Classification of C*-algebras
admitting ergodic actions of the
two-dimensional torus
by
Raphael H¢egh-Krohn and Tor Skjelbred
Matematisk institutt,
Oslo Universitet
Abstract
We g1ve a complete classification under *isomorphism of the
c*-algebras which admit an
ergodic action of the two-dimensional torus.
- 1 -
1.
Ergodic actions of the two-dimensional torus
G be the two-dimensional
Let
of
G
of a
into
Aut(A)
where
C*-algebra A .
norm on
E:
A , g
Aut(A)
\.Je assume
E:
G, a
a
'?
~
If
A •
torus~
its kernel is a closed subgroup
~
and
a homomorphism
is the group of *automorphisms
~
-+
to be continuous in the
g (a)
does not act effectively on A ,
H.
G/H
It is easy to see that
is either the two-dimensional torus or the one-dimensional circle,
G/H
and in the second case it is easy to see that since
ergodically
=
A
~,
the complex numbers.
acts
G/H
In the first case
is the two-dimensional torus and acts effectively and ergodically.
~' Corollary 4.~ that if
It was proven in
A
on aeffectively on a
the
acts effectively on A •
~
We may therefore just as well assume that
c*-algebra generated by a faithful
G acts ergodically
= Ap
is
where
projective unitary repre-
sentation
Y
a
= (nl,n2)
where
(2)
c (yl'y 2 )a
a
Yl Y2 = p
and the multiplier
y
= G,
Z2
of the dual group
(1)
ay
-+
and
-
y
= cnl,nzl
and
= e- 21T i:\ n2 fh
c ( Y'Y)
is glven by
cp
p
p
= e 21T iA
The unitary multi-
plier representation (2) is implemented by letting
the unitary operators on
(bf) (x)
= e 21T ixf(x)
L 2 (R)
given by
with
(af)(x)
a
and
= f(x+:\)
b
be
and
and
(3)
Then
tor
ab
a
=
and
pba
b .
and
A
p
lS the
c*-algebra generated by the opera-
By interchanging
a
and
b
we see that
Ap
~
AP
-
2 -
We have the following theorem.
Theorem 1.1
Let
a
be a continuous ergodic effective action of the two-
-dimensional torus on a
AP ;:; B(L 2 (R))
= f(x+J..),
(af)(x)
a
n
g
is the
n
(a 1 b 2 )
=
c*-algebra
Then
A
c;':-algebra generated by
a
n
Moreover, if
= e 21Tixf(x),
(bf)(x)
n
y(g)a 1 b 2 ,
AP = \
A .
(npn 2
y:::
then
1
p
p
A
~
where
p
and
= e 21riJ..
b
where
and
).
= a
or
p
cr .
:::
This theorem gives a complete classification of the ergodic
actions of
G on
c*-algebras.
The first part of the theorem is
~] .
as we have already pointed out a consequence of the results 1n
What remains to be proven is that if
P
-
= cr .
AP
~
=
Acr
then
This problem is very different according as
of unity or not.
If
called the irrational rotation algebras.
studied by M.A. Rieffel
p
is a root
is not a root of unity then the
p
[2J ,
or
P ::: cr
AP
are
These algebras were
M. Pimsner, and D. Voiculescu rL3
J
From their work it follows that for the irrational rotation algebras
AP
~
A0
bedding
bra.
only if
AP
=
p
into an
cr
or
p
=
cr .
The method in [3]
AF-algebra and computing
K0
is by im-
of the
This method utilizes the discrete structure of
AP
AF-algewhen
is not a root of unity, and does not extend to the case when
a root of unity.
Hence
~ve
p
is
p
need only to consider the case when
p
1s a root of unity.
Let therefore
and
x2
be a primitive
p
q-th root of unity, and
the two generators of the character group
the action
ag
of
G
on
AP
G = Z2 ;
x1
then
1s g1ven by
i ,j
Ez
(4)
-
where
a
and
3 -
are the generators of
b
We shall now construct
such that
AP
= pba
ab
.
explicitely as a subalgebra of
AP
C(G) e M
where Mq is the algebra of qxq matrices, and C(G)
q
-the algebra of continuous functions on G.
The embedding of AP
C(G)
ln
~
M
CP
on the centre
E cP ,
z
is acting ergodically
X,
For so::J.e compact space
AP .
of
and there is a group action
c
G
comes about as follows.
q
S
of
G
on
X
CP
= C(X)
such that for
E- x
Because of ergodicity
G
is acting transitively on
X.
We
\•.Je obtain a con-
tinuous mapplng
h: G +X
with
= 8g(z 0 )
h(g)
,
and the induced
homomorphism
h°
= C(X)
CP
t G
+
xq
Given a character
x
xq = h 0 (k)
k(Scr(z 0 ))
where
C(G).
is a function on
= xCg)q
or
X ,
that is,
k(z) = x(g)q
whenever
C>
Using
h0
,
we form the tensor product
C(G) C® AP .
p
Setting
K
=
x1
~a
and
--
show that
1-'-
L
= x2
®b,
we have
.
Letting
Thus
Kq
=1
and similarly,
Lq = 1 .
KL
=
pLK.
We will
- 4 -
The subalgebra generated by
and
K
is isomorphic to
L
hence, we obtain a homomorphism
C(G) ® ~
(C
q
~ C(G) ® AP
cp
which is clearly an isomorphism of
AP = 1®Ap
we have an embedding of
Construction of
2.
Let
L
,
E1
-'-
,E
, •••
q
be the standard basis of
i
K(E.) = p E.
be a torus;
d: G
+
C(G) ® JM
(C
by
v = yq = bq
l
Let
and
C(G)
®
cq ,
and let
(t
JM
q
d(x,y) = (xq,yq) .
and
K
L(Ei) = Ei+l ( i
taken
G = {Cx,v>E<t 2 llxl=lyl=1}
, we define a = X®K
q
and aq = xq®I
and
q
= {(u,v)EC 2 ! lul=lv!=1}
T2
T2
= pba
ab
We-then have
l
KL = pLK .
We then have
In the algebra
1n
AP
Hriting
AP
be the matrices with
mod q) .
C(G)-algebras.
and
b = y®L
o
Let
T2
Define a homomorphism
o
u = xq = aq and
He may then set
0
C(G) ® 11
c q is the algebra of sections in the
trivial algebra bundle G X 11
Let a group H of automorphisms
q
The algebra
of this bundle be generated by
.
= (x,py,KXK
= (p
The group
with
H
G/H = T 2
-1
-1
x,y,LXL
and
h2
where, for
(x,y,X)EGxJMq~
)
-1
1s of order
•
h1
)
and 1s acting freely on the base
Hence
is an algebra bundle over
T2
~-vi th
fibre
1-1
sections of BP is the H-invariant subspace of
q
The algebra of
C(G)
®
<t
JM
q
The
G
- 5 -
H-invariant. Any element of C(G) ~ JMq
q -1~
l
J
may be written uniquely as . E.
and it is invariant
fij (x,y)a b
l 'J =0
q-1
l
J
if and only if it is of the form
E
h .. Cu,v)a b •
i,j=O lJ
a
elements
b
are
Noting that
u = aq
and
sections in
Bp
A
3.
and
is
Automorphisms of
The centi'e of
Any automorphism
by restriction.
<j>:
A +A
P
a
If
it follows that the algebra of
p
AP
Ap
<!>:
v = bq ,
C(T 2 ) , when
lS
Ap
lS
constructed as above.
A
+ A
p
cr
is any other root of unity, an isomorphism
p
induces an automorphism
induces an automorphism
C(T 2 ) + C(T 2 ),
1jJ:
C(T 2 ) + C(T 2 ) , by restriction,
ljl:
C(T 2 ) .
as the centre in both algebras has been identified with
Definition 3.1
Given a 1'automorphism
continuous map inducing
C(T 2 ) + C(T 2 ), let
1jJ :
1jJ
Let
1jJ
Given
morphism
1jJ 2 ' 1/J 1 '
1)J 1 1)J
ln
Lemma 3.1
2
is induced by
Let
condition
l
f.l '
i=1,2,
It follows that
•
1
Hl(T2) + Hl(T2)
,~:
and
ljl.
be the
As
the auto-
(1ji 1 1/J 2 )'
=
GL ( 2 ,'Jl., ) •
Let
be a *automorphism with
Then there is a *automorphism
Proof
f 2f
= f
I
f· T2 + T2
f: T 2 + T 2
f* = 1/J' = I 2
tity mapping of
T2 .
<I>:
A
p
_,. A
p
with
be the homeomorphism inducing
implies that
f
1/J •
The
is homotopic to the iden-
By the homotopy invariance of fibre bundles)
there lS a bundle automorphism
there is a commutative diagram,
F: Bp
+
Bp
convering
f ,
that is,
- 6 -
B
F
Bp
lpr
---+-
p
prpl
~
Tz
p
Tz
---+-
f
where
If
F
is an algebra
E AP,
s
=
pr p OS
s(u,v)
s
then
*isomorphism 1n the fibres.
We define
id .
= F -1 osof.
¢(s)
1s a scalar matrix for each
= F- 1 s(f(u,v)) =
<f>(s)(u,v)
lJi •
extends
s: T 2
is a continuous mapplng
s(f(u,v)
(u,v)
f
*automorphism because the structural group of
Given a matrix
d>:
Ap + A(J
.
inducin.g
where
(J
Proof
~~
-~
ME
on the centre
and define
induced by the homeomorphism
It follows that
I
f
= f "4'~ = M.
if
+
A _1
p
if
obvious way to a
lS a
There is a
C(T 2
with
)
*automorphism
1/J'
= M,
and
and
Ic 1 I = Ic 2 I = 1
f(u,v)
=
ltle must check the relation
an automorphism
p
4>
by con-
(
.
Thus
4>:
Ap
det M
+
Ap
= -1
ab
p a.o-By <f>(ba) .
det M
=
= p ba
.
Ap
An
tfle thus have
1 , and an isomorphism
~
In both cases
;"automorphism of
*isomorphism.
=
1/J
c 1 u a. v y ,c 2 u Bv o)
is continuous c:md lS defined on all of
4>
tjl(ab)
A
BP
by
<I>
with
·
easy computatlon
shows that
<!>:
'
= p detM
Let
1/J
)
•
GL(2,~).
where
Also
C(T 2
may be chosen so that
F
is a
Lemma 3.2
E
This shows
4>
U(q)/T 1
s
with
p
and we have
T2
that
struction is (a subgroup of)
B
Then if
= IJ!(s)(u,v),
We note that
+
C(G) ® :M
(£
q
extends in the
showing that
is
- 7 -
Corollary 3.1
1s a
n: A
Let
p
~:A
*isomorphsm
identity on the centre
Proof
+A a
p
C(T 2 )
be a
~
*isomorphism.
~:A
or
p
+A a-1
and let
(~n!C(T 2 ))'
Then there
which is the
•
Let
*isomorphism with
Then
+A
( ~ I C ( T 2 ) ) ' = H- 1 , where
= I 2 , and hence there is
~:A
T
a
=
+A T
a
or
a
a
A
If
P
A
a
and
~'(isomorphic,
are
-1
*automorphism
We then put
Theorem 3.1
be a
E;
then
p
= t, 4> n
=a
.
or
P = a-1 .
Proof
Ap +A T
~':isomorphism
By Corollary 3.1, there is a
is the identity on the centre and where
=
T
a
or
T
=
a- 1
which
Any
•
such isomorphism is indqced by an isomorphism of algebra bundles,
Bp
BT
+
The Theorem follotvs from Proposition 3":"1.
Proposition 3.1
phic , then - p
=
If the algebra bundles
B
and
p
B0
are isomer-
a •
The proof of this Proposition will occupy the rema1n1ng pages.
Given an algebra bundle
B
over
define a complex number
w(B)
T2
with fibre
with
w(B)q
B.
only on the topological properties of
determine
B
E:
m.
X
S1
S1
+
= {vEG.:llv!
T2
1 ,
by
w(B )
p
The number
induced algebra bundle
=
w(B)
will
The problem
p •
and define the covering projection
=1}
E(s ,v)
as a function of
He will
and vJhich depends
completely, but we do not need this fact.
is to actually compute
Let
=
11q ,
(e 2 'ITi s ,v) .
E*(B)
m.
over
isomorphism.
struction of a trivialization for
x
A trivialization of the
S1
is an algebra bundle
We will make an explicit conB
= BP
.
-
Definition of
w(B) .
JR
( 3. 2)
Here
Consider the diagram:
F
s 1 x]1 q
"'--
x
8 -
E)':(B)
I
~~ JR
pr 1
E~~(B)
and
pr 2
and
(s+1,v)
pr2
~B
lprl
ll(
sl
lprB
E
~T2
are the projections from the fibered product
is the bundle projection of
be the covering transformation of
B': E*(B)
+
E*(B)
Let
B
E,
and let
be the induced automorphism of
and
pr 2
e = F- 1 e' F •
defined by
is the centre of
SL(q,C) .
a.: JR
x
s1
-+
E*(B)
with
We obtain an automorphism
•
=
e(s,v)
e
of
The group of algebra automor-
There is a continuous map
SL(q) ;z
q
such that
e(s,v,X) = (s+l,v,a.(s,v)Xa.(s,v)- 1
The homotopy class of
We define
w(B)
element of
zq
Lemma 3.3
Let
Proof
[a] =
by
F 1 ,F 2:
e.
l
m.
and
[a] E 1T 1 (SL(q)/Zq)
is an element
w(B)- 1 I
q
We will show that
Let
F.l
(l
X
S1
a.
l
X
JM
q
(i
)
[e~J
when viewing
w(B)
-+ E~=(B)
=
zq
as an
is well defined.
be two trivializations
= 1, 2) be defined as above, using
Then
There is a bundle automorphism
D
There is a continuous mapping
such that
a:
JR
x
such that
D(s,v,X) = (s,v,B(s,v) X S(s,v)- 1 ) .
s1
+ sL(q)/Z
q
- 9 -
=
De 2
e1D •
It follows that
=
8 1 (s,v, S(s,v) X B(s,v)- 1
)
and hence
(s+1,v,S(s+1,v)a 2 (s,v) X a 2 (s,v)- 1 S(s+1)v)- 1
( s +1 J v , a 1 ( s , v ) a( s , v ) X S ( s , v) - 1 a 1 ( s , v ) -
)
=
for all
1 )
X
E :Mq.
It folloHs that
A homotopy is given by
1
= e 21T i p ' I q
w( B )
p
Proof
Let
p
where
=1
~
2 •
q
1
= e 21T iplq
pp '
c
1T (SL(q)IZ )
We have, ln the abelian group
Proposition 3.2
~
where
(p ,q)
=
1 •
Then
(mod q ) .
E*(B p )
We shall construct a trivialization of
To this
end, we define a bundle mapplng
F,: JR 2
-
x]M
q
+
G x11
q
F 1 ( s, t ,.X)
= ( e - 21T is I q , e 21T it I q , e 21T it K ,X e - 21T it K)
where
lS
e
K
21T i qK
=
I
q
the
q
and
e
by q
21Tipv:
matrix •vith
K(E.)
J
=
(jlq)Ej
He have
= K.
We compute, to obtain
_._X)_- ( e -21ris/q ,e 21Ti(t+p)/q ,e 21TiCt+p)Kx.e-21Ti(t+p)K)_rc
F ( s,~_,
h1 11
- 1 s, t+pX)
,,
and
F 1 ( s , t +q, X)
=
F 1 ( s , t, X) .
I
- 10 -
Let
F2
be the conposite bundle mapping,
Fl
F2
As
:
JR 2 xJM
=1
(p,q)
,
q
~ GxJM
-+
q
(GxJM )/H
q
= Bp
He have
It follows that there lS a bundle mapping
F3: JR
X
S 1 x 11
q
B
.-!"
p
such that
when
the mapping
E: JE x S 1
F: JR
a bundle isomorphism
bundle automorphism
have, for some
Y
8(s,v,X)
Because
pr 2 F .
Pr
2
=
e
E 11q
=
e' = -pr 2
27Ti t
=v
covers
.
in the base spaces, and hence induces
T2
+
e
x
S1
x 11
q
F- 1 e'F.
+
E," ( B ) •
Given
p
Now we compute the
(s,v,X)
E
JR
X
S 1 x11q' 'Ove
,
(s+1,v,Y)
(see diagram 3.2), we have
It follows that
F30
=
F3
because
F3
= pr 2 F
,
this implies that
We note that
27Tit/q
27TitKy -2v.itK)
_ ( -27Ti(s+1)/q
F 1 ( s+1,t,Y) - e
, e
, e
e
:
where
( e -27Tis/q p -p 1 , e 27Tit/q , e 27TitKy e -27TitK)
e
27T it K
Letting
- 11 -
It folloHs that
and hence
e(s,v,X) = (s+1,v,a(t)X a(t)- 1 )
when
= e 2Tr it
v
w(B ) ,
By definition of
oJ ( B ) - 1 I
p
-p f
= K
an d h ence,
p
W(
we have
= a ( 1 ) a ( 0 ) - 1 = e -2TriK
q
L
p
T
-p'
D I
K..:-
L
= P
-(p' )2
and
w(B ) = w(B ) ,
hence that
1
2TriK L-p
-
= e 21T i
p' Iq I
q ,
B
and B
are isomorphic algebra
P
a
a will both be primitive qth roots of unity, and
a
since
If
w(B)
a = e
If
and hence
q
e
p
p
P
I
1
B ) __ e2Tr ip' I q .
Proof of Proposition 3.1
bundles,
-
Lp
p~
p
-
p~
=a
(mod q)
•
is clearly a topological invariant.
2Triu 2 1q
L
,
.
we obta1n
It follows that
p1
= p 2 (mod q)
, and
- 12 -
References
[~
S. Albeverio and R. Hoegh-Krohn, Ergodic actions by
compact groups on
C*-algebras, to be published in
Mathematische Zeitschrift.
~]
M.A. Rieffel, Irrational rotation
C*-algebras, Short
communication at the International Congress of Mathematicians, 1978.
[3]
M. Pimsner and D. Voiculescu, Imbedding the irrational
rotation
C*-algebra into an
AF-algebra, Preprint series
1n mathematics No 45/1979, Inst. Nat.
de Matematica, Bucuresti.
PCSST, Institutl