CUBO A Mathematical Journal
Vol.21, No¯ 03, (01–08). December 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000300001

The K-theory ranks for crossed products of C∗-algebras by
the group of integers

Takahiro Sudo

Department of Mathematical Sciences,

Faculty of Science, University of the Ryukyus, Senbaru 1, Nishihara, Okinawa 903-0213, Japan.

sudo@math.u-ryukyu.ac.jp

ABSTRACT

We study the K-theory ranks for crossed products of C∗-algebras by the group of

integers. As an application, we obtain certain estimates for the K-theory ranks of

the group C∗-algebras of torsion free, finitely generated, nilpotent or solvable discrete

groups, written as successive semi-direct products.

RESUMEN

Estudiamos los rangos de K-teoŕıa para productos cruzados de C∗-álgebras por el grupo

de los enteros. Como aplicación, obtenemos ciertas estimaciones para los rangos de K-

teoŕıa de las C∗-álgebras de grupos libres de torsión, finitamente generados, nilpotentes

o solubles, escritos como productos semidirectos sucesivos.

Keywords and Phrases: K-theory, C*-algebra, crossed product, Betti number, discrete group.

2010 AMS Mathematics Subject Classification: 46L05, 46L55, 46L80

http://dx.doi.org/10.4067/S0719-06462019000300001


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21, 3 (2019)

1 Introduction

In this paper we study the (free or Z) ranks of the K-theory groups for crossed products of

C∗-algebras by Z the group of integers. Such C∗-algebras and their K-theory play fundamental

roles in the theory of C∗-algebras and K-theory (cf. Blackadar [1], Pedersen [2], Tomiyama [10],

Wegge-Olsen [11]). By using the Pimsner-Voiculescu six-term exact sequence (PV) of the K-

theory groups of the crossed product C∗-algebra A ⋊α Z of a C
∗-algebra A by an action α of Z

by automorphisms (Pimsner and Voiculescu [3], cf. [1]), in Section 2 we estimate the K-theory

group ranks of A ⋊α Z in terms of those of A. This simple result should be new in some insight

and interesting in some sense, as another introductory step in this developed research area. As

an easy, direct application of PV, in Section 3 we obtain certain estimates for the K-theory ranks

of the group C∗-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups,

written as successive semi-direct products by torsion free, abelian groups. There may be more other

applications left to be considered, but not so many probably. May as well refer to [5], [6], [7], [8],

[9] for some related details. In particular, in [5], [7], and [9], the K-theory groups of the C∗-algebras

of the generalized Heisenberg discrete nilpotent groups as typical examples of non-type I discrete

amenable groups are computed by some methods of determining K-theory class generators as

projections or unitaries, of the K-theory groups, but it seems that still, the K-theory groups of the

C∗-algebras of general (torsion free, finitely generated) nilpotent (or solvable) discrete groups are

not yet done completely, because of some difficulties involving successive unknown group actions.

However, this time, without determining their K-theory groups as groups, the K-theory group rank

estimates are obtained by us in such a way mentioned above, as the motivated examples, as given

in Section 3.

2 The K-theory ranks for crossed product C∗-algebras by Z

Let A be a C∗-algebra. We denote by A ⋊α Z the crossed product C
∗-algebra of A by an action

α of Z on A by automorphisms, where αn = α
n = α ◦ · · · ◦ α as the n-fold composition of

α = α1 : A → A for n ∈ Z (cf. Blackadar [1], Pedersen [2], Tomiyama [10]). There is the following
Pimsner-Voiculescu six-term exact sequence of the K-theory abelian groups (K0 additive and

K1 multiplicative) (Pimsener and Voiculescu [3], cf. [1]):

K0(A)
(id−α)∗−−−−−→ K0(A)

i∗−−−−→ K0(A ⋊α Z)

∂

x




ind exp





y
∂

K1(A ⋊α Z)
i∗

←−−−− K1(A)
(id−α)∗
←−−−−− K1(A),

where id : A → A is the identity map and i : A → A ⋊α Z is the canonical inclusion map and the

K-theory group maps (id − α)∗ and i∗ are induced by id − α and i, respectively, and the upward



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The K-theory ranks for crossed products of C∗-algebras . . . 3

and downward arrows as the boundary maps ∂ are the index map as ind and the exponential map

as exp, respectively.

It follows from exactness of the PV diagram above that

Lemma 2.1. For any C∗-algebra A and any A ⋊α Z, we have the following short exact sequences:

for j = 0, 1,

0 −→ Kj(A)/(id − α)∗Kj(A) = Kj(A)/ker(i∗)
i∗−→ Kj(A ⋊α Z)

∂−→ im(∂) = ker(id − α)∗ → 0

with (id−α)∗Kj(A) = ker(i∗) ⊂ Kj(A), where (id−α)∗Kj(A) is the image of Kj(A) under (id −α)∗
and ker(id − α)∗ is the kernel of (id − α)∗ on K0 or K1, and im(∂) is the image of the boundary

map ∂ equal to exp or ind.

Let G be an abelian group. We denote by rankZ G the Z-rank (or free rank) of G, which is

also called the Betti number of G, denoted as b(G). For a C∗-algebra A, set bj(A) = b(Kj(A)) for

j = 0, 1, each of which we call the j-th Betti number of A (cf. [6]). We denote by t(G) the torsion

rank of G, which is defined to be the number of direct sum components of indecomposable, finite

cyclic groups in G. Set tj(A) = t(Kj(A)) for j = 0, 1, each of which we may call the j-th torsion

rank of A.

Recall as a fundament fact in group theory that a finitely generated abelian group H has the

following direct product decomposition:

H ∼= Z
b(H) × Z

p
n1
1

× · · · Z
p

nt(H)

t(H)

,

where p1, · · · pt(H) are primes and n1, · · · , nt(H) are some positive integers and each Zpnj
j

=

Z/p
nj
j
Z for 1 ≤ j ≤ t(H) is the finite cyclic group of order pnj

j
, that is indecomposable, and these

powers of primes are distinct.

Lemma 2.2. For a short exact sequence 1 → H → G → G/H → 1 of finitely generated, abelian

groups, we have b(H) ≤ b(G) and b(G/H) ≤ b(G) and b(G) = b(H) + b(G/H).

Proof. Note that there is no homomorphism from a finite torsion group to a torsion free group.

Hence b(H) ≤ b(G), and b(G/H) = b(G) − b(H) ≤ b(G).

Proposition 1. For any A ⋊α Z, we have that for j = 0, 1,

bj(A ⋊α Z) ≤ b0(A) + b1(A)

and b(Kj(A)/(id − α)∗Kj(A)) ≤ bj(A ⋊α Z).



4 Takahiro Sudo CUBO
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Proof. By using the Lemmas 2.1 and 2.2 above, we obtain

bj(A ⋊α Z) = bj(Kj(A)/ker(i∗)) + bj+1(ker(id − α)∗)

≤ bj(Kj(A)) + bj+1(Kj+1(A))

for j = 0, 1 and j + 1 (mod 2), and bj(A ⋊α Z) ≥ bj(Kj(A)/ker(i∗)).

Let G be an abelian group. Let Gf and Gt denote the free and torsion parts of G respectively,

so that G ∼= Gf × Gt with b(G) = b(Gf) and t(G) = t(Gt).

Lemma 2.3. Let G be a finitely generated, abelian group and H a subgroup. Then there is the

following short exact sequence of groups, preserving the free and torsion parts of H and G :

0 → H = Hf × Ht → G = Gf × Gt → G/H = (Gf/Hf) × (Gt/Ht) → 0

with Gt ∼= Ht × (Gt/Ht) and (Gf/Hf)t × (Gt/Ht) ∼= (G/H)t and (Gf/Hf)f = (G/H)f. It then
follows that

t(H) ≤ t(G) ≤ t(H) + t(G/H).

Proof. Note that there are injective maps from Z to Z and from Zk to Zl with k ≤ l, but there
is no injective map from Z to a finite cyclic group. It follows that an injective map from H to G

preserves their free and torsion parts. Note also that Gt/Ht is a torsion group, but Gf/Hf may

have its free part (Gf/Hf)f and torsion part (Gf/Hf)t.

Remark. The inequality t(G/H) ≤ t(G) does not hold in general. For instance, there is a
quotient map from G = Z to Z2 = Z/2Z, with H = 2Z, so that t(H) = t(G) = 0 < 1 = t(G/H) =

t(H) + t(G/H).

Proposition 2. It then follows that for j = 0, 1 ∈ Z2,

tj(A ⋊α Z) ≤ t(Kj(A)/(id − α)∗Kj(A)) + t(ker(id − α)∗)

with ker(id − α)∗ ⊂ Kj+1(A) as a subgroup, and

t(Kj(A)/(id − α)∗Kj(A)) ≤ tj(A ⋊α Z).

Remark. Let A be a C∗-algebra. Set χ(A) = b0(A) − b1(A), which is called the Euler charac-

teristic of A, where we assume that it is defined to be an integer or ±∞ (or formally ∞ − ∞). If
χ(A) and χ(A ⋊α Z) are finite, then it holds that χ(A ⋊α Z) = 0 by using the PV diagram (see [6]

or [8]).

Let A be a C∗-algebra. We denote by A ⋊α(1) Z · · · ⋊α(n) Z the n-fold successive crossed
product C∗-algebra of A by successive actions α(j) of Z (1 ≤ j ≤ n). It then follows that



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The K-theory ranks for crossed products of C∗-algebras . . . 5

Theorem 2.1. For such an n-fold successive crossed product C∗-algebra of a C∗-algebra A by n

successive actions of Z as above or below, we have

bj(A ⋊α(1) Z · · · ⋊α(n) Z) ≤ 2n−1(b0(A) + b1(A))

for j = 0, 1.

Proof. When n = 2, we have

bj(A ⋊α(1) Z ⋊α(2) Z) ≤ b0(A ⋊α(1) Z) + b1(A ⋊α(1) Z)

≤ 2(b0(A) + b1(A)).

When n = 3, we have

bj(A ⋊α(1) Z ⋊α(2) Z ⋊α(3) Z) ≤ b0(A ⋊α(1) Z ⋊α(2) Z) + b1(A ⋊α(1) Z ⋊α(2) Z)

≤ 2[b0(A ⋊α(1) Z) + b1(A ⋊α(1) Z)]

≤ 22(b0(A) + b1(A)).

The general case follows by induction with respect to n.

3 Examples and more

Example 1. Let C(Tn) be the C∗-algebra of all continuous, complex-valued functions on the n-

dimensional torus Tn, which is also the univesal C∗-algebra generated by mutually commuting n

unitaries. The C∗-algebra is regarded as the successive crossed product C∗-algebra of C by trivial

actions id of Z:

C(Tn) ∼= C
∗

(Z
n
) ∼= C ⋊α(1) Z · · · ⋊α(n) Z

with α(j) = id for 1 ≤ j ≤ n, via the Fourier transform from C∗(Zn) to C(Tn), with Tn as the
dual group of Zn. It then follows that

bj(C(T
n
)) ≤ 2n−1(b0(C) + b1(C)) = 2n−1(1 + 0) = 2n−1

for j = 0, 1. Moreover, the estimate equality holds. Because Kj(C(T
n)) ∼= Z2

n−1

(cf. [11]), which

is also deduced by using the Pimsner-Voiculescu six-term exact sequence repeatedly.

Example 2. Let TnΘ denote the n-dimensional noncommutative torus, which is the C
∗-algebra

generated by n unitaries uj such that ujuk = e
2πiθj,kukuj for 1 ≤ j, k ≤ n, where i =

√
−1 and

Θ = (θj,k) is a n × n skew adjoint matrix over R of reals so that −Θ = Θt the transpose of Θ (cf.
[1], [11]). The C∗-algebra is regarded as the successive crossed product C∗-algebra of C by id of Z:

T
n
Θ

∼= C ⋊id Z ⋊α(2) Z · · · ⋊α(n) Z



6 Takahiro Sudo CUBO
21, 3 (2019)

and by successive actions α(j) for 2 ≤ j ≤ n given by

α(j)uk = Ad(uj)uk = ujuku
∗

j = e
2πiθj,kuk

for 1 ≤ k ≤ j − 1. It then follows that

bj(T
n
Θ) ≤ 2

n−1(b0(C) + b1(C)) = 2
n−1(1 + 0) = 2n−1

for j = 0, 1. Moreover, the estimate equality holds. b Because Kj(T
n
Θ)

∼= Z2
n−1

, which is deduced

by using the Pimsner-Voiculescu six-term exact sequence repeatedly. Note that Example 3.1 is

just the case where Θ is the zero matrix.

Example 3. Let H2n+1 be the discrete Heisenberg nilpotent group of rank 2n + 1, consisting of

the following (n + 2) × (n + 2) invertible matrices:

H2n+1 =














1 a c

0n,1 1n b
t

0 01,n 1









∈ GLn+2(R) | a, b ∈ Zn, c ∈ Z






where 1n is the n × n identity matrix and 0j,k is the j × k zero matrix, and with a, b ∈ Zn as
row vectors and bt the transpose of b. The group H2n+1 is viewed as the semi-direct product

Z
n+1

⋊α Z
n of tuples (c, b, a) identified with the matrices above, where the action α is defined by

matrix multiplication as

αa(c, b) = a(c, b)a
−1 = (c +

n∑

j=1

ajbj, b) ∈ Zn+1,

where a = (a1, · · · , an) = (0, 0n, a) and (c, b) = (c, b1, · · · , bn) = (c, b, 0n), with 0n = (0, · · · , 0)
the zero of Zn. Then the group C∗-algebra C∗(H2n+1) = C

∗(Zn+1 ⋊α Z
n) is regarded as the

crossed product C∗-algebra C∗(Zn+1) ⋊α Z
n, where the action α of the semi-direct product group

is extended and identified with that of the crossed product C∗-algebra, by the same symbol as

α (also in what follows). Note that each element of an amenable (such as nilpotent or solvable)

discrete group Γ is identified with the corresponding unitary under the left regular representation

λ on l2(Γ) the Hilbert space of all square summable, complex-valued functions on Γ (cf. [2]). Let

ej (1 ≤ j ≤ 2n + 1) be the canonical basis for Zn+1 and Zn in Zn+1 ⋊α Zn and let uj = λej
(1 ≤ j ≤ 2n + 1) be the corresponding unitaries in C∗(Zn+1 ⋊α Zn). Then we have that

αa(u1) = λαa(e1) = λe1 = u1,

αa(uj) = λαa(ej) = λaj−1e1+ej = u
aj−1
1 uj

for 2 ≤ j ≤ n + 1. It then follows that

bj(C
∗(H2n+1)) ≤ 2n−1(b0(C(Tn+1)) + b1(C(Tn+1)) = 2n−1(2n + 2n) = 22n

for j = 0, 1. In fact, it is computed in [9, Theorem 4.7] that Kj(C
∗(H2n+1)) ∼= Z

2
n
(2

n
−1)+1 for

j = 0, 1, with 2n(2n − 1) + 1 ≤ 22n for n ≥ 1 (cf. [5], [7]).



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The K-theory ranks for crossed products of C∗-algebras . . . 7

Theorem 3.1. Let G be a successive semi-direct product of torsion free, finitely generated discrete

group, written as G = Zn0 ⋊α(1) Z
n1 · · · ⋊α(k) Znk for some n0, · · · , nk ≥ 1, k ≥ 1. Let C∗(G) be

the group C∗-algebra of G. Then bj(C
∗(G)) ≤ 2n0+n1+···+nk−1 for j = 0, 1.

Proof. Note that

C∗(G) ∼= C
∗(Zn0) ⋊α(1) Z

n1 · · · ⋊α(k) Znk

with C∗(Zn0) ∼= C(Tn0), where the right hand side above is viewed as an n1 +· · ·+nk fold, crossed
product C∗-algebra by the successive actions of Z.

Theorem 3.2. Let G be a torsion free, finitely generated nilpotent discrete group, with b(G) = n.

Then bj(C
∗(G)) ≤ 2n−1 for j = 0, 1.

Proof. It is well known that such a nilpotent discrete group can be written as such a successive

semi-direct product as in the theorem above.

Remark. These theorems above partially answer to a question as given in the Remark of [9,

Theorem 4.7]. Note that any torsion free, finitely generated solvable discrete group may be not

be written as such a successive semi-direct product as above, in the sense as neither always being

split nor being supper-solvable with such a normal series (cf. [4]).

Acknowledgement. The author would like to thank the referee for several critical comments and

suggestions for some improvement as in the introduction.



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21, 3 (2019)

References

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

[2] G. K. Pedersen, C∗-Algebras and their Automorphism Groups, Academic Press (1979).

[3] M. Pimsner and D. Voiculescu, Exact sequences for K-groups and Ext-groups of certain

cross-product C∗-algebras, J. Operator Theory 4 (1) (1980), 93-118.

[4] Derek J. S. Robinson, A Course in the Theory of Groups, Second Edition, Graduate

Texts in Math., 80, Springer (1996).

[5] T. Sudo, K-theory of continuous fields of quantum tori, Nihonkai Math. J. 15 (2004),

141-152.

[6] T. Sudo, K-theory ranks and index for C∗-algebras, Ryukyu Math. J. 20 (2007), 43-123.

[7] T. Sudo, K-theory for the group C∗-algebras of nilpotent discrete groups, Cubo A Math. J.

15 (2013), no. 3, 123-132.

[8] T. Sudo, The Euler characteristic and the Euler-Poincaré formula for C∗-algebras, Sci.

Math. Japon. 77, No. 1 (2014), 119-138 :e-2013, 621-640.

[9] T. Sudo, The K-theory for the C∗-algebras of nilpotent discrete groups by examples, Bulletin

of the Faculty of Science, University of the Ryukyus, No. 104, (September 2017), 1-40.

[10] J. Tomiyama, Invitation to C∗-algebras and topological dynamics, World Scientific (1987).

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


	Introduction
	The K-theory ranks for crossed product C*-algebras by Z
	Examples and more