(1994) SYMMETRIC ENVELOPING ALGEBRAS, AMENABILITY AND AFD PROPERTIES FOR ...

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Mathematical Research Letters 1, 409–425 (1994)
SYMMETRIC ENVELOPING ALGEBRAS,
AMENABILITY AND AFD
PROPERTIES FOR SUBFACTORS
Sorin Popa
Introduction
Theproblemsandresultsthatrelatedirectlycombinatorialaspectswith
analytical ones appear to play a central place in the theory of subfactors
of finite Jones index.
Along these lines, a rather puzzling phenomenon that needs to be clari-
fied is the existence of hyperfinite subfactors that can be approximated by
finite dimensional commuting squares, i.e., which are AFD, but not in a
standard way, i.e. not by commuting squares that are locally isomorphic
to algebras of higher relative commutants.
Approximation by higher relative commutants (standard AFD) turns
out to be equivalent to the amenability of the inclusion N ⊂ M, a func-
tional analytical property which requires existence of hypertraces for all
(smooth) representations of N ⊂ M and which is also equivalent to M
being hyperfinite and (cid:2)Γ (cid:2)2 = [M : N] (see 2.1 in this paper and [Po1]).
N,M
As amenability can also be characterized by merely requiring existence of
hypertraces for the standard representation of N ⊂ M (see 2.1), we may
conclude that standard AFD can be “encoded in a hypertrace”.
Also, (3.1.3 in [Po1]) shows that the AFD (not necessarily standard)
property for an inclusion implies the existence of atomic representations
with hypertraces for N ⊂ M.
The purpose of this paper is to introduce a complementary approach
to finite dimensional approximation of subfactors, which was partly hinted
in ([Po3]). This approach is based on the notion of symmetric envelop-
ing algebras and symmetric Markov traces (and hypertraces) associated to
N ⊂ M. Thus,weconsidertheuniversalC∗-algebraU = C∗ (M,e ,Mop)
bin N
generated by a “normal” copy of M, an antisymmetric copy of it Mop that
Received February 22, 1994.
Supported in part by NSF Grant DMS-8908281.
409
410 SORIN POPA
commutes with M and a projection e that implements both the condi-
N
tional expectation of M onto N and of Mop onto Nop. This algebra has a
∗
natural irreducible representation C (M,e ,JMJ) as left-right multipli-
N
cation by M on L2(M) and e = proj . We prove that U has a unique
N L2(N)
maximal ideal and a unique trace, called symmetric Markov trace. The
resulting type II factor M (cid:1) Mop is called the symmetric enveloping type
1
eN
II factor associated to N ⊂ M. If M is hyperfinite then the enveloping
1
type II factors are “thin”, i.e., they are generated linearly by a product of
1
−
two of its hyperfinite subfactors R ,R , as (span(R R )) . Thin type II
1 2 1 2 1
factors need not be hyperfinite.
In fact, we show that M (cid:1) Mop is hyperfinite iff N ⊂ M is amenable,
eN
so iff N ⊂ M is standard AFD (or, equivalently, if (cid:2)Γ (cid:2)2 = [M : N]).
N,M
We use this to prove that the amenability for inclusions is a hereditary
property i.e., if Q ⊂ P is an extremal inclusion of type II factors which
1
is embeddable in an amenable N ⊂ M (without necessarily having Q and
P with finite index in N, resp. M) then Q ⊂ P is amenable. We then
introduce the notion of symmetric Markov hypertraces for representations
of U on B(H), as states on B(H) that have M,e ,Mop in their centralizer,
N
and prove that N ⊂ M is amenable iff there exist symmetric Markov
hypertraces on any representation of U and also iff U is simple.
Also, we show that in order for N ⊂ M to be AFD it is sufficient to
exist subfactors Q ⊂ N with Q(cid:3) ∩ N ⊂ Q(cid:3) ∩ M finite dimensional and
having inclusion matrix of square norm close to [M : N]. If in addition
Q can be taken so that [N : Q] < ∞ then N ⊂ M follows amenable. Fi-
nally, we show that if N ⊂ M is AFD (not necessarily standard) then
C∗(M,e ,JMJ) ⊂ B(L2(M)) has symmetric Markov hypertraces and
N
provethelaterconditionequivalenttoaConnes-Folnertypecondition. We
use this to construct examples of inclusions of hyperfinite type II factors
1
that do not have symmetric Markov hypertraces and thus are not AFD. In
fact, most of our considerations are accompanied by motivating examples.
1. Symmetric enveloping algebras
Let N ⊂ M be a subfactor of finite Jones index, [M : N] < ∞, which we
willalwaysassumetobeextremal,i.e.,[pMp : Np] = (trp)2[M : N]forany
projection p ∈ N(cid:3) ∩M, or equivalently EN(cid:1)∩M(e0) ∈ C, e0 being a Jones
projection for N ⊂ M. If N ⊂ M ⊂e1 M ⊂e2 ··· is the Jones’ tower of factors
1
associated to the inclusion then its closure M∞ d=ef ∪ Mn with respect
n≥1
to the unique normalized trace tr on ∪ M is a type II factor, called the
n 1
n
enveloping algebra associated to N ⊂ M. M∞ is clearly hyperfinite if and
SYMMETRIC ENVELOPING ALGEBRAS 411
only if M (or N) is hyperfinite.
In the case N ⊂ M is strongly amenable, i.e., M is hyperfinite and the
standard graph of N ⊂ M, Γ , is ergodic and satisfies (cid:2)Γ (cid:2)2 = [M :
N,M N,M
N], then N ⊂ M is antiisomorphic to M1(cid:3) ∩M∞ ⊂ M(cid:3)∩M∞ (cf [Po1]). So
in this case M∞ is generated by M, by a copy of Mop that commutes with
M and by a projection e which implements both the expectation of M
N
onto N and of Mop onto Nop. Equivalently, M∞ has an (anti)symmetry,
denoted x (cid:10)→ xop, such that (xop)op = x, [M,Mop] = 0, (e )op = e and
N N
M∞ = M ∨{eN}∨Mop.
For an arbitrary inclusion of type II factors of finite index one has a
1
similar symmetric enveloping algebra.
Definition 1.1. Let Q E⊂Q P be a Markov inclusion of finite von Neumann
algebras with E the trace preserving expectation (thus, there exists an
Q
orthonormalbasisofP overQ,{m } ,suchthatΣm m∗ isascalarmultiple
j j j j
of the identity (see [Po1])). Let Q ⊂ P e⊂Q (cid:15)P,e (cid:16) be the Jones basic
Q
construction for Q ⊂ P ([J]). We let C∗ (P,e ,Pop) be the universal
max Q
C∗-algebra containing a copy of (cid:15)P,e (cid:16) and having an antiautomorphism
Q
x (cid:10)→ xop, satisfying the conditions
(a) (xop)op = x
(b) [P,Pop] = 0, (e )op = e
Q Q
(c) P,Pop,e generate U as a C∗-algebra.
Q
We then denote by U = C∗ (P,e ,Pop) = C∗ (P,e ,Pop)/∩kerπ,
bin Q max Q
where the intersection is over all representations π of C∗ (P,e ,Pop) for
max Q
which π(P), π(Pop) are von Neumann algebras. We call U the universal
symmetric enveloping C∗-algebra associated to Q ⊂ P.
Theorem 1.2. Let N ⊂ M be an extremal inclusion of type II factors
1
of finite index. Then there exists a unique positive normalized trace on
C∗ (M,e ,Mop) and its associated ideal is the unique maximal ideal of
bin N
C∗ (M,e ,Mop). Equivalently, any C∗-algebra U which contains a copy
bin N 0
of (cid:15)M,e (cid:16) and has an antisymmetry op satisfying the axioms 1.1 a), b),
N
c) and which acts on a Hilbert space so that M, Mop are weakly closed,
has a unique trace.
To prove this theorem, one uses the following:
Lemma 1.3. Let M ⊃e0 N e⊃−1 N ⊃ ··· be a choice of the tunnel for
1
N ⊂ M. If e1 = eN, e2 = (e0)op, e3 = (e−1)op,... in Cm∗ax(M,eN,Mop),
and M = (cid:15)M, e (cid:16),M = (cid:15)M ,e (cid:16),... , then N ⊂ M ⊂ M ⊂ ··· is iso-
1 1 2 1 2 1
morphic to the Jones tower for N ⊂ M and one has C∗ (M,e ,Mop) =
max N
spnMop(∪ M )Mop.
k
k
412 SORIN POPA
With this observation at hand one can check that
xo1pYxo2p (cid:10)→ xo1pEM(cid:1)∩Mk(Y)xo2p, x1,2 ∈ M, Y ∈ Mk
defines an expectation of U onto Mop that factors through any representa-
tion of U. Then one gets that τ(xo1pYxo2p) = τ(xo1pEM(cid:1)∩Mk(Y)xo2p) defines
a trace on any representation of U.
We denote by C∗ (M,e ,Mop) the quotient of U by its trace ideal.
min N
Note that, by using 1.3 and a similar argument as in the above proof, one
can show that there exits a unique conditional expectation of U onto the
C∗-algebra C∗(M,Mop) generated by M and Mop in U. The Stinespring
dilation of this expectation then shows that the subalgebra C∗(M,Mop) of
UcoincideswithC∗ (M,Mop)asdefinedin([EL]).Inparticular,ifN = M
bin
then U (cid:18) C∗ (M,Mop).
bin
By taking the GNS representation corresponding to the trace given by
the above theorem we then get:
Corollary 1.4. There exists a unique (up to conjugacy) type II factor U,
1
with a copy of (cid:15)M,e (cid:16) ⊂ U and with an antiautomorphism x (cid:10)→ xop, such
N
that
(a) (xop)op = x,∀x ∈ U
(b) [M,Mop] = 0, eop = e
N N
(c) M, Mop, e generate U as a von Neumann algebra.
N
Moreover, if M ⊃e0 N e⊃−1 N ⊃ ··· is a choice of a tunnel for N ⊂ M
1
and e1 = eN, e2 = (e0)op, e3 = (e−1)op,··· ∈ U, then N ⊂ M ⊂e1 M1 ⊂ ···
is isomorphic to the Jones tower, M∞ = ∪ Mnw to the usual enveloping
n
algebra of N ⊂ M and one has U = spwM∞Mop, Mn(cid:3) ∩U = (Nn−1)op,
∀n ∈ Z, in particular (M(cid:3) ∩U)(cid:3)∩U = M , ∀n ∈ Z.
n n
Definition 1.5. The type II factor U in the above Corollary is called the
1
symmetric enveloping type II factor associated to N ⊂ M. We denote it
1
by M (cid:1) Mop.
eN
Thus, as the above comments show, M (cid:1) Mop coincides with M∞ in
eN
the case N ⊂ M is extremal and strongly amenable, with N (cid:20)= M. Note
also that it coincides with M⊗¯Mop when N = M.
For arbitrary (e.g., finite dimensional) Markov inclusions Q ⊂ P one
should first determine what traces on C∗ (P,e ,Pop) are of interest for
bin Q
us.
∗
Definition 1.6. AtraceontheuniversalsymmetricenvelopingC -algebra
C∗ (P,e ,Pop)whichextendstheMarkovtraceon(cid:15)P,e (cid:16)andisinvariant
bin Q Q
to op is called a symmetric Markov trace.
SYMMETRIC ENVELOPING ALGEBRAS 413
In some situations we can prove existence of such traces by using 1.2.
Proposition 1.7. Let Q−1 ⊂ P−1 be a Markov inclusion. Assume there
exists an extremal inclusion of type II factors N ⊂ M and an embed-
1
ding of Q−1 ⊂ P−1 into N ⊂ M as a nondegenerate commuting square.
Then Cb∗in(P−1,eQ−1,P−op1) has a symmetric Markov trace. In particular,
if Q−1 ⊂ P−1 is finite dimensional and
P−1 ⊂ P0
∪ ∪
Q−1 ⊂ Q0
is an extremal commuting square with Z(P−1)∩Z(P0) = C,
then Cb∗in(P−1,eQ−1,P−op1) has a symmetric Markov trace.
One should note that for general Q ⊂ P the symmetric Markov trace
on C∗ (P,e ,Pop) (if it exists) is not necessarily unique. For instance
bin Q
if Q = C ⊂ Cn = P then one has a symmetric Markov trace τ on
0
U = C∗ (P,e ,Pop) by simply taking P,Pop, act on L2(P) by left and
bin Q
respectively right multiplication, e to act as the projection onto L2(Q)
Q
and τ0 = dim1PTrB(L2(P)). The image of U under πτ0 is then B(L2(P)) =
Mn×n(C). But if we take Q ⊂ P to be the inclusion Q−1 ⊂ P−1 in a com-
muting square like in 1.7, with P0 = Mn×n(C) and Q0 an exotic Jones-de
la Harpe orthogonal maximal abelian subalgebra in Mn×n(C), with n a
prime number, then the trace τ produced by the proof of 1.7 is different
1
from τ0. In fact πτ1(U) (cid:20)= Mn×n(C).
There is no a priori reason for an arbitrary finite dimensional inclu-
sion Q ⊂ P to have an associated Markov trace on C∗ (P,e ,Pop)(=
bin Q
C∗ (P,e ,Pop) in this case). It is not hard to see though that if C∗ (P,
max Q bin
eQ,Pop) does have one then Cb∗in(pPp,epQp,(pPp)op) and Cb∗in(P ⊗Mn×n
(C),eQ⊗Mn×n(C),(P ⊗Mn×n(C))op) have one. So, the existence of such a
trace really depends only on the bipartite inclusion graph of Q ⊂ P,TQ⊂P.
We can thus call such traces symmetric Markov traces associated to TQ⊂P.
Problem 1.8. Do all bipartite graphs have symmetric Markov traces?
Note that by 1.7 the failure for some bipartite graph T to have such a
trace would imply that the commuting square problem has no solution for
T. We do not however have any example of such a graph.
The representation of C∗ (P,e ,Pop) on L2(P) works in fact for arbi-
bin Q
trary inclusions.
∗ ∗
Corollary 1.9. Let U = C (P,e ,JPJ) be the C -algebra generated in
0 Q
B(L2(M)) by P, J PJ = Pop and e = proj and denote xop d=ef
P P Q L2(Q)
414 SORIN POPA
Jx∗J, x ∈ U . Then U , with (cid:15)P,e (cid:16) ⊂ U and the above op is a rep-
0 0 Q 0
resentation of C∗ (P,e ,Pop). Moreover, if Z(Q)∩Z(P) = C than this
bin Q
representation is irreducible. Also, if Q ⊂ P is an inclusion of type II
1
∗
factors then there exists a unique trace on C (P,e ,JPJ).
Q
In the case M is the hyperfinite type II factor the above results justify
1
considering the following property for type II factors.
1
Definition 1.10. AtypeII factorU iscalledthinifthereexisttwocopies
1
of the hyperfinite factor R ,R ⊂ U such that U = spwR R .
1 2 1 2
Thus, to each extremal inclusion of hyperfinite type II factors N ⊂ M
1
one associates a thin type II factor whose isomorphism class only depends
1
on the isomorphism class of N ⊂ M. Note that we have a natural group
morphism from Aut(N ⊂ M) into the automorphism of M (cid:1)M that com-
eN
mute with op, leave e fixed and M, N invariant, by associating to σ in
N
Aut(N ⊂ M) the automorphism of M (cid:1) M that sends x yope x yope ...
1 1 N 2 2 N
eN
into σ(x )σ(y )ope σ(x )σ(y )ope .... This morphism implements an in-
1 1 N 2 2 N
jective morphism from Aut(N ⊂ M)/Int(N(N)) into Out(M (cid:1) Mop),
eN
where N(N) is the normalizer of N in M.
2. Amenability and standard AFD
A first motivation for studying the symmetric enveloping algebras and
their Markov traces comes from the following:
Theorem 2.1. Let N ⊂ M be an extremal inclusion of type II factors of
1
finite index. The following conditions are equivalent:
(i) N ⊂ M is amenable ([Po1]).
(ii) N ⊂ M has the standard AFD property, i.e., it can be approximated
by finite dimensional commuting squares which are locally isomor-
phic to reduced algebras of higher relative commutants (in the sense
of 4.4.1 in [Po1]).
(iii) ∀ x ,... ,x ∈ M, ∀ε > 0, ∃ a subfactor P ⊂ N of finite index
1 n
such that x ∈ P(cid:3)∩M.
i
ε
(iv) M (cid:1) Mop is hyperfinite.
eN
(v) ThestandardrepresentationNst ⊂ Mst ofN ⊂ M hasa(N ⊂ M)-
hypertrace.
(vi) M is hyperfinite and (cid:2)Γ (cid:2)2 = [M : N].
N,M
(vii) M is hyperfinite and Γ satisfies the Folner condition ([Po2]).
N,M
SYMMETRIC ENVELOPING ALGEBRAS 415
Note that in the case N = M the equivalence (i) ⇔ (ii) simply states
that M is amenable (or injective) iff M ⊗Mop is hyperfinite, which is an
equivalent form of Connes’ fundamental theorem.
(i) ⇒ (ii) is proved in [Po1]. Then (ii) ⇒ (iii) is straightforward, by
(4.4.1 in [Po1]) and by defining the subfactor P as { ⊕ θ (x) | x ∈
j∈I0 i0j
Ni0 s }, where θ : Ni s → Nj s are isomorphisms and the rest of
thkei0noi0tations areifjrom (k4i.4i.1 in [kPjo1j]). To prove (iii) ⇒ (iv) one uses
that if P ⊂ N ⊂ M is so that [M : P] < ∞ then P(cid:3) ∩ (Pop)(cid:3) is finite
dimensional in M (cid:1) Mop (since [Pop(cid:1) ∩M (cid:1) Mop : P] < ∞). Then (iv)
eN eN
⇒ (v) as follows: let U = M (cid:1) Mop be represented by left multiplication
eN
on L2(U) and define Mst = M ∨J MJ , Nst = N ∨J MJ . It is easy
U U U U
to see that the embedding of N ⊂ M into Nst ⊂ Mst is isomorphic to the
standard representation of N ⊂ M as defined in [Po1], taken with infinite
multiplicity. Thus, if Φ : B(L2(U)) → U is a conditional expectation onto
U then Φ(Mst) ⊂ U commutes with Mop, so Φ(Mst) = (Mop)(cid:3) ∩U = M,
and Φ(Nst) commutes with (cid:15)Mop, e (cid:16), so Φ(Nst) = (cid:15)M,e (cid:16)(cid:3) ∩U = N.
N N
Thus τ ◦Φ is a (N ⊂ M)-hypertrace for Nst ⊂ Mst.
Further on, (v) ⇒ (vi) (by 2.4 and 4.4.1 in [Po1]) and (vii) ⇒ (i) is
proved in [Po2]. Finally, (vi) ⇒ (vii) by using that N(cid:3) ∩ N ⊂ N(cid:3) ∩ M
k k
has inclusion graph T with (cid:2)T (cid:2)2 ≥ [M : N] − ε = τ(e )−1 − ε, where
k k 0
e“0axlme0os=t eEqNuk(cid:1)a∩l”N1t(oxN)e(cid:3)0∩foMr x. ∈ Nk(cid:3) ∩N, and that sp(Nk(cid:3) ∩N)e0(Nk(cid:3) ∩N) is
k
The above condition (iv) enables us to prove the hereditarity properties
for the amenability of inclusions that were missing until now from this
theory.
Corollary 2.2. Let M be a hyperfinite type II factor and N ⊂ M an
1
amenable subfactor (i.e. with (cid:2)Γ (cid:2)2 = [M : N]).
N,M
(a) If Q ⊂ P is an inclusion of type II factors embedded in N ⊂ M
1
as a commuting square such that spPN is an algebra (equivalently,
spPN = spNP) then (cid:2)Γ (cid:2)2 = [P : Q]. In particular, if Q ⊂ P is
Q,P
embedded in N ⊂ M as a commuting square and [P : Q] = [M : N]
then (cid:2)Γ (cid:2)2 = [P : Q] = [M : N] = (cid:2)Γ (cid:2)2.
Q,P N,M
(b) If N ⊂ M ⊂ M ··· is the Jones’ tower and p ∈ M(cid:3) ∩ M then
1 i j
(cid:2)Γ (cid:2)2 = [pM p : M p].
Mip,pMjp j i
Note that in the previous Corollary, we do not require [M : P] to be
finite! By taking Q = N in the above we get:
Corollary 2.3. If N ⊂ P ⊂ M are inclusions of hyperfinite type II
1
factors and (cid:2)Γ (cid:2)2 = [M : N] then (cid:2)Γ (cid:2)2 = [P : N] and (cid:2)Γ (cid:2)2 =
N,M N,P P,M
416 SORIN POPA
[M : P].
In the case N ⊂ M has finite depth (respectively has subexponential
growth [Po1]), the similar implication on the finite depth (resp. subexpo-
nential growth) of the intermediate subfactors, as well as the analogue of
2.2(b), are proved in [Bi2] (resp. [Hi]), for arbitrary (not necessarily hy-
perfinite) subfactors.
The proof of 2.2 follows from Theorem 2.1, Connes’ hereditarity for
hyperfiniteness and the following:
Lemma 2.4. Assume Q ⊂ P is an inclusion of type II factors embedded
1
in N ⊂ M as a commuting square.
(a) If spPN = spNP then P (cid:1)Pop is naturally embedded as the weakly
eQ
closed∗-subalgebrainM(cid:1)Mop generatedbyP, Pop ande . More-
N
eN
over, this embedding is unital iff [M : N] = [P : Q], in which case
spPN = spNP is redundant.
(b) If p ∈ N(cid:3) ∩M and (Np ⊂ pMp) not=ation (S ⊂ R) then R (cid:1) Rop is
eS
naturally embedded as the weakly closed ∗-subalgebra of M (cid:1) Mop
eN
generated by ppop(M ⊗Mop)ppop and by (trp)−1ppope ppop.
N
(c) If[M : Q] < ∞then M(cid:1)Mop isnaturallyembeddedasaunitalvon
Neumann subalgebra ofeNM (cid:1) Mop by taking e = λΣm e m∗ =
N j Q j
eQ
λΣmope mop∗, where {m } is an orthonormal basis of N over Q
j Q j j j
and λ = [N : Q]−1. In particular, (Q(cid:3)∩M)∨{e }∨(Q(cid:3)∩M)op ⊂
N
Q(cid:3)∩(cid:15)M,e (cid:16) = Q(cid:3)∩Qop(cid:1).
Q
The following shows that the finite depth condition has no chance of
satisfying similar hereditarity properties.
Example 2.5. Let P be a copy of the hyperfinite type II factor and G
0 1
a finitely generated amenable group with nonergodic Cayley graph with
respect to some set of generators g = id, g ,... ,g (cf. Kaimanovitch-
0 1 n
Vershik, [Po1]). Let σ be a properly outer action of G on P and P =
0 0
P (cid:3) G. Let σ be the inner action Adu , g ∈ G, of G on P. Let (Nσ ⊂
0 σ0 g
Mσ) (cid:18) (P ⊂ P⊗M(n+1)×(u+1)(C))bethediagonallocallytrivialsubfactor
associated to σ, P and g0,g1,... ,gn, like in [Po1]. Let Nσ0 ⊂ Mσ0 be the
subfactor associated in a similar way to σ , P and g ,... ,g . Note that
0 0 0 n
one has an obvious commuting square
Nσ ⊂ Mσ
∪ ∪
Nσ0 ⊂ Mσ0.
SYMMETRIC ENVELOPING ALGEBRAS 417
Yet Nσ ⊂ Mσ has finite depth while Nσ0 ⊂ Mσ0 has infinite depth and
nonergodic graph. In particular its graph is not strongly amenable.
Note that if [M : P] < ∞ in 2.2 then indeed if N ⊂ M has finite depth
then Q ⊂ P has finite depth as well, regardless of M being hyperfinite
or not (see 2.7 below). However, we cannot weaken the condition on the
hyperfiniteness of M in 2.2 if we allow [M : P] = ∞.
Example 2.6. If in 2.5 we take G to be nonamenable and otherwise pro-
ceed with the same construction as in 2.5, then Nσ ⊂ Mσ has finite depth
and Nσ0 ⊂ Mσ0 has nonamenable graph. But this time Mσ is nonhy-
perfinite. Moreover, since ΓNσ0,Mσ0 is nonamenable, Mσ0 (cid:1) Mσ0 is a
eN
nonhyperfinite, yet thin, type II factor (cf. 2.1).
1
Wecanhoweverprovesomeequivalentcharacterizationsoftheamenabil-
ity of an inclusion which have “finite depth” versions as well. To state
them recall from ([Po1]) that a representation N ⊂ M of N ⊂ M is exact
if M = M ∨P, where P = M(cid:3) ∩N. If M is also atomic then each of its
direct summands B(H) is an irreducible M − P correspondence (bimod-
ule). Also, in ([Po1]) we denoted by ⊕B(K) = Nuf ⊂ Muf = ⊕B(H) the
direct sum of all exact atomic representations for which dim H dim H is
M P
finite and by Au = Auf its inclusion matrix. Note that Nst ⊂ Mst is a
N,M
direct summand of this representation, so that Γ is a direct summand
N,M
(or connected component) of Au and (cid:2)Γ (cid:2) ≤ (cid:2)Au(cid:2). We’ve seen in 2.1
N,M
that if M is hyperfinite then the amenability of N ⊂ M is equivalent to
(cid:2)Γ (cid:2)2 = [M : N]. We will show that (cid:2)Au(cid:2)2 = [M : N] is in fact suf-
N,M
ficient for N ⊂ M to be amenable, which in turn implies that any direct
summand of Au has square norm equal to [M : N]. Also, we will consider
atomic representations for which only the weaker condition dim H < ∞ is
P
satisfied.
Proposition 2.7. Let N ⊂ M be an extremal inclusion of hyperfinite type
II factors. The following conditions are equivalent:
1
(i) N ⊂ M is amenable.
(ii) N ⊂ M hasanexactatomicrepresentationN ⊂ M, withdim H <
P
∞ for all direct summands B(H) of M, such that (cid:2)TN⊂M(cid:2)2 = [M :
N].
(iii) There exists a matricial inclusion (see [PiPo2] for the definition)
E
of hyperfinite type II von Neumann algebras Q ⊂ P, with discrete
1
centers, trace preserving expectation E and (cid:2)TQ⊂P(cid:2)2 = [M : N],
and a nondegenerate commuting square embedding of N ⊂ M into
Q ⊂ P.
418 SORIN POPA
Proposition 2.8. Let N ⊂ M be an extremal inclusion of arbitrary type
II factors. The following conditions are equivalent:
1
(i) Γ is finite (respectively is amenable, respectively is strongly
N,M
amenable).
(ii) Nuf ⊂ Muf has an irreducible direct summand N ⊂ M with finite
inclusion matrix (respectively of square norm equal to the index,
respectively ergodic and of square norm equal to the index).
(iii) All irreducible direct summands of Nuf ⊂ Muf have finite inclu-
sion matrix (respectively of square norm equal to the index, respec-
tively ergodic and of square norm equal to the index).
Note that condition (iii) in 2.7 doesn’t require M to have finite index in
the direct summands of P. Also, the trace on P in (iii) may be infinite, but
itwillbeautomaticallyfiniteifP hasfinitecenter. NotethatsinceQ ⊂ P is
matricialandhasanorthonormalbasis{m } satisfyingΣm m∗ = IndE =
j j j j
j
[M : N] (since it contains N ⊂ M), it follows like in [J] that the trace on P
isgivenbyweights(a ) ontheminimalcentralprojectionsofP, satisfying
k k
TTt(a1/2) = [M : N](a1/2) (see also [PiPo2]). The trace is infinite when
k k k k
Σa = ∞. The proof of (ii) and (iii) in 2.7 is easier to explain in the case
k
k
TN⊂M respectively TQ⊂P are finite. By taking the embedding of N ⊂ M
into (M(cid:3) ∩ N)(cid:3) ∩ N = Q ⊂ P = (M(cid:3) ∩ N)(cid:3) ∩ M we see that to prove
(ii) it is sufficient to prove (iii). But if dimZ(P) < ∞ then we see that
(Q ⊂ P)op (cid:18) ((cid:15)P,eP(cid:16)(cid:3) ∩P∞ ⊂ P(cid:3) ∩P∞) so that if N ⊂ M is embedded
Q
in Q ⊂ P then M (cid:1) Mop is embedded in P∞ which is hyperfinite, thus
eN
M (cid:1) Mop is hyperfinite and 2.1 applies.
eN
The proof of 2.8 can be best understood by first noting a simple fact
whichrelatesthegrowthpropertiesoftheinclusionsinacommutingsquare
of finite index. In this statement, by “standard graph” of a Markov inclu-
sion of type II von Neumann algebras with finite dimensional centers we
1
simply mean the bipartite graph that describes the inclusions of the higher
relative commutants in the Jones tower.
Lemma 2.9. Let S ⊂ R be embedded in Q ⊂ P as a nondegenerate
commuting square of Markov inclusions of finite ([PiPo1]) index, where
P,Q,R,S are type II von Neumann algebras with finite dimensional cen-
1
ters. If R and P denote the algebras in the Jones tower for the two inclu-
n n
sionsthenwehave: (a)supdimZ(S(cid:3)∩R ) < ∞iffsupdimZ(Q(cid:3)∩P ) < ∞.
n n
(b) lim dim(S(cid:3)∩R )1/n = lim dim(Q(cid:3)∩P )1/n. In particular, the standard
n n
graphs of S ⊂ R and Q ⊂ P have the same norm. (c) dim Z(S(cid:3) ∩R∞) is
finite iff dim Z(Q(cid:3)∩P∞) is finite.