Table Of ContentA SIMPLE C∗-ALGEBRA WITH FINITE NUCLEAR
DIMENSION WHICH IS NOT Z-STABLE
3
1 ILIJAS FARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
0
2
Abstract. We construct a simple C∗-algebra with nuclear dimension
n
zerothatisnotisomorphictoitstensorproductwiththeJiang–Sualge-
a
J braZ, andahyperfiniteII1 factor not isomorphic toitstensorproduct
withtheseparablehyperfiniteII1 factorR. Theproofsuseaweakening
1
of theContinuum Hypothesis.
2
]
A Elliott’s programofclassification ofnuclear (a.k.a.amenable)C∗-algebras
O
recentlyunderwentatransformativephase(seee.g.,[5]). Followingthecoun-
h. terexamples of Rørdam and Toms to the original program, it was realized
t that a regularity assumption stronger than the nuclearity is necessary for
a
m C∗-algebras tobeclassifiablebyK-theoretic invariants. Conjecturally, regu-
larity propertiesofthreedifferentflavoursareallequivalentandare, modulo
[
the UCT, sufficient for classification (restricting, say, to simple, separable,
1 nuclear C∗-algebras). We shall consider two of these regularity assump-
v
0 tions on a C∗-algebra A. One of them asserts that A is Z-stable, meaning
3 that it is isomorphic to its tensor product with the Jiang-Su algebra Z.
0
An another postulates that A has finite nuclear dimension, this being a
5
strengthening of the Completely Positive Approximation Property (CPAP)
.
1
introduced by Winter and Zacharias in [18] (the CPAP is an equivalent
0
3 formulation of amenability for C∗-algebras, see [3, Chapter 2]). The Toms–
1 Winter conjecture states (among other things) that for separable, nuclear,
v: simple, non-typeI C∗-algebras, havingfinite nuclear dimensionis equivalent
i to being Z-stable (see e.g., [17, 14]). The direct implication is a theorem of
X
Winter ([16]). We show that it badly fails if the separability assumption is
r
a dropped.
Theorem 1. The Continuum Hypothesis implies that there exists a simple
nuclear C∗-algebra with nuclear dimension zero which is not Z-stable.
The paradigm of regularity properties for C∗-algebras parallels certain
older ideas in the study of von Neumann algebras. It has long been known
that amenability for von Neumann algebras is equivalent to hyperfiniteness,
and in the separable, non-type I case, it implies R-stability (the property of
being isomorphic to one’s tensor product with the unique separable hyper-
finite II factor R; von Neumann algebras with this property are commonly
1
called McDuff) (see [13, Chapters XIV and XVI]). It is also not a stretch
Dedicated to Stuart Whiteon theoccasion of his 33rd birthday.
1
2 ILIJASFARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
to find that amenability is equivalent to a von Neumann-theoretic analogue
of nuclear dimension zero [11, Lemma 1.2]. The construction used to prove
Theorem 1 adapts easily to the von Neumann case, allowing us to prove the
following.
Theorem 2. The Continuum Hypothesis implies that there exists a hyper-
finite II factor which is not R-stable.
1
We provethesetheorems by constructingalgebras whosecentral sequence
algebras are abelian. In fact, we strengthen the construction in two di-
rections, one in which the conclusion is strengthened (Theorems 1.2 and
1.5) and another in which weaker set-theoretic axioms are assumed (Theo-
rems 1.3 and 1.6). It seems unlikely that the conclusions of Theorems 1.3
and 1.6 — or even the stronger conclusions of Theorems 1.2 and 1.5 — are
independent from ZFC. The assumption of Theorems 1.3 and 1.6 is the car-
dinal equality b = c, where b denotes the boundingnumber — see §2, where
we also give two new characterizations of b.
A result similar to Theorem 1.5 was proven in [7, Proposition 3.7 (1)],
where a II factor M is constructed using ZFC alone, whose central se-
1
quence is trivial, but every separable II subfactor of M has property Γ
1
(this factor has density character ℵ , with respect to the strong operator
1
topology). Unlike the example in Theorem 1.5 (which shares the aforemen-
tioned properties), the von Neumann algebra in [7, Proposition 3.7 (1)] is
not hyperfinite, due to the use of the free product construction. Likewise, a
minor modification of the construction in [7] gives a monotracial C∗-algebra
whose central sequence algebra is abelian, yet whose non-type I, separable
subalgebras each have nonabelian central sequence algebras; but again, the
freeproductconstructionpreventsthisexamplefrombeingnuclear,letalone
having nuclear dimension zero.
Thisfactor inTheorem2is clearly differentfromthe‘obvious’hyperfinite
factorwhosepredualhascharacterdensityℵ . Theexistenceof‘nonobvious’
1
hyperfiniteII factorswithnonseparablepredualswasfirstprovedbyWidom
1
([15]),andin[8]itwasprovedthatthereare2κ nonisomorphichyperfiniteII
1
factorswithpredualofdensitycharacterκ,foreveryuncountablecardinalκ.
In §1 we provide more information about the algebras we construct and
their central sequence algebras. In §2 we give two new reformulations of the
bounding number b, one in terms of convergent subseries of null sequences
and another in terms of convergence of a sequence of inner automorphisms
of a C∗-algebra. The proofs of main results are completed in §3, using
transfinite recursive construction,
and in §4 we give concluding remarks.
Recall that density character of a topological space X is the minimal car-
dinality of a dense subset. Therefore X is separable iff its density character
is ℵ and it has cardinality c if and only if its density character is ≤ c. In
0
our case, X will either be a C∗-algebra equipped with the norm topology,
or a von Neumann algebra considered with the strong operator topology
A SIMPLE C∗-ALGEBRA WITH FINITE NUCLEAR DIMENSION... 3
(SOT). In fact, the density character of a von Neumann algebra M is the
same whether measured with respectto the strong operator topology or any
other usual, non-norm, von Neumann algebra topology (eg. WOT, weak∗-
topology); it also coincides with the density character of the predual M
∗
under its norm topology. All C∗-algebras are assumed to be unital. Back-
groundcanbefounde.g., in[1](forC∗-algebras andvon Neumannalgebras)
and in [2] (for set theory).
Acknowledgments. The results of this paper were proven duringthe Sep-
tember 2012 Fields Institute Workshop on Applications to Operator Alge-
bras, November 2012 Oberwolfach Workshop on C∗-algebras, Dynamics and
Classification, and in November 2012 back at the Fields Institute. I.F. is
partially supported by NSERC, D.H. is supported by the Fields Institute,
and A.T. is supported by DFG (SFB 878).
1. Locally matricial algebras
Following [9] and [12], we say that a C∗-algebra A is
• approximately matricial (or AM) if it has a directed family of full
matrix subalgebras with dense union.
• locally matricial (or LM) if for any finite subset F of A and any
ε > 0, there exists a full matrix subalgebra M of A such that for
every a ∈ F we have dist(a,M) < ε.
• locally finite dimensional (or LF) if for any finite subset F of A and
any ε> 0, there exists a finite dimensional subalgebra M of A such
that for every a ∈ F we have dist(a,M) < ε.
Nuclear dimension, as defined in [18], is a property that is preserved under
local approximation; it follows that each LF algebra has nuclear dimension
zero (in fact, the converse also holds [18]). LM algebras are not necessarily
AM ([9, Theorem 1.5]), although LM algebras are direct limits of separable
UHF algebras ([9, Lemma 2.12]) LM algebras are simple.
Recall that x , for n ∈ N, is a central sequence of a C∗-algebra A if
n
k[x ,a]k → 0foralla ∈ A. Thecentral sequence algebra ofAisasubalgebra
n
of ℓ∞(A)/c (A) consisting of all equivalence classes of (bounded) central
0
sequences.
Lemma 1.1. If a central sequence algebra of a unital algebra A is abelian
then A is not Z-stable.
Proof. IfB andC areunitalC∗-algebrasthenthecentralsequencealgebraof
B⊗ C (where⊗ is any C∗-tensor product)clearly hasthecentral sequence
ν ν
algebra of C as a unital subalgebra. The conclusion now follows from the
fact that the central sequence algebra of Z has a subalgebra isomorphic
to Z. (cid:3)
Every sequence of elements from the centre of a C∗-algebra is obviously
a central sequence. We call the central sequence of a C∗-algebra A trivial
4 ILIJASFARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
if it only consists of (equivalence classes of) such sequences (i.e. if it equals
(ℓ∞(Z(A))+c (A))/c (A)).
0 0
Theorem 1.2. The Continuum Hypothesis implies that there exists an AM
algebra all of whose central sequences are trivial.
By Lemma 1.1, Theorem 1 will follow once Theorem 1.2 is proved in §3.
Strengthening Theorem 1 in another direction we show that an assump-
tion weaker than both the Continuum Hypothesis and Martin’s Axiom suf-
fices. We write c = 2ℵ0 and b for the bounding number (see §2 for defini-
tions).
Theorem 1.3. If b = c then there exists a LM algebra whose central se-
quence algebra is abelian.
By Lemma1.1 andTheorem 1.2 (oncethelatter isproved) theconclusion
of Theorem 1 also follows from b = c.
A sequence, x for n ∈ N, in a von Neumann algebra M is said to be
n
central if [x ,a] → 0 in the strong operator topology, for every a ∈ M. The
n
central sequence algebra of M is the subalgebra of
ℓ∞(A)/{(x )|SOT−limx = 0}
n n
consisting of equivalence classes of central sequences. A trivial central se-
quenceisoneinthesameequivalenceclassasasequencefromZ(M). Proven
in the same manner, we have the following analogue of Lemma 1.1.
Lemma 1.4. If the central sequence algebra of a von Neumann algebra M
is abelian then M is not R-stable.
We will make great use of the fact that, when M has a faithful trace τ,
then the strong operator topology coincides with the topology induced by
the norm k·k , given by
2
kxk := τ(x∗x)1/2;
2
indeed, adapting our main construction from the C∗-setting to the von Neu-
mann setting will primarily consist in using k·k in place of k·k. Here are
2
the von Neumann versions of Theorems 1.2 and 1.3.
Theorem 1.5. The Continuum Hypothesis implies that there exists a hy-
perfinite II factor all of whose central sequences are trivial.
1
Theorem 1.6. If b = c then there exists a hyperfinite II factor whose
1
central sequence algebra is abelian, and in particular, which is not R-stable.
2. New characterizations of the bounding number
In the present section we provide alternative characterizations of the
bounding number, b. Readers interested only in the proofs of Theorems 1.2
and 1.5 can skip ahead to §3, modulo the facts that b defined below is un-
1
countable (this is not hard to show directly, or alternatively, using b ≤ b
2 1
A SIMPLE C∗-ALGEBRA WITH FINITE NUCLEAR DIMENSION... 5
and the uncountability of b ). Given two functions f,g : N → N, we say
2
thatf eventually dominates g if f(n)≥ g(n) forall butfinitely many n ∈N.
Recall that the bounding number, which we denote by b, is the minimal
cardinality κ such that there exist functions fξ :N → N for ξ < κ, such that
for every g : N → N, there exists some fξ which isn’t eventually dominated
by g.
Letb betheminimalcardinalκsuchthatthereexistsaunitalC∗-algebra
1
A of density character κ and a central sequence of unitaries u , for n ∈ N,
n
such that the following holds. For every subsequence u , for i∈ N, with
n(i)
v := u u ...u ,
k n(1) n(2) n(k)
we have that the sequence of inner automorphisms Adv , for k ∈ N, does
k
not converge pointwise on A.
Let b (LF) be the minimal κ with the above property where we addition-
1
ally require that A is LF.
Let b be the minimal cardinal κ such that there are sequences rξ, for
2 n
n∈ N and ξ < κ, satisfying
ξ
(1) r ∈(0,∞) for all ξ and n,
n
ξ
(2) lim r = 0 for all ξ, and
n n
(3) there is no infinite increasing sequence n(i), for i ∈ N, of natural
ξ
numbers such that r < ∞ for all ξ.
i n(i)
It is well-known that MartPin’s Axiom for σ-centered posets implies b = c
(see e.g., [10]).
Readers familiar with the generalized Galois–Tukey connections will have
notroubleinrecastingtheabovedefinitionsinthisframeworkandobserving
that each part of Lemma 2.2 asserts the existence of a morphism in the
terminology of [2, §4].
Proposition 2.1. We have b = b = b (LF) = b .
1 1 2
The proof that b = b uses the following lemma, which relates null se-
2
quences to functions N → N.
Lemma 2.2.
(1) If r ∈ (0,∞), for n ∈ N, is a null sequence, then there exists a
n
function f : N → N such that for any g : N → N, if g eventually
∞
dominates f then r < ∞.
i=1 g(i)
(2) If f : N → N is aPfunction than there exists a null sequence rn ∈
(0,∞), for n ∈ N, such that for any g : N → N, if ∞ r < ∞
i=1 g(i)
then g eventually dominates f. P
Proof. (1) Let f be any function such that
1
(∀i)(∀n ≥ f(i))[r ≤ ],
n 2i
(which exists because lim r = 0). If g : N → N eventually dominates f,
n n
then eventually r < 2−i and so ∞ r < ∞.
g(i) i=1 g(i)
P
6 ILIJASFARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
(2) Let r be any sequence of positive real numbers converging to 0 sat-
n
isfying
1
(∀i)(∀n ≤ f(i))[r ≥ ].
n
i
Aiming to prove the contrapositive, suppose that g : N → N does not even-
tually dominatef. LetB ⊆ N bethe setof all ifor which f(i) > g(i). Next,
“thin out”B as follows: let b bean increasingsequenceof distinctelements
j
of B such that for each j, over half of the elements of Im(g) before g(b )
j+1
are not before g(b ). Then,
j
∞
r = r + r + r +...
g(i) g(i) g(i) g(i)
Xi=0 iX<b0 b0≤Xi<b1 b1≤Xi<b2
1 1 1
≥ + + +...
b b b
iX<b0 0 b0≤Xi<b1 1 b1≤Xi<b2 2
1 1 1 1 1
≥ (b ) +( b ) +( b ) +...
0 1 2
b 2 b 2 b
0 1 2
1 1
= 1+ + +...
2 2
= ∞,
as required. (cid:3)
Proof that b = b . b ≤ b : Let κ < b be arbitrary. Let rξ be sequences of
2 2 n
positive reals, for n ∈ N and ξ < κ. Assumethat lim rξ = 0 for each ξ < κ.
n n
For each ξ < κ, let f :N → N be as in Lemma 2.2 (1). Since κ < b, there is
ξ
someg :N → Nthat eventually dominates each f , and thereforeby Lemma
ξ
ξ
2.2 (1), r < ∞ for each ξ < κ.
i g(i)
The pProof that b ≤ b is exactly the same, but this time using Lemma
2
2.2 (2) instead of (1). (cid:3)
ToprovetheremainingpartofProposition2.1(namely,thatb = b (LF)=
1 1
b ), we need the following lemmas.
2
Lemma 2.3. Let (r )∞ be a null sequence of real numbers. Then ∞ r
n n=1 n=1 n
converges in R if and only if ( N r mod 1)∞ converges in RP/Z.
n=1 n N=1
P
Proof. The forward implication is trivial. Suppose, on the other hand, that
( N r mod 1)∞ converges in R/Z. Let N be such that |r | < 1/2 for
n=1 n N=1 0 n
nP≥ N0. If N0 is sufficiently large, then for all N ≥ N0,
N r ∈ (−ε,ε)+Z.
n=N0 n
P
If ε< 1/2 then, between these facts, it follows that
N r ∈(−ε,ε).
n=N0 n
P ∞
This shows that the partial sums form a Cauchy sequence, so that r
n=1 n
converges in R. P (cid:3)
A SIMPLE C∗-ALGEBRA WITH FINITE NUCLEAR DIMENSION... 7
The following simple fact will be used both here and in Section 3, to
‘decentralize’ elements or sequences.
Lemma 2.4. If A is a C∗-algebra and α :A → A is an endomorphism, then
α+: A→ M (A) defined by
2
a 0
α+(a) =
(cid:18)0 α(a)(cid:19)
is a ∗-homomorphism. Moreover, for any a ∈ A,
k[α+(a),v]k = ka−α+(a)k,
where
0 1
(1) v := A A .
(cid:18)1A 0A(cid:19)
Proof. Obvious. (cid:3)
Lemma 2.5. Let (r ) be a null sequence of real numbers. Then there exist
n
unitaries v,u , for n ∈ N, in the CAR algebra such that (u ) is a central
n n
sequence and, for any increasing sequence (n(i)) of natural numbers,
r
i n(i)
converges in R if and only if P
Ad(u ...u )(v)
n(1) n(k)
converges. Moreover, the CAR algebra contains subalgebras A ∼= M such
n 2
that u ∈ A and [u ,A ] = 0 if m 6= n.
n n n m
Proof. LetA= M⊗∞ andletαbetheautomorphismofAgivenbyapplying
2
0 1
Ad
(cid:18) 1 0 (cid:19)
to each tensor factor M . Then B = M (A) is isomorphic to the CAR
2 2
algebra, and we shall make use of the morphism α+ : A → B given by
Lemma 2.4.
Define v as in (1). For each n, set
u′ := 1⊗(n−1) ⊗ 1 0 ⊗1⊗∞ ∈ A,
n M2 (cid:18) 0 e2πirn (cid:19) M2
and
u = α+(u′ ).
n n
Evidently, (u′ ) is central in A, and an easy computation shows that (u )
n n
approximately commutes with v, and therefore, with every element of B.
That is to say, (u ) is a central sequence.
n
Now, given an increasingsequence (n(i)) of naturalnumbers,wecompute
Ad(u ...u (v) = exp(2πi i r )v,
n(1) n(i) j=1 n(j)
P
8 ILIJASFARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
and therefore, the sequence (Ad(u ...u (v)) converges if and only if
n(1) n(i)
the sequence
(exp(2πi i r ))
j=1 n(j)
converges in T, which by Lemma 2.P3, occurs if and only if ∞ r con-
i=1 n(i)
verges in R. P
Finally, we set A := α+(1⊗(n−1) ⊗M ⊗1⊗∞). (cid:3)
n M2 2 M2
Proof of Proposition 2.1. We shall now prove that b = b (LF)= b .
1 1 2
b ≤ b : Let A be a unital C∗-algebra of density character κ< b and let
2 1 2
v , for n ∈ N, be a central sequence of unitaries in A. Let a , for ξ < κ, be
n ξ
a dense subset of A and let
rξ = k[v ,a ]k.
n n ξ
ξ
Since (v ) is a central sequence, we have lim r = 0 for all ξ. We can
n n n
therefore choose an increasing sequence of natural numbers n(i), for i ∈ N,
ξ
so that r < ∞ for all ξ.
i n(i)
Let wP= v v ...v . For k <m and all ξ we have
k n(1) n(2) n(k)
k(Adw )a −(Adw )a k= kAdw (a −Ad(v ...v )(a ))k
k ξ m ξ k ξ n(k+1) n(m) ξ
= k[a ,v ...v ]k
ξ n(k+1) n(m)
m−1
≤ k(Adw )a −(Adw )a k
j ξ j+1 ξ
Xj=k
m−1
≤ (rξ +2−jka k),
n(j) ξ
Xj=k
and therefore (Adw )a , for k ∈ N, is a Cauchy sequence. Since the au-
k ξ
tomorphisms (Adw ) are isometries which pointwise converge on a dense
k
subset of A, they pointwise converge to an endomorphism. Since A was
arbitrary we have proved that κ< b implies κ < b , and therefore b ≥ b .
2 1 1 2
b (LF)≥ b : this is trivial.
1 1
b ≥ b (LF): Let κ < b , and let rξ ∈ (0,∞), for n ∈ N, be a null
2 1 2 n
sequence, for each ξ < κ. For each ξ < κ, let A(ξ) be a copy of the CAR
algebra and let
v(ξ),u(ξ) ∈ A(ξ)
n
be unitaries, for n ∈ N, as given by Lemma 2.5. Also, let A(ξ) ⊂ A(ξ), for
n
n∈ N, be the subalgebras given by the same lemma.
Set B = A(ξ), and define
ξ<κ
Q
u := (u(ξ)) ∈ A
n n ξ<κ
for each natural number n and
vˆ(j) := (δ v(ξ)) ∈ A.
ij i∈I
A SIMPLE C∗-ALGEBRA WITH FINITE NUCLEAR DIMENSION... 9
Set
(ξ)
B := A
n i∈I n
for each natural number n, and set Q
A′ := C∗ B ∪ A(ξ) ;
n∈N n i∈I
(cid:0)S L (cid:1)
Weeasily seethatA′ isLFandcontainseach u andeach v(ξ),andthat(u )
n n
is a central sequence in A′. By a downward Lo¨wenheim-Skolem argument,
there exists an LF subalgebra A of A′ with density character κ and which
contains each v(ξ) and each u .
n
Now, suppose that (n(i)) is an increasing sequence of natural numbers
such that
Ad(u ...u )
n(1) n(k)
converges in the point-norm topology. Then in particular,
Ad(u ...u )(v(ξ))
n(i) n(k)
∞ ξ
converges for each ξ < κ, which by Lemma 2.5, means that r < ∞.
i=1 n(i)
The proof is complete. P (cid:3)
Remark 2.6. The proof that b ≤ b (= b) can be adapted (by using k·k
1 2 2
in place of k·k) to show that, if M is a von Neumann algebra with a faithful
trace andwithdensitycharacter < b, andu , forn ∈N, isacentral sequence
n
from M, then there exists a subsequence u , for i ∈ N, such that
n(i)
Adu ...u
n(1) n(k)
converges (in the point-strong operator topology). However, the proof of the
converse does not adapt, since when the construction used to show b ≤
2
b (LF) is adapted to the von Neumann setting, the resulting von Neumann
1
algebra does not have a faithful trace.
3. Proofs of the main theorems
(i)
Lemma 3.1. Let A be an LM algebra, and for each i = 1,...,N, let x ,
n
for n ∈ N, be a central sequence. Then there exists an increasing sequence
n(k)∈ N, for k ∈ N, and y(i) such that:
k
(i) (i)
(1) for each i, ky −x k→ 0 as k → ∞; and
k n(k)
(2) for each i,i′,k,k′ ∈ N, if k 6= k′ then y(i) and y(i′) ∗-commute.
k k′
Proof. Since A is LM, there exists a separable LM subalgebra B which con-
(i) (i)
tains each x . By perturbing the sequences x (by an error that vanishes
n n
at ∞), we have without loss of generality that
B = M ⊗M ⊗...,
m1 m2
(i)
and x ∈ M ⊗···⊗M for each i and n.
n m1 mn
10 ILIJASFARAH,DANHATHAWAY,TAKESHIKATSURA,ANDAARONTIKUISIS
Let ε > 0 be any null sequence. Set n(1) = 1. Using compactness of the
n
unit ball of a matrix algebra, we may iteratively choose n(k) such that
(i)
(2) k[x ,a]k ≤ ε kak
n(k) n
for all a ∈ M ⊗···⊗M . Letting E denote the conditional expec-
n(1) n(k−1) k
tation from M to M′ ∩M , set
n(k) n(k−1) n(k)
(i) (i)
y := E (x ).
k k n(k)
Then, it follows from (2) (and by writing the conditional expectation as an
average over the unitary group of M ) that
n(k−1)
(i) (i)
ky −x k ≤ ε ,
k n(k) n
as required. (cid:3)
We say that a central sequence is hypercentral if it commutes with ev-
ery other central sequence (i.e., if it is a representing sequence of a central
element of the central sequence algebra). (Although the terminology origi-
nated in theory of II factors and our C∗-algebras have a unique trace, we
1
emphasize that only the operator norm is being used here.)
Lemma 3.2. Assume A is an LM algebra of density character < b. If x ,
n
for n ∈ N, is a central sequence which is not hypercentral then there exists
an endomorphism α of A such that liminf kx −α(x )k > 0.
n n n
Proof. Let u , for n∈ N, be a central sequence such that for some ε > 0 we
n
have k[x ,u ]k > ε> 0 for all n. We may assume each u is a unitary since
n n n
every element in a C∗-algebra is a linear combination of four unitaries.
By Lemma 3.1, by passing to a subsequence, there exist sequences y and
n
v , for n ∈ N, such that
n
ky −x k,kv −u k→ 0
n n n n
as n → ∞, and [v ,v ] ∗-commutes with v ,y for n 6= m.
n m m m
Since u is unitary, by functional calculus, we may arrange that v is too.
n n
(Note that modifying v using functional calculus does not change the fact
n
that it ∗-commutes with v ,y for n 6= m.)
m m
By using that the density character of A is less than b = b (by Proposi-
1
tion2.1), wecangotoasubsequenceofv (again denotedv )suchthatthe
m m
automorphisms α := n Adv converge pointwise to an endomorphism,
n j=1 j
and so we may set Q
α := limα .
k
k
Then α(y ) = Adv ◦y , and therefore,
n n n
liminfkx −α(x )k = liminfky −α(y )k
n n n n
n n
= liminfk[y ,v ]k
n n
n
= liminfk[x ,u ]k> 0,
n n
n
