CUBO A Mathematical Journal

Vol.10, N
o
¯ 01, (103–115). March 2008

Continuous or Discontinuous

Deformations of C∗-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 study deformations of C
∗
-algebras that become continuous or discontinuous.

RESUMEN

Estudiamos deformación de C
∗
-algebras que son continuas o discontinuas.

Key words and phrases: C*-algebra, Continuous field, Crossed product.

Math. Subj. Class.: Primary 46L05.



104 Takahiro Sudo CUBO
10, 1 (2008)

INTRODUCTION

Continuous fields of C
∗
-algebras have been of interest in the theory of C

∗
-algebras (see

Dixmier [5, Chapter 10]). In particular, continuous field C
∗
-algebras of continuous trace

with Hausdorff spectrums are well studied to classify them. In this case the continuous

fields of C
∗
-algebras become locally trivial and they are built up by trivial continuous field

C
∗
-algebras that are tensor products of the C

∗
-algebras of continuous functions on their base

spaces with some fixed fibers. Continuous deformations of C
∗
-algebras are in a particular

case of continuous fields of C
∗
-algebras in the sense that their base spaces are the closed

interval [0, 1] and the fibers on the half open interval (0, 1] are the same (cf. E-theory in

Blackadar [1]). It has been known that continuous deformations of C
∗
-algebras may have

non-Hausdorff spacetrums in general ([5, 10]).

It is first obtained in [10] that there exists no continuous deformation from a C
∗
-algebra

generated by isometries to a C
∗
-algebra generated by unitaries, in particular, no continuous

deformation from Cuntz and Toeplitz algebras to the C
∗
-algebras of continuous functions

on the tori. In this paper we investigate some interesting properties for continuous or

discontinuous deformations of C
∗
-algebras beyond the result of [10], but using its ideas.

We find it convenient to divide continuous deformations of C
∗
-algebras into two classes.

One consists of degenerate continuous deformations of C
∗
-algebras and the other does of

nondegenerate continuous deformations of C
∗
-algebras, that we define later. We find that it

is easy to have degenerate continuous deformations of C
∗
-algebras, some of which are useful

to provide some examples with non-Hausdorff spectrums, and it is not easy to construct

nondegenerate continuous deformations of C
∗
-algebras. Indeed, we find that there exists no

nondegenerate continuous deformations in some cases as given below.

In Section 1 we forcus on degenerate or nondegenerate continuous deformations of C
∗
-

algebras. In Secion 2 we give some nondegenerate discontinuous deformations of C
∗
-algebras

by considering crossed product C
∗
-algebras by the integer group Z and the real group R and

by semigroup crossed product C
∗
-algebras by the semigroup(s) of natural numbers, which

would be of interest.

Refer to Dixmier [5], Pedersen [8] and Murphy [7] for details of the C
∗
-algebra theory.

1 Continuous deformations of C∗-algebras

Recall that a continuous deformation from a C
∗
-algebra A to another B means a continuous

field C
∗
-algebra Γ([0, 1], {At}t∈[0,1]) on the closed interval [0, 1] with fibers At given by

A0 = B and At = A for 0 < t ≤ 1, where the continuous field C
∗
-algebra is defined and

generated by giving continuous operator fields on [0, 1] such that their norm at fibers are



CUBO
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Continuous or Discontinuous Deformations of C
∗
-Algebras 105

continuous and the set of (or generated by) their evaluations at each point t ∈ [0, 1] is dense

in At. Refer to [5] for details of continuous fields of C
∗
-algebras.

Definition 1.1 We say that a continuous deformation from a C∗-algebra A to another B

is degenerate if there exist continuous operator fields coming from some generators of A

that are zero at 0 ∈ [0, 1]. We say that a continuous deformation from a C∗-algebra A to

another B is nondegenerate if it is not degenerate, i.e., there exist no continuous operator

fields coming from generators of A that are zero at 0 ∈ [0, 1].

Proposition 1.2 Let A, B be C∗-algebras. Assume that we have the following splitting

exact sequence: 0 → C0((0, 1], A) → E → B → 0, where C0((0, 1], A) is the C
∗-algebra of

continuous A-valued functions on the half open interval (0, 1]. Then the extension E is a

continuous deformation from A to B.

Remark. A continuous deformation from A to B has the same decomposition as the extension

E above, but its extension is not necessarily splitting.

Example 1.3 Let A be a unital C∗-algebra. Then we have the following natural splitting

exact sequence: 0 → C0((0, 1], A) → E → C → 0, where the unit operator field f defined by

f (t) = 1 ∈ A for (0, 1] and f (0) = 1 ∈ C is continuous in E. This continuous deformation is

degenerate if A 6= C and nondegenerate if A = C.

Degenerate continuous deformations

Theorem 1.4 Let A be a C∗-algebra. Suppose that A has a non-trivial projection p, and

let pAp denote the C∗-subalgebra of A generated by the elements pap for a ∈ A. Then there

exists a continuous deformation from A to pAp. Also, if A is unital, then there exists a

continuous deformation from A to pAp ⊕ (1 − p)A(1 − p), where 1 − p can be replaced with

a projection of A orthogonal to p.

Proof. We construct a continuous field C∗-algebra Γ([0, 1], {At}t∈[0,1]) with fibers At given

by At = A for 0 < t ≤ 1 and A0 = pAp as follows. Assume that constant continuous operator

fields f on pAp such as f (t) = f (s) ∈ pAp for t, s ∈ [0, 1] are contained in Γ([0, 1], {At}t∈[0,1]).

And assume that other continuous operator fields of Γ([0, 1], {At}t∈[0,1]) vanish at zero.

More concretely, we can take the other way to prove the statement in the case that A

is a unital C
∗
-algebra as follows. Then any element a ∈ A can be viewed as the following

matrix:

a

(

a11 a12

a21 a22

)



106 Takahiro Sudo CUBO
10, 1 (2008)

for a11 = pap, a12 = pa(1 − p), a21 = (1 − p)ap, and a22 = (1 − p)a(1 − p). Thus, we take

the following matrix functions as continuous operator fields of Γ([0, 1], {At}t∈[0,1]):

a(t)

(

a11(t) a12(t)

a21(t) a22(t)

)

with a(0)

(

pap 0

0 0

)

for t ∈ [0, 1] such that a(1) = a.

For the second assertion, we just replace a22(0) = 0 with a22(0) = (1 − p)a(1 − p). 2

Example 1.5 There exists a continuous deformation from the matrix algebra Mn(C) to

Mm(C) for n ≥ m ≥ 1 by Theorem 1.4 since Mm(C) ∼= pMn(C)p for p a rank m projection

of Mn(C). Also, there exists a continuous deformation from the matrix algebra Mn(C) to

C
k
, where 1 ≤ k ≤ n by choosing k orthogonal rank 1 projections of Mn(C). Note that this

continuous deformation has non Hausdorff spectrum if k ≥ 2.

There exists a continuous deformation from the C
∗
-algebra K of compact operators to

Mm(C) for any m ≥ 1 by Theorem 1.4 since Mm(C) ∼= p(K)p for p a rank m projection of

K. Also, there exists a continuous deformation from the C
∗
-algebra K to C

k
(k ≥ 1) and to

C0(N) the C
∗
-algebra of sequences vanishing at infinity.

Let A be an AF algebra, i.e., an inductive limit of finite dimensional C
∗
-algebras (or

finite direct sums of matrix algebras over C). Then, as shown in Theorem 1.4 there exists a

continuous deformation from A to its C
∗
-subalgebra Mm(C) for some m ≥ 1.

Let A ⊕ B be the direct sum of C∗-algebras A, B. Then there exists a continuous

deformation from A ⊕ B to A.

Theorem 1.6 Let A be a C∗-algebra and B a unital C∗-algebra. Then there exists a con-

tinuous deformation from the C∗-tensor product A ⊗ B with a C∗-norm to A.

Proof. Note that any C∗-tensor product A ⊗ B with a certain C∗-norm is generated by

simple tensors a ⊗ b for a ∈ A and b ∈ B. We construct a continuous field C∗-algebra

Γ([0, 1], {At}t∈[0,1]) with fibers At given by At = A ⊗ B for t ∈ (0, 1] and A0 = A as follows.

Since B is unital, we assume that the constant operator fields on A ∼= A ⊗ C in A ⊗ B are

continuous and other continuous operator fields vanish at zero. 2

Example 1.7 Let C(Tn) be the C∗-algebra of continuous functions on the n-torus Tn

(n ≥ 0), where C(T0) = C. Then there exists a continuous deformation from C(Tn) to

C(T
m

) for n > m ≥ 0 since C(Tn) ∼= C(Tm) ⊗ C(Tn−m).

As for crossed product C
∗
-algebras by groups,



CUBO
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Continuous or Discontinuous Deformations of C
∗
-Algebras 107

Theorem 1.8 Let A be a unital C∗-algebra, Γ a discrete group and A ⋊α Γ the full crossed

product C∗-algebra by an action α of Γ on A. Then there exists a continuous deformation

from A ⋊α Γ to either A or the full group C
∗-algebra C∗(Γ) of Γ. Moreover, there exists a

continuous deformation from the reduced crossed product C∗-algebra A ⋊α,r Γ to either A or

the reduced group C∗-algebra C∗r (Γ) of Γ.

Proof. Note that the full crossed product C∗-algebra A ⋊α Γ is generated by A and C
∗
(Γ),

and A and C
∗
(Γ) are C

∗
-subalgebras of A ⋊α Γ. We assume that the constant operator

fields on A (or C
∗
(Γ)) in A ⋊α Γ are continuous and other continuous operator fields vanish

at zero. Also, we can replace A ⋊α Γ with A ⋊α,r Γ and C
∗
(Γ) with C

∗
r (Γ) respectively. 2

Theorem 1.9 Let A be a unital C∗-algebra, G a locally compact group and A ⋊α G the

full crossed product C∗-algebra by an action α of G on A. Then there exists a continuous

deformation from A ⋊α G to the full group C
∗-algebra C∗(G) of G. Moreover, there exists a

continuous deformation from the reduced crossed product C∗-algebra A ⋊α,r G to the reduced

group C∗-algebra C∗r (G) of G.

Proof. Note that the full crossed product C∗-algebra A ⋊α G is generated by elements af

for a ∈ A and f ∈ C∗(G), and C∗(G) is a C∗-subalgebra of A ⋊α G. We assume that the

constant operator fields on C
∗
(G) in A ⋊α G are continuous and other continuous operator

fields vanish at zero. Also, we can replace A ⋊α G with A ⋊α,r G and C
∗
(G) with C

∗
r (G)

respectively. 2

As for free products of C
∗
-algebras,

Theorem 1.10 Let A, B be unital C∗-algebras. Then there exists a continuous deformation

from the (full or reduced) unital free product C∗-algebra A ∗C B (an amalgam over C) to A.

Proof. Note that the (full or reduced) unital free product C∗-algebra A ∗C B is generated

by A and B, where the unit of A is identified with that of B. We construct a continuous

field C
∗
-algebra Γ([0, 1], {At}t∈[0,1]) with fibers At given by At = A ∗C B for t ∈ (0, 1] and

A0 = A by assuming the constant operator fields on A in A ∗C B are continuous and other

continuous operator fields vanish at zero. 2

Example 1.11 Let C∗(F2) be the full group C
∗
-algebra of the free group F2 with two

generators (see Davidson [4]). Then there exists a continuous deformation from C
∗
(F2) to

C(T) since C
∗
(F2)

∼= C∗(Z) ∗C C
∗
(Z) and C

∗
(Z) ∼= C(T) by the Fourier transform.



108 Takahiro Sudo CUBO
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Nondegenerate continuous deformations

Example 1.12 Let H3 be the real 3-dimensional Heisenberg Lie group and C
∗
(H3) its group

C
∗
-algebra. Since H3 is isomorphic to a semi-direct product R

2
⋊ R, we have C

∗
(H3)

∼=

C
∗
(R

2
) ⋊ R ∼= C0(R

2
) ⋊ R by the Fourier transform. Then it is known that C

∗
(H3) can be

viewed as the continuous field C
∗
-algebra Γ0(R, {At}t∈R) with fibers At = K for t 6= 0 and

A0 = C0(R
2
) since At

∼= C0(R) ⋊αt R ∼= K for t 6= 0 where the action α
t

of R on R is a shift

and A0
∼= C0(R) ⋊α0 R ∼= C0(R

2
) since the action α

0
of R on R is trivial. Therefore, the

restriction of this continuous field C
∗
-algebra to [0, 1] gives a continuous deformation from

K to C0(R
2
).

Let H2n+1 be the real (2n + 1)-dimensional generalized Heisenberg Lie group and

C
∗
(H2n+1) its group C

∗
-algebra. Since H2n+1 is isomorphic to a semi-direct product

R
n+1

⋊ R
n
, we have C

∗
(Hn+1)

∼= C∗(Rn+1) ⋊ Rn ∼= C0(R
n+1

) ⋊ R
n

by the Fourier trans-

form. Then it is known that C
∗
(H2n+1) can be viewed as the continuous field C

∗
-algebra

Γ0(R, {At}t∈R) with fibers At = K for t 6= 0 and A0 = C0(R
2n

) since At
∼= C0(R

n
) ⋊αt R

n ∼=

K for t 6= 0 where the action αt of Rn on Rn is a shift and A0 ∼= C0(R
n
) ⋊α0 R

n ∼= C0(R
2n

)

since the action α
0

of R
n

on R
n

is trivial. Therefore, the restriction of this continuous field

C
∗
-algebra to [0, 1] gives a continuous deformation from K to C0(R

2n
).

More generally,

Proposition 1.13 Let A be a C∗-algebra, G a locally compact group and A ⋊αt G the full

crossed product C∗-algebras by actions αt of G on A for t ∈ [0, 1]. Suppose that the actions

{αt}t∈[0,1] are continuous in the sense that the maps from t ∈ [0, 1] to αt(a) for a ∈ A are

continuous and that A ⋊αt G
∼= A ⋊αs G for t, s ∈ (0, 1] and α

0 is trivial. Then there exists a

continuous deformation from A ⋊α1 G to A ⊗ C
∗
(G). Furthermore, similarly we can replace

A ⋊αt G with their reduced crossed product C
∗-algebras and C∗(G) with its reduced group

C
∗-algebra respectively.

Remark. Even if G = R, the assumption A ⋊αt R
∼= A ⋊αs R for t, s ∈ (0, 1] are not

true in general. For instance, let C(T
2
) ⋊θ R be the crossed product C

∗
-algebra by the

action θ of R on T
2

defined by θt(z, w) = (e
2πit

z, e
2πiθt

w) ∈ T2 where θ ∈ R, which is

also called the foliation C
∗
-algebra of C(T

2
) by R of Connes [2]. Then it is known that

C(T
2
) ⋊θ R

∼= K ⊗ (C(T) ⋊θ Z), where C(T) ⋊θ Z is the rotation algebra corresponding to

θ. Moreover, it is known that C(T) ⋊θ Z
∼= C(T) ⋊θ′ Z if and only if θ = θ

′
or θ = 1 − θ′

(mod 1).

The proposition above gives a general procedure to construct nondegenerate continu-

ous fields by crossed products C
∗
-algebras, but it is not easy to have continuous actions



CUBO
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Continuous or Discontinuous Deformations of C
∗
-Algebras 109

{αt}t∈[0,1] in the sense above and check the isomorphisms of their crossed product C
∗
-

algebras for t ∈ (0, 1].

As for tensor products of C
∗
-algebras,

Proposition 1.14 Let A, B be C∗-algebras. Suppose that the C∗-tensor product A ⊗ B

with a C∗-norm is isomorphic to A. Then there exists a continuous deformation from A⊗B

to A.

Example 1.15 We have K ⊗ K ∼= K. A C∗-algebra A is stable if A ⊗ K ∼= A.

Let A be a simple separable nuclear C
∗
-algebra. Then A ∼= A ⊗ O∞ if and only if

A is purely infinite, where O∞ is the Cuntz algebra generated by a sequence of othogonal

isometries. A C
∗
-algebra A is simple, separable, unital and nuclear if and only if A⊗O2 ∼= O2,

where O2 is the Cuntz algebra generated by two orthogonal isometires with the sum of their

range projections equal to the identity. See Rørdam [9] for these significant results.

2 Discontinuous deformations of C∗-algebras

Nondegenerate discontinuous deformations

Theorem 2.1 Let A be a unital commutative C∗-algebra and A⋊α Z the crossed product C
∗-

algebra of A by a non trivial action α of Z. Then there exists no nondegenerate continuous

deformation from A ⋊α Z to A. If A is nonunital and commutative, then there exists no

nondegenerate continuous deformation from A ⋊α Z to A
+ the unitization of A by C.

Proof. Note that A is a C∗-subalgebra of A ⋊α Z and A ⋊α Z is generated by A and a

unitary corresponding to the action α of Z. Let U be such a unitary. Then we have the

covariance relation: U aU
∗

= α1(a) for a ∈ A. Suppose that we had a continuous field

C
∗
-algebra Γ([0, 1], {At}t∈[0,1]) such that A0 = A and At = A ⋊α Z for 0 < t ≤ 1. We may

assume that (certain) constant continuous operator fields on A (or A
+

if A is non unital)

are contained in Γ([0, 1], {At}t∈[0,1]) (where the argument below is applicable to the case

without constant continuous operator fields). Also, we may assume that the operator field f

defined by f (0) = u a unitary of A (or u a unitary of A
+

if A is nonunital) and f (t) = U for

0 < t ≤ 1 is also contained in it. Then the operator field f bf ∗ for (certain) b ∈ A defined by

f bf
∗
(t) = f (t)bf

∗
(t) = U bU

∗
= α1(b) and f bf

∗
(0) = ubu

∗
= uu

∗
b = b must be continuous.

But this is impossible in general since b 6= α1(b) for some b ∈ A since α is non trivial so that

(b − f bf ∗)(t) = b − α1(b) 6= 0 for t ∈ (0, 1] but (b − f bf
∗
)(0) = b − b = 0. 2



110 Takahiro Sudo CUBO
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Example 2.2 Let C(T) be the C∗-algebra of continuous functions on the torus T and

C(T) ⋊αθ Z the crossed product C
∗
-algebra that is called a rotation algebra, where α

θ

is induced from the action of Z on T by the multiplication e
2πiθt

for t ∈ Z (see Wegge-

Olsen [11]). By Theorem 2.1, there exists no nondegenerate continuous deformation from

C(T) ⋊αθ Z to C(T).

Moreover, let C(T
k
) ⋊αΘ Z be the crossed product C

∗
-algebra (which is one of noncom-

mutative tori) by an action α
Θ

by Z on C(T
k
), where Θ = (θj )

k
j=1 and α

Θ
t (zj ) = (e

2πiθj tzj ) ∈

T
k

for t ∈ Z. Then there exists no nondegenerate continuous deformation from C(Tk) ⋊αθ Z

to C(T
k
).

Furthermore,

Theorem 2.3 Let A be a unital simple C∗-algebra and A ⋊α Z the crossed product C
∗-

algebra of A by a non trivial action α of Z. Then there exists no nondegenerate continuous

deformation from A ⋊α Z to A. If A is nonunital and simple, then there exists no nonde-

generate continuous deformation from A ⋊α Z to A
+ the unitization of A by C.

Proof. Let U be a unitary corresponding to α. Suppose that we had a continuous field

C
∗
-algebra Γ([0, 1], {At}t∈[0,1]) such that A0 = A and At = A ⋊α Z for 0 < t ≤ 1. We may

assume that the operator field f defined by f (0) = u a unitary of A (or u a unitary of A
+

if

A is nonunital) and f (t) = U for 0 < t ≤ 1 is also contained in it. Then the operator field

f uf
∗

defined by f uf
∗
(t) = f (t)uf

∗
(t) = U uU

∗
= α1(u) and f uf

∗
(0) = uuu

∗
= u must

be continuous. Hence it follows that α1(u) = u since the operator field f uf
∗ − α1(u) is

continuous and (f uf
∗ − α1(u))(t) = 0 for t ∈ (0, 1] so that (f uf

∗ − α1(u))(0) = 0. Thus, u

is fixed under α. Therefore, the C
∗
-algebra C

∗
(u) generated by u is fixed under α. Then A

must have C
∗
(u) as a nontrivial quotient C

∗
-algebra, which contradicts to that A is simple.

We use the similar argument for the case of A nonunital and simple. 2

Example 2.4 Let On be the Cuntz algebra generated by n orthogonal isometries {Sj}
n
j=1

such that
∑n

j=1 Sj S
∗
j = 1 (see Cuntz [3] or the text books Davidson [4] or Wegge-Olsen [11]).

Then by Theorem 2.3 there exists no nondegenerate continuous deformation from On ⊗ K to

Mn∞ ⊗ K, where Mn∞ is the UHF algebra. It is known that the C
∗
-tensor product On ⊗ K

isomorphic to the crossed product C
∗
-algebra (Mn∞ ⊗ K) ⋊α Z (see Rørdam [9]).

Moreover,

Theorem 2.5 Let A be an either commutative or simple, unital C∗-algebra and A ⋊α Γ the

(reduced or full) crossed product C∗-algebra of A by a non trivial action α of Γ a discrete



CUBO
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Continuous or Discontinuous Deformations of C
∗
-Algebras 111

group. Then there exists no nondegenerate continuous deformation from A⋊α Γ to A. If A is

nonunital and either commutative or simple, then there exists no nondegenerate continuous

deformation from A ⋊α Γ to A
+ the unitization of A by C.

Proof. Note that the (full or reduced) crossed product C∗-algebra A ⋊α Γ is generated by

A and the unitaries corresponding to generators of Γ and A is a C
∗
-subalgebra of A ⋊α Γ.

Let U be one of the unitaries. We apply the arguments given in the proofs of Theorems 2.1

and 2.3 for the C
∗
-algebra generated by A and U . Note that U may have torsion in the

arguments. 2

As for crossed product C
∗
-algebras by continuous groups,

Theorem 2.6 Let A be an either commutative or simple, unital (or non unital) C∗-algebra

and A ⋊α R the crossed product C
∗-algebra of A by a non trivial action α of R. Then there

exists no nondegenerate continuous deformation from A ⋊α R to A.

Proof. Note that the crossed product C∗-algebra A ⋊α R is generated by elements af for

a ∈ A and f ∈ C∗(R). Since C∗(R) ∼= C0(R) by the Fourier transform, we identify elements

of C
∗
(R) with those of C0(R). Note that the unitization C0(R)

+
by C is isomorphic to

C(T). Now suppose that we had a continuous field C
∗
-algebra Γ([0, 1], {At}t∈[0,1]) such that

A0 = A and At = A ⋊α R for 0 < t ≤ 1. Then we can have a extended continuous field

C
∗
-algebra Γ([0, 1], {Bt}t∈[0,1]) such that B0 = A and Bt the C

∗
-algebra generated by A

and that C(T) for 0 < t ≤ 1 by assuming that the operator field from the unit of C(T) to

the unit of A (or of A
+

if A nonunital) is continuous.

Suppose that A is commutative. Since α is nontrivial, there exists b ∈ A such that

U bU
∗ 6= b. Indeed, if U bU ∗ = b for any b ∈ A, then A and C(T) commute. Hence A and

C0(R) commute. Thus, A ⋊α R
∼= A ⊗ C∗(R) so that α must be trivial. Therefore, we can

adopt the argument given in the proof of Theorem 2.1.

Suppose that A is simple. On the other hand, by the argument given in the proof of

Theorem 2.3, we have U uU
∗

= u, where the operator field from U to u ∈ A is continuous.

Thus, the C
∗
-algebra C

∗
(u) generated by u commutes with C(T) generated by U . Hence

C
∗
(u) commutes with C

∗
(R). Then A has C

∗
(u) as a nontrivial quotient C

∗
-algebra, which

is the contradiction. 2

Remark. We can replace with R with T in the statement above. Note that C∗(T) ∼= C0(Z) by

the Fourier transform and C0(Z)
+ ∼= C((Z)+), where (Z)+ is the one point compactification

of Z and it is identified with a closed subset of T.

Example 2.7 Let C∗(H3) be the group C
∗
-algebra of the real 3-dimensional Heisenberg

Lie group H3. Then C
∗
(H3)

∼= C∗(R2) ⋊ R ∼= C0(R
2
) ⋊ R since H3

∼= R2 ⋊ R. Hence there



112 Takahiro Sudo CUBO
10, 1 (2008)

exists no nondegenerate continuous deformation from C
∗
(H3) to C0(R

2
) of C0(R

2
) ⋊ R.

Furthermore,

Theorem 2.8 Let A be an either commutative or simple, unital (or non unital) C∗-algebra

and A⋊αR
n the crossed product C∗-algebra of A by an action α of Rn such that the restriction

of α to any factor R of Rn is non trivial. Then there exists no nondegenerate continuous

deformation from A ⋊α R
n to A.

Proof. We use the same process as given in the proof of the theorem above. Note that

A ⋊α R
n

is generated by elements af for a ∈ A and f ∈ C∗(Rn), and C∗(Rn) ∼= C0(R
n
)

so that C0(R
n
)
+ ∼= C((Rn)+) ∼= C(Sn), where (Rn)+ is the one point compactification of

R
n

and S
n

is the n-dimensional sphere. Take a unitary U of C(S
n
) that corresponds to a

coordinate projection from S
n

(n ≥ 2) to T and gives a nontrivial action on A. 2

Remark. We can replace with Rn with Tn in the statement above. Note that C∗(Tn) ∼=

C0(Z
n
) by the Fourier transform and C0(Z

n
)
+ ∼= C((Zn)+), where (Zn)+ is the one point

compactification of Z
n

and it is identified with a closed subset of T.

Example 2.9 Let C∗(H2n+1) be the group C
∗
-algebra of the real (2n + 1)-dimensional

Heisenberg Lie group H2n+1. Then C
∗
(H2n+1)

∼= C∗(Rn+1) ⋊ Rn ∼= C0(R
n+1

) ⋊ R
n

since

H2n+1
∼= Rn+1 ⋊ Rn. Hence there exists no nondegenerate continuous deformation from

C
∗
(H2n+1) to C0(R

n+1
) of C0(R

n+1
) ⋊ R

n
.

As for crossed product C
∗
-algebras by semigroups.

Theorem 2.10 Let A be a unital C∗-algebra with no proper isometries and A ⋊α N the

semigroup crossed product C∗-algebra of A by an action α of the additive semigroup N

of natural numbers by proper isometries. Then there exists no nondegenerate continuous

deformation from A ⋊α N to A. If A is non unital and without proper isometries, then there

exists no nondegenerate continuous deformation from A ⋊α N to the unitization A
+ by C.

Proof. Suppose that we had a continuous field C∗-algebra Γ([0, 1], {At}t∈[0,1]) with fibers

At given by A0 = A and At = A ⋊α N for t ∈ (0, 1]. Note that A ⋊α N is generated by A

and a proper isometry. Let S be such a isometry. Then we have the covariance relation:

SaS
∗

= α1(a) for a ∈ A. Since S
∗
S = 1 the unit of A (and A ⋊α N) (or 1 ∈ C of A

+
if A

is non unital) the operator field f defined by f (t) = S
∗
S and f (0) = 1 in A is continuous.

We may assume that the operator field g defined by g(t) = S for t ∈ (0, 1] and g(0) = a an

element of A is continuous. Then it follows that a
∗
a = 1.



CUBO
10, 1 (2008)

Continuous or Discontinuous Deformations of C
∗
-Algebras 113

If a 6= 1, then the last equation is the contradiction since A has no proper isometries.

If a = 1, then note that the operator field h defined by h(t) = SS
∗

for t ∈ (0, 1] and

h(0) = 1 is continuous since the operator field g is so. Hence, the operator field f − h is

also continuous, which is impossible because f (t) − h(t) = 1 − SS∗ 6= 0 for t ∈ (0, 1] but

f (0) − h(0) = 1 − 1 = 0. 2

Example 2.11 It is known that On
∼= Mn∞ ⋊α N (see [9]). Since the UHF algebra Mn∞ has

no proper isometries, we obtain by the theorem above that there exists no nondegenerate

continuous deformation from On to Mn∞ .

Theorem 2.12 Let A be a unital C∗-algebra with no proper isometries and A ⋊α N
× the

semigroup crossed product C∗-algebra of A by an action α of the multiplicative semigroup

N
× of natural numbers by proper isometries. Then there exists no nondegenerate continuous

deformation from A ⋊α N
× to A. If A is non unital and without proper isometries, then

there exists no nondegenerate continuous deformation from A ⋊α N
× to the unitization A+

by C.

Proof. Note that the semigroup crossed product C∗-algebra A ⋊α N
×

is generated by A

and C
∗
(N

×
), and C

∗
(N

×
) is isomorphic to the infinite tensor product of C

∗
(N) over prime

numbers since N
× ∼= ⊕N over prime numbers, where C∗(N) is the C∗-algebra generated by

a proper isometry, which is just the usual Toeplitz algebra. Thus, A ⋊α N corresponding

to A and a certain proper isometry in C
∗
(N

×
) is regarded as a C

∗
-subalgebra of A ⋊α N

×
.

Therefore, we can use the arguments as given in the proof of the theorem above. 2

Example 2.13 Following Laca-Raeburn [6], the Hecke C∗-algebra of Bost-Connes is re-

alized as the semigroup crossed product C
∗
-algebra C

∗
(Q/Z) ⋊α N

×
. Thus, we obtain

by the theorem above that there exists no nondegenerate continuous deformation from

C
∗
(Q/Z) ⋊α N

×
to C

∗
(Q/Z).

Moreover,

Theorem 2.14 Let A be a unital C∗-algebra with no proper isometries and A ⋊α N the

(reduced or full) semigroup crossed product C∗-algebra of A by an action α of a discrete

semigroup N by proper isometries. Then there exists no nondegenerate continuous deforma-

tion from A ⋊α N to A. If A is non unital and without proper isometries, then there exists

no nondegenerate continuous deformation from A ⋊α N to the unitization A
+ by C.

Proof. Note that the (reduced or full) semigroup crossed product C∗-algebra A ⋊α N is

generated by A and isometries corresponding to generators of N . Let S be one of the



114 Takahiro Sudo CUBO
10, 1 (2008)

isometries. We apply the argument given in the proof of Theorem 2.10 for the C
∗
-algebra

generated by A and S. 2

As for free products of C
∗
-algebras,

Theorem 2.15 Let A be a C∗-algebra that contains an either unitary or isometry generator.

Then there exists no nondegenerate continuous deformation from the (full or reduced) unital

free product C∗-algebra A ∗C C(T) to C(T).

Proof. Let U be a unitary generator of A and V the generating unitary of C(T). We assume

that we had a nondegenerate continuous deformation from (full or reduced) free product

C
∗
-algebra A ∗C C(T) to C(T). Then we may assume that the constant operator field f

by V is continuous and the operator field g from U to a certain unitary W of C(T) is also

continuous. Then (f g − gf )(t) = f (t)g(t) − g(t)f (t) = V U − U V 6= 0 for t ∈ (0, 1] but

(f g − gf )(0) = f (0)g(0) − g(0)f (0) = V W − W V = 0 since C(T) is commutative, which

leads to the contradiction.

In the argument above we can replace U with a isometry generator S of A since we can

assume that the operator field from S to a unitary of C(T) is continuous. 2

Example 2.16 Since C∗(F2)
∼= C(T) ∗C C(T), there exists no nondegenerate continuous

deformation from the full group C
∗
-algebra C

∗
(F2) of F2 to C(T).

Similarly,

Theorem 2.17 Let A be a C∗-algebra that contains an either unitary or isometry generator

U , and A⋊α Z be the crossed product C
∗-algebra by a non trivial action α of Z on A. Suppose

that V U V ∗ 6= U , where V is the generaing unitary corresponding to α. Then there exists no

nondegenerate continuous deformation from A ⋊α Z to C(T).

Proof. Consider the operator field from V U V ∗ − U 6= 0 to V W V ∗ − W = V V ∗W − W = 0,

where W is a certain unitary of C
∗
(Z) ∼= C(T) (by the Fourier transform). If we had a

nondegenerate continuous deformation from A ⋊α Z to C(T), this operator field should be

continuous but it is impossible. 2

Received: July 2007. Revised: September 2007.



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10, 1 (2008)

Continuous or Discontinuous Deformations of C
∗
-Algebras 115

References

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[2] A. Connes, Noncommutative geometry, Academic Press, (1990).

[3] J. Cuntz, K-theory for certain C∗-algebras, Ann. of Math., 113 (1981), 181–197.

[4] K.R. Davidson, C∗-algebras by Example, Fields Institute Monographs, AMS.

(1996).

[5] J. Dixmier, C∗-algebras, North-Holland, (1962).

[6] M. Laca and I. Raeburn, A semigroup crossed product arising in number theory,

J. London Math. Soc., (2) 59 (1999), 330–344.

[7] G.J. Murphy, C∗-algebras and Operator theory, Academic Press, (1990).

[8] G.K. Pedersen, C∗-Algebras and their Automorphism Groups, Academic Press

(1979).

[9] M. Rørdam and E. Størmer, Classification of Nuclear C∗-Algebras. Entropy in

Operator Algebras, EMS 126 Operator Algebras and Non-Commutative Geometry

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[10] T. Sudo, K-theory of continuous deformations of C∗-algebras, Acta Math. Sin. (Engl.

Ser.) 23, no. 7 (2007) 1337–1340 (online 2006).

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


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