Commnn. math. Phys. 6, 101--120 (1967)

Asymptotically Abelian Systems S. DOl~LICI~EI~*, 1R,.V. KADISO~**, D. KASTIml~.*** and DEIgEK W. ROBINSON* University of Aix-Marseille France Received ~,'Iay 15, 1967 Abstract. We study pairs {92, ~} for which 9.1 is a C*-aIgebra and ~ is a homomorphism of a locally compact, non-compact group G into the group of *-automorphisms of 92. We examine, especially, those systems {92, ~} which are (weakly) asymptotically abelian with respect to their invariant states (i.e. (¢IA %(B) - - ~ ( B ) A) -+ 0 as g --> c¢ for those states ¢ such that ¢(%(A)) = q~(A) for all g in G and A in 92). For concrete systems (those with 92 acting on a Hilbert space and g -+ ~g implemented by a unitary representation 9 -+ Ug on this space) we prove, among other results, that the operators commuting with 92 and {U~} form a commuting family when there is a vector cyclic under 92 and invariant under {Ug}. We characterize the extremal invariant states, in this case, in terms of "weak clustering" properties and also in terms of "factor" and "irreducibility" properties of {92, Ug}. Specializing to amenable groups, we describe "operator means" arising from invariant group means; a.nd we study systems which are "asymptotically abelian in mean". Our interest in these structures resides in their appearance in the "infinite system" approach to quantum statistical mechanics. Introduction

I n t h e general f r a m e of q u a n t u m mechanics t h e p h y s i c a l o b s e r v a b l e s are d e s c r i b e d as s e l f - a d j o i n t o p e r a t o r s on a H i l b e r t space ~ a n d t h e b o u n d e d observables (corresponding to b o u n d e d operators) therefore g e n e r a t e a C*-algebra acting on ~ . T h e algebraic a p p r o a c h to field t h e o r y [1, 2] proposes to consider as p h y s i c a l o n l y t h e local o b s e r v a b l e s i.e. t h o s e corresponding to m e a s u r e m e n t s p e r f o r m e d w i t h i n finite regions of space d u r i n g a finite t i m e . These observables are described m a t h e m a t i c a l l y as t h e self-adjohlt e l e m e n t s of a n i n c o m p l e t e C*-algebra whose c o m p l e t i o n 92, called the quasi-local algebra, is considered as t h e m a i n * Permanent address: Istituto di Fisiea G. Marconi, Piazzale delle Science 5, Roma. ** Permanent address: Department of Mathematics, University of Pennsylvania Philadelphia, Penn. U.S.A. *** Present address: Institut des l~autes Etudes Scientifiques, 91 - - Bures sur Yvette, France, * Present address: Division Th6orique CERN, Genbve 23.

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S. DOFLICHER,R. V. KiDISON,D. KASTLERand D. W. 1%oB[~so~-:

mathematical entity whose abstract algebraic structure contains the whole physical information (the choice of a particular faithful representation being a matter of arbitrariness without, physical implications). Field theory is thus viewed mathematically as the investigation of the quasilocal algebra 92 which is supposed to satisfy appropriate axioms of physical origin (locality, Lorentz invariance etc.). Whilst the physical (local) observables are described as elements of 91, the physical states are represented as the normalized positive linear forms over 92 (states over 9A). In this framework the invariance groups of the physical theory (group of space and time translations, Lorentz transformations etc.) are homomorphically mapped into the automorphism group of the C*-Mgcbra 92. The automorphism ~ corresponding to the group element g describes physically the "shifting by g" of the local observables. One of the first problems arising in this approach is the hlvestigation of the translationally invariant states over the algebra 92, those being natural candidates for the description of the equilibrium states of statistical mechanics (more generally the states invariant under any group of physical transformations are of interest) [3, 4, 4a]. In this connection recent studies have revealed that a number of general results depend upon only a very weak locality assumption about the quasi-local algebra (entailing only a small part of the structure required by physics) namely a property of asymptotic abelianness (el. Definitions 1 and 2 below) [5, 6, 7, 8, 9, 10]. It seems therefore interesting to investigate mathematically the systems {91, zt} of a C*-algebra 92 and a homomorphic mapping ~ of a group G into the automorphism group of 92 which possess such an asymptotic abelian property. This is the purpose of the present paper. Section I is concerned with general groups G and an abstract notion of mean of elements of 92. Section II describes the ease of an amenable group possessing invariant means and discusses the connection between the above notion and the invariant means over G. Owing to the delay between the conclusion of this research and the writing o2 this paper (due to geographical dispersion of the authors) some results overlapping with ours meanwhile appeared in the literature

[9, 9a]. I. General ease Theorem 1. Let g-~ Ug be a unitary representation o] the group G on the Hilbert space J~f, 92 a C*-algebra acting on ~ such that ~g(A) = UgA U-~i is in 92/or each g in G. We denote ]urthermore by E o the orthogonal projection U ~ =~o for all g CG}; by ~ the von 2~euin ~ on the subspace {~o~ l mann algebra generated by 92 and {U~ ] g ~ G}; by P the central carrier o[

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E o in ~2; and we assume that,/or all V ~ E o ~ f and A, B ~ 92 I n f I(VI A . e~(B) - ~ ( B ) -

gEq

A t~)l = 0 .

(1)

Then (i) E o ~ E o is abelian 1 so that E o is an abelian projection in ~ and ~ P is o/type I with its center isomorphic to E o ~ E o. (ii) I] ~ E o ~ is dense in 2,f (or equivalently, 9 A E o ~ is dense in ~ ; or if P = I ; or i/ T ~ ~2'-> T E o C ~ ' E o is an isomorphism between yon Neumann algebras) then ~ is of type I with its center isomorphic to Eo,~ Eo; and there exist~ a normal positive mapping A-+ M ( A ) /tom ~ onto its center determined uniquely by M ( A ) E o = E o A E o,

A ~N .

(iii) I] E 1 is an abelian projection in ~ with central carrier I, then E 1 contains a vector cyclic under ~ (or 92, i ] E o = El) i/ and only i / ~ ' is abelian and ~ ' (or ELSE1) is countably decomposable. Pro@ Since U~A = U~A U~-1 U~ = B U~ with B (= U~A U~-1) in o,l, the elements of the form A~U~ + . . . + A~Ue,, comprise a strongoperator dense subalgebra of ~ , where A mis in o.1 and g~ in G. As UaE o = E o = E oU~ for g in G, E o ( A ~ + ' ' ' + A ~ U e . ) E o = Eo(A~ -t. . . . + An)E o so t h a t it will suffice to prove t h a t E o A E o B E o = E o B E o A E o for each A a n d B in 0A to establish t h a t E o is an abetian projection in ~ . W i t h V = Eo~°, ~ ~ 5/~, we h a v e

(~ft E o A E o B E o - E o B E o A E o t~f) = (Vl A E o B - B E o A iV). N o w E 0 lies in the yon N e u m a n n algebra generated b y the U~, g in G. Suppose t h a t it lies in the strong-operator closure of the set of real linear combinations of U~'s, g ~ G. F o r each e > 0, we can choose elements g¢ ~ G and real constants 2~, ] = 1, 2 . . . . . n, such t h a t n

ii(Eo - Z ~Ua~)Bv][ = ]l(Eo - 2 ,'I~%s~)Bvit < e/2llA* ~it ~=1 ]=] and n

]I(Eo - ~ ~/Ug) B* ~Ii -- ii(E0 - Z i=1 j=l

~ ugly)B* vJt ~ e/2p!A Vii

where g can still be arbitrarily chosen in G. We then have

i(~fl A E o B - B E o A lq))} < ~ e + I n~f

_

~A

~"

B-,-B

\ ~J

/

A

I This fact and its connection with reduction theory was first, noted in [26] i n the framework of Wightman field theory.

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S. Do~LIC~, R. V. KxolSO~, D. KASTLERand D. W. PmBIXSO)r:

where the second term on the right hand side vanishes by assumption being equal to

Inf l(~vl (A ~g(B') -- ~g(B')A) 1~o)I -with •

eEo

B' =~

2j~j(B) .

i=1

Thus E 0 is abelian, N P is a yon Neumann algebra, eontMning an abelian projection with central carrier the unit and is therefore of type I [12, Theorgme 1 p. 123]. As E o N E o is abelian it is the center of its eommutant ~ ' E 0. Since the latter is isomorphic to N ' P [12, Proposition 2 p. 19], E o N E o is isomorphic with the center of ~ ' P which is obviously also the center of .~ P. I t remains to prove that E 0 is approximable in the strong operator topology b y sums of the form 2 )~JUgj with the 2.j real. 5fore specifically i=1 let 3- be the strong operator closure of convex linear combinations of the Ugj, gj ff G; and let Eg, g E G, be the projector with range those vectors such that Ugh(= 5~-~0)= yJ. The mean crgodie theorem [11; Cot. 2 and 4, p. 662] yields the fact that Eg is the strong operator limit of 1

UJ in Y as n ~ co. If E and F are two projections which are the

n

i=1

strong operator limits of operators of 3-, then their intersection E A F is such a limit, since E A F is the strong operator limit of E F E F . . . E F (Method of Images) and multiplication of operators is strong operator continuous on bounded sets. Since E 0 = A Eg and A E~ is strong eE(~

gEG

operator limit of finite intersections of the projections Eg, E o lies in J-. Thus (i) is proven and the specialization P = I yields the first assertion of (ii). We then obtain the mapping A -+ M ( A ) by composition of the mapping A ~ ~ ~ E o A E o ~ ~ ' E o with the isomorphism T E o C ~ ' E o ~-~ <-~ T C~', both of which are positive and normal. As the range of the first mapping is the center of ~ ' E 0 the range of M is the center of N' (and ~). If M 0 is another mapping from ~ to ~ ' such that Mo(A) = E o A E o, [M0(A ) - M ( A ) ] E o = 0; so that 0 = [M0(A) - M ( A ) ] P = Mo(A ) -- M ( A ) since P = I and M o ( A ) - - M ( A ) ~ A ~ ' . Assuming now that ~00 is a unit vector in the range of the abelian projection E 1 such that ~ o is dense in dgf, E 1 has central carrier I as does the projection E on the closure of ~'~0. Now- E ~ E~, and abe/inn projections can be characterized as being minimal in the set of projections with the same central carrier [25; § 31 p. 332]. Thus E = E~ and ~ is cyclic under ~ ' in E~j4f. Hence I, the projection on the closure of ~ 0 o is an abelian projection in ~ ' [12; Prop. 3 p. 242], so that N' is abelian and is the center of ~ ( = ~ " ) . Thus E ~ ' E ~ is the center of the abelian algebra ExNE~ and N'E~ = E ~ E ~ . Further ~ ' ( = #2'P) is isomorphic with E o ~ E o and is the range of the mapping M.

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With N' abelian, we have just noted that ~ ' E I (= El~2E1) is maximal abelian on E 1~f~. If in addition N' (or EIP2E1) is countably decomposable, ~ ' E 1 has a vector ~01cyclic for E I ~ [12; Coroltaire p. 20]. Since ~'~Pl is dense in E 1~ , ~ 1 is dense in the range of an abelian projection in ~ ' with central carrier I. This projection must be I, since N' is abelian. Remark. In the foregoing proof, if (1) is valid for a single vector ~0 ~ E 0JC we have established that ~o is a trace vector for E 0 ~ E 0. If in addition ~0 is cyclic under 9A, then E o N E o, ~ ' E o and ~ ' are finite yon Neumann algebras. In place of (1), the assumption that Inf I(~o]Ao:g(B) -- o:g(B)A Iq~)l = 0 for a single ~o in E o5~t° such that the closure of 92~o contains E o~ , and all ~o C Eo ~ f would suffice to establish (i), (ii) and (iii). Theorem 1 is especially interesting in connection wi~h "asymptotically abelian systems" defined as follows Definition 1. I] {92, a} is a pair consisting o/ a C*-algebra 92 and a homomorphism g -+ ~ oJ the locally compact non compact group G into the automorphism group o/9A we call {92, o:} an asymptotically abelian system (a weakly asymptotically abelian system) whenever, to each e > 0 (to each e > 0 and state ¢ over 92) with A, B COAthere is a compact K ( G such that g ~ g implies IlAo:g(B)- ~(B)A][ < e (1<¢, A c % ( B ) - ~g(B)A>] < e). @orollary 1. Let g-~ Ug be a unitary representation o] the locally compact non compact group G on a Hilbert space ~ , OAa C*-algebra acting on :/t" such that o:g(A) = UgA U f 1 is in 01/or each g in G and let E o a n d ~ be as in Theorem 1. The conclusions (i) (ii) and (iii) o/ Theorem 1 are valid whenever {92, c¢} is a weakly asymptotically abelian system or whenever this is the case/or {~l, giG0}, where Go is a non compact subgroup o[ G. Theorem 1 is mainly useful for the study of invariant states over C*-algebras, in the following manner: @orollary 2. Let ~1 be a C*-algebra, g -> o:~ a homomorphism o] the group G into the automorphism group o/92, q) a state over OAinvariant under G i.e. such that q~( ~g (A )) = ¢ (A ) ]or all A E 92 and g ~ G, 7~ the *-representation o] 9Aon a Hilbert space ~ with cyclic vector ~ , Ue the unitary representation o / G on ~ determined by (D[ ~o(A) ID) = <~, A>

~o(~o(A)) = U~(g) ~ ( A ) U~(g)-~,

A ~ 92

(2)

the yon lYeumann algebra generated by the set x~v(92) ~ U¢(G) and E o the orthogonal projection in ~ on the set {~, ( 2 F I Ue(g)~ =~o for all g ~ G}. Suppose that,/or all A, B ~ 9A and all vector states q~ over OA/tom

106

S. DOPLICHER,R.. V. K~kDISOI~ ~, D. KASTL]~ and D. W. t~o]~-so~":

invariant under G Inf 1<¢, A . %(B) - % ( B ) . AS] = 0 .

gEG

(3)

Then 2 ' and E o ~ E o are abelian von Neumann algebras and the mapping T C ~'--> TEo is a *-isomorphism/rom 2 ' onto Eo,~E o. Thus, /or each A C 92 there is a unique Me(A) ~ ~ ' such that M~ (A) E o = E o ~ , (A)E o , (4) the mapping 7 ~ ¢ ( A ) ~ M ¢ ( A ) C ~ ' being linear, positive and weakoperator continuous [rom 7~ (02) into :~' and the set {M~ (A) [ A ~ 92} being weak.operator den.se in ~'. It is interesting that under quite general conditions, in addition to the above structure, one has the feature that 5~' C 92". This is the case whenever G is amenable and g-> gg is strongly continuous as will be shown in Section I I below but also in other contexts as shown by Theorem 2. {92, ~} be as in Corollary I above with {92, ~} a weakly asymptotically abelian system and G locally compact, non compact and connected. Then ~ ' is contained in the weak-operator closure 9A" o/9.1 (or equivalently 92' ~ ~ ). Pro@ From [13] some cyclic subgroup Go is not relatively compact; and from the definition {OA,U IGo} is a concrete weakly asymptotically abelian system ~4th ~ invariant under U IGo and cyclic for 9/. Now Go has an invariant mean (see section II below) and N 0, the yon Neumann algebra generated by 92 and U] Go is eontMned in ~ . Thus ~ ' c=~ ~ 92". We note that if ~ ' ( 92" one then has ~ ' = 9A" h 9[' A U (G). ~ ' then consists of those elements in the center of 92" which commute with all Ug, g ~ G. In particular the central reduction of ~ (see below) then yields mutually disjoint representations of 92. A further specialization of particular interest in the study of ground states at finite temperature [14] has equivalent formulations stated in the Theorem 3, Let again {92, ~} be as in Corollary 1 above wit.g {Pg, o~} a weakly asymptotically abetian system, [2 a vector cyclic /or ~ invariant under Ug, and ~R' contained in 92". The ]ollowing are equivalent (i) .(2 is cyclic/or OA' (or separating ]or 92").

(~) {92'~ v(a)}" = ~. (ifi) 9A" f~ v ( a ) ' g 92" A 92'. Pro@ Writing ~ 1 = { 9 2 ' w U(G)}", (ii) reads ~ 1 = ~ and (iii) ~'~ C_9A" ~ 92'. Since :~' ( 92" or 9.1'< ~ one has ~1 ( ~ - Therefore (of. proof of Theorem 1) EoN1E 0 is abelian and since E 0 ~ U(G)", E o ~ E o is a yon Neumann algebra. From (i) E o ~ E o is maximal abelian on E o ~ so that EoN~E o = Eo~Eo. Thus their eommutants N'~E o and N ' E o coincide and N'x = N' or N~ = N (since E o has central carrier I). On the other hand (iii) trivially results from (ii) since ~ ' = 92" ~ 92' ~ U (G)' as

Asymptotically Abeliaa Systems

107

we noticed above. In turn (iii) :implies ~ i =~91 and hence that D is cyclic under :~1. Since ~Q is invariant under U(G) and each Ug induces an automorphism of 91', D is then cyclic for 91' i.e. one has (i). We note that the cyelieity of D for 91' is a property which is conserved under the central reduction of ~ discussed below. This follows from the fact that the reduction of ~ is a reduction of ~1 because of equality (ii) of the Theorem. The extronal invariant states (E-states) over 91, The states over 9t invariant under G (the G-invariant states) form a convex w*-relatively compact subset of the topological dual 91" of 91. Special interest is attached to the extreme points of this convex set to which we give the name of E-states. E-states can be characteEzed in a variety of ways expressed in Theorem 4. With the assumptions and notation o] Corollary 2 the [ollowing are equivalent/or the G-invariant state q) over 91 (i) /or all A, B C 9[ (D I z~e(A ) M e ( B ) 1D) = @5, A} <¢, B). (ii) D is the only G-invariant vector o / J f i.e. Eo = E~ where E~ denotes the orthogonal projection in ~ on the vector ~. (iii) For all A1, A~, B ~ 9_1 (~2[ ze(A~) M®(B) ~e(A2)ltg) = <~, A~A25
(D I ~re(A) M¢(B) If)) = (D 1 J r . ( A ) E o z . ( B ) ID), thus since Q is cyclic for ~r. (91) and (~, A) (q), B) = (D 1~o(A)l-Q) (D[ ~ ( B ) [D)

(5) (6)

(i) is equivalent to E 0 = )9) (t? 1 = E~. On the other hand (iii) is obtained from (i) by setting A = A1A 2 and permuting A 2 with M e ( B ) C ~ ' . Conversely we obtain (i) from (iii) by setting A 1 = A and taking for A~ an approximate identity in 9t. Now (iv) is evidently equivalent to (iii) due to the eyclieity of D for ~v (91). (iv) and (vi) are equivalent because the Mo(A), A C 91, are dense in N'. (vi) is equivalent to (v) because ~ ' is abelian. Finally (vii) is equivalent to (v), for, with E' a projection in ~ ' , m~,~zre g ~5= e ) ~ e and cog,g z e is invariant under ~ . Thus ~ , ~ ~e = a ~ and @ ( A ) D ta.Q) = @(A) E'DtE'.f2 ) = @(A).Q IE'K2) for all A in 91. (7) Since ~ is cyclic under 91, (aI - E')D = 0 and, since D is separating

108

S. I)OPLIOHEt¢,R. V. KADISON,D. KASTLEI¢and D. W. Ro]~INSON:

for ~ ' , E ' = aI. Hence E ' = 0 or I and ~ ' = {bI}. Conversely, if ~ is all bounded operators and co is an invariant linear functional such t h a t 0 < co < q5 from [16; L e m m a 2.2] co = c o T ' ~ ¢ with T' a positive operator in ~ ' . Since ~ ' is the scalars co = aq5 and q5 is extremal among the invariant states of 92. Decomposition o/ an invariant state into E-states. With the assumptions and notation of Corollary 2, suppose, in addition, t h a t ~ ' , which we know to be abelian, is generated by its minimal projections {F;.} (i.e. those non-zero projections in ~ ' which dominate no other non-zero projection in ~ ' ) . I n this case, ~ F x has commutant F ~ ' F ~ = = {aF;~} on F~(J/z); so t h a t ~ F ~ is the algebra, of all bounded operators on F~(J~). The U~F~ induce automorphisms of ~ F ~ and tgx(= F~/2) is invariant under all UgF x and cyclic for z¢(gA)F~. From Theorem 4, (v) and (vii), ~z gives rise to an E-state ~bz of 9.1, where ~ ( A ) = (~(A)Q'~I~Q'~), with Q~ = ~/I[[2~II -- noting that Q~ 4= 0 since .O is separating for ~ ' being cyclic for ~ . We have

and E~]I~II ~ = [IY2]I~ = 1. I n this way we have expressed fi5 as a convex sum of E-states ~5~.. If ~ = ~vay~yt with ~Sy! an E-state for each y and a v > 0, then ~5;(A)= @ ¢ ( A ) T ' Y 2 1 T ' Q ) for some T ' in ~ ' b y [16, L e m m a 2.2] and all A in 92. Thus F ~ T ' ~ = b~Q'~ and q5' _- ~.lb~I,¢~. By extremality #~ = ~b~ofor some )~o, Ib~J = 1, and bz = 0 for all )~ 4 )*o. I t remains to note t h a t if Xzcaqbz = 0, where Xlezl < 0% then c~ = 0 for all 2., in order to establish the uniqueness of our decomposition of into E-states. For this note t h a t ( ( ~ ¢ ( A ~ ) V ~ , + ' ' ' + zo(An)U~,),C2'~]Y2'~ ) = #~(Ax + - • • + A~) so t h a t Zc~ w~, vanishes on the *-algebra generated by av(91) and {U~}. Since Xic~l < 0% 2Jc~coe, is normal and vanishes on the weak-operator closure, ~ , generated by this algebra. Thus 0 = Ec~co~,(F,) = e~ for all 2; and the uniqueness of our decomposition follows. With suitable separability assumptions, the analogous argument using direct integral decompositions of ~ and 54f relative to ~ ' , yields a (unique) direct integral decomposition of an arbitrary G-invariant state of 9.1 as a "convex" integral of E-states. II. The Case of an Amenable Group We first recall a few definitions and results concerning amenable groups. For a locally compact group G let us denote b y ~(G), and ~o(G), the C*-algebras of continuous complex valued functions on G respectively

Asymptotically Abelian Systems

109

bounded, and vanishing at infinity, with the sup norm fl [Io~. We will sometimes denote a function /C ~(G) by the symbol f(O) where the roofed letter indicates a dummy variable. A mean over G is a positive linear form ~ over the C*-algebra C#(G) of unit norm (i.e. ~(1) = 1 where I denotes the constant unit function). The mean U is respectively left. or right-invariant if ~{/(hO)} = ~{/(#)} or ~{/(Oh)} = U{/(O)} for all f ~ ~ (g) and h C G. I t is (two sided) invariant if it is both left, and rightinvariant. The existence of a left-invariant, right-invariant or invariant mean over G are equivalent requirements which characterize a subclass of locally compact groups which we will call amenable (for the properties of amenable groups see [17]). Amenable groups can be characterized by the existence of M-filters as defined in [7; Definition 1]. Every abelian or solvable locMly compact group is amenable. Every compact group is trivially amenable. However, no non-compact semi-sixnple Lie group is amenable [18] (for instance SL(2, R) is not amenable). The Lorentz group is not amenable. The group of euclidean motions in 3-space is amenable since each extension of an amenable group by an amenable group is amenable [17]. If G is amenable and non compact the restriction of every left- or right-invariant mean to ~0 (G) vanishes. In what follows we denote by 9A~ the linear space of all weakly-continuous, weakly bounded functions from G to 9/(i.e. for each ~b in the topological dual ~* of 9A and Z ~ 92~, @5, X(~)) ~ ~(G)). We note that each X ~92~ is norm-bounded (i.e. Sup llX(g)lt = ttXtI~ is gEG

finite) from the uniform boundedness principle [1 1; Chapt. II § 1 p. 49]. We denote by 9/** the von Neumann enveloping algebra of 9~ (the topological dual of Pg* with its strong topology, see [15, Chapter 12]). In the three next paragraphs we describe means of vector or operator valued functions. Then we introduce the notion of u'asympt°tic abelianness relevant to C*-atgebras acted upon by amenable groups which allows us to derive properties analogous to those of the previous section. The mapping M , / r o m 9.1 to 91"*. This paragraph deals with averages of the functions of the type g --> ~ (A), A ~ 9.1,from G to 9Awith invariant means over G. The following lemma first gathers elementary facts on "vectorial means". Lemma 1. Let ~ be a C*-algebra and G a locally compact group. To each mean U over G there exists a unique mapping ~ [rom 9.1a to 91"* such that ( ¢ , ~ ( X ) ) = V { ( ~ , X ( # ) ) } ]oraU

~*.

(11)

This mapping ~ is linear, bounded o/ norm not exceeding 1 (i.e. li (x)II =< lizti /or all X ~gA~e), G positive (i.e. X(g) >= 0 /or all g ( G 8

Commum ma~h. Phys., Vol. 6

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S. DOPLICHm~,1~. V. KADISON,D. KASTLEI~and D. W. RoBI~rSOX:

implies ~(X) > O) and such that,/or all A C .91 ~(A) -- A

(t2)

~ ( A . X) = A . ~(X)t ~ ( X A) ~(X).AJ

(13)

(here A, A . X and X . A are the elements o/ 9/~ respectively defined by A(g) = A, {A . X} (g) = A . X(g), { X . A} (g) = X(g) . A). I] G is amenable and non compact and U is right-invariant, (7(X) vanishes whenever X vanishes weakly at infinity (i.e. whenever, ]or each q~ ~ 9.1", <¢, X(~)) ~ ¢go(G)). I/ G is amenable and ~? is right-invariant then ~ is right-invariam (i.e. ~(X(#h)) = ~(X(#))/or all g ~ G). Pro@ One has

x(#))}l

Sup

gEG

x(g))l =<- II [l il/lloo

therefore q5 ~ 9/* -~ U{@b X(0)> } is a linear form on 9/* of norm ~ llXlloo. For posi~ivity, note that ~{(~5, X(0)>} > 0 if ~5 is a state of 9/ and X ~ 0; while (12) stems form the fact that 7(1) = 1 . For X C9/~ and A E 9/one has V{<¢, A . X(O))} = v{ = <~, A @(X)> and analogously for right-multiplication by A (we denote by A * the transpose of left-multiplication by A and note that multiplication by A ( 9/in 9/** can be defined by double transposition). The last assertion of the Lemma follows immediately from the definition (11) and the fact that ~ I eg0(G) = 0. As an immediate consequence we now have Lemma 2. Let 9/ be a C*-algebra, g ~ G-+ o:g a strongly continuous homomorphism o] the locally compact group G into the automorphism group o] 9/(i.e. g ( G-~ ~g(A) is a norm-continuous ~unction of g [or all A C 91). Let ~] be a mean over G; and let

M , ( A ) = ~(e~(A)),

A ~ga.

(14)

Then M n is a linear positive bounded mapping/tom 9/to 9/** o] norm not exceeding 1. I/ G is amenable and ~1 is right invariant ~f n is such that M,(o~g(A)) = M , ( A ) ,

A Eg/, g ~ G .

(15)

Moreover the means M~ (A), A E 9.1 are G-invariant elements of 9/** in the sense that,/or all g ~ G, and each left invariant mean ~] over G, o~g(i,(A)) = M~(A)

(15a)

where ~g denotes the action of G on 9/** obtained by double transposition. Pro@ M is evidently linear, positive since A > 0 implies ~g(A) > 0 for all g ~ G, and bounded by 1 because Ileg(A)lt = lIAtl, g 6 G. The invariance of M, for a right-invariant ~ follows from the definition (14)

Asymptotically Abelian Systems

1t 1

and the last assertion of Lemma 1. For (15a) note that

) = <~, Mr(A)> for each ~ in 92*. The mapping M r on the dual space 92*. In the same setting as for the last proposition we now describe the transpose of the operator Mn, which we again denote by M,. Lemma 3. Let 9A be a C*-algebra and g--> ~g a strongly continuous homomorphism o/the locally compact group G into the automorphism group o] 92. To each mean ~ over G there exists a unique mapping M , of 92* into 92* such that

(M,~@), A ) = },

A E92, ~ 492*.

(16)

This mapping M , is linear, positive and bounded o/norm not exceeding 1. I / G is amenable and ~] is right-invariant M , is idempotent and projects onto the set o/G-invariant elements o/~A* (i.e. those qDE 92* such that = / o r all A C 92 and g E G). I] V is an arbitrary mean and q5 is G.invariant M r (~)) = ¢. Proo]. One has, for all q5 ~ 92* and A C 92 ]<¢, Mr(A)>] ~ IIq)]I tfM~(A)II < IlqoII . !]AII therefore ~b --> M~j(~5) is a linear mapping in 92* for which I]M~ (~b)ll < I[~bH, ¢ ~ 9 2 " . Further, for ~ > 0 and A > 0 one has M,(A)>=O and <~, M,(A)> > 0 whence M r ( ¢ ) > 0 and M~ is positive. If ~ is rightinvariant M , ( ¢ ) is G.invariant for all ~5 C 92* as a consequence of (16) and (15). On the other hand G-invariant elements of 92* are invariant under M, for an arbitrary mean V due to (16) and the definition of a mean. Thus for arbitrary ¢ one has M , ( M , (~)) = M r ( ¢ ). We note that, since M r projects onto the G-invariant elements of 92*, the fact that M, = 0 for some right-invariant mean ~ implies that there does not exist any G-invariant state (= positive linear form of unit norm) in 92*. Conversely the absence of G-invariant states entails the vanishing of all operators M, (in 9A* or from 92 to 92**) for all right-invariant means over G. A last remark is that, if 92 has a unit, the mapping M~ is of unit norm (obvious by setting A = I in (16)). But if 92 = fir ~ {~I}, with J a two-sided ideal, the M, might be trivial in the sense that M, @) {A ~ XI} = 2 for all states ¢ over 92 and all ~. Means in covariant representations. W e consider now as in Section I eovariant representations (~, U) of the pair (92, G) and describe means of such operator valued functions as g-+ Ug or g ~ Ugh(A)U~ -z. We begin with a lemma analogous to Lemma 1 describing means of operatorvalued functions on G. 8*

112

S. DOI'LlCltER,g. V. I{d~DISON,]). KASTLERand O. W. ROBINSO~X:

Lemma 4. Let Gf ( ~ ) be the algebra o] bounded linear operators on a Hilbert space dd°, G a loea.lly compact group and £r the set o] /unctions g-+X(g) ]rein G to ~f(Jd') such that g ~ - ( 9 1 X ( g ) IV) is a continuous ]unction. on G ]or all 9, ~o E JF and Sup tIX (g)tl = ttX]I~ is finite. To each gE(~

mean ~ over G there exists a unique operator ~1/rom 3f to 2fl (J/¢') such that (~1 ~(X)IV) = n{~l X(~)IV)},

~, V C ~ f .

(17)

The operator (7 is linear, bounded o] norm not exceeding 1 (i.e. lI(l(X)I( g <-_ IlzlI~ /or all X ~ 3Y), positive (i.e. Z(g) > 0 /or all g ~ G implies (X) > O) and such that,/or all A ~ Gf (J~t°), with notations analogous to those o/Lemma 1 ~(A) = A (18) ~ ( A . X) = A . q ( i ) (19) q ( X . A) = q ( X ) - A . Furthermore (7(X), X ~ YK, is contained in the bicomrautant o/the range o/ i (i.e. (I(X) C{X(g) lg ~ G}"). ~ Proo/. Analogous to the proof of Lemma 1. We have [~{(~[ X(~)iV)}I < Sup I(vl X(g)IvDi < Ii~Ii liViI tI/II~ gEG

whence, by giesz's Theorem, the existence of ¢/(X) of norm g IixG follows. Positivity and property (18) are proved as in Lemma 1 and properties (19) in an analogous manner: one has, for left multiplications

(q~t q(A . X)IV) = ~]{(Vl A X (~) IV)} = ~((X* 9] X(~)IV)} = (-'4"99t (7(X) iV) = (q~] A ~ ( X ) I v / ) . The fact that q(X) is contained in ~he bieommutant, of the range of X is then an immediate consequence of (19). Proposition 1. Let G be an amenable locally compact group, g-+ U a a strongly continuous unitary representation o/G on a Hilbert space, E o the orthogonal projector onto the G-invariant vectors o/ ~ (i.e. those V C d/C such that UaV = V /or all V ~ JY). Using the notation of the preceding Lemma we then have that (I ( U ~) = E o/or all right or le/t-invariant means ~7 over G. P~vo]. We first observe that, due to (19), for each h E G and a rightinvariant ~1 ~(U) G = ~ ( V ; G ) = ~ ( V ~ ) = ~ ( V ) . (20) By taking means, using (19) and (18) again, it follows that q{q(U) G } = ~(q(C)) = ~(U)~ = ~ ( V ) , 2 The above Lemma 1 is esseatially identical with Lemma 2 from which it can be deduced by application to ~he universal representation of ~. However we prefer to present straightforward independent proofs.

Asymptotically Abelian Systems

113

thus ~(U) is idempotent. To prove that it is self adjoint we note that, taking adjoints in (20), U~(~(U)* = @(U)*, whence we derive as above =

=

=

Hence ~ (U)* is self adjoint and ~ (U) is a self adjoint projection such that U~q(U) = ~j(U) U~ = ~(U), h E G, i.e. such that its range is contained in that of E 0. Thus q ( U ) E o = "7~(U)and, using (19), (U)Eo =

Eo} =

Eo.

Therefore ~ ( U ) = E 0. Finally, for a left invariant ~], ~]l given by UI(/(0))= V(/(g-1)) is right invariant and ~ ( U ) = ~ I ( U * ) = ~i(U)* =

E*

=

E o.

Corollary. With G, U, ~ and U as in the preceding Proposition and %a continuous character o/ G, ~](~U)= Ez, the orthogonal projection on the space o/vectors ~o ~ 9~ such that Ug~f = Z (g)~ for all g ~ G. Proof. Apply the Proposition to the representation g -~ ~ (g) Us. Lemma 5. With ~Aa U*-algebra and g ~ ~g a strongly continuous homomorphism of the locally compact amenable group G into the automorphism group of 9.1, we recall that a covariant representation (~, U) of the system {~, ~} is a pair consisting o/ a *-representation 7~ o/ OA and a strongly continuous unitary representation U o/G such that 7~( o:g(A )) = U g~ (A ) U g- i /or all A C 9.1 and g C G. Using the preceding notation we then have that, for a left invariant mean U over G, ~{U;~(A) U; -1} = ~r{M,(A)} (= m'~(A)) (21) where the ultraweakly continuous extension of 7~ to 91"* is again denoted by 7~. The m:~(A), A ~ 91, are elements of ~(91)" invariant under G in the sense that U m (A)

u;

= m

(A) .

Proof. Equation (21) results immediately from the comparison of definitions (16) and (17). The rest of the lemma follows from (~1 U / ~ ( M , ( A ) ) U; i lye)= V{(wv;~,~, v;~z~I ~# (A))} =

= U{(V[ ~(%# (d))IV)} = U{(Vl =(~# (A))IV)} Asymptotic abelianness in mean. We begin with the Definition 2. Let {91, c~) be a pair consisting of a C*.algebra ~1 and a strongly continuous homomorphism g -~ c% of the locally compact amenable group G into the automorphism group o/91. Let U be a right or left invariant mean over G. The system {91, o:} is called an (abstract) ~-asymptotieMly abelian system if, for all A, B ~ 91 and each state q~ over 91 U{(qS, A . c~#(B) -- e# (B)- A)} = O. i f 9A is a concrete C*-algebra acting on a Hilbert space ~ and g --> U~ is a

114

S. DO~LICHrR,R. V. KADISON,D. KASTLE~and D. W. ROBrNSO~¢:

strongly continuous unitary representation o/ G on ~ such that c~g(A) = UsA U~-1/or all A C91, g ~ G, {91, ~} is called a concrete u-asymptotically abetian system whenever the condition above holds/or all vector states

o]91. Remark 1. Each covariant representation of an abstract V-asymptotically abelian system {91, ~} yields a concrete ~-asymptotically abelian system. Remark 2. For a non-compact amenable group G weak asymptotic abelianness of the abstract system {91, ~} obviously implies U-asymptotic abelianness for all right or left invariant means V over G. Remark 3. The set of right (left) invariant means V for which the abstract or concrete system {91, at} is ~-asymptotically abelian is a w*-closed (and thus w*-compact) convex subset oi the topological dual space of ~ (G). Lemma 6. Let U be a right or le/t invariant mean over the amenable group G. The concrete system (91, c~} is v-asymptotically abelian i/ and only i/V { U ~ A U~1} (which we denote by m R(A )) is contained iu me ceTtter o] the weak closure 91" o/ 9A /or all A C 9£ Analogously the abstract system {91, a} is v-asymptotically abelian i / a n d only i / M y ( A ) is in the center o/ 91"*/or all A ( ~I. Proo/. From (19), ~{(¢, B ~ ( A ) - ~0(A)B)} = (~b, B M , ( A ) so that the abstract system {~/, ~} is u-asymptotically -M,(A)B); abelian if. and only if MR(A ) lies in the cen~er of 91'* for each A in 91. Replacing M R above by ran, we conclude that the concrete system {91, ~} is u-asymptotically abelian if and only if m,(A)~91'r~91" Ior each

AE~. Theorem 5. Let (~, a} be a concrete v-asymptotically abelian system acting on the Hilbert space J~ with V a te/t invariant mean over the group G. We denote by E o the projector onto the subspace o/vectors ~v ~ ~ such that U ~ = ~ /or all g ~ G and by ~ the yon Neumann algebra generated by 91 and U (G). Then (i) E o ~ E o is abelian with the consequences stated in (i), (ii), (~) o/ Theorem 1. (ii) I / E o has central carrier I in 9~ the mapping m, o/Lemma 6 co. incide~, on 91 with the mapping M o/ Theorem ] (thus the restriction o/ m, to 91 does not depend upon the choice o / V within the set considered in Remark 3 above) 3 (i~) M and m, coincide also on 91" i/ and only i/ the conditions o/ Theorem 3 are realized or equivalently i] the system {91", a~} is V-asymptotically abelian. 3 This explains why the M could be obtained in [7] by use of an arbitrary M-filter (most statements and proofs in [7] are - - except for the last two sections -valid for amenable groups rather than only abelian groups).

Asymptotically Abelian Systems

115

Pro@ By Lemma 6 we have that m , ( A ) . B - B . m~(A) = 0 for all A, B C 91. Now E o m ~ ( A ) B E o = E o ~ ( U s A U~-I)BEo = ~(Eo A U~-I) B E , = E o A E o B E o from (19) and Proposition 1. We conclude that E o A E o B E o -- E o B E o A E o = 0 whence (i). In order to prove (ii) we observe that, for A ~ 91, using (19) again, m~(A)E o = E o A E o = M ( A ) E o .

Thus m , ( A ) = M ( A ) , since ran(A) and M ( A ) are in ~ ' . To prove (iii) we first observe that Lemma 6 entails the equivalence of condition (iii) of Theorem 3, namely 91" (~ U (G)' g 91', and the u-asymptotic abelianness of the system {91", g}. However the latter implies that m, = M over 9t" using the equation above. Now this last property implies in turn property (iii) of Theorem 3 since M (91") %_~ ' and m r (9A") = 91" ~ U (G)'. Definition 3. Let 91 be a C*-algebra, G a locally compact amenable group, g-> ega strongly continuous homomorphism o/ G into the automorphism group o/ 91. A state qb over 91 is called U-weakly clustering whenever U{(~b, Ace(B))} = ~ ( A ) . V{(q~, a~ (B)}}/or all A, B C 91 with U a right or le/t invariant mean over G. A G-invariant state q5 is called weakly clustering whenever there exists a right or le/t invariant mean U over a /or which it is ~7-weakly clustering i.e. such that ( ~ , A . Mn(B)} = (q~, A) (~b, e } .

(22)

Remark. For a general state ~ and denoting by the same symbol the ultraweakly continuous extension of q5 to 91"*, u-weakly clustering can be formulated as the requirement that (~5, M,,(B)~) = (~5, M~(B))~ for all self adjoint B E 91 (the last condition is known to be equivalent to <#, A . Mn(B)) = (~b, A) @5, Mn(B)) for all A 6 91"* [see, for example, 27, Lemma]. We can now rephrase Theorem 4 in the following way: Theorem 4a. With {91, o~} an ~-asymptotieally abelian abstract system and q5 an invariant state over 9.1 the [ollowing are all equivalent to (v), (vi), (vii) of Theorem 4; (i) qb is weakly clustering. (ii) ~ is the only G-invariant vector in 2/¢'. (iii) Condition (22) holds/or all right or left invariant means over G. (iv) M : ~ ( A ) = (qS, A > . I / o r all A Egt and all ~7 such that {91, ~} is u-asymptotically abelian.

III. Examples A. A System where the Group is compact Let 9.1be the full algebra of n x n complex matrices, G be the compact group of unitary n × n matrices with Haar measure # and define

116

S. DOPLICItER,R. V. KADISOI%D. KhSTLERand D. W. ROB~-SO~:

ocv(A ) = UA U -~, A ~ 91, U ~ G. We then have

f

¼Tr(A)Z

so that (cf. Lemma 6) the system {91, g} is #-asymptotically abeHan. Remark. If we let H be the direct sum of G and any locally compact, non compact group, then composing ~ with the quotient mapping modulo this group yields a system which is U-asymptotically abelian for each invariant mean ~ over H but not weakly asymptotically abelian.

B. The Algebra o/Canonical Anticommutation Relations: a System which is weakly but not Norm asymptotically Abelian With it the C*-algebra of c.a.r, on L~(R n) (generated by the smeared out bounded creation field operators ~o([), [ in L2) , let ~ be the automorphism of 91 determined by ~@(/))---~0(/~), / in L~. where /x(Y) = / ( y -- x). Then ~ is a strongly continuous representation of the translation group of R" by *-automorphisms of it. We show first that the system {it, ~} is weakly asymptotically abelian. We recall that 91 is the linear space sum of 91~ and ito where it~ is the norm closure of the linear span of monomials in ~#(/) and ~y*(/) having an even number of terms with 910 the same for monomials having an odd number of terms. From the commutation relations it follows that l i r a II[~(A),B]_It = 0 for each A in its and all B in 9/, while

l i r a II[~(A), S]+II = 0 for all A and S in it0. If we show that ~(a~(A)) -~ 0 as x-+ oo 4 for each A in 910 and each state ~ of it, then ~([a~(A), B)]_) = ~B(ax(A)) -- ~B(a~(A), where ~:8(C) = q~(CB) and ~:~(C) = of(Be), tends to 0 as x-+ co for A in 91o, while [a~(A), B]_ = [a~(A0), B]_ + [a~(A,)B]_ with A 0 in 910 and At in 91o. Of course it will suffice to deal with self-adjoint A. Passing to the representation associated with ~, we may assume that 9I acts on ~F with cyclic vector D and that F(A) = (As9 I ~2). Suppose that ~x~(A)~ tends weakly to F in 2/Y where x~ -~ co with n. Then

II~P = "li~ (~(A)'°" I m=oolimo~,~(A).Q) = n=~olim(m=oolim(ex~(A) e~(A)f2 if2)) lim q~(~x"(A) °~x'(A)) = li~n (m=~

= -- Iim ( lim ~(oc~(A) %~(A)) = -ItVlr. n=oo

~ o o

I t follows that ~0 = 0 and that lira (o~(A)DID) = lira ~0(e,~(A)) = 0. 33-+ oo

~--~oo

This fact follows from an argumen~ of :F~.Powntcs (Princeton thesis to o.ppear). We wish to thunk R. Powers for helpful discussion concerning this example.

AsymptoticMly Abelian Systems

117

If {91, ~} was norm asymptotically abelian, ]l~o([~) ~o(g) - ~o(g) y~(1~)]1-~ O as x - , oo while IIF(]x)~(g)÷ ~o(g)~o([x)[I = 0 for all x; so ~ha~ it~'(/.~) ~°(g)tl--> 0 as x ~ o o . If 1~ and g have compact disjoint supports, however, il~(l~) ~(g)ll = tllil Ilgtl + 0 if both [ and g are non zero. Thus {91, c~} is not norm asymptotically abelian. In the last examples all M,~ are zero, so that there are no G-invariant states (cf. the remark following Lemma 3).

C. The C*.Algebra o/ a locally compact Abelian Group Acted upon by the dual Group With G a locally compact non discrete abelian group and ~ its non compact duaI group topologized as usual, the C*-algebra of G is the algebra ~fo(O) of continuous functions on ~ vanishing at infinity with pointwise multiplication, complex conjugation and the Sup norm I! It~For each k C ~ we define { ~ (1)} (p) = ] (p -- k), [ ~ ~0 (~), P C ~. Since <#0(G) = 91 is abelian we have a trivially asymptotically abelian system {91, c~}.Now the set A (0) of Fourier transforms of Lt-functions on G is a dense sub-*-algebra of 9[ and therefore, to prove that all means on 91 and on 91" vanish, it suffices to verify that they vanish on A (G). Now for [ C A (G) and ,u a bounded measure over G one has

@, M~(/)} (= ( M , ( ~ ) , t)) = v{@~,~ (1)}} = 0; since (/~, ~Z ([)} (= ] * #) is an element of A (G)C ~0(G). D. The Twisted Convolution Alqebra L 1(E, o) Acted upon by the Translations o] the underlying Symplectic Space E Let E be a finite-dimensional real vector space equipped with a nondegenerate bilinear form o. The twisted convolution algebra L 1(E, a) over E (see [19]) is obtained by taking the set of Lebesgue-hltegrable functions over E with its Ll-norm II II1 and the following *-operation and product X : I* (~) = t ( - ~), i, g C 51 (E, a) (23) (1 x g) (~) = fe -~°(~,~) 1(~) g(y~ -- ~) d~, ~f ~ E L 1(E, a) is shown to be a Banach *-algebra possessing a unique irreducible representation up to unitary equivalence. Since this representation is faithful, the completion L 1(E, o) = 91 of L 1(E, o) in the operator norm is a C*-algebra which is shown to be isomorphic to the C*-algebra of compact operators on the irreducible representation space. Furthermore the *-operation and product (23) can be extended to the set ~1 (E, G) of bounded complex measures over E by defining, for each continuous function / on E with compact support @*, l} -- (/~, 1"}, /~, ~, ~ M~(E, cr) @ x v , l } = f d l ~ ( ~ ) f d v ( ~ ) e-'~(n,¢)/(~+ ~).

(24)

118

S. DOrLIC~EI¢, R. V. KADISOI%D. KXS~LERand D. W. RoBInSOn:

MI(E, a) with the [i ]I1"n°rm of measures thus becomes a *-Banach algebra in which L I (E, a), identified with the set of measures absolutely continuous with respect to Lebesgue-measure, is a (two-sided) *-ideal. Furthermore the definitions (24) are compatible with the extension of the operation (23) on L 1(E, a) to its C*-eompletion 92. We now assign to each u E E the *-automorphism a~ of 92 defined by ~ ( a ) = ~ x a x ~_~ a E 92

(25)

where 8u is the Dirac measure on E at u. Since, according to (24), one has 6. = ~_~ = ~ (26) ~u X (9~ = ei~(u,v)(~u+v

{

u -+ ~ is a homomorphic mapping of the additive group of E into the group of *-automorphism of 92. Further, one easily calculates from (24) tha$, for ] E L1 (E, a),

{~.~(1)} (~) = ~-~(~,~)I(~)

a.e. in

~ ~E.

(27)

We now show that /or each a ~ 92 and each continuous linear [orm q~ over 92 one has

<~, ~,(a)> ~ <~, a>

as

u-+ 0

(2S)

and -+ 0

as

u-+ c¢,

(29)

that is, ~ % ( ~ ) . Since each continuous linear form on 9/ is the difference of two positive forms and since L1 (E, a) is dense in 92 it suffices to prove (28) and (29) for a positive form q~ and for a = 1, /E LI(E, a). Now each positive form ~ on LI(E, a) corresponds to a function ~ E L~ (E); so that, by (27),

<~, ~ ( l ) > = f e-~c~,~)/(~) ~(~) d~ = l~(~u) (30) where ] ~ is the Fourier-transform of the Lyfunction /~. As a consequence of the weak continuity of the mapping u E E--> -+ ~u (a) E 92 expressed by (28) it is known that this mapping is also continuous in the norm topology of 92 [28; 10.2. Corollary]. On the other hand it follows immediately from (28) that one has the following asymptotic abelian property: a, b E92 <~),~(a).b-b.~,(a)>~O, as u-+oo, q) E92* (31) and that, for every mean ~ on E and all a ~ 91 and ~b E 92* V(<~b, ~a(a)>) = (~b, i , ( a ) > = <21£~(q~),a> = 0 . Therefore all means on 92 (or on 92*) vanish.

(32)

Asymptotically Abelian Systems

I 19

Acknowledgements. Our thanks are due to G. GALLAVOTTI,D. I:~UELLE and A. V~BEV~E for enlightening discussions and to R. HnAG, N. I-tUG~HOnTZand M. WI~INK for communicating to us an early version of their work. Grants from the Minist&re de l'Education Nationale and from OINR and NSF made possible the collaboration during June and July 1966 when these results were obtained. We wish to express our gratitude to Dr. L. MOC~TA~ for the kind hospitality of the I.H.E.S. and to the University of Pennsylvania for the use of its facilities during the preparation of this paper. References 1. HAAG,R., and D. K~STLER: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964). 2. DOPLICHER,S. : An algebraic spectrum condition. Commun. Math. Phys. 1, 1 (1965). 3. H.~AG,R. : Mathematical structure of the BCS model. Nuovo Cimento 25, 281 (1962). 4. ARAb;I,It., and E. J. WooDs: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637 (1963). 4a. ROBInson, D. W. : The ground state of the Bose gas. Commun. Math. Phys. 1, 159 (1965). 5. DOPLICHER,S., D. KASTLEt~.,and D. W. RoBInson: Covariance algebras in field theory and statistical mechanics. Commun. Math. Phys. 8, 1 (1966). 6. RU]:LL~, D. : States of physical systems. Commun. Math. Phys. 8, 133 (1966). 7. KASTLEI~, D., and D. W. Ro~Lwso~: Invariant states in statistical mechanics. Commun. Math. Phys. 8, 151 (1966). 8. Robinson, D. W., and D. RVELLE: Extremal invariant states. I.tt.E.S. Ann. Inst. H. Poincar~, to appear. 9. L)~Fo~]), 0., and D. RUELLE: Integral representations of invariant states on B*-a]gebras. J. Math. Phys., to appear. 9a. ST~R~R, E. : Large groups of automorphisms of C*-algebras. Commun. Math. Phys. 5, 1 (1967). 10. RVELLE, D. : The states of classical statistical mechanics, Preprint 1966. Ii. DUNFORD, N., and J. T. SC:gWARTZ:Linear operators, Part I, New York: Interscience Publ. 1958. 12. DIxMI~, J. : Les alg~bres d'op6rateurs dans I'espaco hilbertien. Paris: GauthierVillars, 1967. 13. K~zSON, R. V. : A topological Burnside theorem, to appear. 14. HA~¢, R., N. HUGE~--~OL~Z,and M. W ~ N ~ K : On the equilibrium states in quantum statistical mechanics. Preprint 1966. 15. ])IX~IER, J. : Les C*-alg~bres et leurs representations. Paris: Gauthier-Villars 1964. 16. DYE, H. A. : The Radon Nicodym theorem for finite rings of operators. Trans. Am. Math. Soc. 72, 243 (1952). 17. P ~ n , J. P.: Sur une classe de groupes loealement compacts remarquabtes du point de vue de I'Analyse harmonique. Th~se de 3brae Cycle, ~ancy (1965). 18. T)-KEI~OUCH~,0. : Sur une elasse de fonctions continues de type positif s u r u n groupe localement compact. Math. J. Okayama Univ. 4, 143 (1955) 19. K ~ s T ~ , D. : The C*-atgebras of ~ free Boson field. Commun. math. Phys. 1, 14 (1965). 20. FOL~E~, E. : On groups with full Banach mean value. Math. Scand. 8, 243--254 (1955).

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21. ULA•ICKI, A. : Amenable topological groups. Univ. of Seattle (1965). 22. R~ITER, H. : On some properties of locally compact groups. Univ. oi Utrecht (1965). 23. - - Sur la propri~t6 (P~) et les fonetions de ty'pe positif. C.I~. Aead. Sci. Paris, 258 (1964); pp. 5, 134---5, 135. 24. - - The convex hull of transl~tcs of a function in L 1, L. London Math. Soe. 8~,

5, (1960). 25. I(ADISOI% i~. V.: Unita,ry invariants for representations of operator ~Igebras. Ann. Math, 66, 304 (1957). 26. REICH, g . , and S. SCtt.LIED:ER:?Jber den Zerfall der Feldoperatoralgebra im Falle einer Vakuumentartung. Nuovo Cimento 26, 32 (1962). 27. K~.DISOt%1~. V. : The trace in finite operatar algebras. Proe. Am. Math. Soc. 12, 973 (1961). 28. HILLE, E., and R. S. PI-IILLIPS: Functional analysis and semi groups. Am. Math. Soe. Coll. Pub. Providence 1~.I. (1957).