Crossed Products and MF algebras

0.11 MB·English

by  Weihua Li

Crossed products and MF algebras
Weihua Li, Stefanos Orfanos
January 22, 2013
3
1
0
Abstract
2
We prove that the crossed product A ⋊ G of a unital finitely generated MF algebra A by a
n α
a discrete finitely generated amenable residually finite group G is an MF algebra, provided that the
J action α is almost periodic. This generalizes a result of Hadwin and Shen. We also construct two

9 examplesofcrossedproductC∗-algebraswhoseBDF Ext semigroupsarenotgroups.
1

]
A Keywords: MF algebras; crossed products; BDF Ext semigroups; amenable groups; residually finite
O groups.
.
h
2000Mathematics SubjectClassification: 46L05
t
a
m
[ Introduction

1
v The purposeofthis noteis togeneralizetworecentresultsconcerningcrossedproducts. The firstis:
5
8 Theorem 1 (Hadwin–Shen [4]). Suppose that A is a finitely generated unital MF algebra and α is a
5 homomorphismfromZintoAut(A)suchthatthereisa sequenceofintegers0≤n <n <··· satisfying
4 1 2
.
1 lim kα(n )a−ak=0
j
0 j→∞
3
1 forany a∈A. ThenA ⋊ ZisanMFalgebra.
α
:
v
andthe secondis:
i
X
Theorem2(Orfanos[5]). LetA beaseparableunitalquasidiagonalalgebraandG adiscretecountable
r
a amenableresiduallyfinitegroupwithasequenceofFølnersetsF andtilingsoftheformG =K L . Assume
n n n
α:G→Aut(A)isa homomorphismsuchthat
max kα(l)a−ak→0as n→∞
l∈L ∩K K−1F
n n n n
forany a∈A. ThenA ⋊ G isalsoquasidiagonal.
α
whichwere bothmotivatedby
Theorem 3 (Pimsner–Voiculescu [6]). Suppose that A is a separable unital quasidiagonal algebra and
α is a homomorphism from Z into Aut(A) such that there is a sequence of integers 0 ≤ n < n < ···
1 2
satisfying
lim kα(n )a−ak=0
j
j→∞
forany a∈A. ThenA ⋊ Zisalsoquasidiagonal.
α
1
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 2
MFalgebrasareimportantintheirownrightbutalsoduetotheirconnectiontoVoiculescu’stopological
free entropy dimension for a family of self-adjoint elements x ,...,x in a unital C∗-algebra A ([8]).
1 n
The definition of topological free entropy dimension requires that Voiculescu’s norm microstate space
of x ,...,x is “eventually" nonempty, which is equivalent to saying that the C∗-subalgebra generated
1 n
by x ,...,x inA isanMFalgebra. SoitiscrucialtodetermineifaC∗-algebraisMF,inwhichcaseits
1 n
Voiculescu’stopologicalfreeentropydimensionis well-defined.
Inthenextsectionwedescribeanotherconnection,thatbetweenMFalgebrasandtheBrown–Douglas–
FillmoreExt semigroup(introducedin[2]). Wewillthenexhibittwonewexamplesofcrossedproduct
C∗-algebras whose Ext semigroupfailstobeagroup.
Background
Westartwith afewwell-knowndefinitionsandfacts.
Definition 1. A discrete countable group G is amenable if there is a sequence of finite sets {F }∞
n n=1
(called a Følner sequence) such that lim F = G and lim |F △F s|/|F | = 0 for any s ∈ G. The group
n n n n
n→∞ n→∞
G is residually finite if for every e 6= x ∈ G, there is a finite index normal subgroup L of G such that
L 6= xL. Stated differently, finite index normal subgroups of G separate points in G. A tiling of G is a
decomposition G = KL, with K a finite set, so that every x ∈ G is uniquely written as a product of an
elementin K andanelementin L.
Lemma 1. Assume G is a discrete countable group. Then G is amenable and residually finite if and only
if G has a Følner sequence {F }∞ for which there exists a separating sequence of finite index normal
n n=1
subgroups L and a sequence of finite subsets K ⊃ F such that G has a tiling of the form G = K L for
n n n n n
all n≥1.
Theorem4. Adiscretegroup G isamenableifandonlyif C∗(G)∼=C∗(G). LetA bea C∗-algebra. If G is
r
amenableandifthereisahomomorphismα:G →Aut(A),thenA ⋊ G ∼=A ⋊ G.
α,r α
Quasidiagonal operators were firstconsideredbyHalmos.
Definition 2. A separable family of operators {T ,T ,...} ⊂ B(H) is quasidiagonal if there exists a
1 2
sequence of finite rank projections {P }∞ such that P → I in SOT and kP T − T P k → 0 for all
n n=1 n n j j n
j ≥ 1 as n → ∞. A separable C*-algebra is quasidiagonal if it has a faithful ∗-representation to a
quasidiagonal setofoperators.
Theorem 5 (Rosenberg [7]). Let G be a discrete group. If C∗(G) is a quasidiagonal algebra, then G is
r
amenable.
MFalgebraswere introducedbyBlackadarandKirchberg in [1].
Definition 3. Aseparable C∗-algebraA isanMFalgebraifthere is anembeddingfromA to
∞ ∞
M (C)/ M (C)
N N
t t
t=1 t=1
Y X
for positive integers {N }∞ . If A= A ∞ is an elementofthe above C*-algebra,define its norm by
t t=1 t t=1
kAk=limsupkAtkM (C).
t→∞ Nt (cid:8) (cid:9)
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 3
Theorem 6 (Blackadar–Kirchberg [1]). A separable C∗-algebra A is MF if and only if every finitely
generatedC∗-subalgebraofA isMF.SubalgebrasofMFalgebrasarealsoMF.Everyquasidiagonalalgebra
A isMFandtheconverseistrueif,inaddition,A isnuclear.
An exciting result connecting quasidiagonal and MF algebras on one hand, and Brown–Douglas–
Fillmoretheoryofextensionsontheother, is thefollowing.
Theorem7. LetA beaunitalseparableMFalgebra. IfA isnotquasidiagonal,then Ext(A)failstobe
agroup.
Example1(Haagerup–Thorbsørnsen[3]). ThereducedgroupC∗-algebraofthefreegrouponngener-
ators,namelyC∗(F ),isanMFalgebrabutitisnotquasidiagonalornuclear(sinceF isnotamenable).
r n n
Therefore, Ext(C∗(F ))is notagroup.
r n
Moreexamplesofthis flavorwereexhibited in[4].
Preliminary Facts
Here we state a few facts that will be used extensively in the rest of this note. In what follows,
C(X ,...,X )willdenotethesetofallnon-commutativecomplexpolynomialsinX ,...,X ,X∗,...,X∗.
1 m 1 m 1 m
The first result gives equivalent definitions for an MF algebra. Refer to [4] and the references therein
foraproof.
Proposition 1. Suppose A is a unital C∗-algebra generated by a family of elements a ,...,a in A.
1 m
Thenthefollowingareequivalent:
(i) A isanMFalgebra.
(ii) Foranyε>0andanyfinitesubset{f ,...,f }ofC(X ,...,X ),thereisapositiveinteger N anda
1 J 1 m
family ofmatrices{A ,...,A }inM (C),suchthat
1 m N
max kf (A ,...,A )k −kf (a ,...,a )k <ε.
j 1 m M (C) j 1 m A
1≤j≤J N
(cid:12) (cid:12)
(cid:12) (cid:12)
(iii) Suppose π : A → B(H) is a faithful ∗-representation of A on an infinite dimensional separable
complex HilbertspaceH. Then thereisafamily{[a ] ,...,[a ] }∞ ⊂B(H)such that
1 n m n n=1
(a) For each n≥1,{[a ] ,...,[a ] }⊂B(H)isquasidiagonal;
1 n m n
(b) f [a ] ,...,[a ] → f a ,...,a as n→∞,forany f ∈C(X ,...,X );
1 n m n B(H) 1 m A 1 m
(c) (cid:13)[ai(cid:0)]n→π(ai)in∗-S(cid:1)O(cid:13)Tas n→(cid:13)∞(cid:0),forevery1(cid:1)≤(cid:13) i≤m.
(cid:13) (cid:13) (cid:13) (cid:13)
Let G beadiscretecountableamenableresiduallyfinitegroup,equippedwithFølnersets F ,finitesets
n
K and finite index normal subgroups L such that F ⊂ K and G = K L is a tiling of G for every
n n n n n n
n≥1. Considerthefamily{ξ : y ∈K }⊂ℓ2(G),with ξ = φ (x)δ and
yL n yL n x
n n
x∈yL
Xn
|K ∩F x|
n n
φ (x)= .
n È |Fn|
The followingtwolemmascanbefoundin [5]. Thesecondis aconsequenceofthefirst.
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 4
Lemma2. Assume G andξ areasabove. Thefollowingaretrue:
yL
n
(i) For every n≥1,{ξ : y ∈K }isanorthonormalfamilyofvectors.
yL n
n
(ii) For anys∈G,
|F △F s|
2 n n
λ(s)ξ =ξ + φ (x)−φ (sx) δ ,with φ (x)−φ (sx) ≤ .
yLn syLn n n sx n n |F |
x∈yL x∈yL n
Xn(cid:0) (cid:1) Xn(cid:12) (cid:12)
(cid:12) (cid:12)
Lemma3. Assume G andξ areasabove. If P ξ=〈ξ,ξ 〉ξ and P = P ,then
yL yL yL yL n yL
n n n n n
y∈K
Xn
(i) For every n≥1, P isa self-adjointprojectioninB(ℓ2(G))andrank P =|K |.
n n n
|F △F s|
(ii) P → I inSOTas n→∞;andforanys∈G,kP λ(s)−λ(s)P k2≤4 n n →0as n→∞.
n n n |F |
n
Main Result
Let A = 〈a ,...,a 〉 be a unital finitely generated MF C∗-algebra. We first establish three claims
1 m
relatedtoourmainresult.
For y ∈ K , s ∈ S ={s ,s ,...,s }∪{e}⊂ G and 1≤ i ≤ m, consider the elements α(y−1s−1)a ∈ A,
n 1 2 k i
the quasidiagonal set α(y−1s−1)a :1≤i≤m,s∈S,y ∈K ⊂B(H) obtained from part (a) of
i n n
Proposition 1(iii), andn”a sequence of—finite rank projections Qnoin B(H) such that Qn → I in SOT
as n → ∞ and Q asymptotically commutes with all elements in the above-mentioned set. For every
n
1≤i≤m,positive integer nands∈S,define
A(s)= Q α(y−1s−1)a Q ⊗P ,A =A(e) ,and U =Q ⊗P λ(s)P .
i n i n n yLn i i s n n n
y∈K
Xn ” —
Claim1. Foranyε>0andanyfinitesubset{q ,...,q }ofC(X ,...,X ),thereisapositiveinteger
1 J 1 (k+1)m
nsuchthat,for N =rank(Q ⊗P ),
n n
max q A ,...,A(sk),...,A ,...,A(sk) − q a ,...,α(s−1)a ,...,a ,...,α(s−1)a <ε.
1≤j≤J j 1 1 m m M (C) j 1 k 1 m k m A
(cid:12) N (cid:12)
(cid:12)(cid:13) (cid:16) (cid:17)(cid:13) (cid:13) € Š(cid:13) (cid:12)
(cid:13) (cid:13) (cid:13) (cid:13)
Proof.(cid:12)(cid:13)Forevery1≤ j≤J, (cid:13) (cid:13) (cid:13) (cid:12)
(cid:12) (cid:12)
q A ,...,A(sk) =max q Q α(y−1)a Q ,...,Q α(y−1s−1)a Q
(cid:13) j€ 1 m Š(cid:13)MN(C) y∈Kn¨(cid:13) j(cid:16) n” 1—n n n” k m—n n(cid:17)(cid:13)MN/|Kn|(C)«
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
by Lemma 2(i). Now use the quasidiagonality of α(y−1s−1)a :1≤i≤m,s∈S,y ∈K , with n
i n n
sufficientlylarge,toobtain,forevery1≤ j≤J, n” — o
q Q α(y−1)a Q ,...,Q α(y−1s−1)a Q
j n 1 n n n k m n n M (C)
(cid:12)(cid:12)(cid:13) (cid:16) ” — ” — (cid:17)(cid:13) N/|Kn|
(cid:13) (cid:13)
(cid:12)
(cid:13) (cid:13) ε
(cid:12) − q α(y−1)a ,..., α(y−1s−1)a < ,
j 1 n k m n B(H) 2
(cid:12)
(cid:13) (cid:16)” — ” — (cid:17)(cid:13) (cid:12)
(cid:13) (cid:13)
(cid:12)
(cid:13) (cid:13)
(cid:12)
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 5
andpart (b)ofProposition 1(iii) toget
ε
q α(y−1)a ,..., α(y−1s−1)a − q α(y−1)a ,...,α(y−1s−1)a < .
j 1 n k m n B(H) j 1 k m A 2
(cid:12) (cid:12)
(cid:12)(cid:13) (cid:16)” — ” — (cid:17)(cid:13) (cid:13) € Š(cid:13) (cid:12)
(cid:13) (cid:13) (cid:13) (cid:13)
The(cid:12)(cid:12)(cid:13)lastnormisequalto qj a1,...,α(s−k1)am(cid:13) sinc(cid:13)e α(y−1)is a∗-automorphism. (cid:13) (cid:12)(cid:12)
A
(cid:13) (cid:13)
Claim 2. For any s ,...,s(cid:13)∈€G, any ε>0, andŠa(cid:13)nyfinite subset {p ,...,p }of C(X ,...,X ), thereis a
1 (cid:13)k (cid:13) 1 J 1 k
positiveinteger nsuchthatfor N =rank(Q ⊗P ),
n n
max p U ,...,U − p λ(s ),...,λ(s ) <ε.
1≤j≤J j s1 sk M (C) j 1 k B(ℓ2(G))
(cid:12) N (cid:12)
(cid:13) (cid:13)
Proof. Itfollowsfrom(cid:12)(cid:12)L(cid:13)emm€a3(ii)andŠt(cid:13)hedefiniti(cid:13)(cid:13)on o(cid:0)ftheprojection(cid:1)s(cid:13)(cid:13)Q . (cid:12)(cid:12)
(cid:13) (cid:13) n
(cid:12) (cid:12)
Let ε > 0 and s ∈ F ⊂ G. Choose appropriately large positive integer n so that |F △F s| < ε2|F |/4.
n n n n
Assumethatforevery1≤i≤m,the actionis almostperiodic, inthe sensethat
max kα(l)a −a k→0as n→∞.
i i
l∈L ∩F K K−1
n n n n
Claim 3. ForA(s),A , U asaboveandsufficiently largepositiveinteger n, U∗A U −A(s) <ε.
i i s s i s i M (C)
N
(cid:13) (cid:13)
Proof. Withoutlossofgenerality,startwith aunitvectorη∈Q H andc(cid:13)ompute,for y(cid:13)∈K ,
n (cid:13) (cid:13) n
2
2
U∗A U −A(s) η⊗ξ = U∗A η⊗P ξ + (φ (x)−φ (sx))δ −A(s)η⊗ξ
(cid:13)(cid:16) s i s i (cid:17) yLn(cid:13) (cid:13)(cid:13)(cid:13) s i n syLn xX∈yLn n n sx i yLn(cid:13)(cid:13)(cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13)(cid:13)  2 ε2  (cid:13)(cid:13)
≤ U∗A η⊗(cid:13)ξ −A(s)η⊗ξ + ,byLemma2(ii). (cid:13)
s i syLn i yLn 4
(cid:13) (cid:13)
Let sy =zl′ = lz with z(cid:13)∈ K and l,l′ ∈ L . Then l =(cid:13)syz−1 ∈ L ∩F K K−1 and ξ = ξ , which
(cid:13) n n (cid:13) n n n n syLn zLn
gives
2 2
U∗A η⊗ξ −A(s)η⊗ξ = U∗Q α(z−1)a η⊗ξ −Q α(y−1s−1)a η⊗ξ
s i zLn i yLn s n i n zLn n i n yLn
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) ” — ” — (cid:13)2
(cid:13) (cid:13) (cid:13) (cid:13)
= Q α(z−1)a η⊗P ξ + (φ (sx)−φ (x))δ −Q α(y−1s−1)a η⊗ξ
(cid:13) n i n n yLn n n x n i n yLn(cid:13)
(cid:13) x∈yL (cid:13)
(cid:13) ” — Xn ” — (cid:13)
(cid:13)(cid:13)  2 ε2  2 (cid:13)(cid:13)ε2
≤ Q(cid:13) α(z−1)a − α(y−1s−1)a η⊗ξ + ≤ α(z−1)a − α(y−1s−1)a +(cid:13) .
n i n i n yLn 4 i n i n 4
By(cid:13)(cid:13)part(cid:16)(”b) ofProp—ositio”n 1(iii), — (cid:17) (cid:13)(cid:13) (cid:13)(cid:13)” — ” — (cid:13)(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
2 ε2
α(z−1)a − α(y−1s−1)a < α(z−1)a −α(y−1s−1)a 2 + .
i n i n B(H) i i A 4
(cid:13) (cid:13)
” — ” — (cid:13) (cid:13)
Finally, α(z−(cid:13)(cid:13)1)a −α(y−1s−1)a 2 = a −α(cid:13)(cid:13)(l)a 2 <(cid:13) ε2 bythealmostperiod(cid:13)icity ofthe action.
i i A i i A 4
Overall,forsufficientlylarge n,
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
2
U∗A U −A(s) η⊗ξ <ε2.
s i s i yLn
(cid:13)(cid:16) (cid:17) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 6
Weare nowreadytostatethe mainresult.
Theorem8. LetA =〈a ,...,a 〉beaunitalfinitelygeneratedMFalgebraandG =〈s ,...,s 〉adiscrete
1 m 1 k
finitelygeneratedamenableresiduallyfinitegroupwithasequenceofFølnersets F andtilingsoftheform
n
G =K L . Assumeα:G →Aut(A)isahomomorphismsuchthatforevery1≤i≤m,
n n
max kα(l)a −a k→0as n→∞.
i i
l∈L ∩F K K−1
n n n n
ThenA ⋊ G isalsoMF.
α
Proof. Following the proof in [4], we will show that A ⋊ G is an MF algebra by using Proposition 1.
α
More specifically,we will showthat foranyε>0andanyfinite subset{f ,...,f }ofC(X ,...,X ),
1 J 1 m+k
there isapositive integer N andafamilyofmatrices {A ,...,A ,U ,...,U }inM (C),suchthat
1 m s s N
1 k
max f a ,...,a ,λ(s ),...,λ(s ) − f A ,...,A ,U ,...,U <ε.
1≤j≤J j 1 m 1 k A⋊αG j 1 m s1 sk M (C)
(cid:12) N (cid:12)
(cid:13) (cid:13)
(cid:12)(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) € Š(cid:13) (cid:12)
Let {f ,...,f(cid:12)(cid:13)} ⊂ C(X ,...,X ), and A ,..(cid:13).,A ,U ,(cid:13)...,U ,N as in Claims 1-3.(cid:13)We firs(cid:12)t prove that
1 (cid:12)J 1 m+k 1 m s s (cid:12)
1 k
forevery1≤ j≤J andsufficientlylarge n,
f a ,...,a ,λ(s ),...,λ(s ) ≥ f A ,...,A ,U ,...U .
j 1 m 1 k A⋊αG j 1 m s1 sk M (C)
N
(cid:13) (cid:13)
Consider an (cid:13)(cid:13)enu(cid:0)meration of all polynomial(cid:1)s(cid:13)(cid:13)in C(X1,.(cid:13)(cid:13)..,€X(k+1)m) (respectively,Šin(cid:13)(cid:13) C(X1,...Xk)) with
rational coefficients. Foreach t =1,2,...,we can findapositive integer n and N = rank(Q ⊗P ),
t t n n
t t
suchthat,byClaim1(respectively, Claim2)andforall1≤ j≤ t,
1
q A ,...,A(sk),...,A ,...,A(sk) − q a ,...,α(s−1)a ,...,a ,...,α(s−1)a <
j 1 1 m m M (C) j 1 k 1 m k m A t
(cid:12)(cid:12)(cid:13) (cid:16) (cid:17)(cid:13) Nt (cid:13) € Š(cid:13) (cid:12)(cid:12)
(cid:13) (cid:13) (cid:13) (cid:13)
(re(cid:12)(cid:13)spectively, (cid:13) (cid:13) (cid:13) (cid:12)
(cid:12) (cid:12)
1
p U ,...,U − p λ(s ),...,λ(s ) < ).
(cid:12)(cid:13) j s1 sk (cid:13)MNt(C) j 1 k Cr⋆(G)(cid:12) t
Let,forevery1≤i≤(cid:12)(cid:12)(cid:13)ma€nds∈S, Š(cid:13) (cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) (cid:12)(cid:12)
(cid:13) (cid:13)
(cid:12) (cid:12)
∞ ∞ ∞ ∞
∞
A(s)= A(s) ∈ M (C)/ M (C),A =A(e) ,andU = U ∞ ∈ M (C)/ M (C),
i i t=1 Nt Nt i i s s t=1 Nt Nt
t=1 t=1 t=1 t=1
n o Y X Y X
(cid:8) (cid:9)
and C denote the C∗-algebra generated by A ,...,A(sk),...,A ,...,A(sk) . Consequently, there are
1 1 m m
embeddingsρ1:A →C andρ2:Cr∗(G)→Cn,givenby o
ρ α(s−1)a =A(s) andρ (λ(s))=U
1 i i 2 s
with the property that (ρ ,ρ ) is€a covarianŠt homomorphism (by Claim 3). Therefore, there exists a
1 2
∗-homomorphism ρ : A ⋊ G → C, with ρ α(s−1)a = A(s) ,andρ(λ(s)) = U . It follows that for
α i i s
all1≤ j≤J,
€ Š
f a ,...,a ,λ(s ),...,λ(s ) ≥ f A ,...,A ,U ,...,U
j 1 m 1 k A⋊αG j 1 m s1 sk C
(cid:13) (cid:13)
(cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) = l(cid:13)(cid:13)imt→€s∞up(cid:13)fj A1,...,Am,Us1,Š..(cid:13)(cid:13).,Usk (cid:13)MNt(C).
€ Š
(cid:13) (cid:13)
(cid:13) (cid:13)
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 7
Itremainstoshowthat forevery1≤ j≤J andsufficientlylarge n,
f a ,...,a ,λ(s ),...,λ(s ) ≤ f A ,...,A ,U ,...,U +ε.
j 1 m 1 k A⋊αG j 1 m s1 sk M (C)
N
(cid:13) (cid:13)
(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) € Š(cid:13)
There exist(cid:13)apositive integer D andfamilie(cid:13)s ofmono(cid:13)mials p(d) ∈C(X ,...,X )a(cid:13)ndpolynomialsq(d)∈
j 1 k j
C(X ,...,X )for1≤d ≤D and1≤ j≤J,suchthat
1 (k+1)m
D
f a ,...,a ,λ(s ),...,λ(s ) = p(d) λ(s ),...,λ(s ) q(d) a ,...,α(s−1)a ,...,a ,...,α(s−1)a
j 1 m 1 k j 1 k j 1 k 1 m k m
d=1
(cid:0) (cid:1) X (cid:0) (cid:1) € Š
bythecovariance relation forcrossedproducts. Similarly,forsufficientlylarge n,wecanget
D
f A ,...,A ,U ,...,U = p(d) U ,...,U q(d) A ,...,A(sk) +r A ,...,A(sk),U ,...,U
j 1 m s1 sk j s1 sk j 1 m j 1 m s1 sk
d=1
€ Š X € Š € Š € Š
with
ε
max r A ,...,A(sk),U ,...,U <
1≤j≤J j 1 m s1 sk M (C) 3
N
(cid:13) (cid:13)
€ Š
byClaim3andrepeateduseofth(cid:13)(cid:13)eapproximate covariance rela(cid:13)(cid:13)tion AiUs =UsA(is)+r(Ai,A(is),Us).
Wethenhave f a ,...,a ,λ(s ),...,λ(s ) − f A ,...,A ,U ,...,U
j 1 m 1 k A⋊αG j 1 m s1 sk M (C)
N
(cid:13) (cid:13)
(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) € Š(cid:13)
D (cid:13) (cid:13) (cid:13) (cid:13)
< p(d) λ(s ),...,λ(s ) q(d) a ,...,α(s−1)a
j 1 k C∗(G) j 1 k m A
Xd=1(cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) (cid:13)(cid:13) € Š(cid:13)(cid:13)
(cid:13) (cid:13) (cid:13)D (cid:13) ε
− p(d) U ,...,U q(d) A ,...,A(sk) +
j s1 sk M (C) j 1 m M (C) 3
Xd=1(cid:13) € Š(cid:13) N (cid:13) € Š(cid:13) N
(cid:13) (cid:13) (cid:13) (cid:13)
D (cid:13) (cid:13) (cid:13) (cid:13)
≤ p(d) λ(s ),...,λ(s ) q(d) a ,...,α(s−1)a − q(d) A ,...,A(sk)
j 1 k C∗(G) j 1 k m A j 1 m M (C)
Xd=1(cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) (cid:18)(cid:13)(cid:13) € Š(cid:13)(cid:13) (cid:13)(cid:13) € Š(cid:13)(cid:13) N (cid:19)
D (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) ε
+ p(d) λ(s ),...,λ(s ) − p(d) U ,...,U q(d) A ,...,A(sk) + .
j 1 k C∗(G) j s1 sk M (C) j 1 m M (C) 3
Xd=1(cid:18)(cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) (cid:13)(cid:13) € Š(cid:13)(cid:13) N (cid:19)(cid:13)(cid:13) € Š(cid:13)(cid:13) N
Finally, no(cid:13)te that q(d) A ,...,A(cid:13)(sk) (cid:13) ≤ q(d) a ,...,(cid:13)α(s−1)a(cid:13) +1 for n suffi(cid:13)cientlylarge,
j 1 m M (C) j 1 k m A
N
sodefine (cid:13) (cid:16) (cid:17)(cid:13) (cid:13) € Š(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
D D
M = max p(d) λ(s ),...,λ(s ) , q(d) a ,...,α(s−1)a +D
1≤j≤J( j 1 k C∗(G) j 1 k m A )
Xd=1(cid:13)(cid:13) (cid:0) (cid:1)(cid:13)(cid:13) Xd=1(cid:13)(cid:13) € Š(cid:13)(cid:13)
anduseClaims 1and2w(cid:13)ithalarger nifnecessa(cid:13)ry,toobtai(cid:13)n,forall1≤d ≤D and(cid:13)1≤ j≤J,
ε
q(d) a ,...,α(s−1)a − q(d) A ,...,A(sk) < ,and
j 1 k m A j 1 m M (C) 3M
N
(cid:13) (cid:13) (cid:13) (cid:13)
€ Š € Š
(cid:13) (cid:13) (cid:13) (cid:13) ε
(cid:13) p(d) λ(s ),...,λ(s )(cid:13) (cid:13)− p(d) U ,...,U(cid:13) < .
j 1 k C∗(G) j s1 sk M (C) 3M
N
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) € Š(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
CrossedproductsandMFalgebras(W.Li,S.Orfanos) 8
Remark. Every discrete group is the inductive limit of its finitely generated subgroups. In particular,
a discrete maximally almost periodic group is the inductive limit of its residually finite subgroups.
Moreover, the inductive limit of MF algebras is an MF algebra. Therefore, Theorem 8 remains true if
one assumes G to be any discrete countable amenable residually finite group, and can be extended to
all discrete countable amenablemaximallyalmostperiodic groupswhose finitelygeneratedsubgroups
satisfy the approximate periodicity condition. One couldalso assume that A is separable, rather than
finitelygenerated,becauseofTheorem6.
Examples
We may now use this result to construct more exotic examples of crossed product C∗-algebras whose
BDF Ext semigroupis notagroup.
Example 2. Considerthe integerHeisenberggroup,whichcanbe definedabstractlyas
H =〈s,t|[[s,t],s],[[s,t],t]〉.
orinaconcreteway,as thesubgroupofSL (Z)generatedby
3
1 1 0 1 0 0 1 0 1
s= 0 1 0 and t = 0 1 1 ,withu=[s,t]=s−1t−1st = 0 1 0 .
     
0 0 1 0 0 1 0 0 1
     
     
Everyelementof H canbeuniquelywritten inthe formsktlum,for k,l.m∈Z. The sets
n n
K = sktlum:− <k,l,m≤
n
2 2
§ ª
have subsets F = sktlum:− n <k,l ≤ n,−n <m≤ n that form a Følner sequence, and if we
n 2 2 2 2
define Ln=〈sn,tn,unn〉then H =pKnLn is atipling forevery n≥o1. Indeed, Ln isnormalsince
(sktlum)(snptnqunr)(sktlum)−1=snptnqun(r+qk−pl)∈ L
n
andforany k,l,m∈Z,onecan findunique−n <k′,l′,m′≤ n and p,q,r ∈Z,suchthat
2 2
sktlum=sk′+nptl′+nqum′+n(r−pl′)=(sk′tl′um′)(snptnqunr)∈K L .
n n
Assume now that A is a unital finitely generated non-quasidiagonal MF algebra (e.g. A = C∗(F ))
r 2
andletan actionα:H →Aut(A)beinducedby
α(s)a=α(t)a=e2πθia
where a ∈ A and 0 ≤ θ ≤ 1. Consider a positive integer n with the property that nθ approximates
an integer. Then α is approximately periodic on L ∩F K K−1. It follows that (A,H,α) satisfies the
n n n n
conditions of Theorem 8, and thus the crossed product A ⋊ H is a non-quasidiagonal MF algebra,
α
therefore its Ext semigroupfails tobe agroup.
Example 3. Considerthe Lamplightergroup,which canbedefinedabstractly as
Λ=〈s,t|s2,[tjst−j,tlst−l]: j,l ∈Z〉.
REFERENCES 9
or otherwise, as the semidirect product Z ⋊Z, where the action is by shifting the copies of Z
Z 2 2
along Z. By denoting tjst−j = s , we may write each element of Λ uniquely as s s ···s tj0 with
j j j j
j < j <···< j and j inZ. Let (cid:0)L (cid:1) 1 2 k
1 2 k 0
n n n n
F =K = s s ···s tj0 : − < j ≤ , − < j < j <···< j ≤ ,and0≤k<n .
n n j j j 0 1 2 k
1 2 k 2 2 2 2
§ ª
The Følner condition follows immediatety from F s = F and |F △F t| = 2|F |/n. Let L be the
n n n n n n
subgroupofΛgeneratedbytnands s forall−n < j≤ n. Notethat L containsallelementss s with
j j+n 2 2 n j l
l− j divisible by n, and in fact, L is normal in Λ, since, for s s ···s tj0 ∈Λ and s s ···s tl0 ∈ L ,
n j j j l l l n
1 2 k 1 2 m
wehave
s s ···s tj0 s s ···s tl0 s s ···s tj0 −1= s s ··· s s s s ···s tl0
j j j l l l j j j j j +l j j +l l +j l +j l +j
1 2 k 1 2 m 1 2 k 1 1 0 k k 0 1 0 2 0 m 0
w€hich is in L .ŠM€oreover, Λ =ŠK€ L is a tilingŠfor e€very n ≥Š1, s€ince anyŠs€s ···s tj0 ∈ Λ can bŠe
n n n j j j
1 2 k
decomposedinto
s s ···s tr0 s ···s s ···s tj0−r0 ∈K L
r r r r −r r −r j −r j −r n n
1 2 m 1 0 m 0 1 0 k 0
with 0 ≤ m ≤ k < €n, −n < r ,r ,Š.€..r ≤ n, j − r divisible by n, aŠnd the cardinality of the set
2 0 1 m 2 0 0
(c+nZ)∩ r ,...,r ,j ,...,j to be an even number (or zero) for each c = 1,...,n. To verify the
1 m 1 k
uniquessofsuchadecomposition,onecaneasilycomputethat K−1K ∩L ={e}forall n≥1.
(cid:8) (cid:9) n n n
Having studied the group in detail, let us now consider A to be any unital finitely generated non-
quasidiagonal MFalgebraandletanactionα:Λ→Aut(A)beinducedby
α(s)a=a∗ ,andα(t)a=e2πθia
where a∈A and0≤θ ≤1. Again,forapositiveintegernwiththepropertythat nθ approximatesan
integer, α is approximately periodic on L ∩K K K−1. It followsthat (A,Λ,α) satisfies the conditions
n n n n
ofTheorem 8,andthus the crossedproduct A ⋊ Λis anon-quasidiagonal MFalgebra,henceits Ext
α
semigroupis notagroup.
References
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Math.Ann.307(3)(1997)343–380.
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(1977)265–324.
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red 2
group,Ann.Math.(2)162(2)(2005)711–775.
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product C∗-algebras,J.OperatorTheory4(1)(1980)93–118.
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D.Hadwin), J.OperatorTheory18(1987)15–18.
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