CUBO A Mathematical Journal
Vol.14, No¯ 02, (183–196). June 2012

K-theory for the C∗-algebras of continuous functions on
certain homogeneous spaces in semi-simple Lie groups

Takahiro Sudo

Department of Mathematical Sciences,

Faculty of Science, University of the Ryukyus, Senbaru 1,

Nishihara, Okinawa 903-0213, Japan.

email: sudo@math.u-ryukyu.ac.jp

ABSTRACT

We study K-theory for the C∗-algebras of all continuous functions on certain homoge-

neous spaces in the semi-simple connected Lie groups SLn(R) by the discrete subgroups

SLn(Z), mainly. As a byproduct, we also consider a certain nilpotent case similarly.

RESUMEN

Estudiamos la K-teoŕıa para las C∗-álgebras de todas las funciones continuas sobre

ciertos espacios homogéneos, principalmente en los grupos de Lie conexos semi- sim-

ples SLn(R) y subgrupos discretos SLn(Z). Como subproducto consideramos un caso

nilpotente en forma análoga.

Keywords and Phrases: C*-algebra, K-theory, homogeneous space, semi-simple Lie group,

discrete subgroup.

2000 AMS Mathematics Subject Classification: Primary 46L80, 22D25, 22E15.



184 Takahiro Sudo CUBO
14, 2 (2012)

1 Introduction

This work is started with an attempt to find a candidate for the K-theory groups for the full

or reduced group C∗-algebras of the discrete groups SLn(Z). Our idea comes from the fact that

K-theory for the group C∗-algebra of the discrete groups Zn of integers is the same as that for

the C∗-algebra of all continuous functions on the tori Tn viewed as the quotient Rn/Zn, via the

Fourier transform, and that this picture should have some similar meanings in more general or

noncommutative setting, at least in K-theory level.

Refer to [5] for some basics of K-theory and C∗-algebras.

After a quick review in Section 2 about the abelian case of commutative connected Lie groups,

we consider in Section 3 homogeneous spaces in SL2(R) a semi-simple connected Lie group and

compute the K-theory groups of the C∗-algebras of all continuous functions on those spaces. More-

over, we consider the case of SLn(R) (n ≥ 3) in Section 4. The results obtained would be useful

for further research in this direction. Furthermore, as a byproduct, we consider a certain nilpotent

case of discrete Heisenberg groups.

2 Abelian case

For convenience, recall that we have the following short exact sequence of abelian (or commutative

Lie) groups:

0 → Zn → Rn → Tn → 0.

Consider their group C∗-algebras C∗(Zn), C∗(Rn), and C∗(Tn). By Fourier transform, they are

isomorphic respectively to C(Tn), C0(R
n), and C0(Z

n) the C∗-algebras of all continuous functions

on Tn, on Rn and Zn vanishing at infinity. Their K-theory groups are well known as follows ([5]):

Kj(C
∗
(Z

n
)) ∼= Kj(C(T

n
)) ∼= Z

2
n−1

, (j = 0, 1);

K0(C
∗(R2n)) ∼= K0(C0(R

2n)) ∼= K0(C) ∼= Z, K1(C
∗(R2n)) ∼= K1(C) ∼= 0,

K0(C
∗(R2n−1)) ∼= K0(C0(R

2n−1)) ∼= K1(C) ∼= 0, K1(C
∗(R2n−1)) ∼= K0(C) ∼= Z,

K0(C
∗(Tn)) ∼= K0(C0(Z

n)) ∼= ⊕
Z
n

Z, K1(C
∗(Tn)) ∼= K1(C0(Z

n)) ∼= 0,

where ⊕k means the k-times direct sum. Observe that K-theory of the group C∗-algebra of the

discrete group Zn is the same as that of the C∗-algebra of all continuous functions on the quotient

T
n = Rn/Zn.

3 Homogeneous spaces in SL2(R)

Consider the following inclusion and its homogeneous space denoted as:

0 → SL2(Z) → SL2(R), SL2(R)/SL2(Z) ≡ H2.



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K-theory for the C∗-algebras of continuous functions ... 185

Let SL2(R) = KAN be the Iwasawa decomposition. More precisely, we have the following homeo-

morphism:

SL2(R) ≈ KAN = SO(2)A2N2,

where SO(2) =

{(
cos θ − sin θ

sin θ cos θ

)

| θ ∈ R

}

∼= S
1 = {eiθ | θ ∈ R},

A2 =

{(
a1 0

0 a2

)

| a1a2 = 1, a1 > 0, a2 > 0

}

,

N2 =

{(
1 b

0 1

)

| b ∈ R

}

.

It follows that

SL2(Z) ≈ KZAZNZ = SO(2)ZA2,ZN2,Z,

where SO(2)Z ∼= S
1
Z
= {eiθ ∈ Z2 | θ ∈ R} = {(±1, 0), (0, ±1)},

A2,Z =

{(
1 0

0 1

)}

,

N2,Z =

{(
1 b

0 1

)

| b ∈ Z

}

.

It follows from considering quotient spaces that the homogeneous space H2 is homeomorphic to

the following product space:

H2 ≈ (⊔
4
R)+ × R × T,

where ⊔kR means the disjoint union of k copies of R, and X+ means the one-point compactification

of X, and SO(2)/SO(2)Z ≈ (⊔
4
R)+, and A2 ≈ R, and N2/N2,Z ≈ T.

Let C0(H2) be the C
∗-algebra of all continuous functions on H2 vanishing at infinity. We

compute its K-theory groups as follows. First of all, we have

Kj(C0(H2)) ∼= Kj(C0((⊔
4
R)+ × R × T))

∼= Kj+1(C((⊔
4
R)+ × T)),

by the Bott periodicity, where j + 1 (mod 2). Consider the following short exact sequence of

C∗-algebras:

0 −−−−→ C0((⊔
4
R) × T)

i
−−−−→ C((⊔4R)+ × T)

q
−−−−→ C(T) −−−−→ 0.

Note that this extension of C∗-algebras splits, clearly. We then have the following six-term exact

sequence of K-groups:

K0(C0((⊔
4
R) × T))

i∗
−−−−→ K0(C((⊔

4
R)+ × T))

q∗
−−−−→ K0(C(T))

x









y

K1(C(T))
q∗

←−−−− K1(C((⊔
4
R)+ × T))

i∗
←−−−− K1(C0((⊔

4
R) × T)),



186 Takahiro Sudo CUBO
14, 2 (2012)

with

Kj(C0((⊔
4
R) × T)) ∼= ⊕

4Kj(C0(R × T))

∼= ⊕
4Kj+1(C(T)) ∼= Z

4

for j = 0, 1, where ⊕k means the direct sum of k copies. The commutative diagram also splits into

two short exact sequences of K0 and K1-groups, by the splitting short exact sequence of C
∗-algebras.

Therefore, we obtain

0 → Z4 → Kj(C((⊔
4
R)

+
× T)) → Z → 0

for j = 0, 1. Since extensions of groups by Z also split, certainly known, we obtain that Kj(C((⊔
4
R)+×

T)) ∼= Z5 for j = 0, 1. Hence we get

Theorem 3.1. Let H2 = SL2(R)/SL2(Z) = KAN/KZAZNZ be the homogeneous space via the

Iwasawa decomposition. Then H2 is homeomorphic to the product space (⊔
4
R)+ × R × T, and

Kj(C0(H2)) ∼= Z
5, (j = 0, 1).

Moreover, we obtain

Proposition 3.2. Let K/KZ = SO(2)/SO(2)Z = KAN/KZAN be the homogeneous space of the

compact group SO(2). Then K/KZ is the compact space (⊔
4
R)+, and

K0(C(K/KZ)) ∼= Z and K1(C(K/KZ)) ∼= Z
4.

Proof. Consider the following short exact sequence of C∗-algebras:

0 −−−−→ C0(⊔
4
R)

i
−−−−→ C((⊔4R)+)

q
−−−−→ C −−−−→ 0.

Note that this extension of C∗-algebras splits. We then have the following six-term exact sequence

of K-groups:

K0(C0(⊔
4
R))

i∗
−−−−→ K0(C((⊔

4
R)+))

q∗
−−−−→ K0(C)

x









y

K1(C)
q∗

←−−−− K1(C((⊔
4
R)+))

i∗
←−−−− K1(C0(⊔

4
R)),

with

Kj(C0((⊔
4
R))) ∼= ⊕

4Kj(C0(R)) ∼= ⊕
4Kj+1(C)

for j = 0, 1. The commutative diagram also splits into two short exact sequences of K0 and

K1-groups. Therefore, we obtain that K0(C((⊔
4
R)+)) ∼= Z and K1(C((⊔

4
R)+)) ∼= Z4.

Remark. Note that the quotient space N/NZ is isomorphic to T as a group. Thus, Kj(C(N/NZ)) ∼=

Z for j = 0, 1.

Furthermore, we have



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K-theory for the C∗-algebras of continuous functions ... 187

Proposition 3.3. The homogeneous space SL2(R)/K = AN is homeomorphic to the product space

R × T, and Kj(C0(AN)) ∼= Z for j = 0, 1.

Proof. We have

Kj(C0(R × T)) ∼= Kj+1(C(T)) ∼= Z

for j = 0, 1.

Notes. It is shown by Natsume [2] that for C∗(SL2(Z)) the full group C
∗-algebra of SL2(Z),

K0(C
∗(SL2(Z))) ∼= Z

8, K1(C
∗(SL2(Z))) ∼= 0,

and the same holds by replacing C∗(SL2(Z)) with its reduced group C
∗-algebra of the regular

representation of SL2(Z).

More precisely, since SL2(Z) is isomorphic to the amalgam Z4 ∗Z2 Z6 of cyclic groups with

orders 2, 4, 6, we have C∗(SL2(Z)) isomorphic to the amalgam C
∗(Z4)∗C∗(Z2) C

∗(Z6) of their group

C∗-algebras, so that

Kj(C
∗(Z4) ∗C∗(Z2) C

∗(Z6)) ∼= (Kj(C
∗(Z4)) ⊕ Kj(C

∗(Z6)))/Kj(C
∗(Z2))

for j = 0, 1. In particular, K0(C
∗(SL2(Z))) ∼= Z

8 ∼= Z10/Z2. Also,

Kj(C
∗(Z4) ∗ C

∗(Z6)) ∼= Kj(C
∗(Z4)) ⊕ Kj(C

∗(Z6))

for j = 0, 1, where C∗(Z4)∗C
∗(Z6) is the full free product of C

∗-algebras. More generally, for A∗B

the full free product of C∗-algebras A and B, we have ([1])

Kj(A ∗ B) ∼= Kj(A) ⊕ Kj(B), (j = 0, 1).

Corollary 1. We have

K0(C0(H2)) ⊕ K1(C0(H2)) ∼= K0(C
∗
(Z4) ∗ C

∗
(Z6)) ⊕ K1(C

∗
(Z4) ∗ C

∗
(Z6)),

as a group, but

K0(C0(H2)) ⊕ K1(C0(H2)) 6∼= K0(C
∗(SL2(Z))) ⊕ K1(C

∗(SL2(Z))).

Remark. Since 10 > 8, it may say to be possible that K-theory data of the homogeneous space

C∗-algebra contains that of the group C∗-algebra of SL2(Z). In fact, in the group non-isomorphic

equation above, the right hand side can be a quotient of the left hand side. This picture might be

extended to the more general setting.



188 Takahiro Sudo CUBO
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4 Homogeneous spaces in SLn(R)

Consider the following inclusion and its homogeneous space denoted as:

0 → SLn(Z) → SLn(R), SLn(R)/SLn(Z) ≡ Hn.

Let SLn(R) = KAN be the Iwasawa decomposition. More precisely, we have the following homeo-

morphism:

SLn(R) ≈ KAN = SO(n)AnNn,

where An =














a1 0

...

0 an









| Πnj=1aj = 1, aj > 0






,

Nn =




















1 b12 · · · b1n
...

...
...

... bn−1,n

0 1















| bi,j ∈ R, (i < j)






.

It follows that

SLn(Z) ≈ KZAZNZ = SO(n)ZAn,ZNn,Z,

where SO(n)Z consists of all matrices of SO(n) with components of integers, An,Z of only the

n-th identity matrix, and Nn,Z of all matrices of Nn with components of integers. It follows from

considering quotient spaces that the homogeneous space Hn is homeomorphic to the following

product space:

Hn ≈ (SO(n)/SO(n)Z) × R
n−1

× T
(n−1)n

2 ,

where An ≈ R
n−1 and Nn/Nn,Z ≈ T

(n−1)n

2 .

Recall that as a topological space,

SO(n)/SO(n − 1) ≈ Sn−1,

where Sn−1 is the n − 1 dimensional sphere. Indeed, SO(n) acts transitively on Sn−1 by matrix

multiplication, and the isotropy group for the n-th standard basis vector in Sn−1 is SO(n − 1),

from which the homeomorphism is obtained. However, these quotient spaces do not split in general

into the product spaces:

SO(n) ≈ SO(n − 1) × Sn−1,

but this is certainly true if and only if there is a continuous section from Sn−1 to SO(n). This

is just the cases where n = 4 or n = 8, a well-known, non-tirvial, important result in algebraic

topology. Note that what is necessary in what follows may be the isomorphisms in topological

K-theory level:

Kj(SO(n)) ∼= K
j
(SO(n − 1) × Sn−1)



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K-theory for the C∗-algebras of continuous functions ... 189

(or mere replacements).

We have shown that SO(2)/SO(2)Z ≈ S
1/S1

Z
. If we assume the homeomorphisms for SO(n),

inductively we have

SO(n)/SO(n)Z ≈ (SO(n − 1)/SO(n − 1)Z) × (S
n−1/Sn−1

Z
),

where Sn−1
Z

means the set of all integral points in Sn−1, and the equivalence relation on Sn−1 by

Sn−1
Z

is defined as: for ξ, η ∈ Sn−1, we have ξ ∼ η if and only if ξ = η, or ξ, η ∈ Sn−1
Z

. Therefore,

we obtain

SO(n)/SO(n)Z ≈ (S1/SZ) × · · · × (S
n−1/Sn−1

Z
).

However, this may not be true in general, but even in such a case, we may replace SO(n)/SO(n)Z

by the product space in the right hand side, as a reasonable candidate, and we continue. But what

is necessary in what follows may be the isomorphisms in topological K-theory level:

Kj(SO(n)/SO(n)Z) ∼= K
j((SO(n − 1)/SO(n − 1)Z) × (S

n−1/Sn−1
Z

))

(or mere replacements).

We also have

Sn−1
Z

= {(±1, 0, · · · , 0), (0, ±1, 0, · · · , 0), · · · , (0, · · · , 0, ±1) ∈ Rn}.

Hence we identify Sn−1
Z

with ⊔nZ2 the n-fold disjoint union of Z2 = Z/2Z. Therefore, we get

Sn−1/Sn−1
Z

≈ Sn−1/ ⊔n Z2.

Let C0(Hn) be the C
∗-algebra of all continuous functions on Hn vanishing at infinity. We

compute its K-theory groups as follows. First of all, we have

Kj(C0(Hn)) ∼= Kj(C0((SO(n)/SO(n)Z) × R
n−1 × T

(n−1)n

2 ))

∼= Kj+n−1(C(SO(n)/SO(n)Z) × T
(n−1)n

2 )),

by the Bott periodicity, where j + n − 1 (mod 2).

Now let Sn = SO(n)/SO(n)Z and Tn = T
(n−1)n

2 . Since C(Sn × Tn) ∼= C(Sn) ⊗ C(Tn) a

C∗-tensor product, the Künneth formula implies

K0(C(Sn × Tn)) ∼= (K0(C(Sn)) ⊗ K0(C(Tn))) ⊕ (K1(C(Sn)) ⊗ K1(C(Tn))),

K1(C(Sn × Tn)) ∼= (K0(C(Sn)) ⊗ K1(C(Tn))) ⊕ (K1(C(Sn)) ⊗ K0(C(Tn))).

For j = 0, 1, we have

Kj(C(Tn)) = Kj(C(T
(n−1)n

2 )) ∼= Z
2
2−1(n−1)n−1

= Z
2
2−1(n−2)(n+1)

.

Let Sk/Sk
Z
= Vk for 1 ≤ k ≤ n − 1 and (S

1/S1
Z
) × · · · × (Sk/Sk

Z
) = Uk. Since we have

C((S1/SZ) × · · · × (S
n−1/Sn−1

Z
)) ∼= C(S1/SZ) ⊗ · · · ⊗ C(S

n−1/Sn−1
Z

),



190 Takahiro Sudo CUBO
14, 2 (2012)

the Künneth formula implies that, for instance,

K0(C(U3)) ∼= ⊕(i1,i2,i3)∈I3Ki1(C(V1)) ⊗ Ki2(C(V2)) ⊗ Ki3(C(V3)),

K1(C(U3)) ∼= ⊕(j1,j2,j3)∈J3Kj1(C(V1)) ⊗ Kj2(C(V2)) ⊗ Kj3(C(V3)),

where

I3 = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)},

J3 = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)},

where note that for each tuple in I3, the number of 0 is 3 or 1 odd, while for each tuple in J3, the

number of 0 is 2 or 0 even, and the cardinal numbers of I3 and J3 are computed as:

|I3| = 3C3 + 3C1 = 1 + 3 = 2
2, |J3| = 3C2 + 3C0 = 3 + 1 = 2

2,

where nCk means the combination of k elements in n elements. As one more example, similarly,

|I4| = 4C4 + 4C2 + 4C0 = 1 + 6 + 1 = 2
3,

|J4| = 4C3 + 4C1 = 4 + 4 = 2
3.

Therefore, more generally, we have

K0(C(Uk)) ∼= ⊕(i1,··· ,ik)∈IkKi1(C(V1)) ⊗ · · · ⊗ Kik(C(Vk)),

K1(C(Uk)) ∼= ⊕(j1,··· ,jk)∈JkKj1(C(V1)) ⊗ · · · ⊗ Kjk(C(Vk)),

where if k is even, then

|Ik| = kCk + kCk−2 + · · · + kC0 = 2
k,

|Jk| = kCk−1 + kCk−3 + · · · + kC1 = 2
k.

and if k is odd, then

|Ik| = kCk + kCk−2 + · · · + kC1 = 2
k,

|Jk| = kCk−1 + kCk−3 + · · · + kC0 = 2
k,

and in both cases, Ik and Jk consist of tuples with elements 0 or 1 chosen accordingly to the above

combinatorial sums.

Note that the quotient space Vk−1 is just

Vk−1 = S
k−1/ ⊔k Z2 = (S

k−1 \ (⊔kZ2))
+ ≡ V+k

the one-point compactification V+k of the open subspace Vk of S
k−1obtained by removing points

of ⊔nZ2 from S
k−1.

Consider the following short exact sequence of C∗-algebras:

0 −−−−→ C0(Vk)
i

−−−−→ C(V+k )
q

−−−−→ C −−−−→ 0.



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K-theory for the C∗-algebras of continuous functions ... 191

Note that this extension of C∗-algebras splits, clearly. We then have the following six-term exact

sequence of K-groups:

K0(C0(Vk))
i∗

−−−−→ K0(C(V
+
k
))

q∗
−−−−→ K0(C)

x









y

K1(C)
q∗

←−−−− K1(C(V
+
k
))

i∗
←−−−− K1(C0(Vk)),

and the commutative diagram also splits into two short exact sequences of K0 and K1-groups. It

follows that

K0(C(V
+
k ))

∼= K0(C0(Vk)) ⊕ Z, K1(C(V
+
k ))

∼= K1(C0(Vk)).

Moreover consider the following short exact sequence of C∗-algebras:

0 −−−−→ C0(Vk)
i

−−−−→ C(Sk−1)
q

−−−−→ ⊕2kC −−−−→ 0

corresponding to attaching 2k points to 2k holes in Vk to make S
k−1. We then have the following

six-term exact sequence of K-groups:

K0(C0(Vk))
i∗

−−−−→ K0(C(S
k−1))

q∗
−−−−→ ⊕2kK0(C)

x









y

⊕2kK1(C)
q∗

←−−−− K1(C(S
k−1))

i∗
←−−−− K1(C0(Vk)).

Furthermore consider the following short exact sequence of C∗-algebras:

0 −−−−→ C0(R
k−1)

i
−−−−→ C(Sk−1)

q
−−−−→ C −−−−→ 0,

where note that Sk−1 ≈ (Rn−1)+. Note that this extension of C∗-algebras splits, clearly. We then

have the following six-term exact sequence of K-groups:

K0(C0(R
k−1))

i∗
−−−−→ K0(C(S

k−1))
q∗

−−−−→ K0(C)
x









y

K1(C)
q∗

←−−−− K1(C(S
k−1))

i∗
←−−−− K1(C0(R

k−1))

and the commutative diagram also splits into two short exact sequences of K0 and K1-groups. It

follows that for k ≥ 2,

K0(C(S
k−1

)) ∼= K0(C0(R
k−1

)) ⊕ Z ∼=

{
Z if k even,

Z
2 if k odd;

K1(C(S
k−1)) ∼= K1(C0(R

k−1)) ∼=

{
Z if k even,

0 if k odd.



192 Takahiro Sudo CUBO
14, 2 (2012)

Therefore, we obtain that if k is even, then

K0(C0(Vk))
i∗

−−−−→ Z
q∗

−−−−→ ⊕2kZ
x









y

0
q∗

←−−−− Z
i∗

←−−−− K1(C0(Vk))

and if k is odd, then

K0(C0(Vk))
i∗

−−−−→ Z2
q∗

−−−−→ ⊕2kZ
x









y

0
q∗

←−−−− 0
i∗

←−−−− K1(C0(Vk)).

In both cases, the K0-class corresponding to the unit of C(S
k−1) is mapped injectively under the

map q∗, while the K0-class corresonding to the Bott projection in a matrix algebra over C(S
k−1)

for k odd is mapped to zero under q∗. It follows that if k is even, then K0(C(Vk)) ∼= 0, while if k

is odd, then K0(C(Vk)) ∼= Z. Therefore, we obtain that if k is even, then K1(C0(Vk)) ∼= Z
2k, and

if k is odd, then K1(C0(Vk)) ∼= Z
2k−1. Hence we get

K0(C(Vk−1)) ∼= K0(C(V
+
k ))

∼=

{
Z if k even,

Z
2 if k odd;

K1(C(Vk−1)) ∼= K1(C(V
+
k ))

∼=

{
Z
2k if k even,

Z
2k−1 if k odd.

Note that the case where k = 2 is considered in the previous section.

Summing up the argument above, we obtain

Theorem 4.1. Let Hn = SLn(R)/SLn(Z) = KAN/KZAZNZ be the homogeneous space via the

Iwasawa decomposition. Then Hn is homeomorphic to the product space (SO(n)/SO(n)Z)×R
n−1×

T
(n−1)n

2 , and

K0(C0(Hn)) ∼= K1(C0(Hn))

∼= ⊕j=0,1(Kj(C(SO(n)/SO(n)Z)) ⊗ Z
2
(n−2)(n+1)2−1

).

Proof. If n is even, then

K0(C0(Hn)) ∼= K1(C(SO(n)/SO(n)Z) ⊗ C(T
(n−1)n

2 ))

∼= (K0(C(Tn)) ⊗ Z
2

(n−2)(n+1)
2

) ⊕ K1(C(Tn)) ⊗ Z
2

(n−2)(n+1)
2

),

K1(C0(Hn)) ∼= K0(C(SO(n)/SO(n)Z) ⊗ C(T
(n−1)n

2 ))

∼= (K0(C(Tn)) ⊗ Z
2

(n−2)(n+1)
2

) ⊕ K1(C(Tn)) ⊗ Z
2

(n−2)(n+1)
2

),



CUBO
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K-theory for the C∗-algebras of continuous functions ... 193

where Tn = SO(n)/SO(n)Z for short, and in particular, we get K0(C0(Hn)) ∼= K1(C0(Hn)).

If n is odd, then we can deduce the same conclusions by the same calculation as above.

Remark. The results obtained above and below in K-theory might contain (some of) K-theory

data for the (full or reduced) group C∗-algebra of SLn(Z) or the (full or reduced) free product

C∗-algebra corresponding to the generators of SLn(Z). It is known that if n ≥ 3, then SLn(Z) is

not an amalgam, but a certain multi-amalgam of subgroups, by Soulé [4].

Moreover, we obtain

Proposition 4.2. Let K/KZ = SO(n)/SO(n)Z = KAN/KZAN be the homogeneous space of the

compact group SO(n). For convenience, as a candidate, we replace K/KZ with the compact product

space:

(S1/S1
Z
) × (S2/S2

Z
) · · · × (Sn−1/Sn−1

Z
),

which is identified with

(S1/ ⊔2 Z2) × (S
2/ ⊔3 Z2) × · · · × (S

n−1/ ⊔n Z2)

≈ (S1 \ ⊔2Z2)
+
× (S2 \ ⊔3Z2)

+
× · · · × (Sn−1 \ ⊔nZ2)

+,

or we may assume that we replace the topological K-theory of K/KZ with that of the product space.

Then

K0(C(K/KZ)) ∼= ⊕(i1,i2,··· ,in−1)∈In−1(Ki1(C(V1)) ⊗ · · · ⊗ Kin−1(C(Vn−1))),

K1(C(K/KZ)) ∼= ⊕(j1,j2,··· ,jn−1)∈Jn−1(Kj1(C(V1)) ⊗ · · · ⊗ Kjn−1(C(Vn−1))),

with Vk = S
k/Sk

Z
, where if n is odd , then

|In−1| = n−1Cn−1 + n−1Cn−3 + · · · + n−1C0 = 2
n−1,

|Jn−1| = n−1Cn−2 + n−1Cn−4 + · · · + n−1C1 = 2
n−1.

and if n is even, then

|In−1| = n−1Cn−1 + n−1Cn−3 + · · · + n−1C1 = 2
n−1,

|Jn−1| = n−1Cn−2 + n−1Cn−4 + · · · + n−1C0 = 2
n−1,

and in both cases, In−1 and Jn−1 consist of the tuples with elements 0 or 1 chosen accordingly to

the above combinatorial sums.

Moreover, we obtain

K0(C(Vk−1)) ∼=

{
Z if k even,

Z
2 if k odd;

K1(C(Vk−1)) ∼=

{
Z
2k if k even,

Z
2k−1 if k odd.



194 Takahiro Sudo CUBO
14, 2 (2012)

Remark. For example, as n = 5 we compute

K0(C(V1)) ⊗ K1(C(V2)) ⊗ K1(C(V3)) ⊗ K0(C(V4))

∼= Z ⊗ Z
3 ⊗ Z6 ⊗ Z2 ∼= Z

3·6·2 = Z36,

where (0, 1, 1, 0) ∈ I4.

Note that the quotient space N/NZ is homeomorphic to T
(n−1)n2

−1

as a space. Thus,

Kj(C(N/NZ)) ∼= Z
2
(n−2)(n+1)2−1

for j = 0, 1.

Furthermore, we have

Proposition 4.3. The homogeneous space SLn(R)/K = AN is homeomorphic to the product space

R
n−1 × T(n−1)n2

−1

, and Kj(C0(AN)) ∼= Z
2
2−1(n−2)(n+1)

for j = 0, 1.

Proof. We have

Kj(C0(R
n−1 × T

(n−1)n

2 )) ∼= Kj+n−1(C(T
(n−1)n

2 )) ∼= Z
2

(n−2)(n+1)
2

for j = 0, 1.

5 Nilpotent case

Recall that the discrete Heisenberg group HZ2n+1 of rank 2n + 1 is defined by

HZ2n+1 =














1 at c

0n 1n b

0 0tn 0









∈ GLn+2(Z) | a, b ∈ Z
n, c ∈ Z






where 1n is the n × n identity matrix, 0n is the zero in Z
n, a, b, 0n are column vectors, and x

t

means the transpose of x. The Heisenberg Lie group HR2n+1 with dimension 2n + 1 is defined by

replacing Z with R in the definition above. Then we have the homogeneous space:

HR2n+1/H
Z

2n+1 ≈ T
2n+1

as a space.

Let C∗(HZ2n+1) be the group C
∗-algebra of HZ2n+1. It is shown by the author [3] that for

j = 0, 1,

Kj(C
∗
(HZ2n+1))

∼= Z
3
n

.

It follows that



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K-theory for the C∗-algebras of continuous functions ... 195

Proposition 5.1. We have

Kj(C(H
R

2n+1/H
Z

2n+1))
∼= Z

2
2n

for j = 0, 1, but for n ≥ 1,

Kj(C(H
R

2n+1/H
Z

2n+1)) 6
∼= Kj(C

∗
(HZ2n+1)).

Proof. Because 22n 6= 3n for n ≥ 1.

Remark. We have 4n > 3n, so that it may say to be possible that K-theory data of the homogeneous

space C∗-algebra contains that of the group C∗-algebra. In fact, in the group non-isomorphic

equation above, the right hand side can be a quotient of the left hand side. This picture might be

extended to the more general setting.

Conjecture. Let Γ be a nilpotent discrete group with rank n. Then we have

rankZKj(C
∗(Γ)) ≤ 2n−1

for j = 0, 1, where rankZ(X) means the Z-rank of X.

Remark. The equality holds if Γ = Zn and the estimate is ture if Γ = HZ2n+1 as checked above.

It is certainly known that a discrete nilpotent group Γ can be viewed as a subgroup of matrices,

i.e. to be linear. Also, it can be viewed as a successive semi-direct products by the abelian groups

Z
kj of integers for some kj ≥ 1 (1 ≤ j ≤ n). In this case, Γ is a subgroup of the connected, simply

connected nilpotent Lie group G obtained as a a successive semi-direct products by Rkj, so that

the homogeneous space G/Γ is homeomorphic to:

G/Γ ≈ T
∑

n
j=1

kj.

Our conjecture says that

rankZKj(C
∗(Γ)) ≤ 2−1+

∑
n
j=1

kj.

Acknowledgement. I would like to thank Shuichi Tsukuda and Michishige Tezuka for providing

some useful (also in the future) information about spheres splitting in algebraic topology,

Received: July 2011. Revised: December 2011.

References

[1] B. Blackadar, K-theory for Operator Algebras, Second Edition, Cambridge, (1998).

[2] T. Natsume, On K∗(C
∗(SL2(Z))), J. Operator Theory 13 (1985), 103-118.



196 Takahiro Sudo CUBO
14, 2 (2012)

[3] T. Sudo, K-theory of continuous fields of quantum tori, Nihonkai Math. J. 15 (2004), No.

2, 141-152.

[4] C. Soulé, The cohomology of SL3(Z), Topology, 17 (1978), 1-22.

[5] N. E. Wegge-Olsen, K-theory and C∗-algebras, Oxford Univ. Press, 1993.


	Introduction
	Abelian case
	Homogeneous spaces in SL2(R)
	Homogeneous spaces in SLn(R)
	Nilpotent case