Articulo 16.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (261–274). June 2010

Real and stable ranks for certain crossed products of Toeplitz
algebras

Takahiro Sudo

Department of Mathematical Sciences,

Faculty of Science, University of the Ryukyus,

Nishihara, Okinawa 903-0213, Japan

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

ABSTRACT

We consider the algebraic structure of certain crossed products of the Toeplitz algebra

and its tensor products. Using the structure, we estimate the stable rank and real rank

of those crossed products. In particular, we obtain a real rank estimate for extensions of

C∗-algebras.

RESUMEN

Consideramos la estructura algebraica de ciertos productos cruzados de algebra de Toeplitz

y sus productos tensoriales. Usando la estructura estimamos el rango estable y el rango

real de estos productos cruzados. En particular, obtenemos una estimativa del rango real

para extensiones de C∗-algebras.

Key words and phrases: C*-algebra, Crossed products, Stable rank, Real rank, Toeplitz algebra.

2000 Math. Subj. Class.: Primary 46L05, 46L80

1 Introduction

Crossed products of C∗-algebras (by automorphisms) have been very interesting research objects in the

C∗-algebra theory. See [9] as a reference. As well, crossed products of C∗-algebras by endomorphisms



262 Takahiro Sudo CUBO
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have been studied (rather recently). A typical and important example is given by the rotation C∗-

algebra, that is defined as the crossed product of C(T) by the rotation action of the group Z of integers,

where C(T) is the C∗-algebra of all continuous functions on the 1-torus T, and is also the universal

C∗-algebra generated by a unitary. More generally, noncommutative tori are defined as successive

crossed products by Z. On the other hand, another example is given by the group C∗-algebra of the

semi-direct product Zn ⋊ Z, that is viewed as the crossed product of C(Tn) by the adjoint action of

Z, where Tn is the n-torus. More generally, the group C∗-algebras of successive semi-direct products

by Z are viewed as successive crossed products by Z.

Our first motivation is to replace C(T) with the Toeplitz algebra F, that is the universal C∗-

algebra generated by an isometry, and replace Z with the semigroup N of natural numbers (with

zero), and consider the crossed product of F by N. Furthermore, we replace C(Tn) with ⊗nF the n-

fold tensor product of F and consider the crossed product of ⊗nF by N. While the crossed products of

C(Tk) by Z, that are viewed as noncommutative manifolds, have been studied well, the replacements

by isometries: the crossed products of ⊗nF by N, have not been studied explicitly yet.

Under those circumstances, in this paper we study certain crossed products of the Toeplitz algebra

and its tensor products. The algebraic structure of those crossed products is given explicitly (and

inductively) in Section 1. It is found that the crossed products have quotients that are isomorphic to

the group C∗-algebras of generalized discrete ax + b groups that are defined and studied in [14], so

that they may be viewed as the C∗-algebras of generalized discrete ax + b semigroups in a sense. (Our

first effort was to find such an analogue to the group C∗-algebras of the Heisenberg discrete group,

but this has not been successful yet.) The K-theory groups of the crossed products are computed by

using the Pimsner-Voiculescu exact sequence. Using the structure (and in part the K-theory results)

obtained, we estimate the stable rank and connected stable rank of the crossed products in Section

1, and estimate their real rank as well as the real rank of the group C∗-algebras of the generalized

discrete ax + b groups in Section 2. Note that the stable and real ranks are viewed as noncommutative

complex and real dimensions respectively. For estimating the real rank, we obtain a new real rank

estimate for extensions of C∗-algebras. It turns out that this estimate is quite useful for estimating

and determining the real rank of extensions of C∗-algebras. The stable rank, connected stable rank,

and real rank formulae obtained for those crossed products and the real rank formulae for those group

C∗-algebras are new, and the ranks are estimated with the dimension of the spaces of 2-dimensional

irreducible representations that correspond to certain subquotients of the group C∗-algebras. In

addition, a partial duality result on crossed products of C∗-algebras by N is obtained, which may be

of some independent interest and would be useful for further research in a direction.

Notation. We denote by sr(A) the stable rank of a (unital) C∗-algebra A, and by csr(A) its

connected stable rank. By definition, sr(A) ≤ n if and only if Ln(A) is dense in A
n, where (aj ) ∈ Ln(A)

if there exists (bj) ∈ A
n such that

∑n

j=1 bj aj = 1 ∈ A. Also, csr(A) ≤ n if and only if Lm(A) is

connected for any m ≥ n. Refer to [10]. We denote by RR(A) the real rank of A. By definition,

RR(A) ≤ n − 1 if and only if Ln(A)sa is dense in (Asa)
n, where Ln(A)sa and Asa are the sets of all

self-adjoint elements of Ln(A) and A respectively. Refer to [3].

Recall from [14] that the generalized discrete ax + b group that is a semi-direct product Zn ⋊ Z



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is defined by the following (n + 1) × (n + 1) matrices:

(

⊕neπit s

0n 1

)

∈ GLn+1(Z),

where ⊕neπit means the n × n diagonal matrix with diagonal entries eπit for t ∈ Z, and s ∈ Zn (a

column vector), and 0n = (0, · · · , 0) ∈ Z
n (a row vector).

2 Structure and Stable rank

Let F be the Toeplitz algebra, that is defined to be the universal C∗-algebra generated by a (proper)

isometry s. Write F = C∗(s).

Definition 2.1. We define the crossed product of F by an action of N to be the universal C∗-algebra

generated by F and an isometry t such that the action α of N on F is given by α1(x) = txt
∗ for x ∈ F.

Denote it by C∗(H1,1) = F ⋊α N and call it the C
∗-algebra of the discrete ax + b semigroup H1,1 since

F ∼= C∗(N) the C∗-algebra of N, so that we may write H1,1 = N ⋊ N just as a symbol like a semi-direct

product.

It is well known that F has the decomposition into the exact sequence:

0 → K → F → C(T) → 0,

where K is the C∗-algebra of compact operators on a separable Hilbert space. Furthermore, this K is

isomorphic to the commutator ideal of F. Refer to [6].

Theorem 2.2. The C∗-algebra C∗(H1,1) = F ⋊α N has the decomposition into the exact sequence:

0 → K ⋊α N → F ⋊α N → C(T) ⋊α N → 0,

Moreover, K ⋊α N ∼= K ⊗ C(T) and C(T) ⋊α N ∼= C(T) ⋊α Z a crossed product of C(T) by a unitary

action of Z, which is isomorphic to the group C∗-algebra of the discrete ax + b group Z ⋊ Z.

Proof. Let x ∈ F = C∗(s). Since xx∗ − x∗x is a compact operator, t(xx∗ − x∗x)t∗ = (txt∗)(tx∗t∗) −

(tx∗t∗)(txt∗) is also compact. Therefore, K is invariant under the action α of N. Hence we obtain the

exact sequence.

Furthermore, we have

K ⋊α N
∼= K ⋊α Z ∼= K ⊗ C(T),

where the first isomorphism follows from that the action α on K is an automorphism as discussed

above, and the second isomorphism follows from that any automorphism on K is implemented by a

unitary, i.e. an adjoint action by a unitary, so that K ⋊α Z ∼= K ⊗ C
∗(Z), where C∗(Z) is the group

C∗-algebra of Z, that is isomorphic to C(T) by the Fourier transform.

Also, we have C(T)⋊α N ∼= C(T)⋊α Z since the action α on C(T) by N must be an automorphism

implemented by a unitary, which is isomorphic to the group C∗-algebra of the discrete ax + b group

Z ⋊ Z. 2



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Remark. We may view this extension property as the definition for C∗(H1,1) = F⋊αN. By universality,

there is a quotient map from F ⋊α N to C(T) ⋊α Z. The similar remark as this can be made for the

structure results given below.

Proposition 2.3. The K-theory groups of C∗(H1,1) are obtained as:

Kj (C
∗(H1,1)) ∼= Z (j = 0, 1).

Proof. Since C∗(H1,1) = F ⋊α N, we have the Pimsner-Voiculescu exact sequence of K-groups for

crossed products of C∗-algebras by N (as well as Z):

Z
(id−α)∗
−−−−−→ Z −−−−→ K0(C

∗(H1,1))
x









y

K1(C
∗(H1,1)) ←−−−− 0 ←−−−− 0

(see [12] and [2]), where K0(F) ∼= Z and K1(F) ∼= 0. Since the map (id−α)∗ is trivial, where id is the

identity map on F, we obtain Kj(C
∗(H1,1)) ∼= Z for j = 0, 1. 2

Remark. The six-term exact sequence for the exact sequence obtained above is

Z
i∗

−−−−→ K0(C
∗(H1,1))

q∗
−−−−→ Z2

x









y

Z
2 q∗←−−−− K1(C

∗(H1,1))
i∗

←−−−− Z

where Kj (K⊗C(T)) ∼= Kj(C(T)) ∼= Z for j = 0, 1, and Kj(C(T) ⋊α Z) ∼= Z
2 (j = 0, 1) by the (usual)

Pimsner-Voiculescu exact sequence. Consequently, the maps i∗ induced by the inclusion i : K⊗C(T) →

C∗(H1,1) are zero, so that the maps q∗ induced by the quotient map q : C
∗(H1,1) → C(T) ⋊α Z are

injective.

Theorem 2.4. The stable rank of C∗(H1,1) is 2. The connected stable rank of C
∗(H1,1) is 2.

Proof. By [10, Theorems 4.3, 4.4, and 4.11], we have the following estimates:

sr(C∗(H1,1)) ≤ max{sr(K ⊗ C(T)), sr(C(T) ⋊ Z), csr(C(T) ⋊ Z)},

and max{sr(K ⊗ C(T)), sr(C(T) ⋊ Z)}≤ sr(C∗(H1,1)).

Furthermore, by [10, Theorems 3.6 and 6.4 and Proposition 1.7] sr(K ⊗ C(T)) = sr(C(T)) = 1. Note

that C(T) ⋊ Z ∼= C∗(Z ⋊ Z). By the stable rank and connected stable rank formulae in [14, Remark

3.4] with a correction (see the remark below) we have

sr(C∗(Z ⋊ Z)) = 2, and csr(C∗(Z ⋊ Z)) ≤ 2.

The same estimates (from above, ≤ 2) for C(T) ⋊ Z are also obtained by using [10, Theorem 7.1 and

Corollary 8.6].

On the other hand, by [13, Theorem 3.9] we have

csr(C∗(H1,1)) ≤ max{csr(K ⊗ C(T)), csr(C(T) ⋊ Z)}.



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By [13, Theorem 3.10], csr(K ⊗ C(T)) ≤ 2. Since K1-group of C
∗(H1,1) is not trivial as shown above,

we have csr(C∗(H1,1)) ≥ 2 (cf. [4, Corollary 1.6]). 2

Remark. The stable rank estimate in [14, Remark 3.4] after a correction is

sr(C0(R
n+1) ⊗ M2(C)) = ⌈⌊(n + 1)/2⌋/2⌉+ 1 ≤

sr(C∗(Zn ⋊ Z)) ≤ csr(C0(R
n+1) ⊗ M2(C)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1,

and the connected stable rank estimate in it after that is

csr(C∗(Zn ⋊ Z)) ≤ csr(C0(R
n+1) ⊗ M2(C)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1,

where C∗(Zn ⋊ Z) is the group C∗-algebra of the generalized ax + b group defined in [14], and ⌊x⌋

means the maximum integer ≤ x, and ⌈y⌉ is the least integer ≥ y. In particular, if n is odd, then

sr(C∗(Zn ⋊ Z)) = ⌈⌊(n + 1)/2⌋/2⌉+ 1. Furthermore, if n = 4m, then the inequality does not become

equality, but if n = 4m + 2, then the inequality becomes equality. As for the correction, in fact,

C0(R
n−j+1) in [14, Theorem 3.3] should have been replaced with C0(R

n−j+2) (1 ≤ j ≤ n).

As a note, the Toeplitz algebra F has stable rank 2 and connected stable rank ≤ 2. This follows

from using [10], [8], and [13] as above, where the result of [8] says that if the index map in the six-term

exact sequence of K-groups for a C∗-algebra extension E is nonzero, then E can not have stable rank

1.

Let F ⊗ F be the C∗-tensor product of F, which is also defined to be the universal C∗-algebra

generated by ∗-commuting isometries s1, s2, which means that each sj commutes with both si and

s∗i (i 6= j).

Definition 2.5. We define the C∗-algebra of the (generalized) ax + b semigroup H2,1 = N
2

⋊ N (just

as a symbol like a semi-direct product) to be the universal C∗-algebra generated by F ⊗ F and an

isometry t ⊗ t such that the (product) action α ⊗ α of N on F ⊗ F is given by (α ⊗ α)1(x ⊗ y) =

(t ⊗ t)(x ⊗ y)(t ⊗ t)∗ = txt∗ ⊗ tyt∗ for x ⊗ y ∈ F ⊗ F. Denote it by C∗(H2,1) = (F ⊗ F) ⋊α⊗α N the

crossed product of F ⊗ F by α ⊗ α of N.

In what follows, we often omit the symbol for actions in crossed products.

Proposition 2.6. The C∗-algebra C∗(H2,1) = (F ⊗ F) ⋊α⊗α N has the structure as follows:

0 → (F ⊗ K) ⋊ N → C∗(H2,1) → (F ⊗ C(T)) ⋊ N → 0

and the quotient and closed ideal have the decompositions as follows:

0 → (K ⊗ C(T)) ⋊ N → (F ⊗ C(T)) ⋊ N → C(T2) ⋊ N → 0,

and 0 → (K ⊗ K) ⋊ N → (F ⊗ K) ⋊ N → (C(T) ⊗ K) ⋊ N → 0.

Furthermore, (K⊗C(T)) ⋊ N ∼= K⊗(C(T) ⋊ Z), (C(T)⊗K) ⋊ N ∼= (C(T) ⋊ Z)⊗K, and (K⊗K) ⋊ N ∼=

K⊗C(T), and C(T2)⋊N ∼= C(T2)⋊Z, which is isomorphic to the group C∗-algebra of the (generalized)

discrete ax + b group Z2 ⋊ Z.



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Proof. The quotient and closed ideal, and their decompositions are deduced from the invariance of

the action α ⊗ α. Note that

(K ⊗ C(T)) ⋊ N ∼= (K ⊗ C(T)) ⋊ Z, and

(C(T) ⊗ K) ⋊ N ∼= (C(T) ⊗ K) ⋊ Z.

Furthermore, (K ⊗ K) ⋊ N is isomorphic to the following:

(K ⊗ K) ⋊ Z ∼= (K ⊗ K) ⊗ C(T) ∼= K ⊗ C(T).

2

Proposition 2.7. The K-theory groups of C∗(H2,1) are obtained as:

Kj (C
∗(H2,1)) ∼= Z (j = 0, 1).

Proof. Since C∗(H2,1) = (F ⊗ F) ⋊α N, we have the Pimsner-Voiculescu sequence:

Z
(id−α)∗
−−−−−→ Z −−−−→ K0(C

∗(H2,1))
x









y

K1(C
∗(H2,1)) ←−−−− 0 ←−−−− 0

where K0(F ⊗ F) ∼= Z and K1(F ⊗ F) ∼= 0 by the Künneth formula (see [2]). Since the map (id − α)∗

is trivial, where id is the identity map on ⊗2F, we obtain Kj (C
∗(H2,1)) ∼= Z for j = 0, 1. 2

Theorem 2.8. The stable rank of C∗(H2,1) is 2. The connected stable rank of C
∗(H2,1) is 2.

Proof. By [10, Theorems 4.3, 4.4, and 4.11], we have the following estimates:

sr(C∗(H2,1)) ≤ max{sr((F ⊗ K) ⋊ N), sr((F ⊗ C(T)) ⋊ N), csr((F ⊗ C(T)) ⋊ N)},

and max{sr((F ⊗ K) ⋊ N), sr((F ⊗ C(T)) ⋊ N)}≤ sr(C∗(H2,1)),

and moreover,

sr((F ⊗ C(T)) ⋊ N) ≤

max{sr(K ⊗ (C(T) ⋊ Z)), sr(C(T2) ⋊ Z), csr(C(T2) ⋊ Z)},

and max{sr(K ⊗ (C(T) ⋊ Z)), sr(C(T2) ⋊ Z)}≤ sr((F ⊗ C(T)) ⋊ N),

and sr((F ⊗ K) ⋊ N) ≤

max{sr(K ⊗ C(T)), sr((C(T) ⋊ Z) ⊗ K), csr((C(T) ⋊ Z) ⊗ K)},

and max{sr(K ⊗ C(T)), sr((C(T) ⋊ Z) ⊗ K)}≤ sr((F ⊗ K) ⋊ N).

Furthermore, by [10, Theorems 3.6 and 6.4] sr(K ⊗ (C(T) ⋊ Z)) = sr(C(T) ⋊ Z) = 2. Note that

C(T2) ⋊ Z ∼= C∗(Z2 ⋊ Z). By the stable rank and connected stable rank formulae in [14, Remark 3.4]

with a correction (see the remark above) we have

sr(C∗(Z2 ⋊ Z)) = 2, and csr(C∗(Z2 ⋊ Z)) ≤ 2.



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On the other hand, by [13, Theorem 3.9] we have

csr(C∗(H2,1)) ≤ max{csr((F ⊗ K) ⋊ N), csr((F ⊗ C(T)) ⋊ N)},

and moreover,

csr((F ⊗ K) ⋊ N) ≤

max{csr(K ⊗ (C(T) ⋊ Z)), csr(C(T2) ⋊ Z)}≤ 2,

and csr((F ⊗ C(T)) ⋊ N) ≤

max{csr(K ⊗ C(T)), csr((C(T) ⋊ Z) ⊗ K)}≤ 2.

Hence, it follows that csr(C∗(H2,1)) ≤ 2. Therefore, sr(C
∗(H2,1)) = 2 is obtained from the first part

of this proof. Since K1-group of C
∗(H2,1) is not trivial as shown above, we have csr(C

∗(H2,1)) ≥ 2

(cf. [4, Corollary 1.6]). 2

Let ⊗nF be the n-fold C∗-tensor product of F, which is also defined to be the universal C∗-algebra

generated by n ∗-commuting isometries sj (1 ≤ j ≤ n), which means that each sj commutes with

both si and s
∗
i for any i 6= j.

Definition 2.9. We define the C∗-algebra of the (generalized) ax + b semigroup Hn,1 = N
n

⋊ N

(just as a symbol like a semi-direct product) to be the universal C∗-algebra generated by ⊗nF and

an isometry ⊗nt such that the (product) action ⊗nα of N on ⊗nF is given by (⊗nα)1(⊗
n
j=1xj ) =

(⊗nt)(⊗nj=1xj )(⊗
nt)∗ = ⊗nj=1txj t

∗ for ⊗nj=1xj ∈ ⊗
nF. Denote it by C∗(Hn,1) = (⊗

nF) ⋊⊗nα N the

crossed product of ⊗nF by ⊗nα of N.

Proposition 2.10. The C∗-algebra C∗(Hn,1) = (⊗
nF) ⋊⊗nα N has the structure:

0 → ((⊗n−1F) ⊗ K) ⋊ N → C∗(Hn,1) → ((⊗
n−1

F) ⊗ C(T)) ⋊ N → 0,

the exact sequence at the level 1 (that we call so), and the quotient and closed ideal have the decom-

positions as follows:

0 → ((⊗n−2F) ⊗ K ⊗ C(T)) ⋊ N → ((⊗n−1F) ⊗ C(T)) ⋊ N

→ ((⊗n−2F) ⊗ C(T2)) ⋊ N → 0, and

0 → ((⊗n−2F) ⊗ (⊗2K)) ⋊ N → ((⊗n−1F) ⊗ K) ⋊ N

→ ((⊗n−2F) ⊗ C(T) ⊗ K) ⋊ N → 0,

the exact sequences at the level 2. Inductively, the exact sequences at the level k (1 ≤ k ≤ n) have

quotients and closed ideals that are given by

((⊗n−kF) ⊗ (⊗lK) ⊗ C(Tk−l)) ⋊ N

(0 ≤ l ≤ k) where ⊗0K = C and C(T0) = C. In particular, the exact sequences at the level n have

quotients and closed ideals that are given by

((Kl) ⊗ C(Tn−l)) ⋊ N ∼=











K ⊗ C(T) (l = n),

K ⊗ (C(Tn−l) ⋊ Z) (1 ≤ l ≤ n − 1),

C(Tn) ⋊ Z (l = 0),

and C(Tn−l) ⋊ Z is isomorphic to the group C∗-algebra of the (generalized) discrete ax + b group

Z
n−l

⋊ Z.



268 Takahiro Sudo CUBO
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Proof. Note that ((Kl) ⊗ C(Tn−l)) ⋊ N is isomorphic to the following:

((Kl) ⊗ C(Tn−l)) ⋊ Z ∼= (K
l) ⊗ (C(Tn−l) ⋊ Z) ∼= K ⊗ (C(T

n−l) ⋊ Z).

2

Proposition 2.11. The K-theory groups of C∗(Hn,1) are obtained as:

Kj (C
∗(Hn,1)) ∼= Z (j = 0, 1).

Proof. Since C∗(Hn,1) = (⊗
nF) ⋊α N, we have the Pimsner-Voiculescu sequence:

Z
(id−α)∗
−−−−−→ Z −−−−→ K0(C

∗(Hn,1))
x









y

K1(C
∗(Hn,1)) ←−−−− 0 ←−−−− 0

where K0(⊗
nF) ∼= Z and K1(⊗

nF) ∼= 0 by the Künneth formula (see [2]). Since the map (id − α)∗ is

trivial, where id is the identity map on ⊗nF, we obtain Kj(C
∗(Hn,1)) ∼= Z for j = 0, 1. 2

Theorem 2.12. The stable rank of C∗(Hn,1) is ⌈⌊(n + 1)/2⌋/2⌉+ 1 if n 6= 4m, and if n = 4m, then

m + 1 ≤ sr(C∗(Hn,1)) ≤ m + 2.

The connected stable rank of C∗(Hn,1) is estimated as:

2 ≤ csr(C∗(Hn,1)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1.

Proof. Using the structure obtained for C∗(Hn,1) above and [10, Theorems 4.3, 4.4, and 4.11] repeat-

edly as before for estimating the stable rank, we obtain that sr(C∗(Hn,1)) is estimated by











sr(K ⊗ C(T)) = 1,

sr(K ⊗ (C(Tn−l) ⋊ Z)) ≤ 2, csr(K ⊗ (C(Tn−l) ⋊ Z)) ≤ 2,

sr(C(Tn) ⋊ Z), and csr(C(Tn) ⋊ Z)

(1 ≤ l ≤ n − 1). Note that C(Tn) ⋊ Z ∼= C∗(Zn ⋊ Z). By the stable rank and connected stable rank

formulae in [14, Remark 3.4] with a correction (see the remark above) we have

⌈⌊(n + 1)/2⌋/2⌉+ 1 ≤ sr(C∗(Zn ⋊ Z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1,

and csr(C∗(Zn ⋊ Z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1

where the stable rank estimate becomes equality if n 6= 4m. Therefore, we obtain the stable rank

estimates as in the statement.

On the other hand, using the structure obtained for C∗(Hn,1) above and [13, Theorem 3.9] re-

peatedly as before for estimating the connected stable rank, we obtain that csr(C∗(Hn,1)) is estimated

by










csr(K ⊗ C(T)) ≤ 2

csr(K ⊗ (C(Tn−l) ⋊ Z)) ≤ 2 (1 ≤ l ≤ n − 1),

csr(C(Tn) ⋊ Z).



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Hence, it follows that csr(C∗(Hn,1)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1. Since K1-group of C
∗(Hn,1) is not trivial

as shown above, we have csr(C∗(Hn,1)) ≥ 2 (cf. [4, Corollary 1.6]). 2

Remark. As a note, the stable rank and connected stable rank of ⊗nF are estimated as:

max{2, sr(C(Tn))}max{2,⌊n/2⌋ + 1}≤

sr(⊗nF) ≤ csr(C(Tn)) ≤⌈(n + 1)/2⌉ + 1, and

csr(⊗nF) ≤ csr(C(Tn)) ≤⌈(n + 1)/2⌉ + 1

using the structure of ⊗nF as above.

3 Real rank

Theorem 3.1. For an exact sequece of C∗-algebras: 0 → I → A → A/I → 0, we obtain the following

real rank estimate:

RR(A) ≤ max{RR(I), RR(A/I), csr(A/I) − 1}.

Proof. Let n be the maximum given above. We may assume that n is finite since if it is infinite, the

estimate is automatic. Let (aj )
n
j=0 be an element of A

n+1 with aj = a
∗
j . Let U be an open neigh-

borhood of (aj )
n
j=0. Let π : A → A/I be the quotient map. Write π for the map A

n+1 → (A/I)n+1

extended by π. Then there exists an element (b′j )
n
j=0 of the intersection π(U ) ∩ Ln+1(A/I)sa such

that (π(aj ))
n
j=0 is approximated closely by (b

′
j )

n
j=0. Note that Ln+1(A/I)sa is a subest of Ln+1(A/I).

Since csr(A/I) ≤ n + 1, there exists an invertible matrix S′ of GLn+1(A/I)0 the connected component

of GLn+1(A/I) with the identity matrix such that S
′(b′j )

n
j=0 = (1, 0, · · · , 0). Then there exists a lift

S(bj )
n
j=0 of S

′(b′j )
n
j=0, where S ∈ GLn+1(A)0 and (bj ) ∈ U such that S(bj )

n
j=0 = (1 + c0, c1, · · · , cn) ∈

(I∼)n+1 with each cj ∈ I. Set S(bj )
n
j=0 + (S(bj)

n
j=0)

∗ = (dj )
n
j=0 ∈ (I

∼)n+1sa . Since RR(I) ≤ n, we may

assume that (dj )
n
j=0 ∈ Ln+1(I

∼)sa, where I
∼ is the unitization of I. Indeed, the set of the elements

(dj )
n
j=0 ∈ A

n+1
sa such that S(bj)

n
j=0 is mapped by π to an open neighborhood of (1, 0, · · ·0) is open

relative to (I∼sa)
n+1, i.e., its intersection with (I∼sa)

n+1 is open in (I∼sa)
n+1 since any element of In+1 is

mapped to (0)nj=0 by π. Note that Ln+1(I
∼)sa ⊂ Ln+1(A

∼), where A∼ = A if A is unital and A∼ is the

unitization of A if A is non-unital. Note also that (bj )
n
j=0 + S

−1(S(bj)
n
j=0)

∗ = S−1(dj )
n
j=0 ∈ Ln+1(A

∼)

that is invariant under multiplication by elements of GLn+1(A)0. By taking a deformation of S (or

S−1) to the identity matrix in GLn+1(A)0, it is concluded that (bj )
n
j=0 + (b

∗
j )

n
j=0 is in Ln+1(A)sa, and

belongs to U , as desired. 2

Remark. This real rank estimate for extensions of C∗-algebras, obtained above will be very useful for

computing the real rank of the extensions, as shown below. The estimate corresponds to the following

of Rieffel [10, Theorem 4.11]:

sr(A) ≤ max{sr(I), sr(A/I), csr(A/I)}

which is often used in Section 1.

Theorem 3.2. The real rank of the Toeplitz algebra F is 1.



270 Takahiro Sudo CUBO
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Proof. Since 0 → K → F → C(T) → 0, the estimate obtained in the theorem above implies

RR(F) ≤ max{RR(K), RR(C(T)), csr(C(T))} = max{0, 1, 1} = 1.

On the other hand, by [5, Theorem 1.4],

RR(F) ≥ max{RR(K), RR(C(T))} = 1.

2

Remark. The same result as above is obtained as a corollary of [5, Theorem 1.2], which says that for

an extension of C∗-algebras: 0 → K → A → A/I → 0, we have RR(A) = RR(A/I). Also, the result

[7, Proposition 1.6] implies that for an extension of C∗-algebras: 0 → I → A → A/I → 0, we have

RR(A) ≤ max{RR(M (I)), RR(A/I)},

where M (I) is the multiplier algebra of I. It follows from this estimate that F has real rank 1 since

M (K) ∼= B the C∗-algebra of bounded operators has real rank 0 [3]. However, the above estimate of

[7] is not always useful since it involves the multiplier algebra, and it is hard to know its structure in

general so that it is difficult to estimate its real rank in general.

Theorem 3.3. The real rank of C∗(Z ⋊ Z) of the discrete ax + b group is 1.

Proof. It is shown in [14] that C∗(Z ⋊ Z) has a composition series {Ij}
3
j=1 of closed ideals, with

I3 = C
∗(Z ⋊ Z) such that

I3/I2 ∼= C(T) ⊕ C(T), I2/I1 ∼= C0(R) ⊗ M2(C),

and I1 ∼= C0(R
2) ⊗ M2(C).

Using the real rank estimate obtained above,

RR(I3) ≤ max{RR(I2), RR(I3/I2), csr(I3/I2) − 1}, and

RR(I2) ≤ max{RR(I1), RR(I2/I1), csr(I2/I1) − 1}.

Also, by [5],

RR(Ij ) ≥ max{RR(Ij−1), RR(Ij /Ij−1)}

for j = 2, 3. By [13], csr(I3/I2) = csr(C(T)) = 2. By [11, Theorem 4.7],

csr(C0(R) ⊗ M2(C)) ≤⌈(csr(C0(R)) − 1)/2⌉ + 1 = 2.

By [3, Proposition 1.1], RR(I3/I2) = RR(C(T)) = 1. By [1],

RR(C0(R) ⊗ M2(C)) ≥⌈dim[0, 1]/(2 · 2 − 1)⌉ = 1, while

RR(C0(R) ⊗ M2(C)) ≤⌈dim S
1/(2 · 2 − 1)⌉ = 1, and

RR(C0(R
2) ⊗ M2(C)) ≥⌈dim[0, 1]

2/(2 · 2 − 1)⌉ = 1, while

RR(C0(R
2) ⊗ M2(C)) ≤⌈dim S

2/(2 · 2 − 1)⌉ = 1,

where C([0, 1]), C([0, 1]2) are quotients of C0(R), C0(R
2) respectively, and S1, S2 are the one-point

compactifications of R, R2 (i.e., 1 and 2-dimensional spheres) respectively (see also [15] and [7, Propo-

sition 5.1]). Therefore, it follows that RR(Ij ) = 1 = RR(Ij /Ij−1) for j = 1, 2, 3. 2



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Real and stable ranks for certain
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Theorem 3.4. The real rank of C∗(H1,1) is 1.

Proof. Using the real rank estimate, the structure for C∗(H1,1) obtained above, and the theorem

above, we obtain

RR(C∗(Z ⋊ Z)) = 1 ≤ RR(C∗(H1,1)) ≤

max{RR(K ⊗ C(T)), RR(C∗(Z ⋊ Z)), csr(C∗(Z ⋊ Z)) − 1} = 1,

where RR(K ⊗ C(T)) ≤ 1 by [1]. 2

Theorem 3.5. The real rank of ⊗nF is n.

Proof. By using the real rank estimate and the (n-fold) structure for ⊗nF, that can be obtained

inductively as above from the structure for F, it follows that the real rank of ⊗nF is estimated by

{

RR(C(Tn)) = n, csr(C(Tn)) − 1 ≤⌈(n + 1)/2⌉,

RR(C(Tk) ⊗ K) ≤ 1, and csr(C(Tk) ⊗ K) − 1 ≤ 1

(0 ≤ k ≤ n − 1), where C(T0) = C. The conclusion is deduced as before. 2

Theorem 3.6. The real rank of C∗(Zn ⋊ Z) of the generalized discrete ax + b group is

RR(C∗(Zn ⋊ Z)) = ⌈(n + 1)/3⌉.

Proof. It is shown in [14] that C∗(Zn ⋊ Z) has a composition series {Ij}
n+1
j=1 of closed ideals, with

In+1 = C
∗(Zn ⋊ Z) such that In+1/In is isomorphic to the 2

n-fold direct sum of C(T), and each

subquotient Ij /Ij−1 for 1 ≤ j ≤ n (with I0 = {0}) is isomorphic to the combination nCn−j−1 fold

direct sum of the following extension Ej :

0 → C0(R
n−j+2) ⊗ (⊕n−j+1M2(C)) → Ej →⊕

n−j+1M2(C) → 0.

Using the real rank estimate obtained above, we obtain

RR(In+1) ≤ max{RR(In), RR(C(T)), csr(C(T)) − 1}, and

RR(Ij ) ≤ max{RR(Ij−1), RR(Ej ), csr(Ej ) − 1}, and

RR(Ej ) ≤ max{RR(C0(R
n−j+2) ⊗ M2(C)), RR(M2(C)), csr(M2(C)) − 1}

= max{⌈(n − j + 2)/(2 · 2 − 1)⌉, 0, 0} = ⌈(n − j + 2)/3⌉.

Also, by [5],

RR(In+1) ≥ max{RR(In), RR(C(T))}, and

RR(Ij ) ≥ max{RR(Ij−1), RR(Ej )}, and

RR(Ej ) ≥⌈(n − j + 2)/3⌉.

By [13],

csr(Ej ) ≤ csr(C0(R
n−j+2) ⊗ M2(C)) ≤⌈⌊(n − j + 3)/2⌋/2⌉+ 1.



272 Takahiro Sudo CUBO
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Furthermore,

RR(Ij ) ≥ RR(I1) ≥ RR(C0(R
n+1) ⊗ M2(C))

≥ RR(C([0, 1]n+1) ⊗ M2(C)).

By [1], it follows that

RR(C([0, 1]n+1) ⊗ M2(C)) = ⌈(n + 1)/(2 · 2 − 1)⌉ = ⌈(n + 1)/3⌉.

It follows that RR(Ij ) = ⌈(n + 1)/3⌉ for 1 ≤ j ≤ n + 1. 2

Theorem 3.7. The real rank of C∗(Hn,1) is ⌈(n + 1)/3⌉.

Proof. Using the structure obtained for C∗(Hn,1) above and the real rank esitame for extensions of

C∗-algebras obtained above, we obtain that RR(C∗(Hn,1)) is estimated by











RR(K ⊗ C(T)) ≤ 1,

RR(K ⊗ (C(Tn−l) ⋊ Z)) ≤ 1, csr(K ⊗ (C(Tn−l) ⋊ Z)) − 1 ≤ 1,

RR(C(Tn) ⋊ Z), and csr(C(Tn) ⋊ Z) − 1

(1 ≤ l ≤ n − 1). Note that C(Tn) ⋊ Z ∼= C∗(Zn ⋊ Z). Moreover, we have obtained above that

RR(C∗(Zn ⋊ Z)) = ⌈(n + 1)/3⌉, and

csr(C∗(Zn ⋊ Z)) ≤⌈⌊(n + 2)/2⌋/2⌉+ 1.

Therefore, we obtain the real rank formula as in the statement. 2

Remark. More applications by using the real rank estimate for extensions of C∗-algebras, obtained

above, could be expected, when the extensions are given, where the real ranks of their closed ideal

and quotients are computable. Furthermore, if a C∗-algebra has a composition series of closed ideals

such that the real ranks of its subquotients are computable, then the real rank of the C∗-algebra can

be estimated by using the real rank formula.

4 A partial duality

Definition 4.1. Let A ⋊α N be the crossed product of a (unital) C
∗-algebra A by an action α of N

by an isometry s1. Define the second (or dual) crossed product of A ⋊α N to be the crossed product

A ⋊α N ⋊β N by a (dual) action β of N such that β is trivial on A, and β on C
∗(s1) generated by s1

is implemented by an isometry s2.

Proposition 4.2. The second crossed product B = A ⋊α N ⋊β N has the following decomposition:

0 → (A ⋊α N) ⊗ K → B → (A ⊗ C(T)) ⋊α⊗β N → 0,

where the action β on C(T) is the adjoint action implemented by a unitary.



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Real and stable ranks for certain
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Proof. Note that

A ⋊α N ⋊β N
∼= (A ⋊id N) ⋊(α,β) N,

where id is the identity action of the second N, and the action β on C∗(s1) is exchanged by the action

α on C∗(s2) implemented by s1. Since A ⋊id N ∼= A ⊗F, we have (α, β) = α ⊗ β a product action, so

that there exists the following exact sequence:

0 → (A ⊗ K) ⋊ N → (A ⊗ F) ⋊ N → (A ⊗ C(T)) ⋊ N → 0

where the action β on C(T) becomes a unitary action. Furthermore, we obtain

(A ⊗ K) ⋊ N ∼= (A ⋊ N) ⊗ K.

2

Remark. K-theory for the closed ideal and quotient in the above exact sequence for the second crossed

product B by N can be computed by the Pimsner-Voiculescu exact sequence:

D −−−−→ D −−−−→ D ⋊ N
x









y

D ⋊ N ←−−−− D ←−−−− D

where D ⋊ N is the crossed product of a unital C∗-algebra D by an action of N by a corner endo-

morphism ([12]). Furthermore, by using the six-term exact sequence for extensions of C∗-algebras,

K-theory of B can be determined when K-theory for A is computable.

Example 4.3. Let On be the Cuntz algebra generated by n isometries sj such that
∑n

j=1 sjs
∗
j = 1

(see [2] for instance). It is well known that On ∼= Mn∞ ⋊α N where Mn∞ is the UHF algebra of type

n∞, that is an inductive limit of tensor products ⊗kMn(C) (∼= Mnk (C)). Then the second crossed

product B = Mn∞ ⋊ N ⋊β N ∼= On ⋊ N has the decomposition:

0 → On ⊗ K → B → (Mn∞ ⊗ C(T)) ⋊α⊗β N → 0.

Note that sr(On ⊗ K) = 2 since sr(On) = ∞ by [10, Proposition 6.5]. Also, by [10, Theorem 5.1],

Mn∞ ⊗C(T) has stable rank 1 because it can be viewed as an inductive limit of matrix algebras over

C(T). However, B has stable rank ∞ since the quotient of B has stable rank ∞ because the quotient

has On as a quotient. Indeed, Mn∞ ⊗ C1 is invariant under the action of N. This shows that the

second crossed product B can not be stable.

Received: March 2009. Revised: July 2009.

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