Table Of ContentHNN EXTENSIONS OF VON NEUMANN ALGEBRAS
5 YOSHIMICHIUEDA
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n Abstract. Reduced HNN extensions of von Neumann algebras (as well as C∗-algebras) will be
a
introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several
J
concretesettings,detailedanalysisonthem willbealsocarriedout.
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1. Introduction
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A There are two fundamental constructions in combinatorial or geometric group theory, which are
thoseoffreeproductswithamalgamationsandofHNN(G.Higman,B.H.NeumannandH.Neumann
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[13]) extensions. The interested reader may consult [16] as a standard reference on the topics. Even
.
h in the framework of von Neumann algebras (as well as C -algebras), reduced free products with
∗
t
a amalgamations ([41][44][28] and also [38]) have been seriously investigated so far and played key
m rˆoles in several resolutions of “existence” questions in the theory of von Neumann algebras (see, e.g.
[ [28][32][35][30] and also [37]). However, HNN extensions have never been discussed so far in the
framework.
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Historically, many ideas in group theory, especially part of dealing with countably infinite discrete
v
groups, have been applied directly and/or indirectly to many aspects in the theory of von Neumann
9
3 algebras (as well as C∗-algebras) since the beginning of the theory. In fact, many explicit examples
4 of von Neumann algebras that opened new perspectives in the theory came from group theory (see
2 e.g. [18][17][11][3][4][5] and also recent breakthroughs [29][22][23][24]), and it is still expected to find
1
much more “monsters” (i.e., concrete examples with very special properties) living in the world of
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non-amenable von Neumann algebras. To do so, it seems still to be one of the important guiding
0
/ principles to seek for new ideas in group theory. Following this principle, we will introduce reduced
h
HNN extensions in the framework of von Neumann algebras (as well as C -algebras)and take a very
t ∗
a first step towards serious and systematic investigation on them with aiming that their construction
m
willplayakeyrˆoleinfutureattemptsofconstructingnewmonstersintheworldofnon-amenablevon
: Neumann algebras.
v
Let us explain the organization of this article. In §2, we will review free products with amalga-
i
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mations of von Neumann algebras with special emphasis of the admissibility of embedding maps of
r amalgamated algebras in the construction. Although this slight generalizationof the previously used
a
one is of course a folklore, we will briefly review it to avoid any confusion since the admissibility
of embedding maps plays a key rˆole in our construction of HNN extensions. In §3, reduced HNN
extensions of von Neumann algebras will be introduced, and then their characterization (or their
“construction-free”definition) given in terms of expected algebraic relations and “moment-values” of
conditionalexpectationsasinthecaseoffreeproductswithamalgamations. Inthegroupsetting,one
standard way of constructing HNN extensions is the use of “shift automorphisms” on “infinite free
products with amalgamations” over isomorphic but not necessary common subgroups (in fact, two
different embeddings of amalgamated groups are needed). This amalgamation procedure brings us
“difficulty” in constructing “shift automorphisms” in connection with conditional expectations since
the universal construction is not applicable in the von Neumann algebra setting. Hence, a different
idea is needed to construct the desired ones, and indeed it is based on an observation coming from
our previous work [39] on a different topic. Roughly speaking, our construction can be understood
as an “amalgam” (but not a “combination”) of those of covariant representations without unitary
SupportedbyGrant-in-AidforYoungScientists(B)14740118.
AMSsubjectclassification: 46L10, 46L05(primary),46L54, 46L09(secondary).
1
2 Y.UEDA
implementations in the crossed-product construction (see [36, Vol.II; Eq. (10) in p.241]) and of free
products with amalgamations. Our construction seems somewhat natural from the group theoretic
viewpoint. In fact, the notion of HNN extensions is known to be necessary to describe a subgroup
of a given free product group with amalgamation over a non-trivial subgroup. The §4 will concern
modular theoretical aspects of reduced HNN extensions. More precisely, we will give a complete de-
scription of modular automorphisms and also show that the continuous core of any reduced HNN
extension becomes again a reduced HNN extension. In §5, we will discuss the factoriality and inves-
tigate the ultraproducts of reduced HNN extensions. The results correspond to what we obtained in
our previous work [40] on free products with amalgamations. In §6, we will investigate reduced HNN
extensionsofvonNeumannalgebrasinseveralconcretesettings. Thefirstoneisnaturallyarisenfrom
non-commutative 2-tori, the second from the tensor product operation, and the third from regular
and singular MASAs in the crossed-products by (non-commutative) Bernoulli shifts. The third one
seems important for further investigation because any given surjective (partial) ∗-isomorphism be-
tween regular and singular MASAs in question can never be extended to any global ∗-automorphism
onthe given“base”algebras. In§7,reducedHNN extensionsofC -algebraswillbe introducedinthe
∗
samemannerasinthevonNeumannalgebrasetting,andthensomebasicfactswillbegiven. Further
analysis on them will be presented elsewhere.
Part of this article was presented in the conference “Recent Advances in von Neumann Algebras”
celebrated to Professor Masamichi Takesaki’s 70th birthday, at UCLA in May, 2003. We would like
to express our sincere thanks to the organizers; Professors Yasuyuki Kawahigashi, Sorin Popa, and
DimitriShlyakhtenko,whokindly gaveusthe opportunitytopresentthisworkintheconference,and
also would like to celebrate Professor Masamichi Takesaki’s 70th birthday.
Acknowledgment. We would like to express our sincere appreciation to Yasuo Watatani who first
remindedus,throughmanyexcursiveconversations,HNNextensionsofgroupsandtheimportanceto
look for ideas in geometric grouptheory, when we had workedout free products with amalgamations
over Cartan subalgebras. We thank Tomohiro Hayashi for fruitful conversations at the final stage of
this work, and also Masaki Izumi for his useful comment, to which the present form of Corollary 3.6
is indebted. We thank also the anonymous referee for valuable comments.
2. Preliminaries on Free Products with Amalgamations
Let D and N (s ∈ S, an index set) be σ-finite von Neumann algebras, and we have a normal
s
∗-isomorphismι :D →N for each s∈S. Suppose further that the von Neumann subalgebra ι (D)
s s s
ofN istherangeofafaithfulnormalconditionalexpectationE foreverys∈S. Eveninthissetting,
s s
we will be still able to construct the reduced free product with amalgamation
(N,E)=⋆ (N ,E :ι ).
D s s s
s S
∈
ThediscussionsinthisarticlewilltreatthetypeIIandtypeIIIcasesincommonsothattheapproach
in [38] to the amalgamated free product construction will be convenient since complete treatment
of modular theory was given there. To construct reduced HNN extensions, the admissibility of the
embeddings ι (s ∈ S) in the construction plays a key rˆole. Hence, following [38] we would like
s
to recall (without details) the amalgamated free product construction with special emphasis on the
embeddings ι ’s to avoid any confusion.
s
Fix s∈S fora while,andlet H ,N ,J ,P♮ and L2(D),D,J ,P♮ be the standardforms. See
s s s s D D
[36, Vol.II; Chap. IX, §1] for deta(cid:0)iled account o(cid:1)f stand(cid:16)ard forms. Using (cid:17)the mapping
(1) ξ ∈PD♮ 7−→ ωξ D ◦ι−s1◦Es 12 ∈Ps♮
we can extend the embedding ιs : D → N(cid:0)(cid:0)s t(cid:12)(cid:12)o t(cid:1)he Hilbert(cid:1)space level and still denote it by the
same symbol ιs : L2(D) → Hs. Here, ψ12 ∈ Ps♮ denotes the unique implementing vector of a normal
positivelinearfunctionalψ onN . This embeddingsatisfiesthe followingexpectedproperties: (i)For
s
ξ ∈ L2(D) and d1,d2 ∈ D, we have ιs(d1JDd∗2JDξ) = ιs(d1)Jsιs(d2)∗Jsιs(ξ), i.e., ιs(d1·ξ·d2) =
ι (d )·ι (ξ)·ι (d )withthe usualnotationsinthe bimodule theory. (ii)Foreachξ ∈P♮ ,the vector
s 1 s s 2 D
HNN EXTENSIONS 3
ι (ξ) becomes the canonical implementing one in P♮ of the state ω ◦ι 1 ◦E , a consequence
s s ξ D −s s
from (1).
(cid:0) (cid:12) (cid:1)
Fix a faithful normalstateϕ onD anddenote by ξ its implementin(cid:12)g vectorinP♮ . As mentioned
ϕ D
above,the vectorι (ξ )becomes the unique implementing one ofthe state ϕ◦ι 1◦E in the natural
s ϕ −s s
cone P♮. We denote the kernel of E by N as usual, and introduce the operation x ∈ N 7→ x :=
s s s◦ s ◦
x−E (x)∈N . WealsowriteH :=H ⊖ι L2(D) ,anditisclearthatthesubspaceH isinvariant
s s◦ s◦ s s s◦
under the left and right actions of D via the embedding map ι . Thus the natural D-D bimodule
s
(cid:0) (cid:1)
structure of the Hilbert space H :
s
d1·ξ·d2 :=ιs(d1)Jsιs(d2)∗Jsξ, ξ ∈Hs, d1,d2 ∈D
is inherited to the subspace H . When emphasize this bimodule structure, we will use the symbols
s◦
( H ) , ( H ) (or H , H for short) instead of H , H , respectively. Notice here
D ιs sιs D D ιs s◦ιs D ιs s◦ιs ιs s◦ιs s s◦
that we have the natural bimodule isomorphism
DL2(D)D⊕D(ιsHs◦ιs)D ∼=D(ιsHsιs)D
given by ξ⊕η 7−→ι (ξ)+η. Let us construct the Hilbert space
s
H:=L2(D)⊕ ⊕ H ⊗ H ⊗ ···⊗ H ,
ιs1 ◦ιs1 ϕιs2 ◦ιs2 ϕ ϕιsn ◦ιsn
s16=s2X6=···6=sn
on whichthe desiredalgebraN acts. This naturally becomes a D-D bimodule, and the left andright
actions are denoted by λ and ρ, respectively. For each s ∈ S, we can construct the ∗-representation
λ : N → End(H ) and the anti-∗-representation ρ : N → End( H) by the same way as in [38,
s s D s s D
p.361–362]. To do so, we need only some basic properties on relative tensor products (see [36, Vol.II;
Chap. IX, §3]) and the bimodule isomorphism H ∼= L2(D)⊕( H ) precisely explained above.
ιs sιs ιs s◦ιs
Let us consider two von Neumann algebras
′′ ′′
N := λ (N ) , L:= ρ (N ) on H,
s s s s
! !
s S s S
[∈ [∈
and define ψ :=ω , as a vector state, with regarding ξ ∈L2(D) as a vector in H.
ξϕ N ϕ
Facts 2.1. ([38, p.3(cid:12)62–365])
(cid:12)
(A) λ ◦ι coincides with the left action λ of D for each s∈S.
s s
(B) ρ ◦ι coincides with the right action ρ of D for each s∈S.
s s
(C) The vector ξ is cyclic for both N and L.
ϕ
(D) The commutant N on H contains L. (More on this is true, that is, the commutation theorem
′
N =L holds, see [38, Appendix II].) Hence, the state ψ is faithful.
′
(E) For each x ∈N with s 6=s 6=···=6 s , we have
◦j s◦j 1 2 n
ψ(λs1(x◦1)···λsn(x◦n))=0.
(F) The modular automorphism σψ (t∈R) satisfies
t
σtψ ◦λs =λs◦σtϕ◦ι−s1◦Es and σtψ ◦λ=λ◦σtϕ.
Hence, there is a (unique) ψ-preserving conditional expectation Eψ : N → λ(D) thanks to
Takesaki’s theorem ([36, Vol.II; Theorem 4.2 in Chap. IX]).
As in [38, lines 8–3 from the bottom in p.364], the above (C),(E) imply the freeness (with amalga-
mation over λ(D)) among the von Neumann subalgebras λ (N ) (s ∈ S) with respect to Eψ in the
s s
sense of Voiculescu [41, §5]:
Eψ(λ (x )···λ (x ))=0
s1 ◦1 sn ◦n
whenever x ∈N with s 6=s 6=···=6 s . Similarly one has
◦j s◦j 1 2 n
Eψ(λ (x))=λ (E (x))=λ ι 1◦E (x) , x∈N .
s s s −s s s
The conditionalexpectationEψ canbe shownto be i(cid:0)ndependent f(cid:1)romthe choiceofϕ (see the propo-
sition below for more precise), and hence we rewrite E := Eψ. The pair (N,E) constructed so far
4 Y.UEDA
is the desired one of von Neumann algebra and conditional expectation, and it is characterized by
freeness with amalgamation as follows.
Fact 2.2. ([41,§§5.6];alsosee[38, Proposition2.5].) Let P be a von Neumann algebra with a normal
∗-isomorphismπ :D →P. Supposethattherearenormal∗-isomorphismsπ :N →P withπ ◦ι =π
s s s s
and a faithful normal conditional expectation F :P →π(D) such that
• the π (N )’s generate the whole P;
s s
• F ◦π =π◦ι 1◦E for every s∈S;
s −s s
• the π (N )’s are free with amalgamation with respect to F.
s s
Then, there is a unique surjective normal ∗-isomorphism Π:N →P such that Π◦λ =π for every
s s
s∈S and Π◦E =F ◦Π.
Sinceψ = ψ λ(D) ◦E =ϕ◦λ−1◦E,weseethatσtψ =⋆Dσtϕ◦ι−s1◦Es (t∈R),wheretherighthand
s S
side is unders(cid:16)too(cid:12)d as(cid:17)free product of ∗-automorphisms (con∈structed based on the characterizationby
(cid:12)
freeness, see e.g. [38, p.366]), i.e.,
⋆Dσtϕ◦ι−s1◦Es (λs(x)):=λs σtϕ◦ι−s1◦Es(x) , x∈Ns.
(cid:18)s∈S (cid:19) (cid:16) (cid:17)
Thanks to Connes’cocycleRadon-Nikodymtheorem(see [36, Vol.II; Chap.VIII, §3]), this formula of
modular automorphisms is still valid even for every semifinite weight:
Proposition 2.3. ([38, Theorem 2.6]) For a faithful normal semifinite weight φ on D we have
σtφ◦λ−1◦E =⋆Dσtφ◦ι−s1◦Es (t∈R).
s S
∈
3. Construction and Characterization
Onewouldencounter“difficulty”indealingwithconditionalexpectations(inconnectionwith“shift
automorphisms”) if straightforward adaptation of one of the group theoretic constructions of HNN
extensions (see e.g. [34, Chap. I, §1.4]) was attempted in the von Neumann algebra setting. This
forced us to seek for another route towards the construction of reduced HNN extensions. The rough
idea is still essentially the same, but our method is completely different, avoiding the use of “shift
automorphisms” on “infinite free products with amalgamations.” The method is based on a simple
facton“matrixmultiplications”thatweobservedinourpreviousinvestigationonthereducedalgebra
of a certain amalgamated free product by a projection, see [39, §7].
LetN be aσ-finite vonNeumannalgebraandD be adistinguishedvonNeumannsubalgebrawith
a faithful normal conditional expectation EN : N → D. Let us suppose that we have an (at most
D
countably infinite) family Θ of normal ∗-isomorphisms θ : D → N with faithful normal conditional
expectations EN :N →θ(D).
θ(D)
Set Θ := {1 := Id }⊔Θ, a disjoint union. Let us define the normal ∗-isomorphism ι : D ⊗
1 D Θ
ℓ (Θ )→N ⊗B ℓ2(Θ ) by
∞ 1 1
(cid:0) (cid:1)
x⊗e if θ =1,
11
ι (x⊗e ):=
Θ θθ
(θ(x)⊗eθθ if θ ∈Θ
for each x∈D, where the e ’s denote the canonicalmatrix unit system in B ℓ2(Θ ) . Namely, in
θ1θ2 1
the operator matrix representation, we have
(cid:0) (cid:1)
1 D
... ...
ιΘ = θ , ιΘ(D⊗ℓ∞(Θ1))= θ(D) .
.. ..
. .
HNN EXTENSIONS 5
We also define the faithful normal conditional expectation E :N ⊗B ℓ2(Θ ) →ι (D⊗ℓ (Θ ))
Θ 1 Θ ∞ 1
by
(cid:0) (cid:1)
EN
D
...
EΘ := EN = ⊕EθN(D)⊗IdCeθθ ◦(IdN ⊗Eℓ∞),
θ(D) ... θX∈Θ1 !
where E is the unique conditional expectation from B ℓ2(Θ ) onto ℓ (Θ ). Let us denote by ι
ℓ 1 ∞ 1 1
∞
the inclusion map of D⊗ℓ (Θ ) into N ⊗B ℓ2(Θ ) , and define the faithful normal conditional
∞ 1 1 (cid:0) (cid:1)
expectation E :N ⊗B ℓ2(Θ ) →D⊗ℓ (Θ ) by
1 1 ∞ 1(cid:0) (cid:1)
EN (cid:0) (cid:1)
D
...
E := = EN ⊗Id ◦(Id ⊗E )= EN ⊗E .
1 EN D ℓ∞(Θ1) N ℓ∞ D ℓ∞
D
... (cid:0) (cid:1) (cid:0) (cid:1)
We then construct the reduced free product with amalgamation:
(N,E)= N ⊗B ℓ2(Θ ) ,E :ι ⋆ N ⊗B ℓ2(Θ ) ,E :ι .
1 Θ Θ 1 1 1
(cid:0) (cid:0) (cid:1) (cid:1)D⊗ℓ∞(Θ1)(cid:0) (cid:0) (cid:1) (cid:1)
The embedding maps of N ⊗B ℓ2(Θ ) onto the 1st/2nd free components are denoted by λ and
1 Θ
λ , respectively,andthe embedding map ofD⊗ℓ (Θ ) into N by λ, i.e., λ=λ ◦ι =λ ◦ι . The
1 ∞ 1 Θ Θ 1 1
(cid:0) (cid:1)
desired HNN extension of N by Θ with respect to EN and the EN ’s will be constructed inside a
D θ(D)
corner subalgebra of N.
Let us define
u(θ):=λ (e )λ (e )
1 1θ Θ θ1
with identifying e =1⊗e , and the following equation is a key to our construction:
θ1θ2 θ1θ2
u(θ)λΘ(θ(d)⊗e11)u(θ)∗ =λΘ(d⊗e11), d∈D,
whichsimply comesfromλ (θ(d)⊗e )=λ(d⊗e )=λ (d⊗e )foreachd∈D,θ ∈Θ . We also
Θ θθ θθ 1 θθ 1
define the projection
p:=λ(e )=λ (e )∈N,
11 Θ 11
andthenintroducetheunitalnormal∗-isomorphismπfromN intothecornersubalgebrapNpdefined
by
π(n):=λ (n⊗e ), n∈N.
Θ 11
Thepartialisometriesu(θ)’s,canbethoughtofasunitariesinthecornerpNpsincetheirleftandright
supports are the projection p, and the above-mentioned key equation is translated into the following
algebraic relation:
(2) u(θ)π(θ(d))u(θ)∗ =π(d), d∈D.
Set
M :=π(N)∨{u(θ):θ ∈Θ}′′ ⊆pNp.
Let us consider a faithful normal semifinite weight on D⊗ℓ (Θ ):
∞ 1
φ:=ϕ⊗ Tr
ℓ∞(Θ1)
(cid:16) (cid:12) (cid:17)
with a faithful normal state ϕ on D, where Tr is t(cid:12)he (non-normalized) canonical normal trace on
B ℓ2(Θ ) .
1
(cid:0) (cid:1)
6 Y.UEDA
Lemma 3.1. We have
(3) σtφ◦λ−1◦E(p) = p,
(4) σtφ◦λ−1◦E(π(n)) = π σtϕ◦EDN (n) ,
(5) σtφ◦λ−1◦E(u(θ)) = u((cid:16)θ)π Dϕ◦(cid:17)θ−1◦EθN(D) :Dϕ◦EDN
t
(cid:16)h i (cid:17)
for each t∈R, n∈N, θ ∈Θ.
Proof. The equations(3), (4)are straightforwardfromProposition2.3, while the lastone needs some
additional efforts. In fact, we have
σtφ◦λ−1◦E(u(θ))=λ1 σtφ◦ι−11◦E1(1⊗e1θ) λΘ σtφ◦ι−Θ1◦EΘ(1⊗eθ1)
(cid:16) (cid:17) (cid:16) (cid:17)
=λ (1⊗e )λ Dϕ◦θ 1◦EN :Dϕ◦EN ⊗e
1 1θ Θ − θ(D) D θ1
t
(cid:16)h i (cid:17)
=λ (e )λ (e )λ Dϕ◦θ 1◦EN :Dϕ◦EN ⊗e
1 1θ Θ θ1 Θ − θ(D) D 11
t
(cid:16)h i (cid:17)
=u(θ)π Dϕ◦θ−1◦EθN(D) :Dϕ◦EDN ,
t
(cid:16)h i (cid:17)
where the second equality comes from the so-called “balanced weight technique” due to Connes (see
[36, Vol.II; Chap. VIII, §3, p.111–113]). (cid:3)
Since π(N)=pλ N ⊗B ℓ2(Θ ) p, the restriction of the normal conditional expectation
Θ 1
(cid:0) (cid:0) EΘ(cid:1):(cid:1)N →λΘ N ⊗B ℓ2(Θ1)
that preserves φ◦λ−1 ◦E (and hence E = E ◦(cid:0)EΘ hold(cid:0)s) to M(cid:1)(cid:1)gives a faithful normal conditional
expectation
EM :=E :M →π(N).
π(N) Θ M
We have φ◦λ−1◦E M = ϕ◦ EDN ◦π−1 ◦EπM(cid:12)(cid:12)(N), and hence, by Takesaki’s theorem [36, Vol.II;
Theorem 1.2 in Chap. VIII, §1] we get
(cid:0) (cid:1)(cid:12)
(cid:12)
(6) σtϕ◦EDN◦π−1◦EπM(N) =σtφ◦λ−1◦E M, t∈R
since σtφ◦λ−1◦E(M)=M for every t∈R thanks to Lemma(cid:12)(cid:12)3.1.
Definition 3.1. (Reduced HNN extensions) We call the pair M,EM constructed so far the
π(N)
reduced HNN extension (or HNN extension, in short) of N by Θ w(cid:16)ith respect(cid:17)to EN and the EN ’s,
D θ(D)
and denote it by
M,EM = N,EN ⋆ Θ, EN .
π(N) D θ(D)
When no confusion occurs,(cid:16)we will wr(cid:17)ite M(cid:0) = N⋆(cid:1)DΘ(cid:18)for snhort. Tohθe∈Θg(cid:19)iven von Neumann algebra N
D
is called the base algebra, and each u(θ) the stable unitary of θ ∈Θ.
Definition 3.2. (Reduced words) An element (in M)
w =u(θ )ε0π(n )u(θ )ε1π(n )···π(n )u(θ )εℓ
0 1 1 2 ℓ ℓ
with n ,n ,...,n ∈ N, θ ,θ ,...,θ ∈ Θ, ε ,ε ,...,ε ∈ {1,−1} (possibly with w = u(θ )ε0) is
1 2 ℓ 0 1 ℓ 0 1 ℓ 0
called a reduced word (or said to be of reduced form) if θ =θ and ε 6=ε imply that
j 1 j j 1 j
− −
• n ∈N :=KerEN with θ :=θ =θ , when ε =1, ε =−1;
j θ◦ θ(D) j 1 j j 1 j
− −
• n ∈N :=KerEN, when ε =−1, ε =1.
j ◦ D j 1 j
−
We should point out that our definition of reduced words agrees with so-called Britton’s lemma
in combinatorial group theory (see [16, p.181]), where a reduced word is named as a normal form,
and the sets of representatives of right cosets of distinguished subgroups should be regarded as the
counterparts of N and the N ’s in our consideration.
◦ θ◦
HNN EXTENSIONS 7
Remark 3.2. It is plain to see that u(θ )ε1···u(θ )εℓ is of reduced form in the above sense if and
1 ℓ
only if so is θε1···θεℓ in the free group F(Θ) over the generating set Θ.
1 ℓ
Definition 3.3. (Conditions needed for characterization) We introduce the following two con-
ditions:
(A) u(θ)π(θ(d))u(θ) =π(d) for every d∈D, θ ∈Θ.
∗
(M) For every reduced word w, one has EM (w)=0.
π(N)
Theorem3.3. Thepair M,EM constructedabovesatisfiestheconditions (A),(M).Ontheother
π(N)
hand, the conditions (A)(cid:16), (M) char(cid:17)acterize the pair M,EM completely under the assumption
π(N)
that π(N) and the u(θ)’s generate M as von Neuma(cid:16)nn algebra(cid:17). Strictly speaking, the conditional
expectation of the pair in question is completely determined by those conditions.
Proof. Let us denote the 1st/2nd free components of N by N , N , respectively, for short, i.e.,
Θ 1
N := λ N ⊗B ℓ2(Θ ) , N := λ N ⊗B ℓ2(Θ ) , and set N := N ∩KerE = λ (KerE )
Θ Θ 1 1 1 1 Θ◦ Θ Θ Θ
and N :=N ∩KerE =λ (KerE ) as usual.
1◦ (cid:0) 1 (cid:0) 1(cid:1)(cid:1) 1 (cid:0) (cid:0) (cid:1)(cid:1)
The condition (A) was already verified, see the equation (2), and thus it suffices to check the
condition (M) for the first half of the assertions. Let us choose a word
w =u(θ )ε0π(n )u(θ )ε1π(n )···π(n )u(θ )εℓ,
0 1 1 2 ℓ ℓ
and then we have
w =(λ (e )λ (e ))ε0λ (n ⊗e )(λ (e )λ (e ))ε1
1 1θ0 Θ θ01 Θ 1 11 1 1θ1 Θ θ11
···λ (n ⊗e )(λ (e )λ (e ))ε0.
Θ ℓ 11 1 1θℓ Θ θℓ1
Here, we briefly explain how to manipulate this word in a typical case: If ε =−1, ε =1, then
j 1 j
−
λ1 e1θj 1 λΘ eθj 11 εj−1λΘ(nj ⊗e11) λ1 e1θj λΘ eθj1 εj
− −
(cid:0)=λ(cid:0)Θ e1θj(cid:1)1 λ(cid:0)1 EDN((cid:1)n(cid:1)j)⊗eθj 1θj λΘ e(cid:0)θj1 (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
− −
(cid:0) +λΘ(cid:1) e1(cid:0)θj 1 λ1 eθj 11 λΘ(cid:1) n◦j(cid:0)⊗e11(cid:1) λ1 e1θj λΘ eθj1
− −
with n = n − EN(n ). If n(cid:0) ∈ N(cid:1), th(cid:0)en this(cid:1) bel(cid:0)ongs to N(cid:1) N(cid:0) N (cid:1)N N(cid:0) si(cid:1)nce the first term
◦j j D j j ◦ Θ◦ 1◦ Θ◦ 1◦ Θ◦
disappears in this case. On the other hand, if n is arbitrary but θ 6= θ , then it belongs to
j j 1 j
−
N N N +N N N N N . In this way, one can easily observes that, if the word w is of reduced
Θ◦ 1◦ Θ◦ Θ◦ 1◦ Θ◦ 1◦ Θ◦
form, then it belongs to the linear span of alternating words in N and N of length greater than 2.
Θ◦ 1◦
Therefore, we have E (w)=0, which asserts the condition (M).
Θ
Next, we will show the latter half of the assertions. To do so, it is enough to explain how one can
compute the moment-value:
EM u(θ )δ0π(x )u(θ )δ1π(x )···π(x )u(θ )δm
π(N) 0 1 1 2 m m
of any given x ,x ,...,x (cid:16)∈ N, θ ,θ ,...,θ ∈ Θ, δ ,δ ,...,δ ∈ Z \(cid:17){0}, by using only the
1 2 m 0 1 m 0 1 m
conditions (A), (M). In fact, if the resulting value could be expressed uniquely in terms of only
the data of (θ ,δ ),x ,(θ ,δ ),...,x ,(θ ,δ ) together with EN and the EN ’s, then the desired
0 0 1 1 1 m m m D θ(D)
assertion would follow. Our technique is the essentially same as in the case of free products with
amalgamations. Namely, we use the decompositions:
n=EN(n)+n or EN (n)+[n] n∈N,
D ◦ θ(D) ◦θ
where we define [n] := n−EN (n). By the repeated use of the decompositions together with the
◦θ θ(D)
condition (A), we can make the moment-value in question a (finite) sum of the form:
π(n(w))EM (w)
π(N)
w: reducedwordor1
X
with coefficients n(w) being words in D and θ(D) (in N), and all the coefficients n(w) and all the
words w (the moment-value EM (w) takes 0 if w is of reduced form or otherwise, w = 1) appeared
π(N)
8 Y.UEDA
in the above expression are uniquely determined from the given data (θ ,δ ), x , (θ ,δ ), ..., x ,
0 0 1 1 1 m
(θ ,δ ) together with EN and the EN ’s. Therefore, our desired assertion follows. (cid:3)
m m D θ(D)
Let u(g), g ∈F(Θ), be the natural group isomorphism from the free group F(Θ) into the unitary
group U(M) given by the correspondence θ ∈Θ7−→u(θ)∈U(M). Let us denote by ℓ( · ) the usual
wordlengthfunction with respectto the generatingsetΘ. The computationgivenin the aboveproof
implies the following corollary:
Corollary 3.4. Let w =u(θ )ε0π(n )u(θ )ε1π(n )···π(n )u(θ )εℓ be a word in M, and set g :=
0 1 1 2 ℓ ℓ
θε0θε1···θεℓ, a word in F(Θ) (obtained by replacing all n ’s by the identity 1). Then we have
0 1 ℓ j
ℓ(g)6=0=⇒EM (w)=0.
π(N)
In particular, the unitaries u(θ)’s form a free family of Haar unitaries, so that they generate the free
group factor L(F(Θ)).
The following corollary is also straightforwardfrom Theorem 3.3:
Corollary 3.5. Let G∗ θ = hG,t : tθ(h)t 1 = h, h ∈ Hi be an HNN extension of base group G
H −
with stable letter t by group isomorphism θ from H into G. Then, the group von Neumann algebra
L(G∗ θ) can be identified with the reduced HNN extension of the base algebra L(G) with the stable
H
unitary λ(t), where all the necessary conditional expectations are chosen as the canonical tracial state
preserving ones.
We then discuss what phenomenon occurs when D and the θ(D)’s are assumed to be all mutually
inner conjugate. Let M,EM be as above with identifying n =π(n), n∈N. We here suppose that
N
every θ ∈ Θ has a unitary w ∈ N with the following properties: (i) Adw ◦θ ∈ Aut(D); and (ii)
θ θ
EN =Adw ◦EN◦(cid:0)Adw . D(cid:1)efine the actionγ ofF(Θ)on D in sucha waythatγ =Adw ◦θ, and
θ(D) θ∗ D θ θ θ
consider the free product with amalgamation:
L,FL := N,EN ⋆ D⋊ F(Θ),ED⋊γF(Θ) ,
D D γ D
D
whereED⋊γF(Θ) isthecan(cid:0)onicalc(cid:1)ondi(cid:0)tionale(cid:1)xpec(cid:16)tation. Thefaithfulno(cid:17)rmalconditionalexpectation
D
fromLontothe1stfreecomponentN thatpreservesFLisdenotedbyFL. Then,wehavethefollowing
D N
simple corollary:
Corollary 3.6. In the above setting, the correspondence:
n∈M 7→n∈L, n∈N;
u(θ)∈M 7→λγ(θ) w ∈L, θ ∈Θ
∗ θ
gives a ∗-isomorphism between M and L that intertwines EM and FL. Here, λγ :F(Θ)→D⋊ F(Θ)
N N γ
(⊆L) denotes the canonical unitary representation.
Proof. Itisplaintoverifythatthepair L⊇N,FNL withtheunitariesλγ(θ)∗wθ,θ ∈Θ,satisfiesthe
conditions(A),(M)withrespecttoΘan(cid:0)dEDN, EθN(D(cid:1)) . Infact,thecondition(A)followsfromthe
θ Θ
above (i), while the (M) from the fact that x∈nN =oKe∈rEN if and only if w xw ∈N =KerEN
θ◦ θ(D) θ θ∗ ◦ D
thanks to the above (ii). (cid:3)
Remarks 3.7. (1) [HNN extensions arising from inner conjugate Cartan subalgebras]
Assume that N is a non-type I factor with separable predual (or more generally, a von Neumann
algebra with separable predual having no type I direct summand) and further that D and the θ(D)’s
are all Cartan subalgebras in N. By the uniqueness of normal conditional expectations onto those
Cartan subalgebras, if those Cartan subalgebras are all mutually inner conjugate, then Corollary 3.6
enables us to apply our previous results [38][39][40] to the HNN extension M =N⋆ Θ without any
D
change. However, we have no general result without this inner conjugacy assumption among Cartan
subalgebras in question.
(2) A special case of Corollary 3.6 was one of the starting points of the present work. In fact,
in the setting of Corollary 3.6, the group theoretic construction based on shift automorphisms on
HNN EXTENSIONS 9
infinite amalgamated free products is valid when all the w = 1 (so that γ = θ). Concerning this,
θ θ
we point out that the amalgamated free product appeared in Corollary 3.6 has the crossed-product
decomposition:
M =N(Θ)⋊F(Θ)
by the free Bernoulli shift on
N(Θ):= ⋆ N,EN :γ .
D D g
g F(Θ)
∈ (cid:0) (cid:1)
(See e.g. [12, §3], where only the case of D = C1 was treated, but the argument works even in this
case.) More on this will be discussed in the next section with full generality.
4. Modular Theory
Let
M,EM = N,EN ⋆ Θ, EN
π(N) D θ(D)
betheHNNextensionofbas(cid:16)evonNeum(cid:17)ann(cid:0)algebra(cid:1)ND (cid:18)withnstableuonθ∈itΘar(cid:19)iesu(θ),θ ∈Θ. Here,wewill
use the construction and the notations of HNN extensions given in the previous section; however, in
whatfollows,wewillidentifyn=π(n),n∈N,soπwillbeomitted. Thenexttheoremisimmediately
derived from Lemma 3.1 with the aid of Connes’ cocycle Radon-Nikodym theorem (see [36, Vol.II;
Chap. VIII, §3]).
Theorem 4.1. For a faithful normal semifinite weight ψ on D, we have
σtψ◦EDN◦ENM (u(θ))=u(θ) Dψ◦θ−1◦EθN(D) :Dψ◦EDN , t∈R.
t
h i
This theorem implies the following criterion for the existence of traces on HNN extensions:
Corollary 4.2. If N has a faithful normal semifinite trace τ and if the given EN and the EN ’s
D θ(D)
satisfy the relation:
τ = τ ◦EN = τ ◦θ 1◦EN , θ ∈Θ,
D D D − θ(D)
then so does M, and more pre(cid:0)cis(cid:12)(cid:12)ely(cid:1) τ ◦ENM (cid:0)be(cid:12)(cid:12)com(cid:1)es a trace. In particular, if N is semifinite with
a faithful normal semifinite trace τ and if the given conditional expectations are τ-preserving and
τ| =(τ| )◦θ holds, then τ ◦EM is a trace.
θ(D) D N
A crossed-product decomposition fact for HNN extensions was given in Remarks 3.7, (2) under a
very special assumption. Here, we give such a fact with full generality.
Corollary4.3. LetusdenotebyN(Θ)thevonNeumannsubalgebrageneratedbyalltheu(g)Nu(g)∗,
g ∈F(Θ). Then we have the crossed-product decomposition of M:
M =N(Θ)⋊ F(Θ)
Adu
with the natural adjoint action Adu:g ∈F(Θ)7→Adu(g)∈Aut(N(Θ)).
Proof. First of all, we should remark that Theorem 4.1 shows that there is a unique faithful normal
conditionalexpectationfromM ontoN(Θ)thatpreservesEM. Thus,thedesiredassertionisderived
N
fromCorollary3.4togetherwiththewell-knowncharacterizationofdiscretecrossed-productsinterms
of conditional expectations. (cid:3)
Remark 4.4. Theorem 4.1 says that each subalgebra u(g)Nu(g)∗, g ∈F(Θ) with g 6=e (e denotes
the identity), is not necessary to be globally invariant under the modular automorphism associated
with ψ◦EM.
N
Theorem 4.1 enables us to show that the continuous core of the HNN extension M in question
becomesagainanHNNextension. Forabetterdescription,itisconvenienttousearecentformulation
of continuous cores due to S. Yamagami [45]. (See also A. J. Falcone and M. Takesaki [7], and the
reader may consult [36, Vol.II; Chap. XII, §6] for more detailed account.) Following the formulation,
the continuouscoreP ofagivenP canbe understoodasanabstractvonNeumannalgebragenerated
e
10 Y.UEDA
bytwokindsofsymbolsx∈P andψit withafaithfulnormalsemi-finite weightψ onP,whichsatisfy
the relations:
ψitxψ−it =σtψ(x), ψitψis =ψi(t+s), φitψ−it =[Dφ:Dψ]t
forfaithfulnormalsemi-finiteweightsφ,ψ onP. ItisknownthatsuchavonNeumannalgebraP can
be realizedas the crossed-productP ⋊ R, where ψit denotes the canonicalunitary implementation
σψ
λψ(t) of R inside the crossed-product, and φit =[Dφ:Dψ] ψit =[Dφ:Dψ] λψ(t) in general.e
t t
In our setting, the inclusion relations M ⊇ N ⊇ D, M ⊇ N ⊇ θ(D), θ ∈ Θ, with the faithful
normal conditional expectations EM : M → N, EN :→ D, EN : N → θ(D) give us the following
N D θ(D)
natural embeddings and mapping:
identify
d∈D ↔ d∈D ⊆N ⊆N,
D ֒→N by
(ϕit ∈D ide↔ntify ϕ◦EN it ∈N;
D
e
e e
d∈D 7→θ(d)∈(cid:0) θ(D)⊆(cid:1)N ⊆N,
e e
θ :D →N by it
ϕit ∈D 7→ ϕ◦θ 1◦EN ∈N;
− θ(D) e
e e e identi(cid:16)fy (cid:17)
n∈Ne ↔ n∈M ⊆M, e
N ֒→M by
(φit ∈N ide↔ntify φ◦EM it ∈M,
N
f
and the conditional exepectaftions (cid:0) (cid:1)
e f
\ ]
EN :N →D, EN :N →θ(D)=θ D , EM :M →N
D θ(D) N
(cid:16) (cid:17)
constructed in such a way that
d e e e e e d f e
EN =EN, EN ϕ◦EN it = ϕ◦EN it;
D N D D D D
\ (cid:12) \(cid:16)(cid:0) (cid:1) (cid:17) (cid:0) it (cid:1) it
EN d(cid:12)=EN , EdN ϕ◦θ 1◦EN = ϕ◦θ 1◦EN ;
θ(D) N θ(D) θ(D) − θ(D) − θ(D)
EM(cid:12)(cid:12) =EM, EM (cid:18)φ(cid:16)◦EM it = φ◦(cid:17)E(cid:19)M it(cid:16) (cid:17)
N M N N N N
for faithful normalp(cid:12)ositive linear functio(cid:16)n(cid:0)als ϕ∈D(cid:1) ,(cid:17)φ∈(cid:0)N , whe(cid:1)re one shouldremind the following
d(cid:12) d
∗ ∗
formula:
θ ϕ◦EN it =θ ϕit = ϕ◦θ 1◦EN it.
D − θ(D)
For a faithful normal state(cid:16)(cid:0)ϕ on D,(cid:1)we(cid:17)h(cid:16)ave, i(cid:0)n th(cid:1)e(cid:17)cont(cid:16)inuous core M, (cid:17)
e e
(7) ϕ◦EN ◦EM itu(θ)=u(θ) ϕ◦θ 1◦EN ◦EM it, t∈R,
D N − θ(D) Nf
thanks to Theorem(cid:0)4.1. The gen(cid:1)eral assertion g(cid:16)iven below is a simple(cid:17)application of the formula (7)
and Theorem 3.3, i.e., the characterizationof HNN extensions.
Theorem 4.5. The pair M,EM is again the HNN extension of the base algebra N by the family
N
Θ := θ : θ ∈Θ with th(cid:16)e stable(cid:17)unitaries u(θ), θ ∈ Θ, with respect to the conditional expectations
f d e
EN annd E\N , θo∈Θ, that is,
eD e θ(D)
\
d M,EM = N,EN ⋆ Θ, EN .
N D θ(D)
θ Θ
(cid:16) (cid:17) (cid:16) (cid:17) D (cid:18) n o ∈ (cid:19)
We will often denote this idenftificdation by Me d=N⋆eΘ feor short.
D
Proof. The condition (A) follows from thefformuela (e7). Indeed, for each d ∈ D, t ∈ R, the formula
e
(7) enables us to compute
u(θ)θ d ϕ◦EDN it u(θ)∗ =d ϕ◦EDN ◦ENM it =d ϕ◦EDN it.
(cid:16) (cid:0) (cid:1) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1)
e
