (3) is called a (branch)
function system of order N . The coding map

 :   Z N   Z N is denoted as follows:
i 1
Let
, im ), m, k  and i1 ,
 i RR
m
m  ,
Denote
R(n) | I  (i1 ,
, im  Z N } .
Jorgensen and Bratteli [BJ] have shown that if
the function system { j } jZ N
is regular then
O N (e n ) , for all n   is an irreducible invariant
subspace
of
the
corresponding
permutative
representation  on O N .
That is to say, if the
function system is regular, then the decomposition
theory of  is completely known.

they showed that when
Furthermore,
is regarded as a
 i  Qm . For this m , find p(m)  Qm . Then
there exist I p (m ) and J p (m) such that
S I p ( m) S J* p ( m) (e1 )  e p(m)
representation of UHF N , then without assuming
 SI p ( m ) S J*p ( m ) (c1 )   m cm  a linear combination of
regularity, UHF N (en )  the closure of the linear
{c j } j , where  m  C
span {S I S J* (en ) | I  (i1 ,
 | Q1 || Qm |
i1 ,
, ik , j1 ,
, ik ), J  ( j1 ,
, jk ), k ,
, jk  Z N }
But, we also have
( UHF N is the uniform hyperfinite algebra of order
N ) is invariant and irreducible for all n   .
In [Hua], it is shown that if K  ON (en ) is a
S J p ( m) S I*p ( m) (e p(m) )  e1
 S J p ( m ) SI*p ( m ) (cm )  m c1  a linear combination of
where  m  C
proper invariant subspace of O N with a reduced
{c j } j ,
basis (Definition 6, [Hua]) then ON (n) possesses a
 | Qm || Q1 |
period.
Eqs.
(5)
cm 
1
develop
(Theorem 9, [Hua]).
the
In this paper, we will
decomposition
between
invariant
subspaces of an O N ( en ) with a certain period and
an
interesting
Eq. (5)
connection
between
m
Eq. (6)
and
(6)
imply
| Q1 || Qm | ;
hence
S I p ( m ) S J* p ( m ) (c1 ) . So the proof is complete.
Q.E.D.
invariant
subspaces of O N and the irreducible factors p (x)
over Z [x] of x k  1  0 , where k is the number
of periodic points. (Definition 6, [Hua]).
Our next goal is to find the decomposition
theory for such a reducible O N (e n ) .
Before going
to the theoretical proof, let us look at an example.
2. Main Results
Proposition 1 Suppose O N ( en ) is reducible with
Suppose there exist a1 , a 2 , a3 , a 4  ON (n) such that
a proper subspace K and there is a j0 such that
some j0  Z N .
S j0 (ai )  ai 1 ,  i  1, 2,3
and
S j0 (a4 )  a1
We claim that
and  (a k )  a1 ,
K1  ON (ea1  ea2  ea3  ea4 )
then K  ON (1en1      en ) for some ,
K 2  ON (ea1  ea2  ea3  ea4 )
i  C,  i {1, , } , and {n1 , , n }  {a1 , , ak } .
K3  ON (ea1  ea3 ),
 j0 (ai )  ai 1 ,
 1  i  k 1
for
and
K 4  ON (ea1  ea3 )
Proof: Let {ci }i be a reduced basis for K ,
where ci  (c1i , c 2i , , c mi , )  .
be an i0   such that
i
c a0
1
Since there must
 0 , without loss of
generality, we assume i0  1 .
Let c1 
where Q1   , and c  0,  j  Q1 .
1
j

c1j e j
,
jQ1
We claim
are proper invariant subspaces of O N (e n ) .
In
order to see K 2 is proper it suffices to show that
H 2  span{S mj0 (ea1  ea2  ea3  ea4 ) | m  0,1,2,3} is a
proper subspace of K  span{ea1 , ea2 , ea3 , ea4 }  C 4 .
 ( j )   (i),  i, j  Q1 . If not, then there exists a
But it is easy to see that H 2 is the 1-dimensional
finite string L such that S L* (ei )  0  S L* (e j ) .
space
Observe that S L S L* (C1 )  C1 , which leads to a
contradiction.
Since
Q1  {a1 , , a k } .
cm 
 cim ei
iQm
Let
, where
a1  Q1 , we have
m  2 be given, put
Qm  
and
cim  0 ,
of
K
generated
by
ea1  ea2  ea3  ea4 .
Similarly,
dim H 3  2
dim H 4  2
,
and
the
vector
dim H 1  1 ,
,
where
H i  span{S (bi ) | m  0,1, 2,3},  i  1,3, 4 , b1  ea1 
m
j0
ea2  ea3  ea4 , b3  ea1  ea3 , and b4  ea1  ea3 .
Suppose dim K 0  1 , then we
Furthermore, it is not hard to see that K 4 is the
K 0 is irreducible.
orthogonal complement of K 3 , K1  K 2  K 4 and
have
K 1 is actually the orthogonal complement of K 2 in
S j0 ( xi eai )    ( xi eai )
the
subspace
Hence K  K4  K3 
K4 .
k
k
i 1
i 1
K1  K2  K3 with K 1 , K 2 , and K 3 pairwisely
 x j   x j 1 , j  2,
, k and x1   xk
orthogonal.
 x2   x1 , x3   2 x1 ,
, xk   k 1 x1 , x1   k x1
Moreover, there is an interesting
correspondence between K 1 , K 2 , K 3 and the
  k  1 (Since x1  0)
so we conclude that if  is a k  th root of unity,
irreducible factors of x 4  1 in Z[x] .
x  1  ( x  1)( x  1)( x  1)
4
2
K   ON (i 1 i 1eai )
k
then
We make the following correspondence
is an irreducible
K1  x  1  p1 ( x)
subspace.
K 2  x  1  p 2 ( x)
K   K  '  {0} if  and  ' are disjoint k  th
K 3  x  1  p3 ( x)
Now
it
2
Later on, we will show that in general there is
such correspondence between the irreducible factors
easy
to
see
0

.
1
0

'
( ' )
root of unity since
with dim( K i )  deg( pi ( x)) .
is
 1 dim( K )  k  dim H 0
k
that
Because
, we conclude that
 k 1 K  ON (en ) .
Q.E.D.
of x k  1 in Z[x] and some special invariant
subspaces of O N (e n ) .
The next result carries out our earlier promise.
It is obvious that K 1 and K 2 are irreducible,
but is K 3 irreducible?
The answer is no, and we
introduce the following theorem that decomposes
O N (e n ) into irreducibles.
Corollary 3
{a1 , a 2 , , ak }  
 i  1,
p1 ( x)
Theorem 2
Suppose O N (e n ) is reducible with
period
{a1 , a 2 , , a k } ,
and
 j0 (ai )  ai 1 ,
 i  1,
, k 1 ,  j0 (ak )  a1 .
Then, O N (e n )
Let O N (e n ) be reducible with period
, k 1 ,
pr ( x)
and
 j0 (ai )  ai 1
 j0 (ak )  a1 .
If xk 1 
where pi (x) is irreducible over
Z[ x] , then each pi (x) corresponds to a proper
invariant subspace
K i  ON (  x ij ea j )
jQi
Qi  {1,2,, k} ,
x ij  1 ,
 j  Qi .
decomposes into k disjoint irreducible subspaces as
where
follow:
Furthermore, Ki  K j ,  i  j, and ON (en )  Ki .
r
O N (e n )    k 1 O N (e a1  e a2     k 1e ai1 ) .
Proof:
,
Let
H 0  span{ea1 ,  , eak }  C k .
i 1
In
k
order to show that K 0  O N ( xi eai ) , xi  C is a
i 1
proper subspace of H 0 , it suffices to show that
K 0  {S mj0 (
k
 xi ea ) | m  0,1,, k  1}
i 1
i
is a proper
Proof:
For i {1, 2,
pi (x) be {i ,1 , i ,2 ,
ni
k
j 1
 1
, r} , let the complex roots of
, i ,ni } . Define
K i   O N ( ( i , j )  1 ea )  O N (  x ij ea j )
jQi
where Qi  {1,2,, k} , x ij  0 , j  Qi .
dim( K i )  ni  deg( pi )
and
x  1
i
j
coefficients of pi ( x) are either 1 or -1.
Then
since the
To show
The idea of the proof is to find all
that K i  K j ,  i  j , it suffices to show that if
the K 0 such that dim K 0  1 which implies that
x k  1  p ( x)  q ( x ) , where p( x), q( x)  Z[ x] , then
subspace of H 0 .
the corresponding subspaces
orthogonal.
Kp 
k
 O ( a
p ( a ) 0
Kp
and
are
Kq
Now
N
1
1
ea ) , K q 
x  1  ON (a1  a2  a3  a4  a5  a6 )
k
 O ( 
q(  )0
N
j 1
j 1
ea j ) .
( a
1
1
where  is a complex root of x3  1 .
k
 ea ,   j 1  ea j )
x 2  x  1  ON (a1  1a2  12 a3 
j 1
 1  ( )  ( ) 
2
 ( )
k 1
is a k -th root of unity, Eq. (7) is 0 unless    .
By the complex conjugate root theorem, we know
that  is also a root of q( x)  0 ; hence  cannot
be a root or p( x)  0 ; so    .
k  i 1 deg( pi ),
r
decompose.
Furthermore,
i  1,
the Ki 's,
Hence the proof is complete.
The
following
example
, r,
Q.E.D.
shows
the
 ON (a1  a2  a4  a5 )
where   1 and i
3
i
is a complex number,
 i  1, 2 .
The case k  7 is similar to the case k  3 .
In
fact,
if
k a
( x  1)( x k 1  x k  2 
is
prime,
then x7 1 
 1) is the factorization over
Z[ x] ; hence they are all similar.
k  8 , x8  1  ( x  1)( x  1)( x 2  1)( x 4  1)
x  1  ON (a1  a2 
 a8 )
correspondence between irreducible factors of x  1
x  1  ON (a1  a2  a3  a4 
in Z[ x] and orthogonal invariant subspaces of some
x 2  1  ON (a1  a3  a5  a7 )
special cases.
x 4  1  ON (a1  a5 ) .
k
 15 a6 ) 
ON (a1  2 a2  22 a3    25 a6 )
Eq.(7)
It suffices to show that Eq. (7) is 0, for all  , 
such that p( )  0 and q(  )  0 . Since   
since
x 2  x  1  ON (a1   a2   2 a3  a4   a5   2 a6 )
 ON (a1  a2  a4  a5 )
Consider
k
x  1  ON (a1  a2  a3  a4  a5  a6 )
 a7  a8 )
k  9 , x9  1  ( x  1)( x 2  x  1)( x 6  x3  1)
Example 4
x  1  ON (a1  a2 
k  2 , x 2  1  ( x  1)( x  1)
x 2  x  1  ON (a1  a2  a3  a4    a7  a8 )
x  1  ON (a1  a2 )
x 6  x 3  1  ON (a1  a4 ) .
x  1  ON (a1  a2 ) .
k  10 , x10  1  ( x  1)( x 4  x3  x 2  x  1)
k  3 , x  1  ( x  1)( x  x  1)
3
 a9 )
2
( x  1)( x 4  x3  x 2  x  1)
x  1  ON (a1  a2  a3 )
x 2  x  1  ON (a1  a2 )  ON (a2  a3 )  ON (a3  a1 )
and
ON (a1  a2 )  ON (a1   a2   a3 ) 
x  1  ON (a1  a2 
 a10 )
x  1  ON (a1  a2  a3  a4 
 a9  a10 )
x 4  x3  x 2  x  1  ON (a1  a2  a6  a7 )
2
ON (a1   a2   a3 ) , where  is a complex root of
x 4  x3  x 2  x  1  ON (a1  a2  a6  a7 ) .
2
x3 1  0 .
A Remark on UHFN
k  4 , x  1  ( x  1)( x  1)( x  1)
4
2
If  is a permutative representative of ON
x  1  ON (a1  a2  a3  a4 )
on H , then there is a natural way to view  as a
x  1  ON (a1  a2  a3  a4 )
representation of UHFN .
relation  on  by
x 2  1  ON (a1  a3 ) .
k  5 , x  1  ( x  1)( x  x  x  x  1)
5
4
3
2
n  m  S I S J* (en )  em , for some finite string I , J
x  1  ON (a1  a2  a3  a4 )
with | I || J | .
x  x  x  x  1  ON (a1  a2 ) .
given n  ,
4
3
Define an equivalence
2
k  6 , x 6  1  ( x  1)( x  1)( x 2  x  1)( x 2  x  1)
Theorem 2.7 in [BJ2] shows that
ON (en )  UHFN (em )
References
m
is
the
decomposition
of
into
ON (en )
[BJ] O. Bratteli and P. E. T. Jorgensen, “Iterated
UHFN - irreducible, where m runs through all the
Function
 equivalent class in ON (en ) .
Representations of the Cuntz Algebra”, Mem.
The sum is finite if
and only if there is an a  ON (en ) such that
 I (a)  a for some finite I . Suppose ON (en ) is
ON - reducible with a period {a1 ,
, ak } such that
S j0 (ai )  ai 1 , S j0 (ak )  a1 , for some j0  Z N . By
Theorem 2,
ON (en )   (a1  a2 
  k 1ak ) .
 k 1
and
Permutative
Amer. Math. Soc. 139 (1999) no. 663.
[Cun] J. Cuntz, “ C * -Algebra
Gemerated by
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173-185.
[DaPi] K. Davidson and D. R. Pitts, “Invariant
Subspaces
and
Hyper-reflexivity
for
Free
Semigroup Algebras”, Proc. London Math. Soc.
We claim that
UHFN (a1   a2 
78 (1999), 401-430.
  k 1ak ) 
ON (a1   a2 
UHFN (a1   a2 
[Hua]
  k 1ak )
for all k - th root of unity  .
I  (i1 ,
”On
Permutative
O N ”, Yu-Da Academic
[KaRi] R. Kadison and J.R. Ringrose, “Fundamentals
  k 1ak ) .
of the Theory of Operator Algebras”, Amer. Math.
To show the others direction, let
be given, where
Huang,
Journal, Vol. 9, (2005) 283-290.
  k 1ak ) 
b  S I S J* (a1   a2 
Yuan-Ching
Representation of
It is trivial that
ON (a1   a2 
Soc., (1997).
  k 1ak )
J  ( j1 ,
, ip ) ,
, jq ) .
[Tak] M. Takesaki, “Theory of Operator Algebras I”,
Springer-Verlag, New York-Heidelberg-Berlin,
Firstly, note that
(S j0 )m (a1  a2 
  k 1ak ) 
  m (a1   a2 
(1979).
  k 1ak ) .
Suppose q  p , then let I 0  ( j0 , j0 ,
b  S I S J* (a1   a2 
, j0 )
  k 1ak )
 S I S J* ( q  p  S I0 (a1   a2 
  q  p  S I S J*0 (a1   a2 
  k 1ak ))
  k 1ak )
where J 0 is some finite string with J 0  p  I .
Similarly, if q  p , then we can find I 0 such that
I 0  J with
b   SI0 S J* (a1   a2 
for some
Systems
  k 1ak )
Z . This shows that
UHFN (a1   a2 
  k 1ak ) 
ON (a1   a2 
  k 1ak ) .
Hence
ON (en )  UHFN (a1  a2 
 k 1
  k 1ak )
is the decomposition into UHFN - irreducibles.