The spherical non-commutative tori

THE SPHERICAL NON-COMMUTATIVE TOm

DEOK-HoON Boo, SEI-QWON OH* AND CHUN-GIL PARK**

ABSTRACT. We define the spherical non-commutative torus lLw as the crossed product obtained by an iteration ofl crossed products by actions of Z, the first action on c(s2n+l). Assume the fibres are isomorphic to the tensor product of a completely irrational non- commutative torusAp with a matrix algebraMm(C) (m> 1). We prove that JLw^@Mp(C) is not isomorphic to C(Prim(JLw))^@Ap@

Mmp(C), and that the tensor product of lLw with a UHF-algebra Mpoo oftypepoo isisomorphic to C(Prim(lLw))@Ap @Mm(C) I8l Mpoo if and only if the set of prime factors of m is a subset of the set of prime factors ofp.

Furthermore, it is shown that the tensor product of JLw with theC*-algebra1C(1£) of compact operators on a separable Hilbert space1£isnot isomorphic to C(Prim(JLw ))@Ap@Mm (C)@1C(1l)if Prim(JLw ) ishomeomorphic toLk(n)XTl ' for kandIfnon-negative integers(k > 1), whereLk(n) is the lens space.

1. Introduction

Given a locally compact abelian group G and a multiplier w on G, one can associate to them the twisted group C*-algebraC*(G,w), which is the universal object for unitary w-representations of G.

C* (7£:1+1,w) is said to be a non-commutative torus of rank 1+1 and de- noted byAw. The multiplier w determines a subgroup Sw ofG, called its symmetry group, and the multiplier w is called totally skew ifthe

Received June 25, 1997.

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

Key words and phrases: tensor product, crossed product, K-theory, homoge- neousC*-algebra, twisted groupC*-algebra, non-commutative torus, UHF-algebra, lens space, and Cuntz algebra.

*Supported in part by the Basic Science Research Institute Program, Ministry of Education, Project No. BSRI-97-1427.

**Supported in part by GARCin1996 and the Chungnam National University in 1997.

symmetry group Sw is trivial. And Aw^{is called} completely irrational ifw is totally skew. It was shown in [1] that ifG is a locally compact abe1ian group and wis a totally skew multiplier onG, then C*(G,w) is a simple C*-algebra.

Itis well-known (cf. [5]) thatAw can be obtained by an iteration of l ordinary crossed products by actions of Z, the first action onC(Sl).

And since Aw is the universal object for unitary w-representations of Zl+l, Aw is realized as C*(ua,Ul,'" ,Ul I^UiUj

=

Consider the crossed product lLw obtained by an iteration of l or- dinary crossed products by actions of Z, the first action onC(S2n+l), whereC(Sl) in the crossed product C(Sl) x_a1 Z x_{a2 • • •} x_al Z repre- sentingAw isreplaced byC (S2n+1),and the actions of Z onC (S2n+1)

are induced from the homeomorphisms

for i = 1,2" .. ,l.

Inthis paper, using Pimsner-Voiculescu exact sequence for a crossed product, we compute the K-theory of lLw and we are going to show that the class [1Jr..J E Ko(lLw ) is primitive. Using the fact that the class [1lLJ E Ko(lL_{w )} is primitive, we are going to show that for (}Oi

rational numbers (i = 1"" ,l) the tensor product of lLw (with fibres Ap®Mm(C) for Ap a simple non-commutative torus and m a positive integer (m > 1)) with a matrix algebra Mp(C) is not isomorphic to C(Prim(lLw))^®A p^®Mmp(C) ,and that the tensor product of lLw with a UHF-algebraMp<x> of type poo is isomorphic toC(Prim(lLw)) ®A p® Mm(C) ®Mp<x> ifand onlyifthe set of prime factors of m is a subset of the set of prime factors ofp.

Furthermore, we are going to show that 02d® lLw has the trivial bundle structure ifand onlyifm and 2d - 1 are relatively prime, and that 000®lLw has always a non-trivial bundle structure if m> l.

By comparison of theK-theory, it is shown that the tensor product of lLw with the C*-algebralC(1I.) of compact operators on a separable Hilbert space11.is not isomorphic toC(Prim(lL_w))®A_p®M_m(C)®lC(1I.) if Prim(lLw) ^ishomeomorphic to Lk(n) x 1r^l' for k and l' non-negative integers (k > 1), where Lk(n) isthe lens space.

2. The K-theory of spherical non-commutative tori

One can canonically replaceC(8¹) in the crossed productC(8¹)x_a1 Z Xa2 ••• Xaz Z representingAw by C (S2ⁿ+l) and the actions of Z on C(S2n+l) are induced from the homeomorphisms

(z_0,^Z_b ^{. , .} _{, n}z) E s2n+1 ~ (e2-rr:i8oiZ_0,e2-rr:i8oi z ..._{b ,}e2-rr:i80iz )_n E s2n+1

for i =1,2"" ,l.

DEFINITION 2.1. The crossed product, constructed as above, ob- tained by an iteration of l ordinary crossed products by actions of Z, the first action on C(S2n+1), is said to be a spherical non-commutative torus o/rank l+l, and denoted by lLw, whereOOiin the crossed product representing the non-commutative torus Aw of rank. l+1 are rational numbers for i = 1" .. ,l.

We are going to show that the class [llL_w^] EKo(lLw ) is primitive.

THEOREM 2.2. Let lLw be a spherical non-commutative torus of rank l+1. Then Ko(lLw ) ~ K1(lLw ) ^{r v} Z2^1, and [h_w^] E Ko(lLw ) is primitive.

Proof The proof is by induction on l. Ifl = 1, lLw is isomorphic to C(S2n+l) x_a1 Z. The action01 is induced from the homeomorphism (z_{0, b}^{Z . , .} _{, n}z) E s2n+¹ ~ (e2-rr:i0⁰¹_"'V,^?en e2-rr:i8^01Z_1"^••• e2-rr:iOo¹Z_n) E s2n+¹_.

Note that this action is homotopic to the trivial action, since we can homotope 001 to O. Hence Z acts trivially on the K-theory of C(S2n+l).

The Pimsner-Voiculescu exact sequence for a crossed product gives

and similarly for K1 , where the map ~ is induced by inclusion. Since

(01)* = 1 and since the K-groups of C(8²ⁿ+l) are free abelian, this reduces a split short exact sequence

and similarly for K I . ButK o(C(S2n+I)) ~KI(C(S2n+I)) ~Z (see [8, II.1.34]). SoKj(lL_{w )} are free abelian of rank2.

Since the inclusionC(S2n+I) ^----t lLw sends lC(S2n+l) to llL_{w '} [llL_{w ]} is the image of [1C(S2n+l)], which is primitive in KO(C(S2n+I)) (see [8, II.1.21]). Hence the image is primitive, since the Pimsner-Voiculescu exact sequence is a split short exact sequence of torsion-free groups.

Assume the result is true for all spherical non-commutative tori of rank l = i - 1..Write lLi = C*(lLi-bUi), where lLi = C*(C(S2n+I), UI, ...,Ui). Then the inductive hypothesis applies to lLi- I. Also, we can think of lLi as the crossed product of lLi-I by an action ^{Xi of Z, where the generator of Z corresponds toUi,which acts on C* (UI,U2, ... , Ui-l)by conjugation (sendingUjtouiujui1

=

=

o.

Tf (lL ) ^{I-(ai). Tf} (lL ) _~ ^v (lL ) K (lL ) ^I-(ai). K (lL )

no i-I ) no i-I --+ no i --. 1 i-I ) 1 i - I

and similarly for K^1, where the map cl) is induced by inclusion. Since (ai)*

=

and similarly for K I. So Ko(lLi ) and KI(lLi) are free abelian of rank 2 .2ⁱ- ^I = 2ⁱ. Furthermore, since the inclusion lLi- I ^----tlL_i sends h_i^- ₁

to hi' [hi] is the image of [llL_{i _ 1 ],} which is primitive in Ko(lLi - I ) by inductive hypothesis. Hence the image is primitive, since the Pimsner- Voiculescu exact sequence is a split short exact sequence of torsion-free groups.

Therefore, Ko(lLw) ,...., KI(lLw) ~ Z2^{1, and [llLJ} E Ko(lLw) is primi-

tive. 0

It is well-known (cf. [10, 11]) that the set [M, BPU(k)] of homo- topy classes of continuous maps of a compact Hausdorff spaceM into

the classifying space B PU(k) of the Lie group PU(k) is in bijective correspondence with the set of equivalence classes of principal PU(k)- bundles over M, and that principal PU(k)-bundles over M and k- homogeneous C*-algebras over M are in bijective correspondence, Le., every k-homogeneousC*-algebraA is isomorphic to theC*-algebra of sections of a locally trivialC*-algebra bundle with base spaceM,fibre Mk(C), and structure group Aut(Mk(C)) '" PUCk).

By a change of basis for Zl,one can assume that 0₀₁ =

7:

ki

ki

ki.

ki

And the cyclic group Zj kZ acts freely on the non-commutative torus C* (^Ul,U2, . .. ,uz) of rank l, and the Mackey machine for a crossed product says that the crossed product of the cyclic group ZjkZ on C*(Ul,U2,'" ,uz) is isomorphic to Mk(C) ® C*(U~,U2,··· ,UI). But C* (u~,U2, . .. ,UI)is a non-commutative torus of rankl. C* (u~,U2, . .. , UI) is isomorphic to the C*-algebra of sections of a locally trivial C*- algebra bundle overPrim(C*(u~,U2,'" ,UI)) with fibres Ap^®Mk'(C) forA_pa simple non-commutative torus andk' a positive integer. Hence the spherical non-commutative torus lLw of rank 1+1 is isomorphic to the C*-algebra of sections of a locally trivial C*-algebra bundle over Prim(lLw)⁼ Lk(n)xPrim(C*(u~,U2,'" ,UI))with fibresA p0Mm(C), where Mm(C) :=Mk(C) ®Mk,(C). See [7] for details.

COROLLARY 2.3. Letp be a positive integer. lLw ® Mp(C) is not isomorphic to C(Prim(lLw))^® A p^®Mmp(C) if m >1.

Proof Assume lLw®Mp(C) is isomorphic to C(Prim(lLw)) ^®A p® Mmp(C). Then the unit llLw®Ipmaps to the unit1C(Prim(lLw))®Ap®Imp, where Is denotes the s x s identity matrix. So

Thus there is a projection eE lLw such that p[1lLJ = (mp)[e]. But Ko(lLw)^rv

-zi

Therefore, lLw ® Mp(C) is not isomorphic to C(Prim(lLw)) ^®A p^®

Mmp(C) if m >1. 0

By comparison of the K-theory, one can show that lLw®K:(ll) has a non-trivial bundle structure if Prim(lLw) is homeomorphic toL^k(n)xT^l'

for k and l' non-negative integers (k > 1), where Lk(n) is the lens space.

COROLLARY 2.4. Let lLw be a spherical non-commutative torus of rank l+1 with fibres A p ® Mm(C) for A p a simple non-commutative torus and Prim(lLw) ⁼ Lk(n) x T l' for k and l' non-negative integers (k > 1). Then lLw 0 K:(1l) is not isomorphic to C(Prim(lLw)) ® A p 0 Mm(C)® JC(ll).

Proof. By Theorem 2.2 Ko(lLw ® JC(ll)) ~ Ko(lLw) ~

-zl,

free. On the other hand, Ko(C(Prim(lLw)) ® A p ® Mm (C) ®JC(ll)) ^rv Ko(C(Prim(lLw))®Ap ) ~Ko(C(Lk(n))®C(Tl')®Ap)^rv Ko(C(Lk(n)))

®Ko(C(Tl') ® Ap ) EI1 Kl(C(Lk(n))) ® Kl(C(Tl') ® A p} by Kiinneth Theorem (see [2, Theorem 23.1.3]). But Ko(C(Lk(n))) 0 Ko(C(Tl') 0 A_{p )} ^rv (Z/knZEI1 Z) ® Z2^{1- 1} ~ (Z/kⁿZ)2^{1- 1} EI1 Z2^{1- 1} (see [8, IV.2.11]).

So Ko (C(Prim(lLw)) ®Ap®Mm(C) ®JC(ll)) is not torsion-free. Hence lLw0JC(1l) is not isomorphic to C(Prim(lLw))®Ap®Mm(C) ®JC(ll).D We have obtained that [1lLJ EKo(lLw) is primitive. This result is very useful to investigate the bundle structure of the tensor products of the spherical non-commutative tori with UHF-algebras and Cuntz algebras.

3. The tensor products of spherical non-commutative tori with UHF-algebras

In this section, we investigate the bundle structure of the tensor products of the spherical non-commutative tori lLw with UHF-algebras Mpoo of typep= and Cuntz algebras.

The following is useful.

THEOREM 3.1 [6, Theorem 7.1]. Suppose there exists an intertwin- ing of the sequence of C* -algebra homomorphisms A l ^--7 A₂ ^{--7 •••}

andBl ---TB₂ ---T •• •• Then the inductive limit C* -algebrastimAi and limBi areisomorphic.

THEOREM 3.2. Let lLw be a spherical non-commutative torus of rank 1+1 with Ebres A p^®Mm(C) for A p^a simple non-commutative torusand m apositive integer. LetMpoo be aUHF-algebraoftypepoo . ThenlLw ®Mpoo is isomorphicto C(Prim(lLw)) ®A p®Mm(C)®Mpoo if andonlyifthe setofprime factors of m isa subsetofthe set ofprime factors ofp.

Proof Assume the set of prime factors of m is a subset of the set of prime factors of p. To show that lLw ® Mpoo is isomorphic to C(Prim(lLw))®Ap®Mm^(C)®Mpoo ,it is enough to show that lLw®Mmoo is isomorphic to C®Mmoo whereC:= C(Prim(lLw)) ®A p. But there exist the canonicalC*-algebra homomorphisms:

The inductive limit of the odd terms

is lLw ®Mmoo, and the inductive limit of the even terms

is C ® Mmoo. Thus by Theorem 3.1, lLw ® Mmoo is isomorphic to C(Prim(lLw)) ®A p®Mmoo.

Conversely, assume lLw ®Mpoo is isomorphic toC(Prim(lLw)) ®A p^® Mm(C)®Mpoo. LetC:= C(Prim(lLw))®Ap. Then the unit llLw®lMpoo maps to the unit le ® lM_p⁰⁰ ®Im . So

[llLw ® lMpooJ = [le ® lMpoo ®ImJ.

And [llLw® lMpooJ = [1][,wJ® [lMpoo] and [le ® lMpoo ®Im ]=m([lc] ® [lMpoo]). But Ko(lLw^® Mpoo) '" [~](Ko(lLw))and Ko(C ® Mpoo ® Mm(C)) '" m[~](Ko(C)). Ifthere is a prime factor q of m such that q f ^p, then [lMpooJ f: q[e_oo] for eoo a projection in Mpoo under the assumption that the unit IlL ®lM_{w p}⁰⁰ maps to the unit le®lM_p⁰⁰®I.m •

So there is a projectioneELw such that [llLJ

=

Therefore, Lw®Mpoo is isomorphic toC(Prim(Lw»®Ap®Mm(C)®

Mpoo if and onlyifthe set of prime factors of m is a subset of the set

of prime factors ofp. 0

We have obtained that Lw®Mpoo has the trivial bundle structure if and onlyifthe set of prime factors ofm is a subset of the set of prime factors ofp.

Let us apply the previous results to the tensor products of Lw with Cuntz algebras.

The Cuntz algebra Od,2 ~ d < 00, is the universal C*-algebra generated by d isometries Sb ... ,Sd, Le., sjSj

=

relation sls1+ ... +Sdsd = 1. Cuntz [3, 4] proved that Od is simple and the K-theory ofOd is KO(Od)

=

=

proved that KO(Od) is generated by the class of the unit.

PROPOSITION3.3. LetLw be a spherical non-commutative torus of rankI+¹ with fibres A p^®Mm(C) for A p^a simple non-commutative torus and m a positive integer (m > 1). Let d be a positive integer such that m and d - 1 are not relatively prime. Then Od®Lw is not isomorphic to Od® C(Prim(Lw

»

Proof Letp be a prime such thatp Im andp Id-1. Suppose that Od®Lw is isomorphic to Od®C(Prim(Lw

»

unit lo_a®lLw maps to the unit 10a®G(Prim(lLw»®A_p ®fm, Le.,

So there is a projection e in Od ® Lw such that [loa^{®lLw] =} m[e].

But [lo_a®lLJ = [lOa] ® [llLJ and [lOa] is a generator ofKO(Od) ~ 7l.1(d - l)Z (see [4]). By assumption p I^{d - 1. [lOa]} =I p[el] for el a projection in Od. So [llLJ ⁼ m'[e2] for a projection e2 ELw and an integer m' withp Im'. This is a contradiction. Hence m and d - 1 are relatively prime.

Therefore, Od®Lw is not isomorphic toOd®C(Prim(Lw»^®Ap® Mm(C) if m and d - 1 are not relatively prime. D

More generally, it follows from a theorem of ~rdamthat m and d - 1 are relatively prime if and only if Od ® lLw is isomorphic to Od®C(Prim(lLw

»

The following result is useful to understand the bundle structure of Od®lLw.

PROPOSITION 3.4 [9, Theorem 7.2]. Let A andB be unital simple inductive limits of even Cuntz algebras. Ifa :Ko(A) --T Ko(B) isan isomorphism of abelian groups satisfyinga([lAD= [lB], then thereis an isomorphism4> :A --TB which induces a.

COROLLARY 3.5.

(1) Let p be an odd integer such that p and 2d - 1 are relatively prime. Then 02d is isomorphic to O(2d-l)p+l®Mpoo. That is, 02d is isomorphicto 02d®Mpoo .

(2) 02d is isomorphic to 02d®M(2d)^{00 •}

THEOREM 3.6. Let lLw be a spherical non-commutative torus given as above. Then 02d®lLw is isomorphic to 02d®C(Prim(lLw» ^{® A}p® Mm(C) if and only ifm and2d-1 are relatively prime.

Proof Assume that m and 2d - 1 are relatively prime. Let m = p2^c for some odd integerp. Thenpand2d-1 are relatively prime. Then by Corollary 3.5 02d is isomorphic to 02d ® Mpoo, and 02d is isomorphic to 02d ® M(2d)00 ~ 02d ® M(2d)00 ® M(2C)00 ~ 02d ® M(2^c)'x>, So Ou is isomorphic to 02d®Mpoo ® M(2C)00 ^f'V 02d®Mmoo. Thus by Theorem 3.2 02d®lLw is isomorphic to 02d® Mmoo ®lLw, which in turn is isomorphic to 02d®Mmoo ®C(Prim(lLw»®Ap®Mm(C). Thus 02d®lLw is isomorphic to 02d®C(Prim(lLw

»

The converse is proved in Proposition 3.3.

Therefore, Ou® lLw is isomorphic to 02d® C (Prim(lLw» ® Ap® Mm(C) if and onlyifm and 2d - 1 are relatively prime. 0

Cuntz [4] computed the K-theory of the generalized Cuntz algebra 0 00 , generated by a sequence of isometries with mutually orthogonal ranges, Ko(Ooo) = Z and K1(Ooo) = O. He proved that Ko(Ooo) is generated by the class of the unit.

PROPOSITION 3.7. Let lLw be a spherical non-commutative torus given as above. 000 0 lLw is not isomorphic to 000 ®C(Prim(lLw)) ® Ap0 M7n(C).

Proof Suppose Ooo®lLw is isomorphic to0000C(Prim(lLw))0A_p0 M=(C). Then the unit 10000lL_w maps to the unit 10000G (Prim(lLw»0A_p

0[=. By the same trick as in the proof of Proposition 3.3, one can show that [looo@JLwl = m[e] for a projection e E000®Lw· ^[10000lLJ =

[10 00 ]®[hJ and [10₀₀^] isa primitive element ofKo (000 ) £=!Z^(see[4]).

So [llLwl = m[el] for a projection el ElLw. This contradicts Theorem 2.2.

Hence 000 0Lw is not isomorphic to 000 0 C(Prim(lLw)) 0 A_p 0

M=(C). 0

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Funct. Anal.14 (1973), 299-324.

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[4] ,K-theory for certain C*-algebras, Ann. Math.113 (1981), 181-197.

[5] G. A. Elliott, On the K -theory of the C* -algebra generated by a projective representation of a torsion-free discrete abelian group, Operator Algebras and Group Representations (G. Arsene et al., ed.),vol. 1, Pitman, London, 1984, pp.157-184.

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Math. 443 (1993), 179-219.

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Department of Mathematics Chungnam National University Taejon305-764, Korea