Homogeneous $C^ast$-algebras over a sphere

J.Korean Math. Soc. 34 (1997), No. 4, pp. 859-869

HOMOGENEOUS C*-ALGEBRAS OVER A SPHERE

CHUN-GIL PARK

ABSTRACT. It is shown that forAk,'1n a k-homogeneous C*-algebra over s2n-l X SI such that no non-trivial matrix algebra can be factored out of Ak,'1n and Ak,'1n I8l Mz (C) has a non-trivial bun- dle structure for any positive integer l, we construct an A k,'1n- c(s2n-l X SI) I8l Mk(C)-equivalence bimodule to show that ev- ery k-homogeneous C*-algebra overs2n-l XSI is strongly Morita equivalent to c(s2n-l x SI). Moreover, we prove that the ten- sor product of the k-homogeneous C*-algebraAk,'1n with a UHF- algebra of type poo has the trivial bundle structure if and only if the set of prime factors ofk is a subset of the set of prime factors ofp.

1. Introduction

In [9], the authors showed that the group [S2n, BPU(k)] of ho- motopy classes of continuous maps of s2n into the classifying space BPU(k) of PU(k) classifies k-homogeneous C*-algebras over s2n, and that the set [S2n-l xS

^1,

BPU(k)] classifies k-homogeneous C*-algebras over s2n-l

^X

SI. k-homogeneous C*-algebras over sn were constructed in [5], and k-homogeneous C*-algebras over s2n-l

^X

SI were con- structed in [7].

It is well-known that for M a finite dimensional CW-complex, the Cech cohomology group H3(M, Z) classifies tensor products of k-homo- geneous C*-algebras over M with the C*-algebra K(1i) of compact op- erators on a separable Hilbert space 1i, and (cf. [1]) that the Cech cohomology group H3(M, Z) is isomorphic to the singular cohomology

Received March13, 1997.

1991 Mathematics Subject Classification: Primary 46L87, 46L05; Secondary 55R15.

Key words and phrases: UHF-algebra, homogeneousC*-algebra, strong Morita equivalence, equivalence bimodule, tensor product, andK-theory.

Supported in part by GARC-KOSEF in 1996 and in part by a grant of the Chungnam National University in1997.

group H3(M,Z) if M is triangularizable. The Cech(=singular) coho- mology group H3(8

²ⁿ,

71.)

^rv

{O}, and the Cech(=singular) cohomology group H

³

(8

^{2n- 1 X}

8

¹

,71.)

rv

{O} ifn =I 2. And the Cech(=singular) co- homology group H3(83 x 8\ 71.)

_~

H3(83, 71.) "" 71., which corresponds to the set of tensor products of K(1l)-homogeneous C*-algebras over 8

³

with C (8

¹)

for 1l a separable Hilbert space. Thus the tensor prod- ucts of k-homogeneous C*-algebras over

B2n

(resp.

B2n-1

x 8

¹)

with K(1l) are isomorphic to C(8

²ⁿ

)®K(1l) (resp. C(8

^{2n- 1X}

8

¹

)®K(1l».

In [3], the authors proved that A ® K(1l) is isomorphic to B

®

K(1l) if and only if there exists an A-B-equivalence bimodule defined in the first section. In this case, we say that A and B are strongly Morita equivalent. In [2], M. Brabanter constructed an A-TF-C(']['2)- equivalence bimodule to prove that every rational. rotation algebra is strongly Morita equivalent to C(']['2).

We are going to construct an equivalence bimodule to show that ev- ery k-homogeneous C*-algebra over

B2n

(resp. 8

²ⁿ- 1_X

8

¹)

is strongly Morita equivalent to C(8

²ⁿ)

(resp. C(8

²ⁿ- 1

X

8

¹

».

Thomsen computed in [10] the homotopy group of equivalence classes"

of maps of 8

ⁿ

into the classifying space of the automorphism group of an infinite dimensional UHF-algebra: let U = ®:1 ^{Mp. (C) be a}

UHF-algebra where {Pi liE N} is a sequence of prime numbers. Let P denote all the prime numbers. For each pEP, we define

!(P) = ^{#{i EN I p} = ^pd·

Then for

n

even

[8

^n,

B(Aut(U))) =

IIpEPFZp!(P)'

where Pp = {p

E

PlO < !(P) < co} (interpreted to mean the trivial group if Pp = 0). For n odd, [8

^n,

B(Aut(U»] = {O}. This result clarifies the bundle structure of k-homogeneous C*-algebras over 8

ⁿ

with a UHF-algebra of type p= .

By a different trick, we are going to investigate the bundle structure

of k-homogeneous C*-algebras over 8

²ⁿ

with a UHF-algebra of type

p= . Furthermore, we are going to study the bundle structure of k-

homogeneous C*-algebras over 8

²ⁿ- ¹ X

8

¹

with a UHF-algebra of

typep=.

HomogeneousC"-algebras over sphere 861

2. Strong Morita equivalence for k-homogeneous C*-algebras over s2n and s2n-l

X

SI

For a separable Hilbert space 11., [sn, BPU(1I.)]

=

[sn-l, PU(1I.)]

and PU(1I.) is a K(Z, 2)-space, so every K(1I.)-homogeneous C*-algebra over sn has the trivial bundle structure if n =!= 3. And Woodward [11]

proved that [S3, BPU(k)] injects to H

²

(S3, Zk) $H4(S3,Z)

^f"V

{O}, so every k-homogeneous C*-algebra over S3 is isomorphic to C(S3) 0 Mk(C). Thus every k-homogeneous C*-algebra over sn is stably iso- morphic to c(sn), Le., every k-homogeneous C*-algebra over sn is strongly Morita equivalent to c(sn).

DEFINITION

2.1 [8]. Let A and B be C*-algebras. By an A-B- equivalence bimodule is meant an A-B-bimodule on which are defined an A-valued and a B-valued inner product such that

(1) (x, y)

AZ

= ^{x(y, z)}

^B,

^{'Vx, y,}

^Z

EX,

(2) the representation of A on X is a continuous *-representation by operators which are defined for (., ·)B, and similarly for the right representation of B,

(3) the linear span of (X,X)B, which is an ideal in B, is dense in B, and similarly for (X,X)A.

We say that two C*-algebras A and B are strongly Morita equivalent if there exists an A-B-equivalence bimodule. It is shown in [3] that if they are separable, then they are strongly Morita equivalent if and only if A

®

K(1I.) is isomorphic to B

®

K(1I.).

Note that Brabanter constructed an A T -C('.['2)-equivalence bimod- ule using the structure of the rational rotation algebra AT.

Since the condition over the boundaries SI on sections of a k- homogeneous C*-algebra over S2 is the same as the condition over the boundaries SI on sections of a suitable k-homogeneous C*-algebra over

'.['2,

one can modify the construction given by Brabanter to construct a B T -C(S2)-equivalence bimodule. But we are going to construct a B T -C(S2)-equivalence bimodule using a slightly different trick.

For n a positive integer (n > 1), [S2n, BPU(k)] = [S2n-\ PU(k)] :::

Z, which is a cyclic group. So the group has a generator, and there

is a unitary U(z)

E

PU(k) such that the generating k-homogeneous

C*-algebra over s2n can be realized as the C*-algebra of sections of a

locally trivial C*-algebra over s2n characterized by the inner automor- phism of the unitary U(z)

E

PUCk) over the boundaries s2n-l. Here PUCk)

=

Aut(Mk(C» = Inn(Mk(C». If (k,m) = p(lpl > 1), then we consider the k-homogeneous C*-algebra over s2n as the tensor product of Mp(C) with a

~-homogeneous

C*-algebra over S2n, which

is

char- acterized by the inner

au~omorphism

of the unitary U(z)Z:

E PU(~).

Consider U(z)m as U(z)'P ® Ip

^E

PUCk). Then every k-homogeneous C*-algebra over s2n can be realized as the C*-algebra of sections of a locally trivial C*-algebra over s2n characterized by the inner automor- phism of the unitary U(z)

m E

PU(k) over the boundaries s2n-l. See [5J for details.

Throughout this paper, we will denote the n-dimensional northern hemisphere (resp. the n-dimensional southern hemisphere) bye+. (resp.

e~).

PROPOSITION

2.2 [5J. Every k-homogeneous C*-algebra over s2n

is

isomorphic to one of the following C* -algebras Bk,m

=

C

^g",(e~n

11

e~n,

Mk(C», m

E

Z, given as follows:

if

f

E

Bk,m, then the following condition is satisfied

for all z

E

S2n-\ where U(z)

E

PUCk)

=

Inn(Mk(C»

is

a suitable projective unitary.

[S2n, BPU(ll)J

~

[S2n-l, PU(ll)J '" {O}, so every k-homogeneous C*-algebra over s2n is strongly Morita equivalent to C(S2n). However, we don't have any equivalence bimodule constructed before.

THEOREM

2.3. The k-homogeneous C*-algebra Bk,m over B2n

is

strongly Morita equivalent to C(S2n).

Proof We are going to construct a Bk,m-C(S2n) ®Mk (C}-equivalence bimodule.

Let X = {f

E C(e~n

11

e~n,

Mk(C» I f+(z) = U(z)-mf_(z)},

where U(z) is given as above. The space X is a left Bk,m-module if

module multiplication is defined by matrix multiplication F· x, where

F

= (fij)f,j=l E

Bk,m and x

E

X. If f

E

C(S2n) ® Mk(C} and x

E

X,

then x· f defines a right C(S2n) ® Mk(C)-module structure on X. We

Homogeneous C*-algebras over sphere 863

define Bk,m and C(S2n) l8l Mk(C)-valued inner products (.,

·)Bk,m

and

(., ·)c(S2n)I8lMk(iC)

on X by

if x,

y E

X and we have matrix multiplication on the right. Equipped with this structure, X becomes a Bk,m-C(S2n) l8l Mk(C)-equivalence bimodule. So Bk,m is strongly Morita equivalent to C(S2n) l8l Mk(C), which is strongly Morita equivalent to C(S2n).

Therefore, Bk,m is strongly Morita equivalent to C(S2n).

0

We have obtained that every k-homogeneous C*-algebra over s2n is strongly Morita equivalent to C(S2n).

In [7], the author showed that every k-homogeneous C*-algebra over s2n-l

X

SI is isomorphic to one of the following C*-algebras; k- homogeneous C*-algebras whose restrictions to s2n-l

^"--t

s2n-l

X

SI are k'-homogeneous C*-algebras over s2n-l (k'lk), and which corre- spond to k-homogeneous C*-algebras over s2n with respect to the boundaries B2n-l of B2n-l

x

[0,1] and

e~n

II e=-n. See [7] for details.

It

is well-known (cf. [5, T]) that for a k'-homogeneous C*-algebra A over s2n-l, there is a matrix algebra Ml(C) such that A l8l Ml(C) has the trivial bundle structure. So for a k-homogeneous C*-algebra A over s2n-l

X

SI, there are a k-homogeneous C*-algebra Ak,m over s2n-l

X

SI and a matrix algebra Ml(C) such that Al8lMl(C) ^{is isomor-} phic to A k,ml8lMl(C), ^where Ak,m is a k-homogeneous C*-algebra over s2n-l

X

SI whose restriction to s2n-l has the trivial bundle structure.

From now on we will only consider the k-homogeneous C*-algebras over s2n-l

X

SI whose restrictions to s2n-l have the trivial bundle structures.

PROPOSITION

2.4 [7]. Every k-homogeneous C* -algebra over s2n-l

X

SI, whose restriction to s2n-l has a trivial bundle structure, is iso- morphic to one of the following C* -algebras Ak,m

=

CYm (S2n-l

^X

[0,1]' Mk(C)), m

E

Z, given as follows:

if

f

E

Ak,m, then the following condition is satisfied

f(z, 1)

=

U(z)-m fez, O)U(z)m

for all z

E

S2n-l, where U(z)

E

PUCk) is given in the statement of Proposition

2.2.

Since the condition over the boundaries s2n-l of s2n-l

^X

[0,1] on sections of a k-homogeneous C*-algebra over s2n-l

X

SI is the same as the condition over the boundaries s2n-l of

_e~n

and e:"n on sections of a suitable k-homogeneous C*-algebra over S2n, one can modify the construction given in the proof of Theorem 2.3 to construct an Ak,Tn- C(S2n-l

X

SI)

®

Mk(C)-equivalence bimodule, and one can show that the unitaries U(z)Tn represent the Bk,Tn'

THEOREM

2.5. The k-homogeneous C*-algebra Ak,Tn over s2n-l

X

SI is strongly Morita equivalent to C(S2n-l

X

SI).

We have obtained that every k-homogeneous C*-algebra over s2n-l

X

SI is strongly Morita equivalent to C (S2n-l

^X

SI).

3. The K -theory of k-homogeneous C*-algebras over a sphere Since every k-homogeneous C*-algebra over s2n is strongly Morita equivalent to C(S2n), the K-theory of a k-homogeneous C*-algebra Bk,m. over s2n is isomorphic to the K-theory of C(S2n). Thus Ko (Bk,Tn)

~

712 and K

₁

(Bk,Tn) :: {O} since K o(C(S2n))

^~

712 and Kl(C(S2n))

~

{O}.

LEMMA

3.1. Let Bk,Tn be a k-homogeneous C* -algebra over B2n of which no non-trivial matrix algebra can be factored out. Ko(Bk,Tn) '"

K o(C(S2n» :: 712 and K

¹

(Bk,Tn) '" K

1

(C(S2n» '" {O}. The class [1Bk ,TrJ of the unit 1Bk,m is a primitive element of Ko(Bk,m).

Proof We know that Bk,m is strongly Morita equivalent to C(S2n).

So Ko(Bk,Tn) '" K o(C(S2n)) :: 712 and K

¹

(Bk,Tn) '" K

¹

(c(S2n))

!::>!

{O}.

It is enough to show that the class [1Bk,m]

E

Ko(Bk,Tn) is primitive.

The C(S2n) can be embedded into the Bk,Tn' and the embedding of the C(S2n) into the Bk,m. induces an isomorphism of K o(c(S2n)) to Ko(Bk,m). By the isomorphism [1C(S2n)] maps to [1B

_k

,m] or [1C(S2n)].

IT [1C(S2n)] maps to [1Ak,m], then [1Bk,m] is primitive, since [1C(S2n)]

E

K o(C(S2n» is primitive (see [6]). IT [1C(S2n)] maps to [1C(S2n)), then

[1Bk,m] = k[1C(S2n)], and so there is a suitable matrix algebra Mp(C)

HomogeneousG*-algebras over sphere 865

such that 1Bk,m

(8)

Jp

=

10(82")

(8)

1k

(8)

Jp in the Bk,'m

(8)

Mp(C), since k(10(82")] can be represented by the projection 10(82") (8)1k

E

C(S2n)(8) Mk(C), where 1k denotes the k x k identity matrix and Jp denotes the

p

x

p

matrix with components 0 except for the (l,l)-component 1.

Then

(l Bk,m

⁽⁸⁾

Jp) 18

^{2 "-1 =}

(1 0(82")

⁽⁸⁾

1k

(8)

Jp) 182"-1.

But (lBk,m (8)Jp)IS2,,-1

⁼

10(82"-1)

(8)

Jp, and (lc(S2") (8)1k

(8)

Jp) 18

^{2"-1 =}

10(82"-1)

(8)

1k

⁽⁸⁾

Jp. However, Thomsen [10] computed

when kp and

q

are relatively prime. This means that Bk,m

(8)

Mp(C) is not isomorphic to C(S2n)(8)Mkp, ^since ifit is, then 1l"2n-l(Aut(Mkp(C)(8)

Mqoo ))

is trivial. But the Bk,m corresponds to the Ak,m with re- spect to the conditions on sections over the boundaries S2n-l, and A k,mI82,,-1

^rv

C(S2n-l), since if A k,mls2,,-1

^rv

C(S2n-l) ® Mk(C), then Ak,m is isomorphic to C(S2n-l)

⁽⁸⁾

C(Sl)

⁽⁸⁾

Mk(C), which is a contradiction. So Bk,mI82,,-1

^rv

C(S2n-l). Thus

Hence 1Bk,m

^E

Ko(Bk,m) is primitive.

Therefore, Ko(Bk,m)

~ ,£2,

K

1

(Bk,m)

~

{O}, and the class [lBk,m]

of the unit 1Bk,m is a primitive element of Ko(Bk,m). 0

COROLLARY

3.2. Let I be a positive integer. Bk,m

(8)

Ml(C)

is

not isomorphic to C(S2n)

(8)

Mkl(C),

Proof. Assume Bk,m

(8)

Ml(C) is isomorphic to C(S2n)

(8)

Mkl(C), Then the unit 1Bk,m

(8)

1l maps to the unit 10 (82")

⁽⁸⁾

1kl. So

Thus there is a projection

pE

Bk,m such that l[lBk,m]

=

(kl)[P]. But Ko(Bk,m)

~

Z2 is torsion-free, so [l Bk,m] ⁼ k[P]. This is a contradic- tion.

Therefore, Bk,m

(8)

Ml(C) is not isomorphic to C(S2n)

(8)

Mkl(C), 0

Let Ak;m be a k-homogeneous C*-algebra over s2n-l

X

SI of which no non-trivial matrix algebra can be factored out, Le.,

k

and m are relatively prime. The k-homogeneous C*-algebras Ak,m over s2n-l

^X

SI are strongly Morita equivalent to C(S2n-l

^X

SI), so Ko(Ak,m)

~

K O (C(S2n-l

^X

SI))

~7f}

and K

₁

(Ak,'Tn)

rv

K

₁

(c(S2n-l

^X

SI))

~^z2.

We are going to show that the class [lAk,,.,J E KO(Ak,m) is primitive.

LEMMA 3.3. Let Ak,m be a k-homogeneous C*-algebra over s2n-l

^X

SI given

in

the statement of Proposition

2.4

whose restriction to s2n-l

is

isomorphic to C(S2n-l). Then Ko(Ak,m)

~

K

₁

(Ak,'Tn)

^~

Z2. And the class [lAk,,,,,] ^E Ko(Ak,m)

is

primitive.

Proof We know that Ak,m is strongly Morita equivalent to C(S2n-l xS

1 ).

So Ko(Ak,m)

~

K o (C(S2n-l

^X

SI))

~

Z2 and K

₁

(Ak,m)

~

K

₁

(C(S2n-l

^X

SI))

~z2.

So it is enough to show that the class [lAk,,,,,] ^E KO(Ak,m) is prim- itive. The C(S2n-l

^X

SI) can be embedded into the Ak,m, and the embedding of the C(S2n-l

^X

SI) into the Ak,m induces an isomorphism of K o( C(S2n-l

^X

SI)) to Ko(Ak,m)' By the isomorphism [lC(S2n-I xSI) maps to [lAk,,,,,] or [lC(S2n-1xSI)]. If [lC(S2n-1xSI)] maps to [lAk,,,,,], then [lAk,,,,,] is primitive, since [lC(S2n-1xSI)]

E

K o (C(S2n-l

^X

SI)) is primitive (see [6]). If [lC(S2n-1xSI)] maps to [lC(S2n-I xSI)], then [lAk,,,,,]

⁼

k[lC(S2n-I)], and so there is a suitable matrix algebra Mp(C) such that lAk,,,,, ®Jp

⁼

lC(S2n-1xSI)®Ik®Jp in the Ak,m®Mp(C), since k[lC(S2n-I XS1)] can be represented by the projection lC(S2n-I xSI) ® I k

^E

C(S2n-l

^X

SI) ® Mk(C). Then

But (lAk,,,,, ® Jp)IS2n-I

=

lC(S2n-I) ® Jp, and (lC(S2n-IxS I) ® Ik ®

Jp)IS2n-I

⁼

lC(S2n-I)

®

Ik ^® Jp. However, Ak,m

®

Mp(C) has a non-

trivial bundle structure for Mp(C) any matrix algebra, since

if

it has

not, then the same is true for the k'-homogeneous C*-algebra Bk,m

over s2n corresponding to the Ak,m with respect to the conditions on

sections. Thomsen result says that Bk,'Tn ® Mp(C) is not isomorphic to

C(S2n) ® M kp . Thus

HomogeneousC*-algebras over sphere 867

Hence

1Ak,l'n E

Ko(Ak,m) is primitive.

Therefore, Ko(Ak,m)

~

7}, K l (Ak,m)

~

71}, and the class

[lA_k

,m] of the unit

1Ak,l'n

is a primitive element of Ko(Ak,m). 0

COROLLARY

3.4. Let I be a positive integer. Ak,m

^@

Mz(CC) is not isomorphic to C(S2n-l

^X

SI) @MkZ(CC).

The proof is similar to the proof of Corollary 3.2.

We have obtained that the class

[lAk,l'n] E

Ko(Ak,m) and the class

[lBk,l'n] ^E

Ko(Bk,m) are primitive.

4. The tensor product of a k-homogeneous C* -algebra over s2n-l

^X

SI and s2n with a UHF-algebra

In this section, we investigate the bundle structure of the tensor product of a k-homogeneous C*-algebra Bk,m over s2n with a UHF- algebra Mp<:x> of type poo and the bundle structure of the tensor product of a k-homogeneous C* -algebra Ak,m over s2n-l

^X

SI with Mp<:x>.

The following is useful.

THEOREM

4.1 [4]. Suppose there exists an intertwining of the

se-

quence of C* -algebra homomorpmsms Al

^{- t}

A 2

^{- t • . .}

and B

1 ^{- t}

B2

- t . . ' .

Then the inductive limit C* -algebras fim Ai and fim Bi are

isomorphic.

Let Bk,m be a k-homogeneous C*-algebra over s2n of which no non- trivial matrix algebra can be factored out, Le., k and

m

are relatively prime.

THEOREM

4.2. Let Bk,m be a k-homogeneous C*-algebra over s2n with (k, m)

= 1.

Let Mp<:x> be a UHF-algebra of type poo. Then Bk,m

^@

Mp<:x> is isomorphic to C(S2n) ® Mk(CC)

^@

Mp<:x> if and only if the set of prime factors of k is a subset of the set of prime factors of p.

Proof. Assume the set of prime factors of k is a subset of the set

of prime factors of p. To show that Bk,m

@

Mp<:x>

is

isomorphic to

c(s2n)

@

Mk(CC)

@

Mp<:x>, it is enough to show that Bk,m

@

Mk<:x> is

isomorphic to C(S2n)@Mk<:x>. But there exist the canonical C*-algebra

homomorphisms:

The inductive limit of the odd terms

is Bk

,m

® Mk=, and the inductive limit of the even terms

.

is C(B2n) ® Mkoo. Thus by Theorem 4.1, Bk,m ® Mkoo is isomorphic to C(S2n) ® Mkoo.

Conversely, assume Bk,m®Mp= is isomorphic to C(S2n) ®Mk(C)®

M

p

=. Then the unit 1Bk,m.®lMp = maps to the unit 1C(S2n)®lMp =®h.

So

[lBk,m.

® 1M

p

=)

= [1C(S2n)

®

1M_p

= ® I

k ).

And

[lBk,m.

® 1Mp =) =

^[lBk,m.)

® [lMpoo) and

[lC(S2n)

® IMpoo ® I

k )

=

k([lC(S2n)] ® [lMp =])' But Ko(Bk,m ® M p=)

rv [~](Ko(Bk,m)) and

K o(C(S2n) ® M p= ® Mk(C)

~ k[~](Ko(C(S2n)).

If there is a prime factor q of k such that q fp, then [lMpoo] i- q[poo] for Poo a projection in M

p=

under the assumption that the unit

IBk,m.

® 1Mpoo maps to the unit

lC(S2n)

® IMpoo ® Ik. So there is a projection PI E Bk,m such that

[lBk,m.] =

q[Pl]. This contradicts Theorem 3.3. Thus the set of prime factors of k is a subset of the set of prime factors of

p.

Therefore, Bk,m ® Mpoo is isomorphic to C(S2n) ® Mk(C) ® Mpoo if and only if the set of prime factors of k is a subset of the set of prime

factors of

p.

D

Next, let Ak,m be a k-homogeneous C*-algebra over S2n-l

X

SI whose restriction to s2n-l is isomorphic to C(S2n-l), Le., k and

m

are relatively prime.

THEOREM 4.3. Let Ak,m be a k-homogeneous C* -algebra over B2n-l x SI with (k, m)

=

1. Let M p= be a UHF-algebra of type poo. Then Ak,m ® M p= is isomorphic to C(S2n-l

^X

SI) ® Mk(C) ® M p= if and only if the set of prime factors of k is a subset of the prime factors of p.

The proof is similar to the proof of Theorem 4.2.

Homogeneous C* -algebras over sphere 869

We have obtained that Ak,m ® Mpoo has the trivial bundle structure if and only if the set of prime factors of k is a subset of the set of prime factors of

p,

and that Bk,m ® Mpoo has a trivial bundle structure if and only if the set of prime factors of

k

is a subset of the set of prime factors of

p.

References

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[2]

M. Brabanter, The classification of rational rotation C* -algebras,Arch. Math.

43(1984), 79-83.

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A. Elliott, On the classification of C* -algebras of real rank zero, J. reine angew. Math443(1993), 179-219.

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Krauss and T.

C.

Lawson, Examples of homogeneous C*-algebras, Memoirs of A.M.S. 148(1974), 153-164.

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[7] C.

Park, Homogeneous C* -algebras over a product space of spheres, preprint.

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Math. Soc., Providence, R. I.,1982, pp. 285-298.

[9] M. Takesaki and J. Tomiyama, Applications of fibre bundles to the certain class of C* -algebras, Tohoku Math. J. 13(1961), 498-522.

[10] K. Thomsen, The homotopy type of the group of automorphisms of a UHF- algebra, J. Funet. Anal. 12 (1987), 182-207.

[11] L. M. Woodward, The classification of principal PU(k)-bundles over a 4- complex,J. London Math. Soc. (2) 25 (1982),513-524.

Department of Mathematics Chungnam National University Taejon 305-764, Korea

E-mail: [email protected]