Lecture in: semi - finite von Nema""

PTSot off

"

easy exercise

Algebras

M"c MIT ; ( proof sketch

) :

A von Neumann algebra is a If TEM"

fix 5 EFC

C'

- subalgebra M c BCH) SI.Pros' MJ =P

E MI

IEM,

and [ T, PJ-_o⇒ Is -_

'i'I II x.em

• Mi is closed in the wot Matrix trick : If 5.,.. ,5n Etc

I . in." ' """

I. → Tiff HH.-T)5H→o #

C-HE?[ • M = M

" This sirs ×,c- M St

l 3)"⇐i⇒o * so

.

•( s'

{ sepal sT=Tf¥g

E* : M -

- PCH).

Ex : T discrete gp

Ex : ( X,- ) a r - finite

measure space (x , m ) or - finite

PEX,- ) c>BCLTX,m) ) park ,m ) measure - preserving .

Elan) aft> Mf A :=LKx,m)

Mflgtfg for gettin) . AT = { ¥ , atl finna -

- at EA }

Lacan)'

⇐ Look, n ) ( fepaeusftfpb.us )-

s.Plan) -- THT

)"

= E at ECbs ) Uts# STET

Ex : T a discrete group wwe of ←Autl CAM) ) ' 3

X :P → Uwi )Xs - St = Sst

.

gun by self ) = f - E'

.

arable't) (feratuc-Y-Eeraeiat-cqp.IT" "' 47=67'T

= E of . Utaten

If TN (x,r ) up. and essentiallyAPCBLLTX,m ) ⑤ lit ) -

-

free,ie

.

P× -- Ees one.

+EX.

⇐paeuoeftspbsus ) Tun q,→×p is a factor

-

=¥e, attends) Uts Iff P r(x. n) B ergodic,

ie if E- CXmeasurable

St

A- xp := AT"TcB(vk.NET/D.sE=Ev-seTtu-mlE1--°

or MINE) -0.

-

↳ group - measure space construction .

Note : This can be done if TAG,-)

preserves null sets .

Rewarms : (Murray - vonNeumann ) :

Z ( LT ) = Cl iff M has infinite

non- trivial conjugacyj classes (Icc)

ie LT is a

factor

Traces onVN alss nut .

= { + EM I Trcx-

x ) -63

-

M a ✓N alg a n faithful Mtr-

- { §,

Yi l Xi, y .

- Enter }.

M is a

semi finite trace on

Lemma : ht,and -tr

areideals

-

inn MM.function

Tr :M+→[o ,SI -

Ex : M-

- LUX,n) Tr ( f) = ffdn .

① trlsxtpy )-

-Ltrcxstptrl 's ) ntr-LHXmntcx.ir )

x. PZO x. yc-Mt

.

in,r=ECx,n)nIlXt② If genis increasing

ad ←WS"

E# i. M -- pot ) { 5,3, anONB

GOT to XEM run

Trix ;) →Trlt ) . Tr ( T) -- { LT 7,5, > .

NtrThe space

of Hilbert- Schmidt③ Tr ( x ) -_ o ⇒ x=0 .

④ If × Emt XEO ,F yEt St operators

Mtr = Traceclass operators .

Octrly ) 26 .

⑤ Trix-

x) =TrLxx. )

.

-

Lemma : Tr has a unique hear µ ←, B( LTM,Tr))

extension Fr :m+r→ Cl. gun by xt = #

Remark : Tr gives an inner - product ( the standard representation )

on ntrau- by M°Pc→BCVCM , tr ) )

( x ,g7r=Trl5x ) . xoP.ci#aI.

[(M ,tr)=nTr' " " Thun : M°P=M

'n BCLTM.tn)

.

we write I for a EngrCLYM, Tr )#

Ift tr ( DLP Tun Tr is a (Recall I¥n) '=Ln))

finite trace . Mtr =ntr=M .

Ex : T discrete gp General problem : How much information

c.Bleat )

Tn.LT- G Sm-

b's from T for Trans ) is

Text : Lx - Se,Se > encoded

-

inLr (or

LAWN?

-

Thos 't ) -- Caste - Se,Se >⇐

=p,iii. SEE : )

= Theirs ).

-

PrGym) probability up.

CBLVK.nu#eyn)

T:L-

can)xT→ Q siu - by

THE ↳ ( I ⑤ Se),( I ④ 8D

.¥m!evmNewannalgebrascser(F) there exists a ref finite truce

.

Def !% B aoemunable if there existsmore : TIO ⇒ LECT) Zo ⇒ FCT) Zo-

a left - invariant state on IT.

#If XELT

Then : suppose M and r are ice.

and Es=Lx Ss , Ss>

LPELA ,

tun T is amenable = 4- Xs Se,

Is Se >

=T(X5x2s ) -- Tex )iff A B amenable

.

-

i- gifts Cx )

proof : -

suppose T B annabaand fit th Nou : ELLIE) ( s )

state g : lap→ e that' 3 left - 'w

. =L # Tzu Ss ,Ss >

= ECT) ( t- 's )

-

- ↳ (ECT)) Cst.Defee J : BWP)- E by -

FIT )= g. ECT) in:- THETA -1) ⇐ JCT) .

I. g- ( yet ) = TheTHE ' )

E. B¥ ) → IT ' 3 defied-- g- ( Tae ) )

by ECT) Ks ) -_ LTS.gs , Ssg > EE - FLEE) =5CT¥ TEDIUM) TEIL- IT# THE FITT) 't

-

'

- 5Gt) -- g-Ctx ) TEKIN

#ELITE

EILEENdefine GT : for→ Q by

Elf ) - THA.

her

FTL .lt )) -- Thieme= Fcf )

.

B

-

i

if a discrete group Len : M has a hypertraceon

T-> Ltc Blt'T) plums ) iff M is insect.ve .

"

MT)" proof :

T is amenable iff ⇐) E :BCLYMD→M

there is a state g on Blur)

ht 951=7 (ECT) )

then glxt)-

- Ttx ECT) )St gut )

-

- gltx ) TENTH =T(ECT) x) -- 95×1.

a- LTand 9th -5T .

⇒x : Burn)) → BCH) SEH

- -

Def : M is insecure if there 951=49515,57Max 1-7 Thx ) 5 gives an isometry

is a conditional expectation Lynx) -→7C- q

from BLUM )) to M. of :Buyms5→ BLUM) )

get) -_ LACTIS , 57 : . GCT) ELMOP) '=M .

D

Max 1-7116-75 gives an isometry-

LUM, ⇒ FCcomes -46 : there is a unique

£Separable insect.ie II , factor -

A. Blum))→ BLUM) )-

by GCT)=eYlT)eNote 41M id

Def : A finite VN alg

-

( M,

T) has propertyCT) if

suppose x. y ,zEM TE BLUM)

Here exists a sequence

# IT ) EPI , I > un EUCM) St un -70WOT

⇐ Kenny-

Txt ) e5 , 57 and Hun x - Xanthe-20

-.

= gcyatxzu) -- gttytx ) Ex : R==⑧M=9IM=LzoP4CT ) 2,57

.

set Un -_¥n⑤-④LIg④ la-

Ex : M ⑤ Re has property ). gcvvop Tv

-

v- op ) = JLT)

⇒converse does not hfvlda.es) -

Thu :( Effros ) If LThas for vector)

TEBET ) .

Property ( T ) tunthere is taking V -- Xt V°P=PE

.

acon 's

.

in .Station 14? If T=f Ever

¥ - 9-0. inner - arenas

.¥gLyn

,

zefeffeittifoadlt?Pf : Given Un as in

the def.

define a state of on' Fr is not inner - amenable .

BUT ) by taking any -

wut - cluster point of

÷⇒. ÷÷÷÷.

e-re> Bleu)-

restrict to LIE .-

Def : suppose p NK→ ""

- Hausdorff. and He has a P- inv.

This is an annable action measure,fun T i3 amenable

.

if the exists a sequence

ninepence'

"""F¥÷¥i e.se#eSt Mn is cont .

and

su-Lw)

sidelined- te"" "

s .

"

→ o

-

t teh. This)•

e

E* : K -- E - 3 in is amable

defoe maiden→Problem )

iff f is a-name. by by mnlw) In ,%

,8w

In fact if TAK llnncsw ) - Samnium stalls,e).

IZ O.

Thu : Truk is amenable iff CCEI )-

- Alcott ).

CCK)xM is nuclear iff" Def : A group T is bi - exact

@GesxtJisinieetne.if prop isamenable .

# #

Fix T a group . Ex :p = Fa TANE

considerA={ fel.net/ ¥7 fact : tf wed #r

f - Reheated. At¥¥T say ⇐→w

ca- a

TNA is act- subalgebraof top - Fa

bus!?"

c. ¢ ,# y p↳Fthen stn£→sw

I d-%:!I i . If fc-CUEudlffct.FRSet 019=7 xp

i - f - Relf ) e-COLT)ie CLAUSE) CA

( ( stfu )↳ CLOE) tf LA-LAe =

a:oI Fake prob"?-

CERT is exact sniff

Ex :@word hyperbolic groups . ⇒p is awake.

25×5154 iff penis so-

amok

-

action on a Cpt Hausdorff• z g p T bi - exact-

space .

- -Thnlozova)

Thin ( Ozawa ) If T B bi - exact T is bi - exact iff-

and B CLT diffuse, pxp~thtyc.cm is awake .-§n

Bl n LT i3 insect"

¥pgµ )-

( i's . LT is solid).

E- bi -¥.

Also , LT has property D) it? then.net

T discrete group proof ( sketch ) : ( Boutonnat - carders')

A- { fee-M f - Rueff Htt} Agreement BUT )

CLT u OT ) = A crepeSince B B diffuse we

can ane

Une UCB ) un→ owot

.

T is bi - exact if the actin

consoler a state on BCEp%yLpDTN ( IF U OT ) is amenable

-

+* is awha - accumulation Pt

⇐ ) last )*.

is inseam -

of T i-LTTun.ua >Then ( Ozawa

'049 ) T bi -eat and

- -

Belt diffuse Luo min proseNote : 91L ,

-

- T and 91µF 'T

tu B'n LT is insect.ve

.Also 91µW,,=O .

e.g . Life is solid ( and henceprime)

Proof ( sketch ) :( Boutonnat - carder ;)If fECT) and EET

Agreement Bce't )

th ' Skf -Eff;)=0

Since B B diffuse wecan he Also y( f Pe - pet )

'

-O

une ULB ) Un-70IT

.

In fact g. ( xfe - Pex )-

-O

consider a state on BCEptyyu.gg it ×eccIoINqTcBCtN.

+* is awire - accumulation Pt : . g(q×,y3)=o + c- CCTUOTYXP

of T '→ LTTui.ua > yt.gr)- - Nonlgcxy ) - ycxy.ME/lxtllly-yHz

Non : 91, ,

-

- T and 91µF 'T g. yen?- tag

: 'g. ( Cx ,y3 ) -0

Also 91µW,O

.

y + ← CCTV XP-

ye RT

If FECCTUOT ) and EET Tain * B' ALT,

take anc-Q.LT)e-

-

XECCTUOT ) XP Han - allen - g¢f -Eff:)=0 - →o

Also ylfpe - pet )=0 9. ( a. x )

- fact 91×4 - Pex )-

-O

⇐ 'n'Ine flank )

V-xeccinoonx.TCBCTN-I.no' Kanin# ¥7)

-

'

- 9( Ex,y3)=o XECCTUOTYXP 'T.net/uie,u.Ta57yt.dpkT) =

• glaiorx )

Nonlgcxy ) - ycxy.ME/lxilHy-yHz=

• g. (xai.se/--9lx-a79k.p-TyTT9. YEN*crooner → BHC ) Soth: -

gu,y3 ) -0 -gcx ) =L + So, So >

If + c- ( (TV XM tan p #part proteanof

ye RM 7 in H.

PETCH"

Tan €13141 ,

take anEQ.CM y, :@rumor)xT)←→BCH)

x c- Cctv or ) XP Han - alla ⑦#

-20

9. ( ax ) ypcphpgg.us a state on PMP.

-

⇐ 'n'Ia gcanx )

o:÷÷÷÷÷÷:÷÷: e:""""

= '

a g. ( an:P X )i- If a and expectation

• g. ( xa: -7=91×1 E. pimp→ B'ALT- w

:C → BHC ) Soen is injective

91×1=2×30,30 > ⇒ BI ALT B insect. Ow

tamp⇒port Protectionof

Note : we can prove LTdoes not

7 in a.

PETCH" ha- property ifeng.7.ec?Ie.-T

In fact what the proof ie.T UOP has a

shows is that if LTFT

tf - invariant probability measure.

has property T then there⇒ bi - exact + property

CT)

exits a state g on BWP)⇒ group

is amenable.

St flax )-

- g. (xa ) Def :( Boutonnat- Ioana -

P )

for all a ELT and A group T is properly Proxhal

XE (( poop) xpif there is no T - invariant

-

Prob . measure on T VOM

⇒ g. ( Tefft - f) =O tegument's on OT).

Prop : T properly prox.ua/

f- C- ( ( M UDT ) EET ⇒ Lp does rothwell ) .-

If.T UOP has a prop :P non -

amenable and bi - exact#

tf - invariant probability measure. ⇒ P B properly proximal .

⇒ bi - exact t property(P)

⇒ groupis amenable

.

Lemma : T is properly proximal

iff there exists anaction

TAK k cpt HausdorffDef :( Boutonnat- Ioana -

P )

A group T is properly Proxhal

if there is no T - invariantw/o - invariant

measures,

Prob - measure on poopbut there exists Y C- Prob (K)

( equivalently on Op).

SI.

tf t ET

Prop : T properly prox.nl writing (S2 - Str ) #

O.

⇒ LT does not here LT ). Pf : x, :P ✓op→ Prob (K )

t 1- try

prop : T non -amenable and bi - exact Exif Def : A group T is a convergence

⇒ P B properly prox-al

-

group if pure is anaction

Lemma : T is properly Proximal T Ark w/o invariant measures

iff there exists anaction and SI . we

have

Tak k cpt Hausdorff"north - South dynamics

"

,

w/o - invariantmeasures

,

Ie. If tu ET En-76

but there exists MEprob (K)

thenthere is

a subsequence

Hurd a ,btEk SI . for

an nbhds A of a and B of bSI

.

F t ET

una-sing,- Str ) # O .

we have

Pf : x :P von→ Prob Ck )tu

.#A) CB

t -7 ty for VE large .

IT Ark w/o invariant measures

and set. wehave

"north - South dynamics"

. Ex : SWR NIT-

Ie. If tu ET tu→A

Exit hyperbolic7725

Thenthere is

a subsequence { T relatively hyperbolic . 1171¥?Ez

(turd a,Dr EK SI . for

' P =P,

* The actingonme

an nbhds A of a and B of bboundary of the Bass -Serre tone

prop : convergence groupsare

we have properly proximal .tu#A) CB proof :

fam n to be any

for VE large . measure having no atoms

Thu .

.If T is a lattice in • SLZLFPTLE

'] ) Xfpft,t -133a non - compact semi-simple is not properly prox .- l,

lie group , then T isand is not inner - amenable.

Properly proximal . -

*± . r-

- sea need• "9%27955%7.39 ¥97,• Man amalgamated f be-ueure.mx

)

Also proper prox.ua/ityigtree products(Ding, Hunn a

-alkan Elagavalli) .closed under direct products. y

Open : • A- cylindrically hyperbolic gps ?-

• BY'M > o ? pi:'(Mao ?If T B inner- amenable

i

. Outten ) ?Tun M B not .

Properly proximal . . Icc property CT) groups ?

T a group then T is properly prox .

E'i:c::hen¥±:*"'s ""

• T is bi -exactif

TN CCTVOT) is amenable

it is properly prox .-al if If Igoe Problk

) w/o ato- s.

tunuhh Eff - Sf) =0 .

paceT ) has no

c-→x -

invariant State .

-

Ex : SWR A # ⇐ surly; :)}Ex : If Tak is-

TNK ki,Ku then F KET

a convergenceaction - -

St 81k , , ka) → DEW.

w/o invariant measures --

Then T is properly prox . If pcS↳1Ra lattice

ns↳lR~lPlR'=s↳1R/µgEaE) )gas¥:

"-

sein

X ,7¥10 .

If ye Problk )to ato- S

. ges↳R g=xa= BI

muntin tu - sa) =o . If gn=r±t¥Ii , HI Kenc-→x

-

E→E

Ex : shirt # ⇐she"%g⇒g g. EI if Tiki

→x

-

-- A-F

- ri 'gnE"

→ [ too )]#

if v¢sP{ 191,1933 .

I

has me as are zero from Haar measure.

If Tc SLZIRa lattice

the upshot is that if

msn.MN#PlhI=st3R44EEoE) ) µgncµs↳lR and if

X B Haar measureanIRP'

S↳lR=KAtK --=Sf!Ie gym

. if ith ratioof the ]§

X , Z- first two eigenvalues

in the

g. Estok g=xa= EI KA+K decompositiontend

- to XThenIIf gn=r±l%'Iii !;) HI Kush

-

E-"Eyea.hn KIA - ta)=Ogn¢v] if Tiki

→Anose

Teri'gnE"

→ [ too )]#

if v¢sP{ 191,1933 .

es.

has me as are zero from Haar measure.

The upshot is that if Def : A boundary piece for a

µjncµs↳R and ifgroup p is a

closed

X B Haar measureanIRP'

qq.ie#nrax-oaotmJgs.;Y:*FrigtFIwiII.first two eigenvaluesin the

- Ex : If welet X be

KATK decompositiontend

to XThen

the set of allaccumulation

PEs of fisc SLEEthen

vain KTX - TX )= O

-

this goes a boundary piece .

ness p

ve say g.→AIX if

ifanypt is in XCPP

Def : A boundary Piece for a feII× if im f- Lt ) =0

group p is aclosed ⇐%

"

subset xcptipthat Ix CAT I+=EfeccpmlflE§.-

is left-rightinvariant. Ay { fever I f

-Relf) e- Ix }

Ex : If welet X be

Def ; T B properly proximal relate

the set of allaccumulation

PES of fig cS↳¥then

to X if there is no

this g.us a boundary piece .left - 'warrant State on

-Ax

.

ve say gn→A/X ifI

ifanypt is in XCPP es- i X= BT 'T then this

is proper prox.ua/ity .

f- c- FIX if im fit ) -- O pcs↳lR Vet ktgnon,)EEq

"

-

T-

considering

Exeter I+=EfeccPMlflE§ prbrlz.IR' )

Af { fever I f-Relf) e- Ix } we see tant T

is

Properly proximal relativeto

"÷:÷÷÷:÷÷÷:÷÷::::÷:*::"AX.

relate to boundary

= pieces x and Ye.si/--BThT then this

is proper proximity .

Where XUY -_ BRP.

Thun Loza-a) If P is

pcs↳lR Vet ktgno:*)Econsidering

-

T-

Properly proanal relate to

TN 642,1123 ) x and Y separatelythen

--

we see that Tis

p is properly proximal

properly proximal Mkt't to

relate to XUVY.

Ya -- { Ign ) 1st. 1%;-203

.

I

Hence If Tcs↳lRa'

. T is properly prox . --

relate to boundarylattice turn P is properly

pieces x and YProximal ..

Also; Proper proximally is

where XUY = BRP.

closed under direct products .

ujn.net#T---B

Then Loza-a) If P is An example of a non- properly

Properly proamal relate to prox.ua/ and noninner - auuaskgp.

X and Y separatelythenI

-- [Ex :( Caprace)

there exists a' cgmp

having twolattices T

and 1

T is properly proximalm,a,,,,yµ g , q , *me. -away

,andpg]

is not.

Hence If Tcs↳lRa-

= Tm( I - P - R ) Proper proximitylattice turn T 13 BOND [€s+~-tense

.

Proximal ..

Also; Proper proximally is

closed under direct products .it

'p=Eut#T

An example of a non- properly g

Proximal and noninner - auuasugp. Def : If PALM,Tr )

# wwe M B semi - finite ,th -

[Ex :( Caprace)then exists a

" cGMPa fundamental

domain is

having twolattices T

and A -

St P B inner -amenableand Ig a project.vn PEM St

- 2%61=1is not

-

EET-

Tm(I - P - R ) Proper proximity Def : T and A are

13 stable under Measure egan-tense

von Neumann equivalent ( VNE)+ if thru is a truce preservingaction

Tinn (m ,Tr ) stT

and N separatelyhave finite

trace fundamental domains.

Def : If FILM,Tr ) Re- ÷ (M abelian ⇐ ME).

-we n is semi - fun , :3theIfI⇒Iw③

a fundamental n= Bleat) = BC

a project.vn PEM St¥÷::÷f÷÷.÷÷÷÷÷÷÷:von Neumann equivalent ( VNE) x , pay . ,=Pro 'segg

C-Blt'T )

if there is a truce preservingaction= µ

Tx't re (M ,Tr ) StT Ps P Ps

-

- Prosagg

E B(tr )

and N separatelyhave finite #

trace fundamental domains.

Thn( IPR ) If primer then

Re- ÷ (M abelian ⇐ ME).

P B prop.

prox.

iff N is.

then: If LTE>LA-

T- Eden : translate

Kon? Parr proximity

m=B(VT) I BUN)

to actionsand"B# spaces

- operator

prim orlik "5

pm# tr ) ←Anorm oft)

-

- Ps TPF heat- NAE by computers

,

p -- Prose, get Blt'T ) A

tenth@ ⑤ E )x , pay . .

-

-Pro 's,cg,EBU'T ) #= \

This induced action shows That

Psp Ps'

- Prosagg

E PCM)p is not prop . Prot .

-