Lecture notes in mathematics

In spite of the above simple basis~ the absence of the dual group in the non- commutative case forces us to employ the notationally if not mathematically complicated Hopf-von N e u m a n

Nakagami, Yoshiomi, 1940-

Duality for crossed products of yon Neumann algebras

(Lecture notes in mathematics : 731)

Bibliography: p

includes index

1 Von Neumann algebras Crossed products

2 Duality theory (Mathematics) I Takesaki, Masamichi, 1933- 11 Title III Senes:

Lecture notes in Mathematics (Berhn) 731

OA3.L28 no 731 [QA326] 510'.8s [512'.55] 79-17038

ISBN 0-387-09522-5

Thts work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishe~

@ by Springer-Verlag Berlin Heidelberg 1979

Printed in Germany

Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr

2141/3140-543210

The recent develol~uent in the theory of operator algebras showed the importance

of the study of automorphism groups of yon Neumann algebras and their crossed prod- ucts The m a i n tool here is duality theory for locally compact groups

Let • be a yon Neumann algebra equipped with a continuous action 2 of a locally compact group G For a u n i t a r y representation ~ U ~ u ) of G~ let ~G(U)

be the ~-weakly closed subspace of ~ spanned b y the range of all intertwining oper- ators T from ~ U into ~ It is easily seen that ~ ( U ) ~ ( V ) is contained in

~ ( U ® V) for a n y pair U ~ V of unitary representations of G~ and that ~ ( U ) * =

• ~(~) where ~ means the cOnjugate representation of U This simple fact is the basis for the entire duality mechanism A t this point~ one ~hould recall the form- ulation of the Tannaka-Tatsuuma duality theorem

In spite of the above simple basis~ the absence of the dual group in the non- commutative case forces us to employ the notationally (if not mathematically)

complicated Hopf-von N e u m a n n algebra approach to the duality principle It should however be pointed out that the Hopf - yon Neumann algebra approach simply means a systematic usage of the unitary W G on L 2 ( G × G) given b y (WG~)(s~t) = ~(s~ts) This operator W G is nothing else but the operator version of the group multiplica- tion table In this sense~ W G is a v e r y n a t u r a l object whose importance can not

be overestimated F o r example~ the Tannaka-Tatsuuma duality theorem simply asserts that a non-zero x ~ £(L2(G)) is of the form x = p(t), where p is the right regular representation~ if and only if W~(x ® 1)W G = x @ x

When the crossed product of an operator algebra was introduced by Turumaru~ [76]~ Suzuki~ CE]]~ Nakamura-Takeda, [51,52]~ Doplicher-Kastler-Robinson~ [20] and Zeller-Meier~ [79]~ it was considered as a m e t h o d to construct a n e w algebra from a given ccvariant system~ although Doplicher-Kastler-Robinson's w o r k was directed more toward the construction of covariant representations Thus it was hoped to add more

n e w examples as it was the case for M u r r a y a n d yon Neumann in the group measure space construction In the course of the structure analysis of factors of type III, it was recognized [12] that the study of crossed products is indeed the study of a special class of perturbations of an action ~ on ~ b y means of integrable

1-cocycles M o r e precisely~ the crossed product ~ x G is precisely the fixed point algebra ~ in the von Ne~m~ann algebra ~ = ~ ~ £(L2(G)) under the n e w action

= G ® Ad(k(s)), where k is the left regular representation W i t h this obser-

vation, Connes a n d Takesaki viewed the theory of crossed products as the study of the perturbed action b y the regular representation~ [14]; thus they proposed the comparison t h e o r y of 1-cocycles as a special application of the M u r r a y - yon Neumann dimension t h e o r y for yon Netmm~_u algebras

In this setting, the duality principle for non-commutative groups comes into

p l a y in a natural fashion as pointed out above Suppose at the moment that G is

chOos~ a unitary u such that ~s(U) = (s,p)u, then this u n i t a r y u gives rise to

an action of p on the fixed point algebra T~; or these u's together with ~.~ generate ~ If we drop the commutativity assumption from G, then @ should be replaced b y something else There are a few candidates One is the algebra L~=(G) together w i t h the co-mul~i~.[c~tion ~ G g~ven b y ~G(f)(s,t) = ~ s t ) ; the second

is the Fourier algebra A(G) , the predual of the von Neumann algebra Q(G) gener- ated b y P(G), [ 2 7 , 2 ~ ; the third is the ring of u n i t a r y representations A t any rate, it w i l l be shown that if the action ~ is "good", then the "dual" of G acts

on T~ and ~ is generated b y the "dual" of G and ~ The precise m e a n i n g of an action of the "dual" w i l l be given as a co-action 6 of G as well as a Roberts action of a ring of representations The crossed product of a v o n Neumarm algebra

b y an action of the "dual", a co-action and a Roberts action~is formulated in Chapter IVand duality theorems~ Theorems 1.2.5 and 1.2.7, are proved there The equivalence

of co-actions and Roberts actions is established in ~4 in Chapter IV

In this paper, we present the dualized version of the Arveson-Connes spectral analysis, the integrability of an action, dominant actions and the comparison theory

of 1-cocycles As an application of our theory, the Galois type correspondence be- tween closed n o r m a l subgroups of G and certain yon Neumann algebras containing ~

is established in Chapter VII We must point out that the restriction of the norm- ality for subgroups should be lifted through an application of Fell's theory of Banach *-algebra bundles, [30]~ and its dualized version We shall treat this some- where else

The present notes have grown out of an attempt to give an expository unified account of the present stage of the theory of crossed products for the International Conference on C*-algebras and their Applications to Theoretical Physics, CNRS,

Marseille, June 1977 In theoretical physics, the analysis of the fixed point algebra

is particularly relevant to the theory of gauge groups and/or the reconstruction of the field algebra out of the observable algebra In this respect, the m a t e r i a l pre- sented in Chapter VII as w e l l as those related to Roberts actions are relevant for the reader m o t i v a t e d b y physics It should, however, be m e n t i o n e d that a theory con- cerning C*-algebras is more needed in theoretical physics It is indeed a v e r y active area The authors hope that the present notes w i l l set a platform for the further develol~nent

The present notes are written in expository style, while Chapters III, I V and

V are partially new The references are cited at the end of each section

The authors would like to express their sincere gratitude to Prof D Kastler and his colleagues at CNRS, Marseille, for their w a r m hospitality extended to them

C h a p t e r i A c t i o n , c o - a c t i o n a n n d u a l i t y I

§ I D u a ! i L y f o r c r o s s e d p r o d u c t s (Abel J a n e a s e ) 2

§ 2 'DuaiiLy f o r c r o s s e d pz'oducLs ( G e n e r a l c a s e ) 4

§ 3 R o b e r t s a c t i o n a n d T a n n a k a - T a t s u u m a d u a l i t y 14

§ i S u p p l e m e n t a r y f o r m u l a s 19

C h a p t er IT E l e m e n t a r y !crope:'ties o f c r o s s e d i:roducvs 2 0 § i F i x e d p o i n z s i_n crosse,'i F:'oduets 21

§ ]' C h a r a c % e r - z a - $ i o n o f c r o s s e d p,~oducts 24

§ i{ Com,mutan~,s o f c r o s s e d p_~'oaucT, s 30

C h a p L e r I17 i n t e g r a b i l [ t y an(l d o m i n a n c e 35

§ I O p e r a t o r v a l u e d w e i g h t s B6 § 2 i n t e g r a b i l J t y and o_Derai~or w }.7

§ 3 :n~ef4rab,e a c t ! o n s a n d c o - a c t i o n s 54

§ 4 Domi_ua~ t a c t i o n s and co-:~,ctions 59

Cha.nt~r IV S p e c tr Lain 6?

§ I T h e C o n n e s spectr'±m o F co-act, ions 6~

§ 2 Spec~,rt~v o f acZ'oi:s 69

§ 3 T h e c e n t e r o f a c r o s s e d p r o d u c t a n d F , ~ ) {5

§ L C o - a c n i o n s a n d [{obert ae~,ions ,'8

C h a p t er V P e r t u r b a t i o n o f act,ions a n d c o - a c t i o n s 88

§ 1 C o m p a r i s o n of' l - c o c y c l e s o f a c t i o n a n d c o - a c t i o n -~9

§ 2 D o m i n a n x l - c o c y c l e s 9 2 § j A c t i o n o f G o n ~ h e c o h o m o ! o g y s p a c e 97

C h a p t e r V] R e l a t i v e e o m m u t a n ~ o f c r o s s e d p r o d u c t s 101

§ I R e l a t i v e c o ~ n u t a n t ~ h e o r e m 102

§ 2 Stabil.ity 109

C h a p t e r V l l A p p l i c a u i o n s t o G a ] o i s t h e o r y 111

§ 1 S u u g r o u p s a n d c r o s s e d product, s 112

§ 2 SubaJ.gebras i n c r o s s e d ioroducts 115

§ 3 O a l o J s c o r r e s p o n d e n c e s 119

§ 4 G a l o i s c o r r e s p o n d e n c e s (I]) 125

A p p e n d i x 129

fb(t) = Ăt)f(t -I) ;

f~(t) = f(t -I) , f~(t) = Ăt)f(t -I) •

~(G) = [p(t) : t ~ G}"

'(G) = IX(t) : t ~ G}" = ~(G)'

Ặ) = The modular function of G

ĂG): The Fourier algebra of G, which is identified with R(G) by ~f,g(0(t)) = ( g ~ f)(t), t c G, ịẹ ĂG) = L2(G) ~ * L2(G)

~,5,~ - : Actions of G on a v o n Neumann algebra: ( C ~ & ) ọ,~=(g~C~G) o,~

,%,~, : Co-actions of G on a yon Net~nann algebra: ( 5 ~ g ) o ?-=(&gBC} ) ° P,

~G : The action of G on L'~'(G) with (~G)t = Pt; (~G f)(s't) = f(st)

cz6 : The isomorphismof L ~ ( G ) i n t o L~(G)@L"~(G) with k ~ G ) t = X t : =

%G : The co-action of G on ,%(G) with ,%G(ăt)) = 0(t) @ p(t)

,%~ : The co-action of G on ,~(G)' with respect to 9,(G)' such that 5~(k(t)) = k(t) @ X(t), ef Chapter I, ~ ;~

9, VG,WG,V6_,W ~ : The tmitaries on L [ G × G) = L2(G) @ L2(G) defined by:

e : The action of G on ~P'e with ,~e(xe) = c~(X)e@l for e£.T~

ge: The co-action of G on ~e with ~;e(ye) = ?'~(Y)e@l for e e :.d

A = The modular operator

J~ = The modular tmitary involution

~O = The modular automorphism

× G = The crossed product of ~ by G

×~.G = The crossed product of ~ by G

×6 G = The crossed product of h by G

xs,G = The crossed product of ~ by G

~' = The dual of ~': ~'(y) = Adl~w~(y ~ i) for y ~ • × , G

p = The action of G on ~(,~.)' with respect to Q '(G) : f~(x) = Adl~v~(X ~ i)

~3 = The co-action of G on 5(h)' : a(y) = Adl~4G(y ~ i)

C(G) = The set of continuous functions on G

C (G) = The set of continuous functions in C(G) vanishing at ~

Y(G) = The set of continuous functions in C(G) wanishing outside a compact set supp(c): The support of ~ e A(G), which is the closure of the smallest set outside

~G,~G The wights on @(G), ,~ (G) defined by:

9G(P(f)) = f(e) ; @~(k(f)) = f<e) for A(G)+ ,

K,J:

where A(G)~ is the set of positive definite functions in A(G)

The operators on L2(G) defined by:

(Kf)(t) = A(t)2f(t -I) ; (Jf)(t) = A(t)Nf(t -I)

G: The dual group of an abelian locally compact group G or the set of(unitary) equivalence classes of irreducible (continuous) unitary representations of G

on ~2 spaces

X X , : Normalized characters of G corresponding to p~q~ ~ G p ' q

~ , ~ w ] , : Unitary representations v of G on ~w"

[W,~w} = The unitary representation conjugate to [ ~ }

= The ring of unitary representations of G, (Definition 1.3.1)

End(~) = The set of endomorphisms of ~

[P,~I] = The Roberts actions of ~, (Definition 1.3.2)

~G(Wl,W2) = The set of intertwiners of Wl and w2 in ~

~G(Pl~P2) = The set of intertwiners of Pl and P2 in End(~)

~ ( ~ ) = The set of all Hilbert spaces ~ in ~ such that ~t(~) = ~ for all t

"~ × Q = The crossed product of ~I by

P

gc~ = The ~-valued weight on ~, given by

g6 = The N~-valued weight 'on ~I given by

ACTION7 CO-ACTION A N D DUALITY

Introduction This chapter is devoted to the formulation of the duality for crossed products of yon Neumann algebras involving non-commutative automorphism groups To do this~ we must reformulate the duality theorem for abelian auto-

morphism groups In §17 the usual duality theorem for crossed products involving only abelian groups is presented without proof together with its consequence in the structure theory of von Neumann algebras of type III In §27 we shall review first the duality principle for abelian groups to pave the w a y for noncommutative groups

We then formulate the duality theorem for crossed products incolving non-commutative groups We present a p r o o f which takes care of the both cases~ abelian and non- abelian Here we take the Hopf-von Neumann algebra approach to the duality

principle Namely, showing that a continuous action J of a locally compact group

G on a v o n Neumann algebra ~ corresponds uniquely to an isomorphism w of

into ~ ~ L~(G) such that (~ ® ~) o ~ = (~ ® ~G) o ~, where ~ G is the isomor- phism of L~(G) into L~(G) ~ L ~ ( G ) given by (GGf)(s,t) = f ( s t ) , we introduce a co- action 5 of G on ~ as an isomorphism of ~ into ~ @ g ( G ) such that ( 5 ® ~ ) o 5

= (~ $ 5G) o 5 , where ~(G) is the yon Neumann algebra generated by the right regular representation p of G and 5 is the isomorphism of ~(G) into

g(G) @ g(G) such that 5G(P(S)) = 0(s) ~ p(S), s c G We then prove the duality theorem~ Theorems 2.5 and 2.77 as the non-commutative version of the usual duality theorem mentioned above

Section 3 is devoted to another approach to the duality principle due to

Roberts which follows more closely the spirit of the Tannaka-Tatsuuma duality

theorem referring directly to a ring of representations Here, the notion of

Hilbert spaces in a yon Neumann algebra plays a crucial role~ which is a replace- merit of unitaries in the case of abelian groups We shall see in the subsequent sections that the Hopf-von Neumann algebra approach is convenient in constructing the crossed product while the Roberts approach has an advantage in the analysis of automorphism groups over the former one Section 4 is m e r e l y for convenience of the

We begin with discussion of a duality for crossed products involving only abelian groups first

Let G be a locally compact abelian group with a Haar measure ds~ where we use the additive symbol for the group product We denote by G the dual group

of G with the Placherel measure dp Given an action l) s of G on a yon Neumann algebra [ ~ } ~ the crossed 2roduct of ~ by ~ which will be denoted

by ~Xoz G, is constructed as follows: A representation ~ of ~ on D ® L2(G)

is given by

~ ( x ) ~ ( t ) = Gt(x)~(t), x e ~, t e G, ~ e ~ @ L2(G) ; (i.i)

a unitary representation u of G on ~ ® L2(G) is then given by:

Then ~ G is the yon Neumann algebra on ~ ® L2(G) generated by ~ ( ~ )

u(G) Next~ we construct a unitary representation v of G on ~ ® L2(G)

Hence the automorphism of £(~ ® L2(G~ induced by v(p) leaves the generators of

X G inv~riant up to multiple by scalars~ so that it gives rise to an automor- phism of ~ x~ G, which will be denoted by ~p Thus we obtain an action ~ of

on ~ X G We shall call it the dual action

Theorem l.l In the above situation,

(m x G) x~ G ~ ~ g £(L2(G))

The isomorphism carries the action ~ of G on (~ × G) ×^ G dual to ~ into the action ~ of G on ~ @ £(L2(G)) given by

i) An action of a locally compact group G on a yon Neumann algebra • means

a homomorp?:Jsm a of G into Aut(~) such that s ~ G , ~ (x) ~ ~ is G-weakly con tinuous for each x e ~ The composition [~,G~G} or [ ~ } will often be called

a covariant system The topology in Aut(hl) should however be considered as the

~(s) of a on ~2(~):

(~Xs)~)(t) ~(t-s) ,s,t ~ G , ~ ~L2(G)

The proof will be given in the next section Applying the above theorem to the modular automorphism group~ we obtain the following structure theorem for factors of type IlI

Theorem 1.2 If ~ is a properly infinite yon Neumann algebra, then there exists a unique properly infinite but semi-finite yon Neumann algebra N equipped with a one parameter automorphism group [ S t ] and a faithful 3 semi-finite 3 normal

-t

t r a c e T such that T o ~ = e T and ~ N ×~]P If ~ is a factor, then ~ is ergodie on the center % of N If ~ is of type III, then {%,el does not admit a multiple of L~(]R) with translation as a direct s ~ m ~ u d and N must be

In this section, we shall discuss a general duality theorem for crossed pro- ducts involving non-abelian groups Since we do not have a dual group for a non- abelian group, we must reformulate the duality theorem for abelian groups without making use of the dual group before we move to the non-abelian case

Suppose G is an abelian locally compact group Let ~(G) denote the you Neumann algebra on L2(G) generated by the regular representation p of where

~(s)~(t) = ~(s + t) , s,t c G , ~ ~ L2(G)

G,

Denoting by 3 the Fourier transform of L2(G) onto L2(~), we have

2ff_~(G)3 -1 = ~(G), ~.(G)3 -I = L~'(~) , where we c o n s i d e r L~(G) ( r e s p L~(G)) as a yon Neumann a l g e b r a on L2(G) ( r e s p L2(G)) acting by multiplication Since we want to eliminate G, we identify L~(G) with R(G) via the Fourier transform We then formulate the duality of G and G in terms-of L*(G) and Q(G) Indeed, the Hopf-von Neumann algebra approach tells us that L~(G) and g(G) carry the structures dual to each other Namely, L ~ ( ~ carries the co-multiplication ~G which is defined by:

Let us review what we have done in the above Indeed, the above procedure means that we translated the group structure of G into the yon Neumann algebra L~(G) together with the isomorphism SG' the co-multiplication, of L~(G) into L~(G) ~ L~(G); and that of the dual group G into @(G) with the isomorphism 8 G

of ~(G) into ~ ( G ) @ ~ ( G ) ~ e both s y s b e ~ ~L'~(G),~G ] and [~(G),6G} satisfy the same commutative d i ~ r a m :

7

neuL ~neueu

compact (not necessarily abelian) group with a right Haar measure ds Let

(resp k) be the right (resp left) regular representation of G on L2(G), and let ~(G) = p(G)" (resp ~'(G) = X(G)") We define unitary operators V G and W G

We must now formulate the notion of an action of G on a yon Neumann algebra

in terms of ~L (G),aG} Suppose that an action a of G on ~ in the con- ventional sense is given This means that we have a map : (x,s) ~ ~ × G ~ a s ( X ) ~ with certain properties If we fix an x ~ ~, then we get an H-valued function:

s ¢ G ~ as(X ) ~ ~ on G, which is, in turn, an element ~ ( x ) of ~ @ L ~ ( G ) Namely, we have a map ~ :x ~ ~ ~ ~ (x) c ~ @ L'~(G) from ~ into ~ @ L~(G) The

isomorphism property of each a s reflects to the isomorphism property of ~ , i.e

is an isomorphism of ~ into ~ @ L~(G) The homomorphism property of the map:

s ¢ G ~ a s ~ Aut(~) is translated to the commutativity of the diagram:

TT

I

1

We then discover that the isomorphism ~ has already appeared in the construction

of the crossed product as in (1.1) In order to complete or to start our program,

the equality:

(2.~)

gives rise to an action ~ of G on ~ with ~ = ~

Proof First we shall prove that ~(~) is invariant under I~ ® ~ ;t ¢ G~ Making use of the duality of ~ and ~,,, we define a linear transformation ~f on for each f ~ Ll(G) as follows:

Therefore, we have established that an action ~ of G on ~ may be iden- tified with an isomorphism ~ of ~ into ~ @ L~(G) satisfying (2.4) We shall use the same symbol s for an action and the isomorphism ~ , and call it an action of G on ~ In this respects the co-raultiplication G G is indeed an action of G on L'=(G) induced by p

Definition 2.2 The yon Neumann algebra generated by s(~) and C ® e(G) is called the crossed product of ~ by G with respect to ~ (or simply the crossed product of ~ by ~)s which will be denoted by ~ × G

2) We consider throughout only normal maps for von Neumann algebras

3) Pg = I g(t)P t d t

algebra ~ We should simply replace L~(G) by ~(G) and ~G by 5 G in the commutative diagram for ~ Namely~ we have the following:

Definition 2.3 A co-action of G on ~ is an ~somorphism 5 of ~ into

(2.~) ( ~ ® L ) o ~ = ( ~ ®~G) o ~

The yon Neumann algebra generated by 8(~) and C ® L ~ ( G ) is called the crossed product of ~ by G with respect to 5 (or simply the crossed product of ~ by

~), which will be denoted by ~ X 5 G

Proposition 2.4 (Dual co-action and action), i) Given an action ~ of G

on ~, if we set

(2.6)

then ~ is a co-action of G on ~ × G, which will be c "8/_led dual to

ii) Given a co-action 5 of G on N first, if we set

(2.7) %(x) = A d ( l ® V ~ ) ( x @ 1), x e ~ X5 G ,4)

then ~ is an action of G on ~ ×5 G~ which will be called dual to

Proof i) Since W G e L~(G) g R(G), 1 ®,W~ and ~(x) ® l, x e m,

@.n)

But X is nothing but w~ with the notation in Proposition 2.1 Thus, 5 m~ps

x 5 G into (~ x 5 G) ~ L~(G) Equality (2.:-) can be checked b y looking at the

Theorem 2.ft (Dual~ty for actions) If ~ is an action of G on ~, then

~ t = at ® kt under the above

m ~ ~(o) = ăm) v (c ~ f'(o))

C (G,.~rO denote the C~ algebra of all continuous h.-valued functions

on G vanishing at infinity with respect to the norm topology in ~ O f course,

C (G,~) is naturally identified with the C -tensor product C (G) @~ ~ For any two distinct points s,t in G and x,y e ~, we choose two continuous func- tions f and ~ with compact support such that f(s) = g(t) = i and

f(t) = g(s) = 0 Let a = ~-l(x) and b = ãl(y) Set ~(a)(l S t ) + ăb)(l $ g)

z e ẵ) V (C @ L~(G)) We then have

~ ( s ) = % ( a ) f ( s ) + % ( h ) ~ ( s ) = % ( a ) = ~ ;

z ( t ) = y

Therefore, the partition of unity shows that every element of C (G,~) is well-

Proof of Theorem 2.5 By the previous lemma, ~ ~ £(L2(G)) is generated b y

ăm), C ® g(G) and "C ® L"(G) By definition, (~ ×a G) ×& G is generated b y

o ~(~).= Q(~L) ~ C, ~(C ® ~(G)) = C g' %G(~(C)) and C ~ C ® L'~)(G) in the algebra

In order to put Theorem 2.5 in the form s3amnetric to the next result, we express the action ~ of G on ~ @ £(L2(G)) in the following formula, which is directly checked by looking at the generators:

is a co-action of G on £(R ® L2(G)), whose restriction to Q(G)

the co-action 5 G We shall use this 5 in the following lemma

agrees with

Lemma 2.8 (i) If S is the set of all compact subsets of G ordered by set inclusion, then there is a net [$K : K c S} in A(G) n W(G) such that SK(t) converges to 1 for each t e G

(ii) If yj e £(R @ L2(G)) and ~j ~ E(G,~) for j = 1,2, then for any

E > 0 and qj e R ® L2(G) there exists a ~ c A(G) n ~(G) such that

I((5~(yj) -Yj)~j I qj)l < a ,

Proof (i) Let f be an element in ~(G) with

tends to G~ q~K(t) comverges to i

(ii) The functions (i @ WG)(~ j ~ fo)

w(G × G,~) supported by K 1 × K 2 for some

on Ky~ then

(~5~)K(yj)~j I ~j) = ((yj

fD ~

":K = fo,~K Since fK(t) converges t o 1

O

combinations of elements in £(~) @ C, C × L~(G) and C X,~(G) Since f c ( s ) N ~ ( t ) =

f~:s(g)~(st) and (fo(s))* = 0s_ I (f)~:(s)*, the set £o is a strongly dense * sub- algebra of £(~ @ L2(G)) For any ~: > 0 there exists x a £ such that

I((Yj -xj)~j I ~lj) I< t and il((yj -xj) ~ i)(i ~ WG)(~ j @ fo)ll < a

for j = I,,2 Therefore

]((SCK(Y j) -yj)~j1~]j)[

< I(Sq~K(Y j -xj)~jlT, j)I + l((:~qjK(X j) -xj)~j ['ij)] + I((xj

< a,jh] ,ll]j.! + I((~ K(Xi) -xj)~j I-',j)1 + ~

-yj)~j 1-~j)1

Thus it remains to show t~mt the second term converges to 0 Since 5(Zo) =

Adl~ W (z ° @ i) = z ° ® ~(r) for z ° = y ~ f~(r), it follows that, for any z e £o

of th~ form wn~.k=l Zk with z k = Yk ~ fkP(rk)'

n

Lemma P.9 (i) If &(x) = A d ~} (x @ i) for x { £(.9 ~ L2(G)), then

Proof (i) Let ~, ,~ 6 ~ and f, g ~ ~(G) For any z ~ £ ( ~ ~ L2(G)) and any ~ > 0 there exists a ~, £ A(G) q ~(G) such that

F o r each ~ ~ A(G), set

If x ® i e 5(~)' O (£(R) ® C), then we have, for any y e ~, e e £(q) and

(2.21) ~(5(y)) = (i ® i ® K)(5(y) ® 1)(l ® i ® K) = 5(y) ~ 1

Then3 by dfrect computation, we have

(2.22) rT(I ® f) = i ® X(f) , ~(1 ® X(r)) = 1 ® 1 ® 0(r)

By the previous lemma, ~ ~ g(L (G)) is generated by 5(U), C ® L~(G) ~nd 2

C ® g'(G); thus ~ maps the generators of N ~ £(L2(G)) onto those of

NOTES

The definition of a co-action, Definition 2.3, and the construction of the crossed product N ×~ G were given independently b y Landstad [42,43], N a k a ~ m i [h5, 46] and S t r ~ t i l a - V o i c u l e s c u - Zsid6 [59~60] Dual co-actions and d ~ l actions, Proposition 2.4, were introduced independently in [,13, 6,~0] and the duality theorems for crossed p r ~ u c t s , Theorems 2 5 and 2.7, were proved there The proofs presented here are taken from [k71 which resembles [60] The ~ y to the proof is in Lemma 2.10 Here we take an idea due to Van Hceswijck to [77] On the other hand Landstad, [42], prepared Theorem II.2.1.(ii) in order to prove Theorem 2.7 Various

~3- Roberts action and Tannaka-Tatsuuma duality

In this section~ we shall discuss the duality for the "automorphism actions" of

a locally compact group on a yon Neumann algebra through a formalism given b y Roberts

In order to avoid unnecessary complications~ we consider only compact group~

in this section~ while this restriction can be lifted ~,~ithout serious difficulties

if one really needs to do so We leave the general ease to the reader

Definition 3.1 A collection e of unitary representations of G is called a

OgG(I"r3,TT2)~G(I~2,~I ) c JG(~3,,.~l) ;

~G(~2,~l ) ® ~ a ( ~ , ~ i ) = J a ( ~ ~ ~.S,,.-1 ® ~i) ; (3.3)

L , ~2(J~ P ( 2 , I))~(P2, Pl ) = ~(o 2 o , , ~ , ° o2'P 1 , ° O l )

Definition 3.2 A Roberts action { p , O of a ring ~ of representations of O

on N is a composition [0 ,-iWl,W2 : ~ ) ~ l , ~ ~ (I 9 ] , where p~ c- End(N) and :i~l,~2

rin~ if i) Wl e ~ c W and Zl ® ~2 c ~ for every pair -i,~2 c Z; ii) The trivial representation ~ of G belongs to R If the conjugate representation

of each ~ e Z falls in ~ again, then the ring 9 is said to be self-ad~oint For each ~i' '~2 ~ ~' we denote

Let End(~) be the set of all *-endomorphisms of ~ Here we assume the normality anq the idc-nt.'Ly prcsurv'ng for endomorphisms For each el, P2 e End(~)~ we write

We then have the following relations among these sets:

= 1 Once it tern {~i : i ~ l , , d ] of isometrics with orthogonal ranges and ~ = i ~i~i

is chosen, the map: x :: ~ ~ i = l U j x u : - i is an endcmorphism of ~ and does not depend

on the choice of a basis; hence we denote it b y p~ One can characterize p9 b y the equality:

It is easy to check that for Hilbert spaces ~l and ~2 in ~, the G-weak

economically identified with

y e ~,, is naturally identified with the tensor product ~i ® R2" Here the point

is that the product RIR ,, is a concrete object sitting in ~ while R @ R 2 is

1 abstract In the following situation, this point becomes clearer

Let [~;,G,~} be a covariant system with ~ properly infinitẹ We denote by

~ ( ~ ) the collection of all Hilbert spaces in ~ globally invariant under

[~t : t e G} If ~ e % ( ~ ) , then we have, x,y e R,

(~t(x)'~t(y)) = ~t(Y)'X~t(X ) : Gt(~'~X)

= ~t((x!y)l) = (xly)

Hence the restriction of ~ to ~ is a unitary representation of G on ~ We denote this representation by ~ or {~,~} Then % ( ~ ) turns out to be a co3~Lee tion of representations of G which is~ in turn~ a ring in the sense that

%t%,~,~l(a) = a , a ~ ogẵ ,Õl),, RI, % ~ ~(r,O •

are isometries in ~ with WlW ~ + w2w ~ = ị

globally invariant, and

A straightforward calculation shows that , ~ : ~,~i,~2 e % ( ~ ) } is indeed

a Roberts action of ~ ( ~ ) on D~ ~°~R' I ~ 2 ' ~ R I

We now have the following Tannaka duality theorem in our context:

Theorem 3.4 Assume that G is compact If every irreducible subrepresenta-

e Aut(~./~ ~) lesving every member ~ e ~ ( ~ ) globally invariant must be of the form d r for some r e G 5)

It should be pointed out that the above theorem can be generalized to a locally compact group, if we assume that ~ ( ~ ) contains a member equivalent to the regular representation of G Thus, the Tatsuuma duality theorem in our context remains valid also

Proof For each a,b ~ ~ ~ ~ ~ (~) we set

q,b(t) = a*~t(b) ,

a n d denote b y C (G) the set of all such functions It is easy to check that C (G)

is a * - subalgebra of C(G), because ~ (~) is a self-adjoint ring

Now, we define a m a p U of C (G) into itself b y

It also follows that U is an isometry Moreover, U is multiplicative Indeed~

U(q,bfo, d) = U qe,bd = f~(ac),bd = %(a),b%(e),d "

Since U0 t = pt U on C (G), Lenm~ 3.5 b e l o w tells us that U = k r for some r e G

on Ca(G ) Therefore

~(a)*~s(b ) = a * ~ _ l (b) = Sr(a)*~s(b)

r s for all a,b ~ ~, ~ ¢ ~ ( ~ ) Therefore

~(x) = ~rCx)

for all x £ ,~, ~ ~ ~4c~(.~) and for all x £ ~cY

5) Aut(~,/~%) means the group of all aatomorphisms of • leaving '~ pointwise

I f [~,~} is an irreducible subrepresentation of [~,~], then [&,~] ~ [~,~] for some ~ e ~ ( ~ ) b y assumption Then there exists a basis [al, ,ad] of and an orthonormal basis [Vl, , v ~ of ~ such that

~r(ak) = ~ }~._a.; ~r(Vk) = E k j k V j •

~ t w = ~ a v * Then w e ~ and ~ = ~ Therefore, if a e ~, then

J J

c(wa) = Wc(a) = W&r(a ) = ~r(Wa) ,

so q = ~r on ~ Since G is compact, the collection of these spaces ~ is

Lemma $-5 Let G be a compact group, A and B be a *-subalgebras of C(G) globally iuvariant under Pt' t e G If q is an isomorphism of A onto B such that

~roof Let ~ and ~ be the closures of A and B in L2(G) respectively

By assumption (ii), ~ is extended to a unitary U of ~ onto ~ Let 7 be the multiplication representation of C(G) on Lg(G) We then have, for each f ~A, g e B,

u~(f)u-½ = ~(~f)g

Thus, the unitary U gives rise to an isomorphism of the uniform closure of ~ A ) I~ onto the uniform closure of ~(B) i~ However, the map: f e A ~ ~(f) l~ ~ £ ~ ) is extended to a faithful representation of the closure [A]~ of A in C(G), where the faithfulness of the extended representation follows from the fact that ~ ] ~ Furthermore, the extended representation is indeed given b y restricting the space

w ( [ A ] ) to ~ Thus, the isomorphism of the closure of ~(A) I~, which is -([A]~)) I ~ onto the closure of ~ B ) I~(= ~([B]~)I~) gives rise to an isomorphism

of [A]~ onto [ B ] which extends ~

Noticing that every character of [A] is given by evaluating each function in [A]~ at some point in G, we can find an element r' e G such that (~f)(e) = f(r') for every f e [A] We then have

where 5~ is an iscmorphism of ~(G)' into ~,(G)' ~ ~,(G)' with 6 ~ ( r ) : ~(r){ ~(r),

( ~ (Y) : ~ w & ( Y ~ 1 ) ) ~ e c r o s s e d p r ~ u c t s ~ m~, G a n d ~ x ~ , ~ a r e d e m n e d as the yon Neumann algebras ~'(~) v (C ~ ~(G)') and 5'(b) V (C ~ L~(G], respectively The dual co-action (~')^ and the dual action (5') ^ are defined by

WG, W~_, V G and V~:

((, ~ ~)(w~ ~ l ) ) ( w ~ l) (, ~ ~>G)(w~) -

( L e 5 G) ( W G) "

((~ ~ o)(w~ ~ l))(w~ ~ i) (L ~ 8~)(w~)

- t 7 ~

(L ~ ~o)(wG ) • (L ~ ~G)(va) - ( , ~ ~G) (v~)

( ~ ~ % ) (v G) • ( L ~ ~ ) (v~*)

ELEMENTARY PROPERTIES OF CROSSED PRODUCTS

Introduction The image of the original algebra in the crossed product

is characterized as the fixed point subalgebra under the dual action (resp the dual co-action), Theorem 1.1, in §l Combining this with the duality theorem, the crossed product is characterized as the fixed point subalgebra of the tensor product o f the original algebra with £(L2(G)) under the tensor product of the original (resp co-)action and the regular (resp.co-)aetion, Theorem 1.2

Section 2 is devoted to a characterization of a dual(resp, co-)action The characterization sho~-d be viewed as a sort of an im.primJtivJty theorem

In ~3, we consider the com~utant of the crossed product oy a closed subgroup which is shown to be the fixed point subalgebra of the crossed product of the commutant of the original algebra under the action (or co-action) of the quotient group Here, since we do not treat the Banach algebra bundle of Fell, [ 3 0], we

#l Fixed ~oints in crossed products

Given an action G of G on [~ or a co-action 5 of

^

Theorem 1.1 (a) (~ x~ G)~ = ~(~)

(b) (n x~ a) e 8(h)

Proof (a) It is clear that ~(~) is contained in (~ x~ G) ~ We have only

to show the reverse inclusion

Since { ~ } ~ {~(~)~L ® p] and ~ ® Pt = Adl~o(t)' we may assume that the action ~ is implemented by a unitary representation u of G such that Jt(x) = u(t)xu(t)* for x ¢ ~ Here we identify u with the unitary in £($) @ L~(G) given by (u~)(t) = u(t)~(t) for ~ c $ ® L2(G) Then J(~) = u(~ ® C)u* We put

~ ' ( x ) = ~*(x ® 1)u , x ~ m'

Then ~' is an action of G on ~' with respect to R'(G) By Lemma 1.2.6,

~' @ L~(G) is generated by ~'(~') and C ® L~(G) Therefore

I'll' ~ £ ( I ' 2 ( G ) ) = C~'(ll1') V (C ® L~(G)) V (C ® fC'(G))

Applying Ad to the both sides and considering the commutants, we have

U

( l 1 ) ~(m): (m' ® C)' n (c ® ~ ( G ) ) ' n { u ( t ) e X ( t ) : t ~ G}'

Now suppose that y a ~xc~ G and ~(y) = y ® i We want to show y belongs

to the right hand side of (1.1) It is straight forward to see that ~ × G commutes with ~' ® C and u(t) ® X(t) for all t a G Thus y commutes with

^

(b) We have only to show that (U ×5 G)5 ~ ccntained in 5(~) Since [U,5] T

~5(~), g ® 5 G} and 5G(X) = WG(X ® I)WG, we may assu~ae that 5 is implemented by

a unitary w in ~(R) ~ R(G) such that

5 ( y ) = w × ( y ® i ) w , y ~ ~ ,

(1.2)

( ® i ) ( ~ ® ~ ) ( w ® z ) : (~ ® 5G)(w)

Let K be a unitary on L2(G) defined by (K~)(t) = a(t)i/2~(t -I) for { ~ s2(e)

and ~' = ( l ® ~ ) w ( i ® K ) Since ~ ( n ® C ) ~ c ~ ~ ( ~ O ( S ) ) and w ~ ~ ( R ) ~ ( O ) ,

Combining Theorem i.i and the duality theorem for crossed products, we have a characterization of crossed products

Theorem 1.2 (a) • ×~ G = ( ~ £(LP(G))) ~

(b) ~ ~ = (~ ~ Z(L2(G))) g

Proof (a) ~$~kir~ use of ~he isomorphism v of ~L ~ £(L2(G)) onto

(m ×a G) ×a a obtained in (m.2.12), ~e h~ve

§2 Characterization of crossed products

W h e n one studies an action or a co-action of a locally compact group G on a von Neumann algebra, often it so happens that the action or the co-action is already dual to another co-action or action Thus, we need to know e x a c t l y ~ h e n the given one is dual to something else The following gives a convenient characterization for those actions and co-actions

Proof (i) :> (ii): Trivial

( i i ) ~ (iii): Given a unitary representation u of G in h satisfying (2.1), we shall use the same symbol u for the unitary in ~ ~ L~(G) defined by (u~)(t) = u(t)~(t) for ~ ~ R @ L2(G) Then u(st) = u(s)u(t) means (2.2) Furthermore, (2.1) means precisely (2.]3) N~meZy, if ~ ~ ~ ® L2(G × G) and

(iii)=~ (i): For each x c R 5, we set c ~ ( x ) = u(x g l)u* Then (~ is an

Theorem 2.1 Let 8 be a co-action of G on N The following three condi- tions are equivalent:

i) There exist a v o n Neumann algebra • and an action ~ of G on such that

= (u ® 1 ) ( 1 e WG)[(L ~ = ) o (~ e ,~)(~ e 1 ) ] ( 1 ® W~)(u × ® Z )

= (u e, i)(1 e, WG)[(~ ~ ~)(~(~) ® 1)](i e WG)(u* ~ l )

= (u ® 1)(l ~ wa)[(, ® ~)(x ~ iL~(~×~))](~ ® W~)(u* ® l)

= ( u ~ ) ( ~ ) ( u * ® ~ ) = ~ ( ~ ) e l

G satisfies the associativity condition, (I °_.4); hence it is an action of G on ~5 Next, we want to show that h is generated by ~5 and u(G) By Lemma 1.2,10, the yon Neumann algebra R = ~ ~ £(L2(G)) is generated by ?I X 5 G and C ® g(G)'

By Theorem 1.2, ~ ~8 G = ~ Since Ad u is an isomorphism of [~,~} onto [~,5"}

by (2.3), it follows that ~ is also generated by ~-~ = h 5 @ ~(L2(G)) and

u(C ® 9 ( G ) ' ) u "×' Since u ( l ® X ( r ) ) u * = u ( r ) ~ X ( r ) , ~% must be generated by ~5 and u(G) Finally, we shall show that {~,5} T [R5 x G,~} For each x e R 5 we have

The du~l version of the above theorem is the following:

Theorem 2.2 Let ~ be am action of G on ~ The following three conditions are equivalent :

i) There exist a von Neumann algebra ~% and a co-action 5 of G on

% o ~ = ~ o X t (or ~ o ~ = ( ~ ) oX) on S~(a)

There exists a unitary w in • @ B(G) such that

(w X ®l)(~ ~ ~)(w* ~ i ) = (~ ® ~o)(w*)

~(w*)= (w*~.1)(1~va) (or ( % ® l ) ( w *) = w * ( 1 ~ p ( t ) ) )

Proof (i) ~# (ii) : Clear

(ii) => (iii): We may assume that ~ is standard and ~ is implemented by a unitary representation u of G on ~ Then (2.5) implies Adu(t) ~ )~ on L~(G) By Mackey's imprimitivity theorem, there exist a Hilbert space ~ and an isometry U of ~ ® L2(G) onto ,9 such that

~(f) u ( i ~ O u -i ~(~) u ( i ~ ~,(O)u -I Therefore U(g ® L ~ ( ~ ) ) u - i = ~ and ~t ° Adu = Adu ° (~ ~ ~t ) on ~ ~ ~ ( ~ ) Therefore we may assume that

= R ~ Li~(O) , ~(f) : i(~ Z f , u(t) = v(t) 'g k(t) for some unitary representation v of G on ~, for (i @, X(t))~u(t)~ - £(R)~L~(G) Now, we define a unitary w in £(~) ~ ( G ) by I ~ W o Since W* G satisfies the associativity condition (I.[l.p), w* satisfies (2.6) Since I ~ W G ( C ~ L ~ ( G ) ~ ) ~ ( G ) and w(L'~(G)) c ~, it follows that w ~ ~ ~ ~(G) Since

A d ( ~ ) ( V ~ l ) ( W G { l) = (W ~ 0 ~, 1)(1 ~ V~) ,

the condition (2.7) is obtained

(iii) > (ii) Put w~ = (i @ K)w and ~)(f) = wK(i ® f)w K for f ~ L (G), where (K~)(t) - A(t)i/2{~t-i - " ) f o r g 6 LP(G) Then ~0(f) m ~ £(L2(G)) Since (2.7) implies

Since ~z(m) = ( m x G) d by Theorem i.i, we have ,~o(f) ~ <z(m)

therefore there exists an isomorphism ~ of I]~(G) into ~ such that ~, =

O (~ o n Since (,~, @ g) o ~z = (~ ®,ZG) o ,'~, it follows that ~ ° ~'t = ( ~ g 0t) ° &

Definition 2.~ (a) An action ~ of G on ~ is said to be dual or the dual action of G on ~, if [~,~} ~ [n x 5 G, g} for some In,5}

(b) A co-action 5 of G on ~ is said to be dual or the dual co-action of

G on ~, if [~,5] ~ [~ X~ G, & ~ for some {~,~}

In the rest of this section, we assume that G is discrete T h e n there exists

a bijection between the set of all unitaries w in ~ ~ g(G) satisfying the

T h e o r e m 2.4 (a) If 5 is a co-action of G on ~ implemented b y a unitary

w c h ~ g(G) satisfying the associativity condition so that 5(y) = Adw.(y ® 1),

t h e n the follcwing twc conditions are equivalent:

(i) 5 is dual;

(ii) there exists a unitary representation u of G in h such that

u(t)e(r) : e(rt-1)u(t) for all r,t

(b) If ~ is an action of G on ~, t h e n the following two conditions are equivalent:

(i) ~ is dual;

(ii) there exists a strictly wandering projection e c ~ for ~, i.e

[~t(e) : t c G} is a partition of the identity such that ~ t ( e ~ s ( e ) = 0 for t ~ s

Proof (a) Condition (2.1) is equivalent to

2 u(t)e(r) ® p(z') = ~ e(s)u(t) @ o(st) ,

which is equivalent to u(t)e(r) = e(rt-l)u(t) for all t,r

(b) Condition (2.6) is equivalent to the existence of a partition

[e(t) : t c G] with (2.11) Condition ( 2 7) is equivalent to

~t(e(r)) @ p(r -I) = ~ e(s) ® D(s-lt)

w h i c h is equivalent to ~t(e(r)) = e(tr) Thus we have only to set e = e(r) for

b y the relation

NOTES

The equivalence of (i) and (ii) in Theorem 2.1 is due to Landstad [42]; that

of (i) and (ii) in Theorem 2.2 is due to Landstad [~3], Nakagami [46] and Str~til~ - Voieulescu - Zsid6 [60] Theorems 2.1 and 2.2 are generalized to the

§3 Commutants of crossed products°

In this section we shall define an action 3 of G on the commutant of ~(~) and a co-action a of G on the commutant of 5(N) in order t o show an imprimi-

t l v i t y theorem, which will be applied to the commutants of crossed products

The associativity (I.2.11) for ~ and (I.2.5) for 5 gives us

Now, we shall show an imprimitivity theorem:

T h e o r e m 3.2 (a) (Assume that II is noznal)

(~ x ~ H)' is an action of G with respect to ~(G)'

( 3 6 ) (r~ x H)' : (rp, x~G), v (C ® S ( G / H ) )

The restriction of ~ to and

(b) The restriction of s to (~ x 5 ( H \ G))' is a co-action of G and

Combining this with (3.1), we have

of ~ to ~ is an action of G with respect to Q(G)'

O

~e~t we s ~ Z show (3.5) Since VO ~ ~(G) ~ L~(G),

( ( ~ X 5 H)') ~ It suffices to show the reverse inclusion

then x @ 1 = Adl®~#(x ® l ) belongs to the commutant of

V~(I ® p(r))V G = p(r) ® p(r), 8(~o) is contained in ~o ~ ~ ( G / H )

isomorphism of L~(G/H) onto £~(G/H) with ¢(f) = f ° ~ for

is the canonical msp of G on G/H Here we set

~(~) = (L ~ ~ @ ~ - i ) o ~ ( x ) , x e mo

and as Let ~ ~e an