Commun. math. Phys. 4, 315--323 (1967)

A Remark on a Theorem of B. MISRA

H. J. B o ~ c ~ s Institut fiir Theoretische Physik der Universitit GSttingen

Received September 25, 1966

Abstract. The two sided ideals of the C*-algebra generated by local v. Neumarm a, lgebras are investigated.

I. Introduction

B. MIst~i [1] has shown that the algebra of all local observables is simple when the following conditions are fulfilled:

1. The algebra is given as a concrete C*-algebra in a Hflbert space fulfilling the usual assumptions of locM ring systems.

2. The rings associated with bounded open regions are v. Ncumann algebras.

3. For any bounded open region (9 exists another bounded open region (91 containing (9 such that the ring associated (91 is a factor.

The third condition, however, has not been derived from the other two assumptions even when we assume that the yon Neumann algebra generated by the global C*-algebra is a factor. Since in recent years different representations of the C*-algebra of all local observables have been discussed [2], [3], [4] it is desirable to have a characterization of all two-sided ideals in the general case where 3. is not assumed. We will show that t,he theorem of Misra stays true without assuming 3., i.e. the U*-algebra generated by all local observables is simple if it contains no center. For later use we will also consider some more general algebras.

II. Assumptions and notations

We denote by (9 open bounded regions in the Minkowski.space and "write:

01 × (93 J~ (91 and (93 are spaeelike separated. (91 < (ps if (91 C (93 and there exists an 0 3 C (93 with (91 x (pa. (91 ~ (92 .if there exists a neighbourhood aI r of the origin such that

(91+ x < (92 Ior all x ~ , 4 r. ~ ' e denote by a local ring system {~ ((9)} a family of rings of operators

in a fixed I-[ilbert space 5f' submitted to the following conditions:

316 H.J. BOTCHERS :

.

a) b)

e)

2. with

a)

((.0) is a v. Neumann algebra for all 0 and ('01 C 0~ ---~ ~ (01) C R (02)

= smallest C*-algebra containing {U ~ ((.0)}.

In :~z exists a unitary representation U (x) of the translation groups

.~ ((.0 + x) = U (x) ~ ((.0) U -~ (x) b) The spectrum of U(x) is contained in the closure of the future

lighteone. e) U (x) C ~ , which can be assumed without loss of generality by [5]. 3. If (.01 x (.03 then N ((.02) ( N ((.03)' (local commutativity). 4. For any (.0 we have ~ = {U ~ ( 0 + x)}" (weak additivity).

We denote by a generalized local ring system {5f(0)} a family of rings of operators in a fixed Hilbert space ~ f submitted to the following conditions:

1. 5 P ((.0) is a yon Neumann algebra for all (.0 with a) (91 C (-03 then ~ (01) C Sf (0~) b ) =

c) ~ = smallest C*-algebra containing ( 0 5P(0)}.

2. There exists a local ring system {~ ((.0)} with a) ~ (0 ) C 5P(0) for all (.0. b) If U(x) is the representation of the translation group given by

{~ ((.0)} then ((.0 + x) = U (x) ~ ((.0) U -~ (x).

3. If 01 × (.02 then R (('01) C 5f ((.02)'. Let ~p E W and P0 be the energy operator. We say ~0 is analytic for

the energy if ~0 is in the domain of every power P~ and the sum $n

I!P~]I • ~ has a nonzero radius of convergence. ~t = 0

III. Some properties of local rings

For the investigation of the ideals of tocal ring systems we need certain properties of local rings which we study f~'st.

III.1 Theorem. Assume we have a continuous representation U(t) of a one-parametric group with semi-bounded spectrum. Moreover assume we have two projections E, F such that

U (t) E U- I ( t )F = F U (t) E U-~(t)

lor Itt < 1. If we have E . F = 0 ~hen follows U(t) EU-~(t)F= 0 for all t.

A Remark on a Theorem of B. M;~s~x 317

Pro@ I n order to make the proof t ransparent we make first a special assumption, namely, t ha t the spectrum of U(t) is bounded. I n this case U(t) = exp{ i tP} with P a bounded selSadjoint operator and hence d~ dr" U(t) E U -I (t) is also a bounded self-adjoint operator and

et,, u (t) E U -1 (t) = U ( t ) -~-~- U ( ~ ) E U -1 (3) ~ = 0 U - 1 (t)

can be wri'gten as U(t) {A ÷ - A - } U-l( t ) where A + resp. A~- are the

bounded. Assume we have already proven F ( A + - A ~ ) = 0 for n = 0, 1 . . . . . . N. We want to show tha t this holds also for n = N ÷ 1. Now F ( A + - A ~ ) = 0 implies F A + = F A ~ = 0. FU(t) A + U-l( t ) is a positive operator for ltl < I and since for a rb i t ra ry ~, C ~ f the funct ion (~v, F U (t) A + U -1 (t) ~v) is analyt ic in t, positive for real t with It[ < 1 and zero a t t ---- 0, we see that, this funct ion mus t have a zero of second order and hence by Schwartz inequal i ty

l(~p, F V ( t ) A + U ( - t ) ~ ) ) l < Itl 2 IIv2t[ ~ IIA+Itell P~ ttl < 1

and I(~, F U(t) A ~ U ( - ~)OI =< itl ~ tfvlI ~ f[A;Tiie~m.

But this implies F ~ U (t) E U -1 (t) has a zero of second order a t t = 0 tiN+ 1

and hence Y ~ U(t )EU-I( t ) is zero at t = 0. Since/~U(t) EU-I( t ) d -

is zero at t -= 0 by assumption, F ~ U (t) E U -1 (t) is zero at t = 0 b y

induct ion for all n. Since P was a bounded operator we see tha t Y U(t) E U -1 (t) is an entire analyt ic funct ion and hence identically zero.

Now the general case. Without loss of generali ty we can assume U(t) = exp{ i tP} with P a positive operator. Consider the operator e-PFU(t) EU-I ( t )e -p which is the boundary-value of an analyt ic funct ion hotomorphic in 0 < I m t < 1 and bounded by 1 in this strip. The operator e-PU(t) EU- I ( t )Fe -P is holomorphie in - 1 < I m t < 0 emd bounded by 1. Since now

,~-PF U(t) E U -~ (t) e-P = e-P U(t) E U -~ (t) F e -p

for reat t, -- 1 < t < 1, we see t h a t e-PY U (t) E U -1 (t) e -P is holomoi]3hic ha the uni t circle and bounded by 1. Since it is a positive operator for real t, Itt < l and zero at t = 0, i t mus t have a zero of second order or tle-PFU(t) EU-~(t) e-PI] < Itl ~ for ]t] < 1. Bu t this implies

ltl for It I < 1

Let h be real, then U (h) E U -1 (h) - E is a selLadjoint operator and let G +, resp. G [ be the projections onto the positive resp. negative part . G + and G~- commute with F for sufficiently small h.

318 H.J . BO.a~:gERS:

Le t now t be real then we get

0 ~ E V (t) G + (V (h) Z U ( - h) - E) U ( - t)

= E V (t) G+ U (h) E U ( - h) G2 U ( - t) - • U (t) ¢ 2 E G 2 U ( - t)

and hence E U ( - h ) G+ EG + U(h) = O.

t t

This implies ~'U (-h) G~- E U (h) = O.

I n the same manner we get :

FG[U(h) E U ( - h ) = O . From this follows:

I L e - P ~u( t ) (G+Z + E G+) U(- t ) e-P~'~ c. lti ~ it+ hi ~, j 2

since the positiv and negat iv par t have a zero at t = 0 and t = h. I n the same manner we f ind:

½ e-PF U(t) (G- U(h) E U(--h) + U@ EU (--h) G-) U(--t) 6--P <=

Adding both equations we have

i ~- ~-PEU(t){G- U(h) E U(--h)+ U(h)~ Z(--h)+ G+E+E a+} U(--t) ~-~

~" Iti ~ It + hl ~. But this gives:

e_PFU(t) E+U(h)EU(-h)2 U(--t) e -2 =<c"ttt~It+, ht~+

1 lie_F2 U(t) {(G~ -- G;) (E-- U(h) E V(--h)) + + ¥

+ (E-- U(h)E U(--t~)(a 2 --e;)} V(- - t )~ -%

Since the last term converges weakly to zero for h going to zero we see tha t the remainder has a zero of four th order.

Hence : ile-P.FU(t) EV(-t)e-Ptt ~ It] a .

Assume now we have shown tha~ e-BE U(t)E U(- t )e -P has a zero of

1 d e_PFU(t)EU(_t)e_Phas a zero of first order. order 2n. Then t~_ ~ dt 1 d

Repeat ing ~he same argument we find t~,~, ~ dt e-t~FU(t)EU(- t)e-P

has a zero of second order or e-P-F U(t) E U(- t )e -P has a zero of order 2n + 2 and by induct ion it has a zero of all orders. Bu t this implies e-PFU(t) E U ( - t ) e -P is identically zero for It[ < 1. Since it is for

A Remark on a Theorem of B. M~SS, A 319

arbitrary real t the boundary-value of an analytic function holomorphie in 0 < Imt < 1, it follows by analytic continuation that

e - P F V (t) E V - t ( t ) e - P = 0

for all real t. Since e - p is an invertible operator we get

F U ( t ) E U - I ( t ) ~ O qed.

As a next step we have to generalize a lemma proved in an earlier paper ([4] corollary 7) for our generalized situation. This lemma tells us tha t every operator belonging to a bounded region which maps one vector analytic for the energy onto another vector also analytic for the energy commutes with all translations.

III.2. Lemma. Let {5P(0)} be a generalized local ring system and {~ (0)} the local ring system contained in { ~ ((9)}. Let A ~ 9 ° ((9) for some 0, ~o ~ 9F be a vector analytic for the energy, and assume A ~0 is also analytic for the energy. Then for every (9i >~ (9 exists a projection :in fl ~9°((91 + x) ~ o such that E~0 = ~0 andA • E C N 5P((91 ÷ x) r ~ .

X X

P r o @ Let B 1 . . . B~ C 5°' ((91), xl . • • xn C J¢~, B i ( x ) = U (x) B ~ U - t ( x ) then we have

B I ( x l ) . . . B ~ ( x n ) A = A B l (x l ) . . .B , , (x~) for x 1 . . . x ~ C .

Now B l ( x l ) . . . B~(x,~,)A and A B I ( X l ) . . . Bn(x~) are both boundary- values of holomorphic functions since ~0 and A~v are analytic for the energy. Since these functions coincide for x l . . . x~ ~ ~4 z they coincide everywhere. Hence we get for BE {O SZ'((gi÷ x)}" the relation

B A y J = A B e . Let now E be the projection onto the closure of the vector space {U 5~'((9i÷ x)}"~ then we get B A E = A E . B or

AK ~ {U S~'(S~ + ~)}'. B~t also ~ c {~ s~'((9~ + x)}' and F has the property E ~ = ~0. Since ~ ' ((01) D ~ (02) for (~1 × (9~ we have

{ u s~' ((9~ + ~)}" ) { u .((9~ + ~)}', = . ~ .

Hence E and A E are elements from f] Sz(@ 1 -t- x) r~ N L qed.

In the following argument we have to consider equivalent projections. We say two projections E 1, E 2 from a fixed yon Nemnann algebra R are equivalent when we can find in R a partially isometric operator V with E 1 = V V*, E 2 = V* V. If E 1 is equivalent to E~ then we write E~ ~ E~ rood R. (For a detailed discussion see [7] chap. III .) With this notation we get

III.3. Theorem. Let {5 ~ (d))} and {N ((9)} be as in Lamina III.2. Let E be a projection in ~((9). Assume moreover that there exists a vector ~0 analytic for the energy such that ~oo ~ is dense in the I-I_ilbeI~ space.

a) If d)~ > (9 and F is the smallest projection in the center ~ (~9 ° ((~)) of 5~((9~) ~dth F E = E then E ~ / ~ rood ~((9~). 2a Commun. math. Phys., Vol. 4

320 H.J. B o ~ c ~ s :

b) If 01>>(9 then there exists an F ~ Y(01) ( ~ with F ~ E and F E = E.

This theorem enlightens the well-known result that the local rings are not finite [8], [9], [10] by showing explicitly some projections which are not finite.

Proo]. a) Let ~ be the cyclic vector analytic for the energy. Then by the l~eeh-Schlieder theorem we have for any ~((~), ~((9)yJ = ~ ([11], [5] Lemma 5). Now define the projection F by F g ~ = -50(~0JE~p. We have F ~ ~ (•J'. But since 01 > (9 there exists an (9~ × 0 and (~ < ~)1 hence

~" gt ~ = gz ( (~ jE y, = gr ((9j ~ ( ( 9 J E ~ = 5~ (0J E R (G'+=)~ = 5~ ((%)E g / .

Therefore we get

~ ( ( g j ' F / f = ~ ( ( g j ' ~ ( ( 9 j EY+ ~ = ~ ( ( g J ' ~ ( (gJ E g f

- gf((9J Z g ~ ( O J ' g t ~ = 9 ~ ( e j E g ~ .

This means F J ~ is invariant under g~((gJ' or FEgv(~0i) hence F ~ ( ~ ( ( P J ) . Since we have 5~((~JF?p= gf((91)Ey~=j~gC follows F g / = 5f((9J'F~v ~ E g F = S((9J'E~p modgP((9J ([7] chap. I i I § 1 co- roltaire de Th6or6me 2). I t is easy to see that F is the smallest projection in 3 (5°((9J) with the property F E = E.

b) If now (91 m (9 then from (1 - F ) E = 0 follows by theorem III.1. that ( 1 - F ) U ( x ) E U - t ( x ) = O for all x which are timelike. Since (1 -F) U (x) E U -x (x)e -~ o with P0 the energy-operator is the boundary- value of an analytic function, it follows that (1 - ~') U(x) E U -~ (x)e -~° vanishes for all x and hence (1 - F) U(x) E U -~ (x) = 0 for all x, which is equivalent to . F U ( x ) E U - I ( x ) = U(x)EU-~(x) for all x. If now q~ is any vector analytic for the energy and g(x) a function with compact suppor~ in momentumspaee then f d x g ( x ) U ( x ) E U - ~ ( x ) + is again analytic for the energy and we have the relation

F f d x g (x) U (x) E U-~ (x) ~ = f d x g (x) U (x) E U -~ (x) ~ .

Hence by Lemma III.2. there exists for any (P~ >> 0, a projection G with the properties G ~ A 5 # ((~ ÷ x) f~ ~ ,

G f dx g (x) U (x) E U -~ (x) q) = f dx g (x) U (x) E U -~ (x)

for all g (x) with compact support in momentumspace and all ~ analytic for the energy such that F G ~ f 3 g f ( ( P 2 ÷ x ) ~ ' ~ . This implies

z

GEq5 - E @ . On the other hand we have for B ~ { U 5 ~' ((9~= + x)}" the

retation

B F f dx g(x) U(x) E U-~(x)¢ = F B f dx g(x) U(x) E U-~(x)¢

= B f dxg(x) U(x) E U - ~ ( x ) ¢

A R e m a r k o n a T h e o r e m of B. h h s ~ 321

which implies F G = G. This means E ~ G mod 5f((~) since E ~ F >- G ->- E. Choosing 6~ >> ~0~ >> # in an arbi t rary position we get the desired result since G ff 5~((~) f~ ~ '~ .

IV. The structure of two sided ideals

Now we are prepared to study the two sided ideals of local ring-systems. :First we need a

IV.1. Lemma. Let A n be an increasing sequence of yon Neumann algebras, ~r~ C ~ for m < n. Denote by [R the normclosure of U ~ . Let

n

be a nonzero norm-closed twosided ideal of 9~ then ~ f~ ~ contains a nonzero element for some n.

Proo/. Let A = A* ~ ~ and llAlI = 1. Then for some n exists an 1

operator B ~ 9t~ with B = B* and I[A - BII g y . Since ~ , is a yon

Neumann algebra there exist projections E~, n = - 4 , - 3 . . . . ÷ 4 ,

B = ~ d y E f -~- 4 n I I

E ~ E ~ = O f o r n # m s u c h t h a t - _-<-~. From IA[ = 1

follows tha t not all E n = 0 for Inl ~ 3. Combining both equations we get

A - Z T E ~ 1 ~ - f . Denote by I I a faithful representation of 9~/U; then

we have H(E~) < ~- . Since we have again H(E~)H(E~)

= 3 ~ H ( E ~ ) follows H(N~) = 0 for n > 2 or E~ ~ g since H was a faithful representation of 91/g qed.

IV.2. Lemma. Let 92~ and N be as in the preceding lemma, and U a norm dosed twosided ideal then g coincides with the normclosure of

Pro@ Let A = A* ~ g and i]All = 1. Give e > 0 then exists a ~ , and an operator B ~ 9~ ~ I such tha t liA - BII G 2e. This holds since we can find a B~:@, . with I [ A - B ~ I [ G e and a B ~ with i l B - B1]] G e (see the proof of Lemma IV.I.) . But this implies A is a norm limit of elements in U/~ { U ~ } qed.

The combination of the last two lemmas with the results of section I I I gives us

IY.3. Theorem. Let {Jr ((~)} be a generalized local ring system and {g~((~)} the local ring system contained in {~(G)}. Assume we have a vector v d analytic for the energy such tha t ~ yJ is dense in ~ . I f ~ is a non-trivial two-sided ideal in ~ then

a) ~ ~ { ~ f~ ~} is a non-trivial ideal;

b) g is generated by g ~ ~ " ~ ~ i.e. I is the smallest norm-closed ideal in ~ containing ~ f~ ~ " f~ ~ . 23*

322 H.J. Bo~c~n~s:

Proo I. Let ~C~ be a two-sided ideal, then by IV.1. and IV.2. f~ {U ~(@)} is not empty and ~ is the norm closure of this set. Let

now A C ~ ~ 5 f (@), Then also its symmetric and skew-symmetric parts are in ~ f~ ~9~(0). Hence it is sufficient to consider the self-adjoint elements.

+ M

Let A = A* = f 2~ dEz ~ ~ P, ~ (@). Since ~ (O) is a v. Neumarm - - M

--¢ M

algebra we find that f + f dBz is also in ~ ~ ~9 ~ (0). Denote by M (0) - - M + 8

the set of projections in ~ f~ Zf (@). Then the ideal generated by M (@) is + M

norm dense in ~ ~ ~f(0) because if A = A* = f 2 dEz C %] In cj (@) then - - M

E_~ + (1 - E+~) is contained in M(&). Hence A{E_~ + (1 - E+J is in the ideal generated by M(@). But HA - (B_~ + (1 - E~))Atl <= e which means that A is in the norm closure of the ideal generated by M (@). Since now ~ is a two-sided ideal and ~ (@) ~ yon Neumann algebra, it follows from E ~ F mode( (9) and E ~ M((9) that also E ~ M(@). Now by theorem III.3. follows that for @1 >~ @ there exists a projection F in Sf(@J f~ ~ o with F E = E and l~ ~ E modSZ(@l). Hence ~ ~ ~ ~ is a non-trivial ideal since 1 ~ g. This proves a). Let now ~) be the two- sided ideal generated by ~ ~ ~ " f~ ~ then ~ < g. But i3 M(@) generates

~. I f E ~ M(@) there exist F ~ E mod:~((gJ and F E = E with F ~ ~ ~ o f~ ~ . Hence E ~ ~ which implies U M((~) <8 or ~ <8 and thus

= ~ which proves statement b) and the theorem.

¥. Application to local ring systems

If we restrict ourselves to local ring systems then it is possible to remove the assumption about the existence of a vector which is cyclic for ~ o . Theorem III.3. becomes:

¥,1. Theorem. Let {,~ (@)} be a local ring system a n d E b e ~ projection in ,~ ((9)

a) If (9~ > @ and F is the smallest projection in the center ~ (~( (g j ) with F E = E then E ~ F m o d ~ ((91),

b) Is @~>~ @ then F ~ ~(~(@J) ~ ~(9{), where ~(9~) denotes the center of the C*-algebra 9{.

Proof. Let G~ be a family of projections in ~ such that G~G# = 0 for a 4 fi, ~ G~ = 1 and in G~Y# exists a vector ~ analytic for the

energy such that ~ = G~J/f. By virtue of theorem III.3. we have FG~ ,~ EG~ m o d ~ ((91) " G~. LetF~ be the smallest projection in ~ (,~ ((gJ) with ~7¢G~= G~ then we get $'~F ~ ~ B mod~( (~ j " F~ ([7] ch~p. I § 2 prop. 2), But since now UF~5/z = J ( f follows F ~ E mod~(@J. This

A Remark on a Theorem of B. ]VI~s~ 323

]?roves s tatement a). Let now &l >~ @, then by theorem III .3 . b) we have 2 . G~ 6 ~ 0 . Hence F = ~ FE~ ~ ~ " which proves b).

This res~flt makes it possible to generalize also theorem IV.3. we get: V.2. Theorem. Let {~ ((~)} be a local ring system and ~ be the center

of 9~. Denote by ~ norm-closed two-sided ideals of 9~ then a) U is not the zero ideal if and only ff U r~ 8 is not the zero ideal b) ~ is generated by U r~ 8. c) The m a p ~ - > ~ ;~ 3 is one-to-one mapping from the two-sided

ideals of 9~ onto the ideals of 8. Proof. Since we have used in the proof of theorem IV.3. only the fact

tha t to every projection E~Sz((~) and ~0~>~(~ exists a projection F ~ 5 f (&~) ~ ~ o with F ~ E and _FE --- E the statements a) ~nd b) are a simple consequence of IV.3. and V.1. Now statement c) follows from the fact tha t 8 commutes with 9~. Hence ff ~ is an ideal in ~ the ideal generated by ~ is 9 ~ - ~ which implies tha t ~ = 8 r'~ 9 ~ - ~ or together with b) the map ~ -~ ~ f~ ~ is one-to-one.

Aclcnowledgement. This work was partially completed during my stay at Palmer Physical Laboratory Princeton University. I like to thank Prof. M. GOLD]~ERGEI¢ for his kind hospitality.

References

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aux bords (mimeographed lecture notes, Orsay 1961). [7] DIx~IE~, J.: Les alg~bres d'operateurs duns t'espuc~ I~[ilbertien. Paris:

Gauthier -Villars 1957. [8] KADISON, R.: J. Math. Phys. 4, 1911 (1963). [9] DOPLIC~E~, S. : Tesi di Laurea Roma (1963), unpublished.

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