Equivariant 02 - absorpHon

Theorem for Exact Groups

Yuhei Suzuki(Hokkaido University)

70 9 1 7 @ Zoom (RMS)

Today5-_lopics@Non-commutatlueamenabkaction_H.wto formdate? ( 170 s ~ 20 2 0)

- Examples? (1 1 7 - )

② Classification- equine .

vers Ion of G - Absorptiontheorem s for Extent groups

.

Beyond amenable groups

Gn興• algebrasidwaysunitalHdmostcaseslmpk.separab.k.nuckar• GroupsT.G.iq/waysCountable,dscrete.ex

① (NC) amenable actions

Amenable actions (Z

[𧘱叫 Concept t

!"amenabk.sides卯でも non.amenable groups .

図 辰 ~ る職𠠇m : 2 辰 → Rob Fe

w い た統 型 Ist kinIFother - t.pe distributi.ie

-

areP12 G > t T ~卵| 袋で階で amenable

-

Important Applications- Zimmer caydesuperrlgldi.ly- Baum- Comes Conjecture- von Neumann algebra- Orbit equivalences

G : exact Group Familiar gapsare exact

• Linear groups斷は鼹齊愛質問一、 -_- -_- -_-

. MCGは)fairly mild ( Exterior) . Out (辰)Version of amereability i

(aka boundary Amenability) :

Non Commutation i (Sometimes)Cleaner

,simplermorefruitfulle.ge Classification)NC amenable Action ?

BAD NEWS : # Interesting examplein WE Setting . . .

Im (Delan che' 79)cM: Waly Then

T ~ MainenableLEt~ZM.ee糶.NAME#htttT1factor)Ynever

Similar de.tn/resuH expectedin Getting (Delan che )

1𨕫 ( '02 Delanode )A : Et T : non - amenable

ANT = A が 「Lが でき、non - trivial ? ?( FAZ(Alanenable?? )

A.

No ! ! ( S ' 1 7 )画 ( S ' 1 7 )T : exact Then

Trial鬜が

Acorrectformulati.es/S'2o):viaAw=AhA'• useful /Yalein Classification Main Theorem)

• Equivalent toSeveral known Amenability- typedefn s

( はo Buss - Edterhoff - will ett )

②

Q-absorptiontheorem_forexactgwup_s.us1 st Classification /Characterizedon( Beyond anenable grep )

• G-absorpti.in theorem.

画 (Hrchberg' 94)

A : Simpleunitalseparablenudearh.to、 a

a = では僕は劇Cuntz algebra

Gritz algebra 62 ;

Most Plain /homogeneousddg.itExact Edges Lia( Hom (0. .

A) well- understood

the Crucial ingradient in

Phillips's Classification theorem

Today 's Goat (Main Thin)

Develop the equivalent

Version for exact groups

Background on NC dynamid Systems

Motivation「 ^_^-_-

• Important in Classification/StructureTheory of operator dgs

• Powerful approach to provide New

Examples & Presentations

(

1assif.cat/nPnblem_-Inwhatsense...?Obviousfdda:a,の) = (B . p)onj ugacylequiu.is。)

Usually TOO STRONG ! !

Ex) が Z ~ AME ひ (A )

mt の" = aduo ん

[ cocydeperturbo.tl on]

Typically の 女 が

almost Need to solve a boundary eq 、

劇raic.it ひ = N No)*in UCA)

Thmlsa_e.fiInfinite groups Then

[無匙:裵で"鬜 Nakamura

'

s them : 味製 で

Only differ Up to cocydeperturbati.in

Correct Problem : Classis

Up to Cycle Conjugate,Indeed natura I :

(T.T)

o have the same

crossed Products. Conjugate on A w = ぺで

When class ifI able ?

CommonI/Expectation[ff 「 is amenable .

Indeed TRUE forvon Neumann algebras

DICHOTOMY : TAR : class if ableby many People ) ⇐) に t.amenable

ぐ - Setting i similar didotomy expectedeg conjecture by I zum I

a Many Successful Classification

Theorems for an enable groups

• (Partial ) non- Classificationfor non amenable groups

Md𦥯This is WRONG

in Good Way !

Main (52 。) に exact

Then

uptococydeconjugayhi.ie?:燕・・

(。吒梺 STAG as above

(Absorption Than)AisImpleunItalsep.nucTheni@tx.Tn.AEdg.to8ES.lthiPnAamenab1aouteractlond@1aS-r.C.C

Motionthy example /mode I of S :

み、 TAXnameinable ActionE Compact

FixCCxnICdxnIxTdGiedknHiTnQdesiredi6mn@ht.i蠟が

Equivariant GaborMonthm

Previously known foreZ Nakamura

. finitegroups Izum !

t.amh groups (Szabo)

How to prove MT ?How Kirchergburyteeo.la)で)

equivariant rePlacement i

(O)f fkedpohtに equivariant)Elements

How to provide

ve @i ? ?

- AVERAGING

anenable groups:

Over Father sets FC 「

Our situation : nonice FC に

But with coefficients

ヨ

hkeaveraghgl.clで姚明

(G)Erich enough

et Cannes 's 2 ×2TicketEntertaining arguments

We also haveamenable

G - absorpti.in than Action•た秘

( Ae。この鱲きfor exact groups

• Equivariant Analogueof Gembeddngthm