Title Notes on the stable rank estimate for twisted crossed productsof C^*-algebra by Z

Author(s) Sudo, Takahiro

Citation Ryukyu mathematical journal, 16: 113-135

Issue Date 2003-12-30

URL http://hdl.handle.net/20.500.12000/43490

Rights

Ryukyu Math. J., 16(2003),113-135

NOTES ON THE STABLE RANK

ESTIMATE FOR TWISTED CROSSED

PRODUCTS OF C*-ALGEBRAS BY Z

TAKAHIRO SUDO

ABSTRACT. In this paper we study the stable rank estimate fortwisted crossed products of C* -algebras by the integers. Moreover,by using this estimate we obtain several consequences such as thestable rank estimates for twisted crossed products of C* -algebrasby discrete solvable groups.

2000 Mathematics Subject Classification. Primary 46L05, 46L55Key words: Stable rank, Twisted crossed product, C* -algebra

O. INTRODUCTION

This article is a technical note on the stable rank estimate fortwisted crossed products of C*-algebras by the integers Z (see Packer­Raeburn [PRl] for twisted crossed products of C*-algebras). In fact,this estimate has already been proved in [Sd2] by using the sim­ple method quite different from one that will be exhibited in thispaper, and our (most) results in this paper are contained in thoseof [Sd2]. Our first interest as motivation is to estimate the stablerank of twisted crossed products of C*-algebras by (actions of) Zsince Rieffel has obtained the stable rank estimate for (ordinary)crossed products by Z ([Rfl] in which Rieffel introduced the stablerank for C*-algebras and obtained some basic formulas for the stablerank). Also, in [Sdl] the author obtained the stable rank estimate forcrossed products by lR or 1[', and also considered the case of twistedcrossed products by lR or 1[' in [Sd2]. Again, our direct computations

Received November 30, 2003

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that will be exhibited below would be useful for our study in thefuture, which could be a merit for publication in this issue.

This paper is organized as follows. We first review some basic no­tation and definitions for the stable rank of C*-algebras and twistedcrossed products of C*-algebras by locally compact groups in therest of this introduction below. In Section 1, as the main resultswe estimate the stable rank and connected stable rank of twistedcrossed products by Z. The proof is based on the ideal of the Rief­fel's estimate for crossed products by Z. However, technical part ofthe proof for twisted crossed products by Z is more complicated thanthat of crossed products by Z. In Section 2, as applications, by usingthose main results we estimate the stable rank and connected stablerank of twisted crossed products of C*-algebras by discrete solvablegroups in terms of those ranks of the C*-algebras and the rank ofthe groups. In particular, we obtain the stable rank estimate fortwisted group C*-algebras of discrete solvable groups. For the proof,we use the decomposition theorem of twisted crossed products dueto Packer-Raeburn [PR1]. Finally, we review the correspondence be­tween Green's twisted covariance algebras and twisted crossed prod­ucts in [PR1] and the correspondence between the twisted covariancealgebras and twisted group C*-algebras in [PR3].

Acknowledgement. The author would like to thank Department ofMathematical Sciences at University of the Ryukyus as well as thestaff for publication in this issue and for sharing the time and space.

First of all, we review some basic notations and definitions asfollows:

Notations.Let 2t be a C*-algebra. When 2t is nonunital we always consider

its unitization 2t+. Let sr(2t), csr(2t) and gsr(2t) denote the (topo­logical) stable rank, connected stable rank and general stable rankof 2t respectively ([Rfl], [Ns]). By definition,

sr(2t), csr(2t), gsr(2t) E {I, 2, ... ,oo},

and sr(2t) ::s n if and only if the open subspace L n (2t) of 2tn is densein 2tn

, where (aj)j=l E L n (2t) if and only if 2:;=1 ajaj is invertible in2t, and csr(2t) ::s n if and only if the connected component GLm (2t)o

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with the unit matrix of GL m (Qt) the group of invertible m x m matri­ces over Qt acts transitively on L m (Qt) for any m ~ n, or equivalentlyLm(Qt) is path-connected for any m ~ n, and gsr(Qt) :s: n if and onlyif GLm(Qt) acts transitively on Lm(Qt) for any m ~ n. We alwayshave

gsr(Qt) :s: csr(Qt) :s: sr(Qt) + 1

([Rfl, Corollary 4.10 and p.328]). Let C(X) denote the C*-algebraof all continuous functions on a compact Hausdorff space X. LetQt ><la,u G denote the twisted crossed product of a (separable) C*­algebra Qt by a locally compact group G with (a, u) a twisted action,where a is a Borel map from G to the automorphism group of Qtwith the topology of pointwise norm convergence, that is, the mapfor each a E Qt : G :3 9 t---+ ag(a) E Qt is Borel measurable, and u is astrictly Borel measurable map from the product G x G to the unitarygroup of the multiplier algebra M(Qt) of Qt with the strict topology,that is, the maps for each a E Qt : G x G :3 (g, h) t---+ u(g, h)a and(g, h) t---+ au(g, h) E Qt are Borel, such that

{

ae = id, u(e,g) = u(g,e) = 1

a g 0 ah = Adu(g, h) 0 agh

al (u(g, h))u(l, gh) = u(l, 9)u(lg, h)

for g, h, lEG and e the unit of G, where the map a on Qt is canoni­cally extended on M(Qt) in the sense that

for 9 E G, a E Qt and mE M(Qt). If the map u is trivial, then Qt><la,uGis the same as the (ordinary) crossed product Qt ><la G (d. [Pd]). Acovariant representation of a twisted dynamical system (A, G, a, u)(or Qt ><la,u G) is a pair (7r, U) (or its integrated form 7r x U) with 7r

a non-degenerate representation of Qt on a Hilbert space Hand Ua Borel measurable map from G to the unitary group on H (or themap for each ~,17 E H : G :3 9 t---+ (Ug~I17) E C (the inner product ofH) is Borel) such that

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for g, h E G and a E 2L Note that the map u does not necessarilymean a usual 1I' (the torus)-valued multiplier. An important pointfor our study is that the C*-algebra 2l )<la,u G has a dense subspacegenerated by 2l and L 1(G) (the space of all C-valued integrable func­tions on G) as in the case of ordinary crossed products 2l)<la G. Also,the representation Jr x U associated with (Jr, U) is defined by

Jr X U(x) =1Jr(x(t))Utdt

for x ELI (G, 2l) the space of all integrable functions on G takingvalues in 2L The convolution product * and the involution of theBanach *-algebra L 1(G, 2l) in 2l ><Ja,u G are defined by

{

X * y(s) = fG x(t)at(y(st- 1) )u(t, st- 1 )dt,

x*(t) = u(t, C 1 )*at(x(t- 1))* 6 G (C 1 )

for s, t E G and x, y E L 1 (G,2l) and 6 G the modular function onG. See [PRl-2] and [BS] for more details. See also [Bl] and [Pd] forgeneral references for the C*-algebra theory. In what follows G willbe a discrete group. In this case, 6 G is trivial.

1. THE MAIN RESULTS

First of all, we consider the stable rank estimate for twisted crossedproducts of C*-algebras by the integers Z, which is essentially mainin this paper.

Theorem 1.1. Let2l be a unitaIC*-algebra and2l><Ja,uZ the twistedcrossed product of 2l by Z with (a, u) a twisted action. Then

sr(2l )<la,u Z) :::; sr(2l) + 1.

Proof. By assumption of 2l being unital we have 2l = M(2l) so thatu(n, m) E 2l for any n, mE Z.

Let (Jr, U) be the universal covariant representation of the twisteddynamical system (2l, Z, a, u) (which will be defined as the univer­sal direct sum representation of all covariant representations of thesystem, or we may assume that (Jr, U) is the regular representation

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induced by a faithful representation p of 2l on a Hilbert space H suchthat

(7r(a)~)(t) = p(at(a))~(t),

(UsO(t) = p(u(t,s))~(ts)

for s, t E G and ~ E L2 (G, H) the Hilbert space of all square inte­grable functions on G taking values in H, since Z is amenable (d.[PRI, Theorem 3.11]). Then 2l )<Io,u Z is identified with the C*­algebra generated by the set {7r(a), Un Ia E 2l, n E Z}. Moreover,the set of linear spans:

span{7r(a)U~Um Ia E 2l, n, mE Z}

is dense in 2l )<Io,u Z (d. [PRI, Definition 2.4] and [PR2, Corollary1.5]). To see that, first observe that for a, b E 2l, n, m E Z,

7r(a)Un7r(b )Um = 7r(a)Un7r(b)U~UnUm

= 7r(a)7r(an(b))7r(u(n, m))Un+m= 7r(aan(b)u(n, m))Un+m,

7r(a)Un7r(b)U:n = 7r(a)Un7r(b)U~UnU:n = 7r(a)7r(an(b))UnU:n

= 7r(a)7r(an(b))7r(u(n - m, m))*Un-mUmU:n

= 7r(a)7r(an(b))7r(u(n - m, m))*Un- m= 7r(aan(b)u(n - m, m)*)Un- m,

7r(a)U~7r(b)Um = 7r(a)7r(a_n(b))U~Um = 7r(aa-n(b))U~Um,

7r(a)U~7r(b)U:n = 7r(a)U~U:nUm7r(b)U:n = 7r(a)(UmUn)*7r(am(b))

= 7r(a)(7r(u(m, n))Um+n)*7r(am(b))

= 7r(a)U:n+n7r(u(m, n))*7r(am(b))

= 7r(a)7r(a_m_n(u(m, n)*am(b)))U:n+n= 7r(aa_m_n(u(m, n)*am(b)))U:n+n·

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Moreover, observe that for k, l E Z,

7r(a)U~Um7r(b )UkUl

= 7r(a)U~(Um7r(b)U:n)UmUkUI

= 7r(a)7r(a-n(am(b)))U~UmUkUI

= 7r(a)7r(a_n(am (b)) )U~ (U(m-k)+kUnUI

= 7r(a)7r(a_n(am(b)))U~(7r(u(m - k, k))*Um-k)UI

= 7r(a)7r(a_n(am(b)))U~7r(u(m - k, k))*7r(u(m - k, l))Um-k+1

= 7r(a)7r(a_n(am(b)))7r(a_ n(u(m - k, k)*u(m - k, l)))U~Um-k+1

= 7r(aa_n(am(b))a_ n(u(m - k, k)*u(m - k, l)))U~Um-k+l,

and (7r(a)U~Um)*

= U:nUn7r(a)* = U:nUn7r(a*)U~Un

= U:n 7r (an(a*))Un = 7r(a_ m (an(a*)))U:nUn,

and UnU:n

= U(n-m)+mU:n = 7r(u(n - m, m))*Un- m.

Next, we first note that

U~+kUm+k = (7r(u(k, n))*UkUn)*7r(u(k, m))*UkUm

= U~Uk7r(u(k, n)7r(u(k, m))*UkUm

= U~7r(a_k(u(k, n)u(k, m)*))UkUkUm= 7r(a_n(a_k(u(k, n)u(k, m)*)))U~Um

Thus, we define the length of a finite sum d = L~=l 7r(aj)U~jUmjby

for aj E 2(, nj, mj E Z, and let L(O) = O. From the above observationand the equality (m + k) - (n + k) = m - n, the length L(d) is well­defined. Moreover, we obtain that L(Ukd) = L(d) and L(U;'d) =

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L(d) for any k E Z. This follows from the following:

(Cd: Uk7f(a)U~Um

= (Uk7f(a)Uk)UkU~Um = 7f(Qk(a))U(k-n)+nU~Um

= 7f(Qda))7f(u(k - n, n))*Uk-nUnU~Um

= 7f(Qk(a))7f(u(k - n, n))*7f(u(k - n, m))Uk- n+m,

= 7f(Qk(a)u(k - n, n)*u(k - n, m))Uk- n+m,

(C2 ): U;7f(a)U~Um

= 7f(Q-k(a))U;U~Um = 7f(Q-k(a))(UnUk)*Um= 7f(Q_k(a))(7f(u(n, k))Un+k)*Um

= 7f(Q-k(a))U~+k7f(u(n, k))*Um

= 7f(Q-k(a))7f(Q-n-k(u(n, k)))*U~+kUm

= 7f(Q-k(a)Q-n-k(u(n, k))*)U~+kUm.

Thus,

L(Ukd) = max {k + mj - nj} - min {k + mj - nj} + 1 = L(d),l~j~1 l~j~1

L(U*d) = max {m - n - k} - min {m" - n" - k} + 1 = L(d).k 1<"<1)) 1<"<1))

-)- -)-

Now suppose that sr(21) ~ s. Let (Cj)j~~ E 2l)<Ja,uZ. Then each Cj

is approximated closely by a finite sum dj = 2.:~=1 7f(ajk)U~jk Umjkwith m"l - n"l < ... < ml - n"1 Set D = (d)S+l and L(D) =)) ) j ) j. ) )=12.:;~i L(dj ). Moreover, suppose that L(D) is smallest among theset {L(WD') IW E ELs+1 (21 )<Ja,u Z), D' E V} by replacing D withW D' for some Wand D' if necessary, where W D' means the leftmultiplication by W to D', ELs +1(21 )<Ja,u Z) is the set of all elemen­tary (s + 1) x (s + 1) matrices over 2l )<Ja,u Z, and V is a small openneighborhood of D.

Now suppose that dj =I- 0 for any j. We then show a contradic­tion in the following. We may assume L(d1 ) ~ ... ~ L(ds+1) by apermutation by elementary matrices if necessary.

When n s +1,l s +\ =I- 0 and m s +1,l s +\ =I- 0, consider the following

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multiplication:

U~S+l,IS+l -njl j Unjll+mS+1,IS+1 -mjlj d j

if n s +l,ls+l ~ njl j and m S +l,ls+l ~ mjl j ,

U~jlj -ns+1,ls+1 Unjll+mS+l,lS+l -mjlj d j

if n s +l,ls+l < njl j and m s +l,ls+l ~ mjlj ,

U~S+l,lS+l -njl j Unjll+mjlj -ms+l,ls+l d j

if n s+l,ls+l ~ njl j and m s +l,ls+l < mjlj

Un*· -n Unll+m·l-m+11 d J·Jlj s+l,ls+l J J j s, s+l

if n s +l,ls+l < njl j and m s +l,lS+l < mjl j

for 1 :S j :S s. When n s +l,ls+l = 0 and ms+l,ls+l =I 0, consider thefollowing multiplication:

Unjll+mS+1,IS+1 -mjlj d j

Unjll+mjlj -ms+1,ls+1 d j

if m S +l,ls+l ~ mjl j ,

if m s +l,ls+l < mjlj

for 1 :S j :S s. When n s +l,ls+l =I 0 and m s +l,ls+l = 0, consider thefollowing multiplication:

T;d; = {if n s +l,ls+l ~ njl j ,

if n s +l,ls+l < njl j

for 1 :S j :S s. Then the highest term of Tjdj with respect to U~Um

in each case (Mj ) (1 :S j :S 3) is U~ Um +1 I by the sames+l,ls+l s, s+l

calculations as (Cd and (C2 ). In each case (Mj ), let 1f(hj ) E 1f(2t)(1 :S j :S s + 1) be the coefficients of Tjdj (1 :S j :S s) and d S + 1

at U~ Um +1 I respectively. Since sr(2t) :S s, there existss+l,ls+l s. s+l

(fj)j=1 E 2ts such that 1f(hs + 1) = I:;=I1f(h)1f(hj) if necessary byreplacing 1f(hj ) with suitable elements by small perturbation. Then,

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take the following operation:

JC J(JJ

n odd,

n even,

where d~+1 = dS +1 ~ l::;=I1r(fj)Tjdj. Then L(dj ) = L(Tjdj ) for1 :::; j :::; s but L(ds+d > L(d~+l)' which is the contradiction.

From the above argument we may assume dj = 0 for some j.Then we replace dj with c1 for a small E > O. Then it is easy tosee (dj)j~~ E Ls+1 (l2( >:1 a ,u Z). Moreover, Ls+1 (2t >:1 a ,u Z) is openand stable under the left multiplication by elementary matrices sothat any element (dj )j~~ of (2t >:1 a,u Z)s+1 can be approximated byelements of L S +1 (2t >:1 a ,u Z). Therefore, sr(2t >:1 a ,u Z) :::; S + 1. D

Remark. Let 2t = C(T"). If we take (11', u) both trivial, then 2t >:1 a ,u

Z ~ C(T"+I). Thus,

sr(2t >:1 a ,u Z) = sr(C(1F+ 1))

{

= sr(C(1F)) + 1= [(n + 1)/2] + 1

< sr(C(1F)) + 1

where [x] means the maximum integer:::; x (d. [Rfl, Proposition1.7]), which says that the estimate of Theorem 1.1 is optimal.

By using the argument of [Rfl, Corollary 8.6] we have

Corollary 1.2. Under the same situation as Theorem 1.1, we obtain

csr(2t >:1 a ,u Z) :::; sr(2t) + 1.

Proof. The proof of Theorem 1.1 implies that the set of the elementsXel (multiplication) for X E ELs+1 (2t >:1 a ,u Z) is dense in (2t >:1 a ,u

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Z)8+1, where e1 = (1,0, ... ,0). Thus, GL 8 +1 (2t )<lQ,U Z)Oe1 is densein (2t )<lQ,U Z)8+1. Hence L 8 +1 (2t )<lQ,U Z) is connected. D

Remark. Since gsr(·) ~ csr(·) by definition, we have

gsr(2t )<lQ,U Z) ~ sr(2t) + 1.

Let 2t = C('Jrl') and (o:,u) both trivial. Then

csr(2t )<lQ,U Z) = csr(C(1F+1))

= [(n + 2)/2] + 1 = sr(C(1I'n)) + 1

(d. [Sh, p.381] and [Rfl, Proposition 1.7]), which says that theestimate of Corollary 1.2 is optimal. When 2t = C(X) with X a con­tractible compact space, we have csr(C(X) 0 C(1I')) = csr(C(1I')) = 2and C(X) 0 C(1I') ~ C(X) )<l Z (the trivial action) ([Eh, Corollary2.12]).

For the nonunital case, we in fact obtain

Theorem 1.3. Let 2t be a nonunital C* -algebra and 2t )<l Q,U Z thetwisted crossed product of 2t by Z with (0:, u) a twisted action. Then

sr(2t )<lQ,U Z) ~ sr(C*(2t, u(Z, Z))) + 1

where C* (2t, u(Z, Z)) means the C* -algebra generated by 2t and allu(s, t) for s, t E Z. Moreover, we obtain

max{sr(2t), sr(C*(u(Z, Z)))}

~ sr(C*(2t,u(Z,Z)))

~ max{sr(2t), sr(C* (u(Z, Z))), csr(C* (u(Z, Z)))},

where C* (u(Z, Z)) means the C* -algebra generated by u(Z, Z).

Proof. The line of the proof is the same as Theorem 1.1. Notethat 2t )<lQ,U Z is identified with the C*-algebra generated by theset {7f(a),7f(u(s,t)),Un la E 2t,s,t,n E Z}. Moreover, the set oflinear spans:

span{7f(a)U~Um, (rrJ=l7f(U(Sj, tj)))U~Um

Ia E 2t, n, m, Sj, tj E Z(1 ~ j ~ kEN)}

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is dense in 21 )qa,uZ, where each 1r(u(Sj, tj)) may be replaced with itsadjoint 1r(u(Sj, tj))*. In fact, note that 1r(a)1r(u(s, t)), 1r(u(s, t))1r(a) E

1r(21) for any a E 21 and s, t E Z since 21 is a closed ideal of the mul­tiplier algebra of 21. Also, C* (21, u(Z, Z)) may be identified withC*(1r(21), 1r(u(Z, Z))) generated by 1r(21) and 1r(u(Z, Z)). Moreover,observe that

1r(a)U~Um1r(b)U;Ul

= 1r(a)U~(Um1r(b)U:n)UmU;Ul

= 1r(a)1r(a_n(am(b)))U~UmU;Ul

= 1r(a)1r(a_n (am(b)) )U~(U(m-k)+kUnUl

= 1r(a)1r(a_n(am(b)))U~(1r(u(m - k, k))*Um-k)Ul= 1r(a)1r(a_n(am(b)))U~1r(u(m - k, k))*1r(u(m - k, l))Um- k+l= U~Un1r(aa_n(am(b))U~1r(u(m - k, k))*1r(u(m - k, l))Um- k+l= U~1r(an(aa_n(am(b)))1r(u(m - k, k))*1r(u(m - k, l))Um- k+l= 1r(a_n(an(aa_n(am(b)))u(m - k, k)*u(m - k, l)))U~Um-k+l

for bE 21 and k, l E Z, and

(A): 1r(u(s, t))U;Ul1r(a)U~Um

= 1r(u(s, t))U;Ul1r(a)UtUlU~Um

= 1r(u(s, t))U;1r(al(a))UlU~Um

= 1r(u(s, t))1r(a-k(al(a)))U;UlU~Um

= 1r(u(s, t))1r(a-k(al(a)))U;UlUtn_l)+IUm

= 1r(u(s, t))1r(a_k(al(a)))U;Ul (1r(u(n -l, l))Un-IUl)*Um= 1r(u(s, t))1r(a-dal(a)))U;U~_l1r(u(n -l, l))*Um

= 1r(u(s, t))1r(a-dal(a)))(Un-lUk)*1r(u(n -l, l))*Um= 1r(u(s, t))1r(a_k(al(a)))

(1r(u(n -l, k))Un- l+k)*1r(u(n -l, l))*Um

= 1r(u(s, t))1r(a-k(al(a)))U~_I+k1r(u(n -l, k))*1r(u(n -l, l))*Um

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where this is to be continued as follows:

= U~_I+kUn-l+k1r(U( s,. t)Q-k(Ql (a)) )U~-l+k

1r(u(n -l, k))*1r(u(n -l, l))*Um

= U~_I+k1r(Qn-l+k(U(S, t)Q-k(QI(a))))

1r(u(n -l, k))*1r(u(n -l, l))*Um

= 1r(Q-n+l+k(Qn-l+k(U(S, t)Q-k(QI(a))))

u(n -l, k)*u(n -l, l)*))U~_I+kUm

for s,t E Z, and

1r(a)U~Um1r(u(s, t) )U~UI

= U~Un1r(a)U~Um1r(u(s, t))U~UI

= U~1r(Qn(a))Um1r(u(s, t))U~UI

= U~1r(Ad(u(m, -m))(u(m, -m)*Qn(a)u(m, -m)))

Um1r(u(s, t))U~UI

= U~1r(Qm 0 Q_m(u(m, -m)*Qn(a)u(m, -m)))Um1r(u(s, t))U~UI

= U~Um1r(Q-m(u(m, -m)*Qn(a)u(m, -m)) )1r(u(s, t) )U~UI

= U~Um1r(Q_m(u(m, -m)*Qn(a)u(m, -m))u(s, t))U:nUmU~UI

= U~1r(Qm(Q-m(u(m, -m)*Qn(a)u(m, -m))u(s, t)))UmU~UI

= 1r(Q-n(Qm(Q- m(u(m, -m)*Qn(a)u(m, -m))u(s, t))))U~UmU~UI

and transform U~UmU~UI as in (A).Moreover, observe that

(I): Uk1r(U(S, t))U~Um = UkUsUtU;+tU~Um

= 1r(u(k, S))Uk+sUtU;+tU~Um

= 1r(u(k, s))1r(u(k + S, t))Uk+s+tU;+tU~Um

= 1r(u(k, s))1r(u(k + s, t))1r(u(k, S+ t))*UkUs+tU;+tU~Um

= 1r(u(k, s))1r(u(k + s, t))1r(u(k, S+ t))*Uk-n+nU~Um

= 1r(u(k, s))1r(u(k + s, t))1r(u(k, S+ t))*

1r(u(k - n, n))*Uk-nUnU~Um

= 1r(u(k, s))1r(u(k + s, t))1r(u(k, S+ t))*1r(u(k - n, n))*Uk-nUm

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and

Uk1r(U(S, t))* = Uk(UsUtU;+t)* = UkUs+tutu;

= 1r(u(k, S+ t))Uk+s+tUtu;

= 1r(u(k, S+ t))1r(u(k + S, t))*Uk+sUtutu;

= 1r(u(k, S+ t))1r(u(k + s, t))*1r(u(k, S))*UkUsU;

= 1r(u(k, S+ t))1r(u(k + s, t))*1r(u(k, S))*Uk.

Moreover,

(II): U'k1r(u(s, t))U~Um = U-k+kU'k1r(U(S, t))U~Um

= 1r(U( -k, k))*U-kUkU'k1r(U(S, t))U~Um

= 1r(U( -k, k))*U_ k1r(u(s, t))U~Um

and transform U_ k1r(u(s, t)) as in (I), and

1r(U(S, t))U~Um1r(u(x, y))U'kUl

= 1r(u(s, t))U-n+nU~Um1r(u(x, y))U'kUl

= 1r(u(s, t))1r(u( -n, n))U_nUnU~Um1r(u(x,y))U'kUl

= 1r(u(s, t) )1r(u( -n, n))U-nUm1r(u(x, y) )U'kUl= 1r(u(s, t))1r(u( -n, n))1r(u(-n, m))U_n+m1r(u(x, y))U'kUl

for x, y E Z, and then transform U- n+m1r(u(x, y))U'kUl as in (I).Moreover,

(1r(U(S, t))U~Um)*

= U:nUn1r(u(s, t))* = U:nUn(UsUtU;+t)*

= U:nUnUs+tutu;

= U:n1r(u(n, S+ t))Un+s+tutu;

= U:n1r(u(n, S+ t))U(n+s)+tutu;

= U:n1r(u(n, S+ t))1r(u(n + S, t))*Un+sUtutu;

= U:n1r(u(n, S+ t))1r(u(n + S, t))*Un+sU;

= U:n1r(u(n, S+ t))1r(u(n + S, t))*1r(u(n, s))*UnUsU;

= U:n1r(u(n, S+ t))1r(u(n + S, t))*1r(u(n, s))*Un

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and then convert U:n to 7r(u( -m, m))*U_ m as in (II) and then trans­form U_ m 7r(u(n, s+t)) as in (1). Then repeat this procedure to obtainthe term of Un - m with a certain coefficient.

Next suppose that sr(C*(21, u(Z, Z))) ::; M as sr(21) ::; s in theproof of Theorem 1.1 and follow the same argument as given there.

Finally, note that

o~ 2l ~ C*(21, u(Z, Z)) ~ C*(u(Z, Z)) ~ 0

since 2l is a closed ideal of C*(21, u(Z, Z)). Then use [Rfl, Theorems4.3, 4.4 and 4.11] which imply that for any exact sequence: 0 ~ J ~123 ~ 123 IJ ~ 0 of C*-algebras,

max{sr(J),sr(I23/J)}::; sr(l23)

::; max{sr(J), sr(I23/J), csr(123/J)}. 0

Remark. When u(n, m) E 1I' for any n, mE Z, we have C*(u(Z, Z)) ~C. Thus we obtain sr(C*(21, u(Z, Z))) = sr(21 + Cl) = sr(21). Theframework for generating elements in the proofs of Theorems 1.1 and1.3 could be useful in other situations. The structure of the algebraC* (u(Z, Z)) might be interesting or rather complicated in general.

By the same way as Corollary 1.2 we obtain

Corollary 1.4. Under the same situation as TheoT'em 1.3 we have

csr(21 ~Q,U Z) ::; sr(C*(21, u(Z, Z))) + 1.

2. ApPLICATIONS

We say that a group f is a successive semi-direct product by(finitely generated) free abelian groups if

for k 1 , k2 ,'" ,kn E Z. Define the rank of f by rank(f) = ,,£7=1 k j .

Note that f is solvable, but a solvable discrete group (without tor­sion) is not always a successive semi-direct product by free abeliangroups. Then

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Theorem 2.1. Let 2t be a unital C* -algebra and f a successive semi­direct pmduct by free abelian gmups. Then

{sr(2t )qa,u f) :; sr(2t) + rank(f),

csr(2t )qa,u f) :; sr(2t) + rank(f).

Pmoj. We have the following decomposition of 2t )qa,u f into succes­sive twisted crossed products:

where the twisted actions (C¥j, Uj) (1 :; j :; n) are the restrictions of(c¥, u) to 'lh respectively. This decomposition follows from the usualargument for generators of twisted crossed products 2t )qa,u f as thecase of the ordinary crossed products 2t)qaf (which is also guaranteedby the decomposition theorem of twisted crossed products used inthe proof of Theorem 2.2 below). Moreover, the twisted crossedproducts by free abelian groups are regarded as successive twistedcrossed products by Z as follows:

where the twisted actions (c¥j, uj) for 1 :; s :; k j are the restrictionsof (C¥j, Uj) to direct factors Z of Zk j respectively, and

Then use Theorem 1.1 rank(r)-times. D

Now recall that a solvable (discrete) group f has a (finite) normalseries {Zj }j=o with Zo = {1} and Zn = f such that Zj-l is normalin Zj, and each subquotient ZjjZj-l is Abelian (1 :; j :; n) (d.[Rg]).

Also recall the following Packer-Raeburn's decomposition theoremof twisted crossed products:

The decomposition theorem of twisted crossed products[PRl]. Let 2t )qa,u G be a twisted cmssed pmduct of a C*-algebra 2tby a locally compact gmup G with (c¥, u) a twisted action. For' N a

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closed normal subgroup of G, there exists a twisted action ((3, v) ofG / N on the twisted crossed product 2t >'1 o,v, N associated with N suchthat

2t >'1 o ,u G ~ (2t >'1 o ,u N) >'1/3,v G/N.

By using this decomposition, we have the following as a general­ization of Theorem 2.1:

Theorem 2.2. Let 2t be a unital C* -algebm and f a discrete solvablegroup with a normal series with subquotients free abelian and finitelygenemted. Then

{sr(2t >'1 o ,u f) ~ sr(2t) + rank(f),

csr(2t >'1 o ,u f) ~ sr(2t) + rank(f),

where rank(r) means the sum of the free mnks of subquotients of f.

Proof. Let {Zj }j=o be a normal series of f with Zo = {1} and Zn =

f such that Zj/Zj-l are free abelian, finitely generated (and closedin f /Zj-l) (1 ~ j ~ n). By using the decomposition theorem oftwisted crossed products [PR1, Theorem 4.1] repeatedly, we have

2t>'1 ou f,

~ (2t >'1 o ,u Zl) >'1 02 ,U2 (f /Zl)

~ ((2t >'1 o ,u Zd >'1 02 ,u2 (Z2/Z d) >'1 03 ,u3 (f/Z2 )

~ (... (((2t >'1 o ,u Zd >'1 02 ,U2 (Z2!Zd) >'1 03 ,u3 (Z3/Z2))

••• ) >'1 on ,un Zn/Zn-l.

Put I.E j = 2tj >'1 0j ,Uj (Zj/Zj_l) (1 ~ j ~ n) with al = a and Ul = U,

where 2t2 = 2t >'1 o ,u Zl, 2t3 = (2t >'1 o ,u Zl) >'1 02 ,U2 (Z2!Zd and 2tj

is defined inductively the same way as in the decomposition. SinceZj/Zj_l is finitely generated and free abelian, Zj/Zj-l ~ Zkj forsome kj , where kj is the rank of Zj/Zj-l. By using the decomposi­tion theorem again repeatedly,

I.E j >'1 0j ,Uj (Zj/Zj-d ~ I.E j >'1 0j ,Uj Zkj

rv (l.E j >'1 0j ,Uj Z) >'1 Zkj-l

~ (... ((l.E j >'1 0j ,Uj Z) >'1 Z) ... ) >'1 Z.

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By using Theorem 1.1 repeatedly, we obtain

sr(21 )<]a,u r) = sr(21n )<]an,un (Zn/Zn-l))

::; sr(21n) + rank(Zn/Zn-l)

::; sr(21n-d + rank(Zn-dZn-2) + rank(Zn/Zn_dn

::; sr(21) +l: rank(Zj/Zj_l)j=l

and rank(r) = 'L7=1 rank(Zj/Zj_d· We use the same argument andCorollary 1.2 for the connected stable rank estimate in the statement.In particular,

csr(21 )<]a,u r) = csr(l13n )<]an,un Zkn)

::; sr(l13n )<]an,Un Zkn-l) + 1. 0

Remark. If r is nilpotent, each subquotient Zj/Zj-1 is central inr / Zj -1. In particular, we may take r as the generalized discreteHeisenberg groups of rank 2n + 1, which are central extensions ofz2n by the center Z, and isomorphic to the semi-direct productszn+1 )<] zn (d. [LPJ). Also, it is well known that a finite group issolvable if and only if it has a composition series (as a refinementof normal series) with its subquotients cyclic groups of prime orders(cf. Theorem 2.4 below). A merit of Theorem 2.1 is that its proof isnot using the decomposition theorem of twisted crossed products.

Recall that the twisted group C* -algebra C* (G, a) of a locallycompact group G with a cocycle (or a multiplier) a E Z2(G, 'lI') thegroup of Borel cocycles is defined to be the enveloping C* -algebra(or C* -completion by the universal representation of the system(te, G, id, a) with id the trivial action) of L 1(G) with the convolu­tion product and involution defined by

{

X * y(s) = Ie x(t)y(st- 1 )a(t, st- 1)dt,

x* (t) = a(t, c 1 )*x(t- 1)* ~e(t-1)

for s,t E G and x,y E L1(G).

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Corollary 2.3. Let f be as in Theorem 2.2 and C*(f, a) the twistedgroup C* -algebra of f with a cocycle a. Then

{

sr(C*(f, a)) :S rank(f),

csr(C*(f, a)) :S max{2, rank(f)},

gsr(C*(f, a)) :S rank(f).

In particular, if a is trivial, then

{

sr(C* (f)) :S rank(f),

csr(C*(f)) :S max{2,rank(f)},

gsr(C* (f)) :S rank(f),

where C* (f) is the group C* -algebra of f.

Proof. By definition, we have C*(f, a) ~ C )<lid,O" f a twisted crossedproduct of C by f with (id, a) a twisted action, where id means thetrivial action of f on C and a is a usual 1I'-valued cocycle. Also notethat C)<l id,O" Z ~ C)<l Z since the (Moore) cohomology group H 2 (Z, 1I')is trivial (cf. [Pk, Example 1.2]), and C )<l Z ~ C*(Z) ~ C(1I')by the Fourier transform. Moreover, we have sr(C(1I')) = 1, andcsr(C(1I')) = 2 ([Rfl, Proposition 1.7], [Sh, p.381]) while gsr(C(1I')) =1 since C(1I') is commutative and gsr(C(1I')) :S 2 (cf. [Rfl, Proposi­tion 10.2] and its remark). 0

Remark. When f = zn, the twisted group C*-algebras C* (zn, a)are called noncommutative tori (cf. [PR3] and [Rf2]). If n = 2,then C* (Z2, a) are the rotation algebras, which are ordinary crossedproducts C(1I') )<le Z of C(1I') by Z, where we have

for (Xl, X2), (YI' Y2) E Z2 and () E R It follows from [BKR] that anysimple noncommutative torus has the stable rank one. Thus, theabove stable estimate for C* (f, a) is some far from the equalities ingeneral. Also, the connected stable rank of simple noncommutativetori is 2, which is deduced from the fact of their having the stablerank one, [Rfl, Corollary 4.10] and the fact that their Kl-groups are

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nontrivial (d. [Eh, Corollary 1.6 and Theorem 2.2]). Furthermore,C* (zn) ~ C('lF). Thus,

sr(C*(Zn)) = sr(C(1I'n)) = [n/2] + 1 ::; n = rank(Zn).

Therefore, it would be desirable to replace rank(f) with [rank(r)/2]+1 in each estimate above. On the other hand, it is known thatCo(X) )<Jo:,u Z is always isomorphic to the ordinary crossed productCo(X) )<Jo: Z (u trivial), where Co(X) is the C*-algebra of all contin­uous functions vanishing at infinity on a locally compact Hausdorffspace X (cf. [Pk, Example 1.3]). Also, M(Co(X)) ~ Cb(X) theC*-algebra of all bounded continuous functions on X, which is alsoisomorphic to C(f3X) with f3X the Stone-tech compactification ofX.

As a corollary, we obtain the following for twisted crossed productsby cyclic groups Zn:

Corollary 2.4. Let 2! be a unital C* -algebra and 2! )<Jo:,u Zn thetwisted crossed product of2! by Zn with (a, u) a twisted action. Then

sr(2! )<Jo:,u Zn) ::; sr(2!) + 1, and csr(2! )<Jo:,u Zn) ::; sr(2!) + 1.

Proof. Note that there is a natural quotient map:

where the twisted action (a, u) of 2! by Z is canonically extendedfrom that by Zno By using Theorem 1.1 we obtain the stable rankestimate in the statement. By using [Eh, Theorem 1.1] and Corollary1.2 we obtain

csr(2! )<Jo:,u Zn) ::; max{csr(2! )<Jo:,u Z), sr(2! )<Jo:,u Z)}

::; sr(2!) + 1. D

Theorem 2.5. Let 2! be a unital C* -algebra and f a discrete solvablegroup with a nOTmal series having subquotients finitely generated andabelian. Then

{sr(2! )<Jo:,u r) ::; sr(2!) + rank(f) + trk(r),

csr(2! )<Jo:,u f) ::; sr(2!) + rank(f) + trk(f),

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where rank(f) + trk(f) means the sum of the free ranks plus thetorsion ranks of subquotients of f, where the torsion rank means thenumber of cyclic direct factors in each subquotient.

Proof. Let {Zj l;=o be a normal series of f as in the statement. Notethat each subquotient Zj/Zj_l is isomorphic to a direct product:

for some mj and n s (1 ::; s ::; kj ). Then rank(Zj/Zj_d = mj

and trk(Zj/Zj_l) = kj , and trk(f) = 2:~=1 trk(Zj/Zj_l). By usingTheorem 1.1 and Corollary 2.4 repeatedly as in the proof of Theorem2.2, we obtain the conclusion. 0

Remark. We may take f as a successive semi-direct product byabelian discrete groups such that

In this case, we may use the same argument as in the proof of Theo­rem 2.2 without using the decomposition theorem of twisted crossedproducts (d. Theorem 2.1).

Corollary 2.6. Letf be as in Theorem 2.5 andC*(f,a) the twistedgroup C* -algebra of f with a cocycle a. Then

{

sr(C*(f, a)) ::; rank(f) + trk(f),

csr(C*(f,a))::; max{2,rank(f) + trk(f)},

gsr(C*(f, a)) ::; rank(f) + trk(f).

In particular,

{

sr(C* (f)) ::; rank(f) + trk(f),

csr(C* (f)) ::; max{ 2, rank(f) + trk(f)},

gsr(C* (f)) ::; rank(f) + trk(f),

where rank(f) + trk(f) means the same as in Theor'em 2.5.

As a little more, we recall that the Green's twisted covariancealgebra C* (G, Q{, TN) of a C*-algebra Q{ by a locally compact group

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G with TN a twisting map on a normal subgroup N of G is definedto be the quotient of the ordinary crossed product 2l ~(3 G with anaction (3 by the ideal J such that TN is a strictly continuous homo­morphism from N to the unitary group of M(2l), (3n = AdTN(n) and(3g(TN(n)) = T(gng- 1) for n EN, 9 E G and the ideal J is defined tobe the intersection of all the kernels of 7r x U where (7r, U) are covari­ant representations of the system (2l, G, (3) preserving TN. Then thefollowing correspondence between twisted covariance algebras andtwisted crossed products is known:

The correspondence [PRl]. With the above notation, we have

C*(G, 2l, TN) ~ 2l ~a,u GIN

where the twisted action (a, u) of GIN on 2l is defined by

agN = (3c(g) , u(gN, hN) = TN(c(gN)c(hN)c(ghN)-l)

for g, h E G and gN, hN E GIN and c : GIN -t G a Borel cross­section.

Remark. When 2l = C, the map a is always trivial so that

C*(G, C, TN) rv C ~id,u GIN ~ C*(GIN, u),

which is the twisted group C*-algebra of GIN with u a cocycle.

By using this correspondence and Theorem 2.5, we obtain

Corollary 2.7. Let C* (G, 2l, TN) be the twisted covariance algebraof a unital C* -algebra 2l by G with a twist TN· If GIN is a discretesolvable group with a normal series with subquotients free abelian andfinitely generated, then

{sr(C*(G,2l,TN))::; sr(2l) +rank(GIN) + trk(GIN),

csr(C*(G, 2l, TN)) :S sr(2l) + rank(GIN) + trk(GIN).

Furthermore, the following correspondence between twisted groupC*-algebras and twisted covariance algebras is known:

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The correspondence [PR3]. For G a locally compact group andits twisted group C* -algebra C* (G, 0-), we have

where Ga means the locally compact group obtained by the followingcentral extension:

1 -+ 1I' -+ Ga -+ G -+ 1

where the multiplication of G a = 1I' x G (setwise) is given by

(z, s)(w, t) = (a-( s, t) zw, st) EGa'

Moreover, we obtain

C*(Ga,C,'PIr) ~ C*(Ga,C*(Na,C,T']['),TNJ

~ C*(Ga,C*(N, 0-), TNJ

~ C*(N,o-) )<Ia,u GIN

where N a means the same as defined as G a, and TNa

is defined by

fOT (z, n) E Na = 1[' x N and x E L 1 (N), and the twisted action(a, u) means the same as given in The correspondence above.

Remark. By definition, we have

for g, h EGa, e the unit of G, and c : Ga l1I' ~ G -+ Ga the Borelsection defined by c(g) = (l,g) EGa'

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[BI] B. Blackadar, K-Theory for Operator algebras, Second Edition, Cam­bridge, 1998.

[BKR] B. Blackadar, A. Kumjian and M. R0rdam, Approximately central matrixunits and the structure of noncommutative tori, K-Theory 6 (1992), 267­284.

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Department of Mathematical Sciences,Faculty of Science, University of the Ryukyus,Nishihara, Okinawa 903-0213, JAPAN.

E-mail address:sudo@math.u-ryukyu.ac.jpVisit us: www.math.u-ryukyu.ac.jp

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