Table Of ContentTHE K-THEORY OF THE FLIP AUTOMORPHISMS
7 MASAKI IZUMI
1
0
2 Abstract. WegiveanalgorithmtocomputetheK-groupsofthe
∗
crossedproductbytheflipautomorphismforanuclearC -algebra
b
satisfying the UCT.
e
F
7
] 1. Introduction
A
O One of the most natural Z -actions appearing in the theory of op-
2
. erator algebras arises from the flip automorphism, and it often char-
h
t acterizes important classes of operator algebras (see [10], [2], [4], [7],
a
m [11]). Moreprecisely, theflipautomorphismactingonthetensorsquare
A⊗2 = A AofaC∗-algebraA, isanextensionofthemapx y y x.
[
In [5],⊗[6], the author developed the theory of finite gr⊗oup7→acti⊗ons
2
with the Rohlin property, and showed that the flip automorphism in
v
6 thecaseoftheCuntzalgebra hastheRohlinproperty. Thefirststep
2
4 O
to obtain this result was to compute the K-groups of the fixed point
9
1 algebra, or equivalently, the crossed product. Since the flip automor-
0 phism is available for every C∗-algebra, it would be desirable to have a
.
1 systematic algorithm to compute the K-groups of the crossed product
0
by the flip automorphism. The purpose of this note is to provide such
7
1 an algorithm in the case of separable nuclear C∗-algebras satisfying the
:
v UCT. The author would not be surprised if some of the contents of this
i note is known to specialists, but he believes that they are still worth
X
to be written.
r
a This note is an extended version of the author’s private note written
around 2001. The author would like to thank Hiroki Matui for useful
discussions, and the anonymous referee for careful reading. This work
is supported in part by JSPS KAKENHI Grant Number JP15H03623.
2. KKZ2-equivalence
Let A be a C∗-algebra, and let n be a natural number greater than
1. Throughout this note, we denote by θ , or θ for simplicity, the
A
flip automorphism of A⊗2. We denote by SA the suspension of A,
2010 Mathematics Subject Classification. 46L80,19K99.
Key words and phrases. flip, K-theory.
1
2 M. IZUMI
which is identified with either C ((0,1),A) or C (R,A) depending on
0 0
the situation. We denote by M and K the n by n matrix algebra
n
and the set of compact operators on a separable infinite dimensional
Hilbert space respectively. The symbol Z stands for the cyclic group
n
Z/nZ. We often use the fact that the crossed product (A A)⋊ Z
γ 2
is isomorphic to M (A), where γ(x y) = y x. ⊕
2
⊕ ⊕
Theorem 2.1. Let A and B be mutually KK-equivalent separable nu-
clear C∗-algebras. Then A⊗2 and B⊗2 are KKZ2-equivalent to each
other with Z -actions given by the flip automorphisms.
2
Proof. Let S2A = S(SA), which is identified with C (C) A. We
0
first claim that (S2A)⊗2 is KKZ2-equivalent to A⊗2. Indeed⊗, we note
that the flip θS2A acting on (S2A)⊗2 is conjugate to θC0(C) ⊗θA acting
on C (C)⊗2 A⊗2, and that θ on C (C)⊗2 = C (C2) comes from
0 ⊗ C0(C) 0 0
0 1
the complex linear map of the underlaying space C2. Thus
(cid:18) 1 0 (cid:19)
thanks to [1, Theorem 20.3.2], we get the claim.
Let φ : S2A K S2B K and ψ : S2B K S2A K be
t t
⊗ → ⊗ ⊗ → ⊗
completely positive asymptotic morphisms giving the KK-equivalence
Z
of A and B. Then φ φ and ψ ψ gives KK 2-equivalence of
t t t t
(S2A K)⊗2 with θS2A⊗K⊗and (S2B ⊗K)⊗2 with θS2B⊗K. Therefore we
⊗ ⊗ (cid:3)
get the statement.
Our idea to compute K ((A⊗2)⋊ Z ) for a given separable nuclear
∗ θ 2
C∗-algebra A is as follows. We choose an appropriate B that is KK-
equivalent to A, and compute K ((A⊗2)⋊ Z ) from K ((B⊗2)⋊ Z )
∗ θ 2 ∗ θ 2
using Theorem 2.1. If B = n B , we have
i=1 i
(B⊗2)⋊ Z L
θ 2
n
= (B⊗2)⋊ Z (B B B B )⋊Z
∼ i θ 2 ⊕ i ⊗ j ⊕ j ⊗ i 2
Mi=1 1≤Mi<j≤n
n
= (B⊗2)⋊ Z M (B B ),
∼ i θ 2 ⊕ 2 i ⊗ j
Mi=1 1≤Mi<j≤n
and
n
K ((A⊗2)⋊ Z ) = K ((B⊗2)⋊ Z ) K (B B ).
∗ θ 2 ∼ ∗ i θ 2 ⊕ ∗ i ⊗ j
Mi=1 1≤Mi<j≤n
The dual action θ acts on K (M (B B )) trivially.
B ∗ 2 i j
⊗
Assume moreover that A satisfies the UCT and its K-groups are
c
finitely generated. Then we can choose each B isomorphic to one of
i
THE K-THEORY OF THE FLIP AUTOMORPHISMS 3
the following building blocks: C, the Cuntz algebra , SC, and the
n+1
O
dimension drop algebra
Dn = {f ∈ C0([0,1),Mn); f(0) ∈ C1Mn}.
Note that their K-groups are
Z, = 0
K∗(C) ∼= (cid:26) 0 , ∗ = 1 ,
{ } ∗
Z , = 0
K∗(On+1) ∼= (cid:26) 0n , ∗ = 1 ,
{ } ∗
0 , = 0
K∗(SC) ∼= (cid:26) Z{,} ∗ = 1 ,
∗
0 , = 0
K∗(Dn) ∼= (cid:26) Z{ }, ∗ = 1 .
n
∗
3. Building bocks
In this section, we compute K ((A⊗2)⋊ Z ) for each of the building
∗ θ 2
blocks.
We denote by λ the implementing unitary for θ in the crossed prod-
uct (A⊗2) ⋊ Z , and by e and e the spectral projections of λ cor-
θ 2 + −
responding to the eigenvalues 1 and 1 respectively. We denote by δ
−
the exponential map in the 6-term exact sequence of the K-groups.
3.1. (C⊗2)⋊ Z . It is trivial to see
θ 2
Z2, = 0
K∗((C⊗2)⋊θ Z2) = K∗(CZ2) ∼= (cid:26) 0 , ∗ = 1 ,
{ } ∗
and the dual action θˆacts on Z2 as the flip of the two components.
Let be the Cuntz algebra, which is the universal C∗-algebra gen-
∞
O
eratedby isometries S ∞ withmutually orthogonalranges. It iswell
known that is K{Ki-}eiq=u1ivalent to C, and Theorem 2.1 implies that
∞
⊗2 with theOflip automorphism is KKZ2-equivalent to C with a trivial
∞
O Z
action, which is further known to be KK 2-equivalent to with a
∞
O
quasi-free action (see [9] for example). On the other hand, Goldstein
and the author [3] showed that all non-trivial quasi-free Z -actions on
2
are mutually conjugate. Note that is isomorphic to
∞ ∞ ∞ ∞
O O ⊗O O
thanks to Kirchberg-Phillips [7].
Problem 3.1. Is the flip on ⊗2 conjugate to a quasi-free Z -action
∞ 2
O
on ?
∞
O
4 M. IZUMI
To solve the above problem affirmatively, it suffices to show that
the flip on ⊗2 is approximately representable as a Z -action (see [3]).
∞ 2
O
More generally we can pose the following problem.
Problem 3.2. Let D be a strongly self-absorbing C∗-algebra with suf-
ficiently many projections (see [11]). Is the flip on D⊗2 approximately
(asymptotically) representable ?
Note that there is a unique Z -action on the Cuntz algebra with
2 2
O
the Rohlin property (see [5]), and it is at the same time asymptotically
representable. Thus the second problem is affirmative for . Since
2
the flip on M⊗2 is inner, the flip on any UHF algebra is apprOoximately
n
representable.
3.2. ( ⊗2 ) ⋊ Z . Let be the Cuntz algebra, which is the uni-
On+1 θ 2 On+1
versal C∗-algebra generated by n+1 isometries S n with mutually
orthogonal ranges satisfying n S S∗ = 1. { i}i=0
i=0 i i
Our computation in the cPase of ( ⊗2 ) ⋊ Z and (D⊗2) ⋊ Z is
On+1 θ 2 n θ 2
based on the following elementary fact.
Lemma 3.3. The elementary divisors of an integer matrix
n n2−n
(cid:18) −n n22+n (cid:19)
− 2
is (n,n) for odd n and (n,2n) for even n.
2
Theorem 3.4. Let the notation be as above.
(1) When n is odd,
K (( ⊗2 )⋊ Z ) = Z Z ,
0 On+1 θ 2 ∼ n ⊕ n
K (( ⊗2 )⋊ Z ) = 0 .
1 On+1 θ 2 ∼ { }
The canonical generators of the right-hand side for K (( ⊗2 )⋊ Z )
0 On+1 θ 2
are given by [e ], [e ].
+ −
(2) When n is even,
K (( ⊗2 )⋊ Z ) = Z Z ,
0 On+1 θ 2 ∼ 2n ⊕ n/2
K (( ⊗2 )⋊ Z ) = 0 .
1 On+1 θ 2 ∼ { }
The canonical generators of the right-hand side for K (( ⊗2 )⋊ Z )
0 On+1 θ 2
are given by [e ], [e ] [e ].
+ + −
−
Proof. Let be the Cuntz-Toeplitz algebra, which is the universal
n+1
E
C∗-algebra generated by n+1 isometries T n with mutually orthog-
{ i}i=0
onal ranges. Let p be the projection defined by
n+1
∈ E
n
p = 1 T T∗.
− i i
Xi=0
THE K-THEORY OF THE FLIP AUTOMORPHISMS 5
Then the closed ideal of generated by p is identified with K, and
n+1
is identified with thEe quotient /K.
n+1 n+1
O E
We define a closed ideal J of by
n+1 n+1
E ⊗E
J = K + K.
n+1 n+1
⊗E E ⊗
Then we have two short exact sequences
0 J 0,
n+1 n+1 n+1 n+1
−→ −→ E ⊗E −→ O ⊗O −→
0 K K J (K ) ( K) 0,
n+1 n+1
−→ ⊗ −→ −→ ⊗O ⊕ O ⊗ −→
and two short exact sequences
(3.1) 0 J ⋊ Z ( ⊗2 )⋊ Z ( ⊗2 )⋊ Z 0,
−→ θ 2 −→ En+1 θ 2 −→ On+1 θ 2 −→
(3.2) 0 (K⊗2)⋊ Z J ⋊ Z M (K ) 0.
θ 2 θ 2 2 n+1
−→ −→ −→ ⊗O −→
Using (3.2), we get the 6-term exact sequence
Z2 K (J ⋊ Z ) Z
0 θ 2 n
−−−→ −−−→
δ,
x
0 K (J ⋊ Z ) 0
1 θ 2 y
←−−− ←−−−
which implies K (J ⋊ Z ) = 0 . The generators of Z2 and Z above
1 θ 2 n
are [(p p)e ], [(p p)e ] and{[p} 1] respectively. In K (J ⋊ Z ), we
+ − 0 θ 2
⊗ ⊗ ⊗
have
[(p p)e ]+[(p p)e ] = [p p] = n[p 1],
+ −
⊗ ⊗ ⊗ − ⊗
which shows that K (J⋊Z ) is isomorphic to Z2 with generators [p 1]
0 2
⊗
and [(p p)e ].
+
⊗
(3.1) implies the 6-term exact sequence
Z2 Z2 K (( ⊗2 )⋊ Z )
−−−→ −−−→ 0 On+1 θ 2
δ ,
x
K (( ⊗2)⋊ Z ) 0 0
1 On+1 θ 2 ←−−− ←−−− y
where we use the facts
K (( ⊗2 )⋊ Z ) = [e ],[e ] = Z2,
0 En+1 θ 2 h + − i ∼
K (( ⊗2 )⋊ Z ) = 0 ,
1 En+1 θ 2 ∼ { }
which follows from Theorem 2.1 as is KK-equivalent to C (see [9]
n+1
E
for example).
Now we compute the image of the generators of K (J ⋊ Z ) in
0 θ 2
K (( ⊗2 )⋊ Z ). It is immediate to get
0 En+1 θ 2
[p 1] = n[1] = n([e ]+[e ]).
+ −
⊗ − −
6 M. IZUMI
To finish the proof, it suffices to show
n2 n n2 +n
[(p p)e ] = − [e ]+ [e ],
+ + −
⊗ 2 2
as it implies
n n2−n
K0((On⊗+21)⋊θ Z2) ∼= coker((cid:18) −n n22+n (cid:19) : Z2 → Z2),
− 2
n n2−n
K1((On⊗+21)⋊θ Z2) ∼= ker((cid:18) −n n22+n (cid:19) : Z2 → Z2),
− 2
and the theorem follows from Lemma 3.3.
We have
(p p)e
+
⊗
n
= e (1 T T∗ +T T∗ 1 T T∗ T T∗)e
+ − ⊗ i i i i ⊗ − i i ⊗ i i +
Xi=0
+ (T T∗ T T∗ +T T∗ T T∗)e
i i ⊗ j j j j ⊗ i i +
Xi<j
n
= e (T T∗ T T∗)e
+ − i i ⊗ i i +
Xi=0
n
(1 T T∗) T T∗ +T T∗ (1 T T∗) e
− − i i ⊗ i i i i ⊗ − i i +
Xi=0(cid:0) (cid:1)
+ (T T∗ T T∗ +T T∗ T T∗)e .
i i ⊗ j j j j ⊗ i i +
Xi<j
For 0 i n, we set
≤ ≤
P = (1 T T∗) T T∗ +T T∗ (1 T T∗) e ,
i − i i ⊗ i i i i ⊗ − i i +
(cid:0) (cid:1)
which are mutually commuting projections. For distinct i,j,k, we have
P P P = 0, and
i j k
P P = (T T∗ T T∗ +T T∗ T T∗)e .
i j i i ⊗ j j j j ⊗ i i +
THE K-THEORY OF THE FLIP AUTOMORPHISMS 7
Thus
n
(1 T T∗) T T∗ +T T∗ (1 T T∗) e
− i i ⊗ i i i i ⊗ − i i +
Xi=0(cid:0) (cid:1)
(T T∗ T T∗ +T T∗ T T∗)e
− i i ⊗ j j j j ⊗ i i +
Xi<j
n−1 n
= (P P P )+P
i i j n
−
Xi=0 jX=i+1
is a projection orthogonal to (T T∗ T T∗)e , and
i i ⊗ i i +
[(p p)e ]
+
⊗
n n−1 n
= [e ] [(T T∗ T T∗)e ] [P P P ] [P ]
+ − i i ⊗ i i + − i − i j − n
Xi=0 Xi=0 jX=i+1
n
= n[e ] [P ]+ [P P ].
+ i i j
− −
Xi=0 Xi<j
We set
(1 T T∗) T +T (1 T T∗)
V = − i i ⊗ i i ⊗ − i i e
i +
√2
(1 T T∗) T T (1 T T∗)
+ − i i ⊗ i − i ⊗ − i i e .
−
√2
Then V V∗ = P and
i i i
(1 T T∗) 1+1 (1 T T∗)
V∗V = − i i ⊗ ⊗ − i i e
i i 2 +
(1 T T∗) 1+1 (1 T T∗)
+ − i i ⊗ ⊗ − i i e
−
2
(1 T T∗) 1 1 (1 T T∗)
+ − i i ⊗ − ⊗ − i i e
+
2
(1 T T∗) 1 1 (1 T T∗)
+ − i i ⊗ − ⊗ − i i e
−
2
= (1 T T∗) 1.
− i i ⊗
Thus we get [P ] = 0.
i
For i = j we define
6
T T +T T T T T T
i j j i i j j i
V = ⊗ ⊗ e + ⊗ − ⊗ e .
ij + −
√2 √2
8 M. IZUMI
Then direct computation yields V V∗ = P P and V∗V = 1, which
ij ij i j ij ij
shows
[P P ] = [1].
i j
Thus we get
n(n+1) n2 n n2 +n
[(p p)e ] = n[e ]+ [1] = − [e ]+ [e ],
+ + + −
⊗ − 2 2 2
(cid:3)
which finishes the proof.
3.3. ((SC)⊗2)⋊ Z .
θ 2
Theorem 3.5. With the above notation, we have
K (((SC)⊗2)⋊ Z ) = 0 ,
0 θ 2
{ }
K (((SC)⊗2)⋊ Z ) = Z.
1 θ 2
The flip θ acts on K ((SC)⊗2) = Z by 1 and the dual action θˆ acts
0 ∼
on K (((SC)⊗2)⋊ Z ) = Z by 1. −
1 θ 2 ∼
−
Proof. It suffices to work on C (R)⋊ Z , where the action α is given
0 α 2
by α(f)(t) = f( t), because we have
−
((SC)⊗2)⋊ Z = S(C (R)⋊ Z ).
θ 2 ∼ 0 α 2
Considering the evaluation map at 0, we get the short exact sequence
0 C ((0, )) C (( ,0)) C (R) C 0.
0 0 0
−→ ∞ ⊕ −∞ −→ −→ −→
and
0 M (C ((0, ))) C (R)⋊ Z CZ 0,
2 0 0 α 2 2
−→ ∞ −→ −→ −→
which implies the 6-term exact sequence
0 K (C (R)⋊ Z ) Z2
0 0 α 2
−−−→ −−−→
δ .
x
0 K (C (R)⋊ Z ) Z
1 0 α 2 y
←−−− ←−−−
Let f C (R) be a real function satisfying f(0) = 1, f(x) = f( x).
0
∈ −
Then, we have
δ([e ]) = [e2πife +e ] = [e2πife +e ] = δ([e ]),
+ + − − + −
which is the generator of K (M (C (R ))). Thus
1 2 0 >0
K (C (R)⋊ Z ) = Z,
0 0 α 2 ∼
K (C (R)⋊ Z ) = 0 .
1 0 α 2 ∼
{ }
The generator of K (C (R) ⋊ Z ) is the preimage of [e ] [e ], on
0 0 α 2 + −
−
which the dual action θˆacts by 1. (cid:3)
−
THE K-THEORY OF THE FLIP AUTOMORPHISMS 9
Remark 3.6. The generator of K (C (R) ⋊ Z ) is the preimage of
0 0 α 2
[e ] [e ] = 2[e ] [1], and it can be written down explicitly as follows.
+ − +
Let −h be continuou−s monotone increasing real function on R satisfying
h( x) = h(x) and
− − π
lim h(t) = .
t→±∞ ±2
We set c(t) = cosh(t), s(t) = sinh(t). Then, the generator is
1+sc+c2λ s2−scλ
[ 2 2 ] [1].
(cid:18) s2−λsc 1−sc+c2λ (cid:19) −
2 2
For later use, we give a more detailed statement actually proved in
the proof of Theorem 3.5.
Lemma 3.7. Let
R : ((SC)⊗2)⋊ Z (SC)⋊ Z = S(CZ )
θ 2 id 2 2
→
be the homomorphism arising from the restriction map to the diago-
nal set, and let η be the natural isomorphism from K (S(CZ )) onto
1 2
K (CZ ) = Z2. Then the map η R : K (((SC)⊗2)⋊ Z ) K (CZ )
0 2 ∼ ∗ 1 θ 2 0 2
is injective and its image is Z([e◦] [e ]). →
+ −
−
There exists a unique separable unital simple purely infinite C∗-
algebra KK-equivalent to SC, which is denoted by . Its tensor
∞
square isMoritaequivalentto ,andθactsoPnK ( ⊗2) = Z
∞ ∞ ∞ 0 ∞ ∼
non-trivPial⊗ly.PTheorem 2.1 and TheoreOm 3.5 imply that ( P⊗2) ⋊ Z
∞ θ 2
P
is isomorphic to , and the dual action θˆ acts on K (( ⊗2) ⋊ Z )
∞ 1 ∞ θ 2
P P
non-trivially.
Problem 3.8. Let A be a unital C∗-algebra Morita equivalent to
∞
O
in the Cuntz standard form, that is, the K -class of the identity is 0.
0
Let α be a Z -action on A acting on K (A) non-trivially. Can we
2 0
say something about the isomorphism class of A ⋊ Z ? Is A ⋊ Z
α 2 α 2
isomorphic to ?
∞
P
3.4. (D⊗2)⋊ Z . Let D be the dimension drop algebra, that is,
n θ 2 n
D = f C ([0,1),M ); f(0) C1 .
n 0 n
{ ∈ ∈ }
We identify SM with f D ; f(0) = 0 . We denote by e
n n ij 1≤i,j≤n
the canonical system of{ma∈trix units of M}. There exists a {shor}t exact
n
sequence
0 SM D C 0,
n n
−→ −→ −→ −→
which implies
K (D ) = 0 ,
0 n ∼
{ }
K (D ) = u = Z ,
1 n ∼ n
h i
10 M. IZUMI
where u(x) = e2πixe +1 e .
11 11
−
Theorem 3.9. Let the notations be as above.
(1) When n is odd,
K ((D⊗2)⋊ Z ) = 0 ,
0 n θ 2 ∼ { }
K ((D⊗2)⋊ Z ) = Z Z .
1 n θ 2 ∼ n ⊕ n
(2) When n is even,
K ((D⊗2)⋊ Z ) = 0 ,
0 n θ 2 ∼ { }
K ((D⊗2)⋊ Z ) = Z Z .
1 n θ 2 ∼ 2n ⊕ n/2
Proof. We realize D D inside C([0,1]2,M M ), and define a
n n n n
⊗ ⊗
closed ideal J of D D by
n n
⊗
J = SM D +D SM = f D D ; f(0,0) = 0 .
n n n n n n
⊗ ⊗ { ∈ ⊗ }
Then we have two short exact sequences
0 J D D C 0,
n n
−→ −→ ⊗ −→ −→
0 SM SM J SM C C SM 0,
n n n n
−→ ⊗ −→ −→ ⊗ ⊕ ⊗ −→
which imply the two short exact sequences
(3.3) 0 J ⋊ Z (D⊗2)⋊ Z CZ 0,
−→ θ 2 −→ n θ 2 −→ 2 −→
(3.4) 0 ((SM )⊗2)⋊ Z J ⋊ Z SM 0.
n θ 2 θ 2 2n
−→ −→ −→ −→
From (3.4) and Theorem 3.5, we get the 6-term exact sequence
0 K (J ⋊ Z ) 0
0 θ 2
−−−→ −−−→
(3.5) δ ,
x
Z K (J ⋊ Z ) Z
1 θ 2 y
←−−− ←−−−
and
K (J ⋊ Z ) = 0 ,
0 θ 2 ∼
{ }
K (J ⋊ Z ) = Z2.
1 θ 2 ∼
We determine the generators of the latter now. Let ι be the inclusion
map ι : ((SM )⊗2)⋊ Z J ⋊ Z and let π be quotient map
n θ 2 θ 2
→
π : J ⋊ Z (SM C C SM )⋊ Z = M (SM ) = SM .
θ 2 n n θ 2 ∼ 2 n ∼ 2n
→ ⊗ ⊕ ⊗
Note that π restricted to J is given by the direct sum of the restriction
maptoy = 0andthatofx = 0. Wesetw = Fe +e withF J+1 as
+ − n
∈
