Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

0.46 MB·

by  Martijn Caspers

ABSENCE OF CARTAN SUBALGEBRAS FOR RIGHT-ANGLED
HECKE VON NEUMANN ALGEBRAS
6
1
0
MARTIJNCASPERS
2

v
o
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0 tAhbesatsrsaoccita.teFdoHreackreigvhot-nanNgeluedmaCnonxeatlegrebsryas,tewmhi(cWhi,sSg)eannerdatqed>b0y,sleeltf-Madjqoibnet
1 operatorsTs,s SsatisfyingtheHeckerelation(√qTs q)(√qTs+1)=0as
well as suitable∈commutation relations. Under the assu−mption that (W,S) is

A] (roedfutycepdeaIIn1d)|fSor|≥ar3anitgewqas p[rρo,vρe−d1b]yanGdaornthcaerrwekis[eGar1q5i]stthhaetdMireqctissaumfacotfoar
O II1-factorandC. ∈ M
In this paper we prove (under the same natural conditions as Garncarek)
.
h that qisnon-injective,thatithastheweak- completelycontractiveapprox-
M ∗
t imation property and that it has the Haagerup property. In the hyperbolic
a
m factorialcaseMqisastronglysolidalgebraandconsequentlyMqcannothave
a Cartansubalgebra. In thegeneral case q neednot bestronglysolid, but
M
[ stillweprovethat q doesnotpossessaCartansubalgebra.
M

2
v 1. Introduction
3
Hecke algebras are one-parameter deformations of group algebras of a Coxeter
9
5 group. Theywerethefundamentforthetheoryofquantumgroups[Jim86],[Kas95]
0 and have remarkable applications in the theory of knot invariants [Jon85] as was
0 shown by V. Jones. A wide range of applications of Coxeter groups and their
.
1 Hecke deformations can be found in [Dav08]. In [Dym06] (see also [Dav08, Section
0 19]) Dymara introduced the von Neumann algebras generated by Hecke algebras.
6 Manyimportantresultswerethenobtained(seealso[DDJB07])fortheseHeckevon
1
Neumann algebras. This gave for example insight in the cohomology of associated
:
v buildings and its Betti numbers. In this paper we investigate the approximation
Xi propertiesofHeckevonNeumannalgebrasaswellastheirCartansubalgebras(here
we mean the notion of a Cartan subalgebra in the von Neumann algebraic sense
r
a which we recall in Section 6 and not the Lie algebraic notion).
Let us recall the following definition. Let q > 0 and let W be a right-angled
Coxetergroupwith generatingset S (see Section 2). The associatedHeckealgebra
is a -algebra generated by T ,s S which satisfies the relation:
s
∗ ∈
(√qTs−q)(√qTs+1)=0, Ts∗ =Ts and TsTt =TtTs,
for s,t S with st = ts. Hecke algebras carry a canonical faithful tracial vector
∈
state(the vacuumstate)andthereforegenerateavonNeumannalgebra under
q
M
its GNS construction. It was recently proved by Garncarek [Gar15] that if (W,S)
is reduced (see Section 2) and S 3, the von Neumann algebra is a factor
q
| | ≥ M
in case q [ρ,ρ 1] where ρ is the radius of convergence of the fundamental power
−
series (2.2∈). If q [ρ,ρ 1] then is the direct sum of a II factor and C. For
− q 1
6∈ M
Date:November 11,2016.
1
2 MARTIJNCASPERS
more general coxeter groups/Hecke algebras (not necessarily being right angled,
or for multi-parameters q) this result is unknown. It deserves to be emphasized
that this in particular shows that the isomorphism class of depends on q; an
q
M
observation that was already made in the final remarks [Dav08, Section 19].
The first aim of this paper is to determine approximationproperties of (as-
q
M
suming the same natural conditions as Garncarek). We first show that is a
q
M
non-injective von Neumann algebra and therefore falls outside Connes’ classifica-
tion of hyperfinite factors [Con76]. Secondly we show that has the weak-
q
M ∗
completely contractive approximation property (wk- CCAP). This means that
∗
there exists a net ofcompletely contractivefinite rankmaps on that converges
q
M
to the identity in the point σ-weak topology. In case q = 1 the algebra is
q
M
the group von Neumann algebra of a right-angled Coxeter group. In this case the
result was known. For instance the CCAP follows from Reckwerdt’s result [Rec15]
and non-injectivity follows easily from identifying a copy of the free group inside
W. In this context we also mention the parallel results for q-Gaussian algebras
with 1 < q < 1: factoriality by Ricard [Ric05], non-injectivity by Nou [Nou04]
−
and the completely contractive approximationproperty by Avsec [Avs11]. We owe
sometotheideasof[Nou04](and[BoSp94])fortheproofofnon-injectivity. [Nou04]
alsousesaKhintchineinequalitythoughourproofismoreinthespiritof[RiXu06].
Anotherimportantresultconcerningtheapproximationpropertiesofoperatoralge-
braswasobtainedbyHoudayerandRicard[HoRi11]whosettledtheapproximation
properties of free Araki-Woods factors. For our Hecke von Neumann algebra
q
M
we summarize:
Theorem A. Let q >0.
(1) Let (W,S) be a reduced right-angled Coxeter system with S 3. Then
| | ≥
is non-injective.
q
M
(2) For a generalright-angledCoxetersystem (W,S)the associatedHecke von
Neumann algebra has the wk- CCAP and the Haagerup property.
q
M ∗
Obviously non-injectivity and the wk- CCAP of Theorem A have different
∗
proofs. However the two proofs each borrow some ideas from [RiXu06] where Ri-
card and Xu proved that weak amenability with constant 1 is preserved by taking
free products of discrete groups. In order to prove non-injectivity we first obtain a
Khintchine inequality for Hecke algebras. We show that this Khintchine inequality
leads to a contradiction in case were to be injective. For the wk- CCAP we
q
M ∗
firstobtaincb-estimatesforradialmultipliersandthenuseestimatesofwordlength
projections (see Proposition 5.12) going back to Haagerup [Haa78].
Our second aim is the study of Cartan subalgebras of the Hecke von Neumann
algebra . Recall that a Cartan subalgebra of a II -factor is by definition a
q 1
M
maximalabeliansubalgebrawhosenormalizergeneratestheII -factoritself. Cartan
1
subalgebras arise typically in crossed products of free ergodic probability measure
preservingactionsofdiscretegroupsonaprobabilitymeasurespace. InfactCartan
subalgebras always come from some orbit equivalence class in the following sense:
foraseparableII factor anyCartansubalgebra givesrisetoastandard
1
M A⊆M
probability measure space X and an orbit equivalence class with cocycle σ such
R
that ( ) (L (X) (X,σ)). We refer to [FeMo77] for details. Cartan
∞
A ⊆ M ≃ ⊆ R
subalgebrascanbeusedtoobtainfurtherrigidityresults,see[PoVa14],[IPV13]for
twoveryprominentillustrations ofthis: the firstshowingthat (suitable)actions of
ABSENCE OF CARTAN SUBALGEBRAS FOR HECKE VON NEUMANN ALGEBRAS 3
freegroupsonprobabilityspacesremember the number ofgeneratorsofthe group;
thesecondshowingthatcertaingroupvonNeumannalgebrascompletelyremember
the group (W -superrigidity). These results typically rely on the uniqueness of a
∗
Cartansubalgebra. Otherapplicationscanbefoundinprimefactorizationtheorems
[OzPo04], [HoIs15] in which the absence of Cartan algebras plays a crucial role.
In[Voi96]Voiculescuwasthefirstonetofindfactors(namelyfreegroupfactors)
that do not have a Cartan subalgebra. His proof relies on estimates for the free
entropydimensionofthe normalizerofaninjective vonNeumannalgebra. Using a
different approach Ozawa and Popa [OzPo10] were also able to find classes of von
Neumann algebras that do not have a Cartansubalgebra (including the free group
factors). These results generalize to a larger class of class of groups. In particular
in [PoVa14] Popa and Vaes (see also Chifan-Sinclair [ChSi13]) proved absence of
Cartansubalgebrasforgroupfactorsofbi-exactgroupsthathavethe CBAP.Isono
[Iso15]thenputtheresultsfrom[PoVa14]intoageneralvonNeumannframeworkin
order to prove absence of Cartan subalgebras for free orthogonalquantum groups.
Isono proved that factors with the wk- CBAP that satisfy condition (AO)+ are
∗
strongly solid, meaning that the normalizer of an injective von Neumann algebra
is injective again. Using this strongsolidity resultby Isono/Popa–Vaeswe areable
to prove the following.
Theorem B. Let q [ρ,ρ 1] with ρ as in Theorem 2.2. Let (W,S) be a reduced
−
∈
right-angledCoxetersystemwith S 3. Assume thatW is hyperbolic. Thenthe
| |≥
associated Hecke von Neumann algebra is strongly solid.
q
M
In turn as is non-injective by Theorem A we are able to derive the result
q
M
announced in the title of this paper for the hyperbolic case.
CorollaryC.Letq [ρ,ρ 1]withρasinTheorem2.2. Forareducedright-angled
−
∈
hyperbolic Coxeter system(W,S) with S 3 the associatedHecke vonNeumann
| |≥
algebra does not have a Cartan subalgebra.
q
M
General right-angled Hecke von Neumann algebras are not strongly solid, see
Remark 6.6. Still we can prove that they do not possess a Cartan subalgebra.
We do this by showing that if were to have a Cartan subalgebra then each
q
M
of the three alternatives in [Vae14, Theorem A] fails to be true, which leads to a
contradiction. ItdeservestobenoticedthattheproofofTheoremDusesTheorem
A as well.
Theorem D. Let q > 0. For a reduced right-angled Coxeter system (W,S) with
S 3 the associated Hecke von Neumann algebra does not have a Cartan
q
| | ≥ M
subalgebra.
Structure. In Section 2 we introduce Hecke von Neumann algebras and some basic
algebraic properties. Lemma 2.7 is absolutely crucial as each of the results in this
paper rely in their own way on this decomposition lemma. In Section 3 we obtain
universal properties of Hecke von Neumann algebras. In Section 4 we prove that
is non-injective. In Section 5 we find approximation properties of and
q q
M M
concludeTheoremA.Section6provesthestrongsolidityresultofTheoremBfrom
which Corollary C shall easily follow. Finally Section 7 proves absence of Cartan
subalgebras in the general case.
4 MARTIJNCASPERS
Convention. Let X be a set and let A,B X. We will briefly write A B for
⊆ \
A (A B).
\ ∩
Acknowledgements. The author wishes to express his gratitude to the following
people: SergeyNeshveyevandtheanonymousrefereeofanearlierversionforpoint-
ing out that Theorem B needs the assumption of hyperbolicity. Lukasz Garncarek
and Adam Skalski for enlightening discussions on Hecke-von Neumann algebras
which eventually led to a simplification in one of the proofs.
2. Notation and preliminaries
Standard result on operator spaces can be found in [EfRu00], [Pis02]. Standard
references for von Neumann algebras are [StZs75] and [Tak79]. Recall that ucp
stands for unital completely positive.
2.1. Coxeter groups. A Coxeter group W is a group that is freely generated by
a finite set S subject to relations
(st)m(s,t) =1,
for some constant m(s,t) 1,2,..., with m(s,t) = m(t,s) 2,s = t and
∈ { ∞} ≥ 6
m(s,s) = 1. The constant m(s,t) = means that no relation is imposed, so
∞
that s,t are free variables. The Coxeter group W is called right-angled if either
m(s,t) = 2 or m(s,t) = for all s,t S,s = t and this is the only case we need
∞ ∈ 6
in this paper. Therefore we assume from now on that W is a right-angled Coxeter
group with generating set S. The pair (W,S) is also called a Coxeter system.
Let w W and suppose that w = w ...w with w S. The representing
1 n i
∈ ∈
expression w ...w is called reduced if whenever also w =w ...w with w S
1 n 1′ m′ i′ ∈
then n m, i.e. the expression is of minimal length. In that case we will write
≤
w = n. Reduced expressions are not necessarily unique (only if m(s,t) =
| | ∞
whenever s =t), but for each w W we may pick a reduced expression which we
6 ∈
shall call minimal.
Convention: For w W we shall write w for the minimal representative w =
i
∈
w ...w .
1 n
To the pair (W,S) we associate a graph Γ with vertex set VΓ = S and edge
set EΓ = (s,t) m(s,t) = 2 . A subgraph Γ of Γ is called full if the following
0
{ | }
property holds: s,t VΓ with (s,t) EΓ we have (s,t) EΓ . A clique in Γ is
0 0
∀ ∈ ∈ ∈
afullsubgraphinwhicheverytwoverticesshareanedge. WeletCliq(Γ)denotethe
set of cliques in Γ. To keep the notation consistent with the literature the empty
graph is in Cliq(Γ) by convention (in this paper we shall sometimes exclude the
emptygraphfromCliq(Γ)explicitly ortreatitasa specialcaseto keepsomeofthe
argumentsmoretransparent). We let Cliq(Γ,l) be the setofcliques with l vertices.
Definition 2.1. A Coxetersystem (W,S) is calledreduced if the complement of Γ
is connected.
2.2. Hecke von Neumann algebras. Let (W,S) be a right-angled Coxeter sys-
tem. Let q > 0. By [Dav08, Proposition 19.1.1] there exists a unique unital
ABSENCE OF CARTAN SUBALGEBRAS FOR HECKE VON NEUMANN ALGEBRAS 5
-algebra C (Γ) generated by a basis T w W satisfying the following rela-
q w
∗ { | ∈ }
tions. For every s S and w W we have:
∈ ∈
e
T if sw > w,
TsTw = sw | | | |
( qTsw+(1 q)Tw otherwise,
−
e
eTew∗ =Tw−1e. e
We define normalized elements T =q w/2T . Then for w W and s S,
e e w −| | w
∈ ∈
T if sw > w,
(2.1) TsTw = Tsw+pT eoth|erw|ise,| | ,
sw w
(cid:26)
where
q 1
p= − .
√q
Thereisanaturalpositivelineartracialmapτ onC (Γ)satisfyingτ(T )=0,w=1
q w
andτ(1)=1. Let L2( ) be the Hilbert space givenby the closure of C (Γ) w6 ith
q q
M
respect to x,y = τ(y x) and let be the von Neumann algebra generated by
∗ q
C (Γ) actinhg oni L2( ). τ extenMds to a state on and L2( ) is its GNS
q q q q
M M M
space with cyclic vector Ω:=T . is called the Hecke von Neumann algebra at
e q
M
parameter q associated to the right-angled Coxeter system (W,S).
Theorem 2.2 (see [Gar15])). Let (W,S) be a reduced right-angled Coxeter system
and suppose that S 3. Let ρ be the radius of convergence of the fundamental
| | ≥
power series:
∞
(2.2) w W w =k zk.
|{ ∈ || | }|
k=0
X
For every q [ρ,ρ 1] the von Neumann algebra is a factor. For q > 0 not in
− q
[ρ,ρ 1] the v∈on Neumann algebra is the direcMt sum of a factor and C.
− q
M
As posessesanormalfaithfultracialstatethefactorsappearinginTheorem
q
M
2.2 are of type II .
1
Fortheanalysisof weshallinfactneed whichisthegroupvonNeumann
q 1
M M
algebra of the Coxeter group W. It can be represented on L2( ). Indeed, let
q
M
T(1) denote the generators of as in (2.1) and let T be the generators of .
w 1 w q
M M
Set the unitary map,
U :L2( ) L2( ):T(1)Ω T Ω.
M1 → Mq w → w
In this paper we shall always assume that is represented on L2( ) by the
1 q
M M
identification (L2( )):x UxU . Note that this way
1 q ∗
M →B M 7→
(2.3) T(1)(T Ω)=T Ω.
v w vw
For w W we shall write P for the projectionof L2( ) onto the closure of the
w q
∈ M
space spannedlinearly by T Ω w 1v = v w (see Remark 2.3 below). For
v −
{ || | | |−| |}
Γ Cliq(Γ) we shall write P for P where w W is the product of all vertex
0 ∈ VΓ0 w ∈
elements of Γ and VΓ for the number of elements in VΓ . Similarly we shall
0 0 0
| |
write P for P where w W is the product of v with all vertex elements of
vVΓ0 w ∈
Γ .
0
6 MARTIJNCASPERS
Remark 2.3 (Creation and annihilation arguments). Note that for w,v W
∈
saying that w 1v = v w just means that the start of v contains the word
−
| | | |−| |
w. Throughoutthe paper we say that s W acts by means of a creationoperator
∈
on v W if sv = v +1. It acts as an annihilation operator if sv = v 1.
∈ | | | | | | | |−
Note that as W is right-angled we cannot have sv = v. For v,w W we may
| | | | ∈
always decompose w =w w such that w = w + w , w v = v w and
′ ′′ ′ ′′ ′′ ′′
| | | | | | | | | |−| |
wv = v w + w . That is w first acts by means of annihilations of the
′′ ′
| | | |−| | | |
letters of w and then w acts as a creation operator on w v. We will use such
′′ ′ ′′
arguments without further reference.
The following Lemma 2.7 together with Lemma 2.4 say that T decomposes
w
in terms of a sum of operators that first act by annihilation (this is T(1)) then a
u′′
diagonalaction(this is the projectionP ) and finally by creation(this is T(1)).
uVΓ0 u′
This decomposition is crucial for each of our main results.
Definition 2.4. Let w W. Let A be the set of triples (w ,Γ ,w ) with
w ′ 0 ′′
∈
w,w W and Γ Cliq(Γ) such that: (1) w = w VΓ w , (2) w = w +
′ ′′ 0 ′ 0 ′′ ′
∈ ∈ | | | |
VΓ + w , (3) if s S commutes with VΓ then ws > w (that is, letters
0 ′′ 0 ′ ′
| | | | ∈ | | | |
commuting with VΓ cannot occur at the end of w but if they are there they
0 ′
should occur at the start of w instead).
′′
Lemma 2.5. For (w ,Γ ,w ) A there exist u,u,u W such that
′ 0 ′′ w ′ ′′
∈ ∈
(1) (1) (1) (1)
(2.4) T P T =T P T ,
w′ VΓ0 w′′ u′ uVΓ0 u′′
and moreover if s W is such that us < u then su > u . We may assume
′ ′ ′′ ′′
∈ | | | | | | | |
that u =w u 1 and u =uw .
′ ′ − ′′ ′′
Proof. Let u W be the (unique) element of maximal length such that w u 1 =
′ −
∈ | |
w u and uw = w u. Set u =w u 1 and u =uw . It then remains
′ ′′ ′′ ′ ′ − ′′ ′′
| |−| | | | | |−| |
to prove (2.4) as the rest of the properties are obvious or follow by maximality of
u. For v W such that VΓ w v = w v VΓ we have,
0 ′′ ′′ 0
∈ | | | |−| |
Tu(1′)PuVΓ0Tu(1′′)(TvΩ)=Tu(1′)PuVΓ0(Tuw′′vΩ)=Tu(1′)(Tuw′′vΩ)
=Tw′w′′vΩ=Tw(1′)PVΓ0(Tw′′vΩ)=Tw(1′)PVΓ0Tw(1′)′(TvΩ).
For v W such that VΓ w v = w v VΓ it follows from a similar compu-
0 ′′ ′′ 0
∈ | | 6 | |−| |
tation that both sides of (2.4) have (T Ω) in its kernel. Indeed,
v
(1) (1) (1) (1)
Tw′ PVΓ0Tw′′(TvΩ)=Tw′ PVΓ0(Tw′′vΩ)=Tw′ 0=0,
and
Tu(1′)PuVΓ0Tu(1′′)(TvΩ)=Tu(1′)PuVΓ0(Tu′′vΩ)=Tu(1′)PuVΓ0(Tuw′′vΩ)=Tu(1′)0=0.
(cid:3)
Remark2.6. InLemma2.5thepropertythat us < u impliesthat su > u
′ ′ ′′ ′′
| | | | | | | |
is equivalent to uu = u + u . The words u and u in Lemma 2.5 are not
′ ′′ ′ ′′ ′ ′′
| | | | | |
unique: in case su = u 1 and s commutes with VΓ then we may replace
′′ ′′ 0
| | | |−
(u,u ) by (us,su ).
′ ′′ ′ ′′
Lemma 2.7. We have,
(2.5) Tw = p#VΓ0Tw(1′)PVΓ0Tw(1′)′,
(w′,Γ0X,w′′)∈Aw
ABSENCE OF CARTAN SUBALGEBRAS FOR HECKE VON NEUMANN ALGEBRAS 7
where A is given in Definition 2.4.
w
Proof. The proof proceeds by induction on the length of w. If w = 1 then
| |
T = T(1) +pP by (2.1). Now suppose that (2.5) holds for all w W with
w w w
∈
w =n. Let v W be such that v =n+1. Decompose v=sw, w =n,s S.
| | ∈ | | | | ∈
Then,
T =T T
v s w
= T(1)+pP p#VΓ0T(1)P T(1)
(2.6) s s  w′ VΓ0 w′′
(cid:16) (cid:17) (w′,Γ0X,w′′)∈Aw
 
=T(1)+ p#VΓ0T(1)P T(1) +p#VΓ0+1P T(1)P T(1) .
sw sw′ VΓ0 w′′ s w′ VΓ0 w′′
(w′,Γ0X,w′′)∈A(cid:16)w (cid:17)
Now we need to make the following observations.
(1) If sw =ws then P T(1) =T(1)P . So in that case,
′ ′ s w′ w′ s
P T(1)P T(1) =T(1)P P T(1).
s w′ VΓ0 w′′ w′ s VΓ0 w′′
Moreover P P equals P in case s commutes with all elements of
s VΓ0 sVΓ0
VΓ and it equals 0 otherwise.
0
(2) In case sw =w s we claim that P T(1)P T(1) =0. To see this, rewrite
′ 6 ′ s w′ VΓ0 w′′
P T(1)P T(1) = P T(1)P T(1) with u,u,u as in Lemma 2.5. As
s w′ VΓ0 w′′ s u′ uVΓ0 u′′ ′ ′′
sw = w s we have su = us and/or su = us (because w = uu with
′ ′ ′ ′ ′ ′
6 6 6
w = u + u, c.f. Lemma 2.5).
′ ′
| | | | | |
(a) Assume su′ 6=u′s. For v∈W with Tu′′vΩ in the range of PuVΓ0,
(2.7) PsTu(1′)PuVΓ0Tu(1′′)(TvΩ)=PsTu′u′′vΩ.
Furthermore, the assertions of Lemma 2.5 imply uuVΓ = u +
′ 0 ′
| | | |
|uVΓ0| and therefore (recalling that Tu′′vΩ is in the range of PuVΓ0)
we get that uu v = u v + u which implies (because su = us
′ ′′ ′′ ′ ′ ′
| | | | | | 6
anduu vstartswithalllettersofu)that(2.7)is0. Forv W with
′ ′′ ′
∈
Tu′′vΩ not in the range of PuVΓ0 we have Tu(1′)PuVΓ0Tu(1′′)(TvΩ) = 0.
(1) (1)
In all we conclude P T P T =0.
s u′ uVΓ0 u′′
(b) Assume su =us but su=us. Then P T(1)P =T(1)P P =0.
′ ′ 6 s u′ u u′ s u
So in all (2.6) gives,
Tv =Ts(w1)+ p#VΓ0Ts(w1)′PVΓ0Tw(1′)′
(w′,Γ0X,w′′)∈Aw
+ p#VΓ0+1T(1)P T(1),
w′ sVΓ0 w′′
(w′,Γ0,w′′)∈Aw,sXw′=w′s,sVΓ0=VΓ0s
and in turn an identification of all summands shows that the latter expression
equals,
p#VΓ0T(1)P T(1).
v′ VΓ0 v′′
(v′,Γ0X,v′′)∈Asw
This concludes the proof. (cid:3)
8 MARTIJNCASPERS
2.3. Group von Neumann algebras. Let G be a locally compact group with
left regular representation s λ and group von Neumann algebra (G) = λ
s s
7→ L { |
s G . We let A(G) be the Fourier algebra consisting of functions ϕ(s) =
′′
∈ }
λ ξ,η ,ξ,η L2(G). There is a pairing between A(G) and (G) which is given
s
h i ∈ L
by ϕ,λ(f) = f(s)ϕ(s)ds which turns A(G) into an operatorspace that is com-
h i G
pletely isometrically identified with (G) . We let M A(G) be the space of com-
pletelyboundedRFouriermultipliersoLfA(G∗). Form MCB A(G)weletT : (G)
m
(G) be the normal completely bounded map dete∈rminCeBd by λ(f) λ(mfL). T→he
L 7→
following theorem is due to Bozejko and Fendler [BoFe84] (see also [JNR09, Theo-
rem 4.5]).
Theorem 2.8. Let m M A(G). There exists a unique normal completely
bounded map M : (L2∈(G))CB (L2(G)) that is an L (G)-bimodule homomor-
m ∞
B → B
phism and such that M restricts to T : λ(f) λ(mf) on (G). Moreover,
m m
7→ L
M = T = m .
k mkCB k mkCB k kMCBA(G)
3. Universal property and conditional expectations
In this section we establish some standard universal properties for subalgebras
of .
q
M
Theorem 3.1. Let q > 0 put p = (q 1)/√q and let (W,S) be a right-angled
−
Coxeter system with associated Hecke von Neumann algebra ( ,τ). Suppose that
q
M
( ,τ ) is a von Neumann algebra with GNS faithful state τ that is generated by
N N N
self-adjoint operators R ,s S that satisfy the relations R R = R R whenever
s s t t s
∈
m(s,t) = 2, R2 = 1 + pR ,s S and further τ (R ...R ) = 0 for every
s s ∈ N w1 wn
non-empty reduced word w = w ...w W. Then there exists a unique normal
1 n
∈
-homomorphism π : such that π(T )=R . Moreover τ π =τ.
q s s
∗ M →N N ◦
Proof. The proof is routine, c.f. [CaFi15, Proposition 2.12]. We sketch it here.
Let (L2( ),π ,η) be a GNS construction for ( ,τ ). As τ is GNS faithful
we may aNssumNe that is represented on L2( )Nvia πN . We dNefine a linear map
V :L2( ) L2( )Nby VΩ=η and N N
q
M → N
V(T Ω)=R η, where w W,
w w
∈
and R := R ...R . One checks that V is isometric by showing that R η
w w1 wn { w |
w W is anorthonormalsystem.1 Puttingπ( )=V( )V concludesthelemma.
∗
As∈VΩ}=η we get τ π =τ. · · (cid:3)
N ◦
1Theproofgoesasfollows. Wemayfinduniquecoefficients cv suchthat
Twn′ ...Tw1′Tw1...Twn = X cvTv.
v∈W
We have c∅ = 1 if w = w′ and c∅ = 0 if w = w′ by comparing the trace of both sides of
6
this expression. In fact the coefficients cv may be found by using the commutation relations
for Ts and the Hecke relation Ts2 = 1+pTs to ‘reduce’ the left hand side of this expression.
As the same relations hold for the operators Rs (by assumption of the lemma) we also get
Rwn′ ...Rw1′Rw1...Rwn =Pv∈WcvRv. So,
hRwη,Rw′ηi=τN(R∗w′Rw)=τN(Rwn′ ...Rw1′Rw1...Rwn)=τNX cvRv=c∅.
v∈W 
ThisprovesthatindeedV isisometric.
ABSENCE OF CARTAN SUBALGEBRAS FOR HECKE VON NEUMANN ALGEBRAS 9
Remark3.2. NotethatthepropertyTs2 =1+pTs,s∈Swithp= q√−q1 isequivalent
totheusualHeckerelation(√qTs q)(√qTs+1)=0thatappearsintheliterature.
−
Weshallsaythat(W,S)isaCoxetersubsystemof(W,S)ifS S andm(s,t)=
⊆
m(s,t) for all s,t S. Here m is the function on S S that determines the
∈ ×
commutation relationfs foer W, c.f. Section 2.1. e e
e e e e
Corollary 3.3. Let q > 0. Let (W,S) be a Coxeter subsystem of a right-angled
f
Coxeter system (W,S). Let and be their respective Hecke von Neumann
q q
M M
algebras. There exists a trace presefrvineg normal conditional expectation :
q
E M →
. f
q
M
Proof. Theorem3.1 implies that is a vonNeumann subalgebraof and the
f Mq Mq
canonical trace of agrees with the one on . Therefore admits a trace
q q q
preservingnormalMconditionalexpfectation valueM, c.f. [Tak03, ThMeoremIX.4.2]. (cid:3)
f f
Consider the Hecke vonNeumannalgebra for the case that S is a one-point
q
M
set, q > 0 and p = q√−q1. In that case we have W = {e,s} and L2(Mq) has a
0 1
canonicalbasis Ω andT Ω. With respectto this basisT takesthe form
s s 1 p
(cid:18) (cid:19)
andone sees(using for examplethe relationT2 =1+pT ) that =CId CT ,
s s Mq 2⊕ s
i.e. it is two dimensional. The following corollary uses the graph product, for
which we refer to [CaFi15]. It is a generalization of the free product by adding
a commutation relation to vertex algebras that share an edge; the free product is
then given by a graph product over a graph with no edges.
Corollary 3.4. Let (W,S) be an arbitrary right-angled Coxeter system and let
q > 0. Let Γ be the graph associated to (W,S) as before. For s S let (s) be
q
∈ M
the2-dimensional Heckevon Neumann subalgebra corresonding tothe one-point set
s . Then we have a graph product decomposition = (s).
q s VΓ q
{ } M ∗ ∈ M
Proof. Let T ,s S be the operators as introduced in Section 2.2. Let
s q
∈ M ∈
T ,s S be the operator T but then considered in the algebra (s) which
s s q
∈ M
in turn is contained in (s) with conditional expectation. Now the map
s VΓ q
∗ ∈ M
Te T determines an isomorphismby Theorem3.1 and the universalproperty of
s s
the7→graph product given by [CaFi15, Proposition 2.12]. (cid:3)
e
Remark 3.5. Note that Corollary 3.4 allows us sometimes to use general results
ofgraphproduct vonNeumann algebrasto study , see [CaFi15]. Note however
q
M
thatpropertiesasstrongsolidity,injectivity,etceterahavenotbeenstudiedbeyond
the case of free products. Moreoverfor many stability results for free products one
needs to impose assumptions on the product algebras which are not satisfied by
(s) (such as being diffuse or being a II -factor, see [CaFi15, Corollary 2.29]
q 1
M
for example). This makes the balance between general graph product results and
applications to rather delicate.
q
M
4. Non-injectivity of
q
M
Recall that a von Neumann algebra ( ) is called injective if there exists
M⊆B H
a (not necessarily normal) conditional expectation : ( ) . This means
E B H → M
that is a completely positive linear map which satisfies x : (x)=x.
E ∀ ∈M E
10 MARTIJNCASPERS
We prove that is non-injective. The proof is based on a Khintchine type
q
M
inequality. For the sake of presentation we shall first prove non-injectivity in the
free case,meaning that m(s,t)= whenevers=t. The proofis conceptually the
∞ 6
same as the general case but the notation simplifies quite a lot, making the proof
much more accessible.
4.1. Non-injectivity of : the free case. In this subsection assume that
q
M
m(s,t) = for all s,t S,s = t. In particular this means that in Lemma 2.7
∞ ∈ 6
every clique appearing in the sum has only 1 vertex or is empty. We proceed now
asin [RiXu06, Section2]. Define the following twolinear subspacesof (L2( )):
q
B M
(4.1) L :=span P T(1)P s S , K :=span P T(1)P s S .
1 s s s⊥ | ∈ 1 s⊥ s s | ∈
n o n o
Note that as s2 = e in fact P T(1)P = T(1)P and P T(1)P = T(1)P . L is a
s s s⊥ s s⊥ s⊥ s s s s 1
spaceofcreationoperatorsandK is a spaceof annihilationoperators. L andK
1 1 1
are not contained in but live in the ambient space (L2( )). The following
q q
M B M
Lemma 4.1 is a special case of [RiXu06, Lemma 2.3 and Corollary 2.4].
Lemma 4.1. We have complete isometric identifications,
L C#S : T(1)P e ,
1 ≃ column s s⊥ 7→ s
K C#S : T(1)P e ,
1 ≃(cid:0) (cid:1)row s s 7→ s
(cid:0) (cid:1)
where the lower scripts indicate the operator space structure of a column and row
Hilbert space with orthonormal basis e ,s S.
s
∈
We define Σ =span T s S and subsequently:
1 s
{ | ∈ }
Σ =span T ... T w W ,
d { w1 ⊗ ⊗ wd | ∈ }
whichiscontainedinthed-foldalgebraictensorcopyofΣ . Thereexistsacanonical
1
map σ :Σ (L2( )):T ... T T . For s VΓ we let A :=CP ,
d d →B Mq w1⊗ ⊗ wd 7→ w ∈ s s
i.e. a 1-dimensionaloperator space. Using for the Haageruptensor product we
h
⊗
set Lk =(L1)⊗hk,Kk =(K1)⊗hk and
d d 1
−
X = L K L A K .
d k h d k k h s h d k 1
⊗ − ! ⊗ ⊗ − − !
k=0 s Sk=0
M M M∈ M
Here the sums are understood as ℓ -direct sums of operator spaces. The multipli-
∞
cation maps L K (L2( )) and L A K (L2( ))
k h d k q k h s h d k 1 q
⊗ − → B M ⊗ ⊗ − − → B M
are completely contractive by the very definition of the Haagerup tensor product.
Extending linearly to X gives a map Π :X (L2( )) with
d d d q
→B M
(4.2) Π (d+1)+d#S.
d
k kCB ≤
In fact the tensor amplification
(4.3) (ι Π ): X (L2( ))
d q min d q min q
⊗ M ⊗ →M ⊗ B M