STATE SPACES OF C*-ALGEBRAS
BY
E R I K M. A L F S E N , H A R A L D H A N C H E - O L S E N and F R E D E R I C W. S H U L T Z
University of Oslo, Norway and Wellesley College, Wellesley Mass., U.S.A.
Contents
w 1. Introduction . . . 267
w 2. States and representations for JB-algebras . . . 270
w 3. JB-algebras of complex type . . . 274
w 4. Reversibility . . . 279
w 5. The enveloping C*-algebra . . . 285
w 6. The normal state space of B(H) . . . 292
w 7. Orientability . . . 295
w 8. The main theorem . . . 299
w 1. Introduction
T h e purpose of this paper is t o characterize t h e state spaces of C*-algebras a m o n g t h e state spaces of all J B - a l g e b r a s . I n a previous paper [6] we h a v e characterized t h e state spaces of J B - a l g e b r a s a m o n g all c o m p a c t c o n v e x sets. Together, these t w o papers give a complete geometric characterization of t h e state spaces of C*-algebras.
l~ecall f r o m [6] t h a t t h e state spaces of J B - a l g e b r a s will enjoy t h e Hilbert ball property, b y which t h e face B(@, a) generated b y an a r b i t r a r y pair @, a of extreme states is (affinely isomorphic to) t h e u n i t ball of some real Hilbert space, a n d t h a t there actually exist such faces of a n y given (finite or infinite) dimension for suitably chosen J B - a l g e b r a s . I n t h e present paper we show t h a t for an a r b i t r a r y pair @, a of extreme states of a C*-algebra, t h e n t h e dimension of B(@, a) is three or one. This s t a t e m e n t , which we t e r m t h e 3.ball property, is t h e first of our axioms for state spaces of C*-algebras. T h e second a n d last axiom is a r e q u i r e m e n t of orientabflity: t h e state space K of a J B - a l g e b r a with t h e 3-ban p r o p e r t y is said t o be orientable if it is possible to m a k e a " c o n s i s t e n t " choice of orienta- tions for t h e 3-balls B(@, a) in t h e w*-eompact convex set K, t h e idea being t h a t t h e orienta-
2 6 8 E . M . ALFSEN ET An.
tion shall never be suddenly reversed b y passage from one such ball to a neighbouring one.
(See w 7 for the precise definition.) Thus we have the following:
M A I N T H E O r E m . A JB-algebra A with state space K is (isomorphic to) the sel/-adjoint part o/a C*-algebra i// K has the 3-ball property and is orientable.
Note t h a t a C*-algebra, unlike a JB-algebra, is not completely determined b y the affine geometry and the w*-topology of its state space. However, the state space does determine the J o r d a n structure, and with this prescribed we have a 1-1 correspondence between C*-structures and consistent orientations of the state space. Thus, for C*-algebras the oriented state ~Tace is a dual object from which we can recapture all relevant structure.
We will now briefly discuss the background for the problem, aIld t h e n indicate t h e content of the various sections.
B y results of Kadison [24], [26], [29], the setf-adjoint p a r t ~sa of a C*-algebra 9~ with state space K is isometrically order-isomorphic to the space A(K) of all w*-continuous affine functions on K. More specifically, 9~,a is an order unit space (a "function s y s t e m "
in Kadison's terminology), and the order unit spaces A are precisely the A(K)-spaces where K is a compact convex subset of a locally convex Hausdorff space; (in fact K can be t a k e n to be the state space of A, formally defined as in the case of a C*-algebra). Thus, the problem of characterizing the state spaces of C*-algebras among all compact convex sets, is equivalent to t h a t of characterizing the self-adjoint parts of C*-algebras among all order unit spaces. This problem is of interest in its own right, and it also gains importance b y the applications to q u a n t u m mechanics, where the order unit space 9~,~ represents bounded observables, while the full C*-algebra ~[ is devoid of a n y direct physical inter- pretation. Note in this connection t h a t the J o r d a n product in 9~sa (unlike the ordinary product in ~) is physically relevant, and t h a t the pioneering work on J o r d a n algebras b y Jordan, yon N e u m a n n and Wigner [19] was intended to provide a new algebraic formalism for q u a n t u m mechanics (cf. also [30]).
I n [25] Kadison proved t h a t the J o r d a n structure in the order unit space 9~,~ is com- pletely determined, in t h a t a n y unital order automorphism of 9~s~ is a J o r d a n automorphism, and he pointed out the great importance of the J o r d a n structure for the study of C*- algebras. An axiomatic investigation of normed J o r d a n algebras was carried out in [7].
Here the basic notion is t h a t of a JB-algebra, which is defined to be a real J o r d a n algebra with unit 1 which is also a Banach space, and where the J o r d a n product and the norm are related as follows:
laaobll < Ilall Ilbll, Ila ll = Ilall Ila ll < Ila +b ll.
(1.1)STATE SPACES OF C*-ALGEBRAS 269 These axioms are closely related to those of Segal [32], and the J B - a l g e b r a s will include the finite dimensional formally real algebras studied b y Jordan, yon N e u m a n n and Wigner (which can be normed in a natural way), as well as the norm closed J o r d a n algebras of bounded self-adjoint operators on a Hilbert space (JC-algebras) studied b y Topping, Stormer and Effros [41], [37], [39], [18]. The main result of [7] shows t h a t the s t u d y of general J B - a l g e b r a s can be reduced to the s t u d y of JC-algebras and the exceptional algebra M~ of all self-adjoint 3 • 3-matrices over the Cayley numbers. (For related results, see [34].)
The geometric description of the state spaces of J B - a l g e b r a s involves, in addition to the Hilbert ball property, three more axioms stated in terms of facial structure. (They are quoted in w 8. See [6] for further details.) These axioms relate the g e o m e t r y of the state space to the projection lattice and the spectral theory of the "enveloping J B W - algebra" (generalizing the enveloping von N e u m a n n algebra of a C*-algebra). The con- nection between faces and projections was first noted b y Effros and Prosser in their papers on ideals in operator algebras [17], [31]. This connection was the starting point for the development of a n o n - c o m m u t a t i v e spectral theory for convex sets [4], [5], which was used extensively in the passage from compact convex sets to J o r d a n algebras in [6].
The transition from J B - a l g e b r a s to C*-algebras presents difficulties of a new kind due to the lack of uniqueness. There is no natural candidate for the C*-product; it m u s t be chosen, and orientability is needed to m a k e this choice possible. The first time a notion of orientation was used for a similar purpose, was in Connes' paper [14], where he gave a geometric characterization of the cones associated with yon N e u m a n n algebras via Tomita- Takesaki theory. Although b o t h t h e setting and t h e actual definition are different in the two cases, t h e y are related in spirit. I n both cases the orientation serves the same purpose, namely to provide the complex Lie structure when the J o r d a n product is given. (See also the papers b y Bellissard, I o c h u m and Lima [8], [9], [10].)
I n the present paper, w 2 provides the necessary machinery of states and representa- tions for JB-algebras. The results here are for the most p a r t analogues of well known results for C*-algebras.
I n w 3 we go into the classification theory and concentrate on J B - a l g e b r a s of "complex t y p e " . T h e y are shown to be precisely those for which the state space has the 3-ball property.
w 4 provides a technical result which is also of some independent interest, namely t h a t a J B - a l g e b r a of complex t y p e acts reversibly in each concrete representation on a complex Hilbert space.
I n w 5 it is shown t h a t each J B - a l g e b r a A admits an enveloping C*-algebra 9~ with a universal property relating J o r d a n and *-homomorphisms. I t is shown t h a t if A is of
270 E . M . A L F S E N E T A L .
complex type, the pure states of 9~ form (except for degeneracy) a double covering of the set of pure states of A.
I n w 6 we discuss the orientation of balls in the normal state space of B(H).
w 7 is a general t r e a t m e n t of orientability for state spaces of J B - a l g e b r a s of complex type.
w 8 contains the main theorem.
The prerequisites include standard theory of C*- and yon N e u m a n n algebras plus the theory of J B - a l g e b r a s as presented in [7]. We will also draw upon the portion of [6]
which establishes properties of state spaces of JB-algebras. The rest of [6] (and thus in- directly the work in [4] and [5]) will be used only when the main theorem of the present paper, characterizing state spaces of C*-algebras among state spaces of JB-algebras, is combined with the main theorem of [6], characterizing state spaces of J B - a l g e b r a s among all compact convex sets, to give a complete geometric description of the state spaces of C*-algebras (Corollary 8.6).
w 2. States and representations for JB-algebras
This section is of preliminary nature, and the results are for the most p a r t analogues oI well known results for C*-algebras.
Note t h a t when we work in the context of J o r d a n algebras, we will use the word ideal to mean a norm closed J o r d a n ideal. Also if A, B are J o r d a n algebras and T: A ~ B is a bounded linear m a p , t h e n we denote the adjoint m a p from B* into A* b y T*. Occasion- ally if T: A**-+B** is a a-weakly continuous linear map, we will denote the adjoint m a p from B*~A* b y T*. Recall t h a t a JBW-algebra is a J B - a l g e b r a which is a Banach dual space, and t h a t the enveloping JBW-algebra of a J B - a l g e b r a A is A** with the (right =left) Arens product (cf. [34] and [6]).
We now consider two JBW-algebras M 1 and M S and a homomorphism ~: M I ~ M ~ which is a-weakly continuous (i.e. continuous in the w*-topology determined b y the unique preduals of M 1 and M2). B y the same argument as for yon N e u m a n n algebras [33; Prop.
1.16.2], the unit ball of T(M1) is a-weakly compact. Hence ~(M1) is a-weakly closed in M 2, and so it is a JBW-algebra. I n other words: A a-weakly continuous homomorphic image o[
a JBW-algebra in a JBW-algebra is a JBW-algebra.
We n e x t relate homomorphisms of J B - a l g e b r a s to a-weakly continuous homomor- phisms of their enveloping JBW-algebras. Here the results and proofs for C*-algebras [33; Prop. 1.17.8 and 1.21.13] can be transferred without significant change. Specifically:
1/9~: A ~ M is a homomorphism /tom a JB-algebra A into a JBW-algebra M, then there
STATE SPACES OF C*-A.LGEBRAS 271 exists a unique a-weakly continuous homomorphism ~: A**-~M which extends q~; moreover
~(A**) is the a-weak closure o/q~(A) in M. (When no confusion is likely to arise, we will denote the extended homomorphism b y ~ instead of ~.)
We will now provide J o r d a n analogues of the basic notions in the representation theory of C*-algebras. Since a JB-algebra might not have any (non-zero) representations into B(H)sa, these notions can not be carried over directly. However, it is reasonable to replace B(H) by any JBW.faetor of type I when we work with general JB-algebras.
(Recall t h a t the JBW-factors of type I are the JBW-algebras with trivial center which contain minimal idempotents, and t h a t they have been completely classified [7; Th. 8.6]
and [37; Th. 5.2]. We return to this classification in w 3.) Note t h a t two representations
~ : ~-->B(H~) ( i = 1 , 2) of a C*-algebra 9/ are unitarily equivalent iff there exists a *-iso- morphism (I) from B(H~) onto B(H2) such t h a t ~2=(PoT~ [15; Cot. III.3.1]. Observe also t h a t a representation ~: ~-~B(H) of a C*-algebra 9/is irreducible iff ~(i~) is weakly ( = a - weakly) dense in B(H) [33; Prop. 1.21.9]. This motivates the following:
Definitions. A representation of a JB-algebra A is a homomorphism ~: A - ~ M into a type I JBW-factor M. We say ~ is a dense representation if q~(A)-=M (a-weak closure).
Two representations ~i: A-~M~ (i = 1, 2) are said to be Jordan equivalent if there exists an isomorphism (I) of M1 onto M2 such t h a t ~2 =(I)~
L E P T A 2.1. Let A be a JB-algebra with state space K and let q~,: A-~M~ ( i = 1 , 2) be dense representations. Then q~l and ~ are Jordan equivalent i// the unique a-weakly continuous extensions ~ : A** ~ M t satis]y ker ~1 = k e r ~2.
Proo]. Suppose t h a t ~i and ~ are equivalent, and let q) be a J o r d a n isomorphism of M 1 onto M 2 such t h a t ~2=(P0~1. Since (I) is a-weakly continuous, we also have ~2 =(I)O~l and so ker ~1 = ker ~2.
Conversely, suppose ker ~1 = k e r ~2. Note t h a t ~1 and ~ are surjective. Thus we can define (I): M I ~ M ~ by @(~l(a))=~2(a) for all aEA**. This (I) determines a J o r d a n equiv-
alence of ~ and q~2" []
We will now relate the representations of a JB-algebra A to the state space K. As usual, the extreme points of K are called pure states, and the set of pure states is denoted
~eK. We recall from [7] how one can associate with any pure state ~ on A a dense represen- tation ~ : A-~Aq with Aq=c(~)oA** and ~Q(a)=c(~)oa for aEA, where c(~) is the central support of ~, i.e. the smallest central idempotent of A** such t h a t (c(~), ~ = 1. (See [7; w 5]
for the existence of c(~), and see [7; Prop. 5.6 and Prop. 8.7] for the demonstration t h a t
272 l~. M. ALFSEI~I E T A L .
~% is a dense representation.) Recall also t h a t a face F of K is said to be split if it admits a, necessarily unique, complementary face F' such t h a t K is direct convex sum of F and
F ' (cf. [1; w 6]).
P B O P O S I T I O ~ 2.2. Let A be a JB-algebra with state space K. I] q~: A ~ M is a dense representation, then there exists ~ ESeK such that ~o is Jordan equivalent with q~o; moreover,
~v* maps the normal state space o] M injectively onto the sm~tllest split ]ace o] K containing ~.
Two such dense representations ~ : 21 -+M~ (i = 1, 2) are Jordan equivalent if] the corresponding split ]aces coincide.
Proo]. Since ker ~ is a a-weakly closed ideal in A**, there exists a central idempotent tEA** such t h a t ker ~ =(1 -c)oA** [34; Lem. 2.1]. Let P, Q: A**~A** be the two a-weakly continuous projections defined b y Pa=coa, Q a = ( 1 - c ) o a for a E21**. Clearly the dual projections P*, O*: A * ~ A * satisfy P*~*=~*, Q*~*=0. Hence q* maps the normal state space of M onto F = K / 1 im P*. Since ~ is surjective, ~o* will be injective. Clearly P + Q = I , from which it easily follows t h a t F = K f] im P* is a split face of K with complementary face F ' = K N im Q*.
We will show F is a minimal split face. To this end we consider an arbitrary split face G such t h a t F N G=4=O, and we will prove F~_ G. L e t G' be the face complementary to G. B y linear algebra, there exists a unique bounded affine function on K which takes the A ~ , i.e. <d, a> = 1 value 1 on G and vanishes on G'. L e t d be the corresponding element of **
for all a fi G and <d, ~> = 0 for all a E G'. Since d is seen to be an extreme point of the positive p a r t of the unit ball of A**, it m u s t be an idempotent. (The standard a r g u m e n t for C ~- algebras applies.) To show t h a t d is central, it suffices b y [7; L e m m a 4.5] to verify the in- equality Uaa <~a for all a ~>0, a E A**. (Recall t h a t Uaa = {dad} where the brackets denote the J o r d a n triple product, and also t h a t U~: 21"*~A** is a positive linear m a p b y [7;
Prop. 2.7].) For given aEA**, a>~O and for each aEG' we have 0 < <u a, a> < II ll < U d l , a> = I1 11 <d, a> = 0.
Hence Uaa vanishes on G'. Applying the same argument with 1 - d in place of d and using [7; Cor. 2.10], we conclude t h a t Uda coincides with a on G. B u t by linear algebra there can only be one such affine function on K, and this function is nowhere greater t h a n a.
Hence U~a<~a, which proves t h a t d is a central idempotent. B y assumption F N G # O , which implies cod#O. Since ~ is injective on eoA**, we also have ~ ( d ) # 0 . Since M is a factor we m u s t have ~ ( d ) = 1, hence c ~<d, which in turn implies F_~ G.
N e x t we claim t h a t the minimal split face F m u s t contain pure states. I n fact, the normal state space of the t y p e I JBW-factor M contains pure states (cf. e.g. [6; p. 159]),
STATE SPACES OF C*-ALGEBRAS 273 therefore F also does. L e t 0 E F n ~eK be arbitrary. By the minimality, F is the smallest split face containing 0" Also c(0 ) =c; for if c(Q)<c then the same argument as above would provide a split face strictly contained in F. Now ker ~ o = k e r ~, and b y L e m m a 2.1, ~Q and ~ are J o r d a n equivalent.
Finally we consider two dense representations ~ : A - + M i (i = 1, 2). Note t h a t b y the above definition of P, the split face F = K ;~ im P* is the annihilator of ker ~ = (1 - c ) o A * * , and vice versa. Hence the split faces corresponding to ~1 and ~2 coincide iff ker ~1 = ker ~ , and b y L e m m a 2.1 this equality holds iff ~1 and ~2 are J o r d a n equivalent. []
I t follows from Proposition 2.2 t h a t for every pure state ~ of a J B - a l g e b r a there exists a smallest split face containing Q. We will denote this split face FQ. (Note t h a t our notation differs from t h a t of [2] where Fo denotes the smallest w*-closed split face con- taining 0.)
Two pure states ~, o of a J B - a l g e b r a will be called equivalent (or "non-separated b y a split face") if FQ = Fr By Proposition 2.2, Q and o are equivalent iff the representations
~Q and ~ are J o r d a n equivalent; hence the terminology.
Recall the brief notation B(0, o) = f a c e {0, o} used for a n y pair ~, o of pure states.
PROPOSITION 2.3. Let 0, o be pure states o t a JB-algebra A . I t ~ and o are equivalent then B(O , 0) is a Hilbert ball o / d i m e n s i o n at least two. I t 0 and o are not equivalent, then B(O, o) reduces to the line segment [~, 0].
Proot. If 0 and o are equivalent, then it follows from the proof of [6; Th. 3.11] t h a t B(0, o) is the state space of a certain spin factor, and so it is a Hilbert ball. This ball m u s t be of dimension at least two, since every spin factor is of dimension at least three.
If 0 and o are not equivalent, then it follows from [6; Prop. 3.1] t h a t B(0, o) = Lo, 0]. []
B y a concrete representation of a J B - a l g e b r a A on a complex Hilbert space H, we shall mean a J o r d a n homomorphism z: A--->B(H)s a with ~ ( 1 ) = 1 . Note t h a t there exist J B - algebras without a n y non-zero concrete representation. (An example is M3 s. See [7; w 9].)
A standard argument for C*-algebras can be applied to show t h a t if a concrete rep- resentation ~: A ~ B(H)s a is dense, t h e n it is irreducible, i.e. there is no proper invariant subspace of H. The converse is false in general. (We shall return to this question in w 3.)
We say t h a t two concrete representations g~: A-~B(H~)sa (i = 1, 2) of a J B - a l g e b r a A are unitarily equivalent (conjugate) if there exists a complex linear (conjugate linear) iso- m e r r y u from H2 onto H 1 such t h a t
~2(a) = u*xcl(a)u for all a E A . (2.1)
274 ~. M. AJ~FSEN ]~T AL.
P R O P O S I T I O ~ 2.4. I / tWO dense concrete representations ~ : A-> B(H~) ( i = 1 , 2) are Jordan equivalent, then they are either unitarily equivalent or conjuqate. The only case in which 7~ 1 and ~2 are both unitarily equivalent and conjugate at the same time, is when dim H 1 = dim H 2 = 1.
Proo/. B y the assumptions there exists a J o r d a n isomorphism (I) from B(H1)sa onto B(H~)~ such t h a t ~2=(I)O7el. B y a known theorem (see [25]) there exists an isometry u: H 2 ~ H 1 which is either complex linear or conjugate linear such t h a t (I)(b)=u*bu for all b e B(H1)sa. Now (2.1) is satisfied.
Assume now t h a t u: H2-+H 1 is a complex linear isometry and t h a t v: H2-->H 1 is a conjugate linear isometry such t h a t ~P(b)=u*bu=v*bv for all beB(H1)sa. Then the two complex linear m a p s a~->u*au and a~-->v*a*v from B(H1) onto B(H~) m u s t coincide. B u t the former of the two is a *-isomorphism while the latter is a *-anti-isomorphism. This is possible only if both algebras are commutative, i.e. dim H 1 = dim H~ = 1. []
An involution of a complex Hilbert space H is a conjugate linear isometry ]: H - ~ H of period two; an example is ?': ~ 2v~v ~-> ~ ~v~v where {~v} is a n y orthonormal basis. I f j: H ~ H is an involution then uj is a conjugate linear isometry for each unitary u, and every conjugate linear isometry v is of this form. (Write u = v j and note t h a t u]=v since
j~=l.)
To a given involution j: H--->H we associate a transpose map a~-->a ~ from B(H) onto itself b y writing a~=ja*j. Clearly the transpose m a p is a *-anti-automorphism of order two for the C*-algebra B(H). If ~: A ~ B(H)s~ is a concrete representation of a J B - a l g e b r a A, then the transposed representation 7~: a~-->ze(a) t (w.r. to j) will be conjugate to ~.
Thus, in the s t u d y of dense concrete representations of J B - a l g e b r a s we encounter two natural equivalence relations: J o r d a n equivalence and u n i t a r y equivalence. E x c e p t for the one-dimensional case, each J o r d a n equivalence class splits in two (mutually con- jugate) unitary equivalence classes.
w 3. JB-algehras o| complex type
The following classification theorem is essentially contained in [37] and [7; w 8].
I~ecall from [42] and [7; w 7] t h a t a spin ]actor is, b y definition, H ( ~ R where H is a real Hilbert space of dimension at least two. Here J o r d a n multiplication is defined so t h a t 1 E R acts as a unit and a o b = ( a l b ) l where a, bEH.
T~EOR]~M 3.1. The type I JWB-]actors can be divided into the ]ollowing classes (up to isomorphism):
STATE SPACES OF C*-ALGEBRAS 275 (i) B(H)s , the symmetric bounded operators on a real Hilbert space H;
(ii) B(H)~a, the sel/-adjoint bounded operators on a complex Hilbert space H;
(iii) B(H)s~, the sel/-adjoint bounded operators on a quaternionic Hilbert space H;
(iv) the spin ]actors;
(v) the exceptional algebra M s o/sel/-adjoint 3 by 3 matrices over the Cayley numbers.
Moreover, these classes are mutually disjoint, with the exceptions that the matrix algebras Mu(R)s, M2((~)s a, and M2(H)s a are all spin ]actors.
Note. The algebra M~ s of self-adjoint 2 b y 2 matrices over the Cayley numbers is also seen to be a spin factor, see the proof of Prop. 3.2 below.
Proo]. L e t M be a t y p e I JBW-factor. Assume t h a t M is not isomorphic to M s or a spin factor. B y [7, Th. 8.6 and Prop. 7.1], [34; Cor. 2.4], and [37; Th. 5.1] we m a y assume t h a t M is concretely represented as an irreducible J W - a l g e b r a of t y p e I~ on a complex Hilbert space H.
L e t ~(M) be the norm-closed real subalgebra of B(H) generated b y M. We claim t h a t M is the self-adjoint p a r t of ~(M), where the bar denotes a-weak closure.
Indeed, let x be a self-adjoint element of R(M). Then x is a a-weal~ limit of sums of terms of the form Yl ... Y~, where each y j E M . Since x = x * , x= 89 is a a-weak limit of sums of terms 89 ... Yn+(Yl ... Y~)*)=89 ." Y,+Yn "" Yl), where each y j E M . B y [37;
L e m m a 3.1] M is reversible, t h a t is Yl ... Y~ + Y~ ... Yl E M. Since M is a-weakly closed x E M, and the claim is proved.
I f M is the self-adjoint p a r t of a v o n N e u m a n n algebra, then this algebra, being ir- reducible, equals B(H). Then we have case (ii).
Otherwise, according to [37; L e m m a 6.1] we have ~(M) ~ i n ( M ) = (0}. Using L e m m a 2.3 and Theorem 2.4 of [39], we get the direct sum decomposition
B(H) = ~(M) | i ~(M).
Thus we can define a a-weakly continuous m a p p i n g (I): B(H)-+B(H) b y setting (b(x + iy) = ( x - iy)* = x* + iy* (x, yE ~(M)).
r is easily seen to be a *-anti-automorphism of B(H), and 0 2 = I .
I n [38] it is proved t h a t there exists a conjugate linear isometry j: H--->H such t h a t
~9(x)=]-lx*j (xEB(H)).
276 1~. M. ~FSEN ET AL.
Since M is the self-adjoint p a r t of ~(M) we find t h a t x ~ M iff x is self-adjoint and O(x) = x , i.e.,
M = {xeB(H)~a[X i = ix}. (3.1)
Because O 2 = I , j2 is a scalar multiple of the identity, say j2 =21, where 121 = 1 . Since j commutes with j~, we find ~2 =2j. But, since j is conjugate linear, )'2=~?'. This implies 2 =~, so we have j2 = + 1.
First, assume ]2= 1. L e t K = {~ E H: ~'~ =~}. Then K is a real ttilbert space, H = K Q i K , and J(~+i~?)=~-i~? whenever ~,
veK.
B y (3.1) x E M iff x is self-adjoint and leaves K invariant, t h a t is, x E M ~ - x I~ E B(K)~. Since a n y x e B(H) is determined b y its restriction to K, M ~= B(K)~ follows. Thus we have case (i).Next, assume ~ = --1. Define k ~i~. I t is easily verified t h a t i, ~, k satisfy the multi- plieation table of the unit quaternions, so H m a y be considered a quaternionic vector space.
Also, i, ] and k are isometries and skew symmetric with respect to the real p a r t of the inner product in H. Thus H is a quaternionic ttilbert space with the inner product
(~ [~)H = Re (~ 1~/) -- (Re (i~ I V ) ) / - (Re (j~ [V))j - (Re (k~ I*/)) k.
B y (3.1) the elements of M are exactly the self-adjoint H-linear operators. Thus we have case (iii), and we h a v e proved t h a t M falls into one of the classes mentioned.
N e x t we prove t h a t M2(tt)~, M2(C)s a and M2(It M are spin factors. Note that, b y definition, a finite dimensional spin factor admits a basis 1, s 1 .... , sN where each s m is a s y m m e t r y (S2m=l), and smos~=0 if m=~n. (Namely, let s 1 .... , sN be an orthonormal basis in H.) Defining
=
(3.2)
we find t h a t 1, sl, s 2 (resp. 1, sl, s2, ~3 resp. 1, 81, ..., 85) is such a basis for M~(R) s (resp.
M2((])~a resp. M~(II)~).
Finally, t h e disjointness of t h e isomorphism classes, with the stated exceptions, follows b y considering orthogonal minimal idempotents e, ] in M and noting t h a t {(e + / ) M ( e + / ) } is isomorphic to M2(R)s , Me({?)sa, Ma(II)~, M and M~ in the respective
c a s e s . [ ]
De/inition. A t y p e I J B W . f a c t o r is said to be real (resp. complex resp. quaternionic) if it is isomorphic to B(H)~ for some real Hilbert space H (resp. B(H),a for some complex resp. quaternionic t t i l b e r t space).
STATE SPACES OF~*-ALGEBRAS 277 In [6; w 3] the normal state spaces of type I JBW-factors are characterized geometric- ally. In particular, if ~, a are distinct extreme points of the normal state space, the face B(~, o) they generate is an exposed face affinely isomorphic to a Hilbert ball (the unit ball in a Hilbert space). The different types of JBW-factors can be distinguished by the dimen- sion of this ball:
PROPOSITIO~ 3.2. Let M be a type I JBW-[actor, and N its normal state space. I[
e, ~ are distinct extreme points of N, we have:
(i) If M is real, then dim B(~, a ) = 2 . (ii) I f M is complex, then dim B(~, a) = 3 . (iii) I f M is quaternionic, then dim B(~, a ) = 5 . (iv) I f M is a spin factor, then dim B(~, a ) = N .
(v) I f M~=M~, then dim B(~, o ) = 9 .
Proof. Let ~ be the (non-central) support projection of ~. Then B(~, a) is isomorphic to the normal state space of {(~ V a)M(~ V ~)} (see the proof of [6; Th. 3.11]). Also, by [6; L e m m a 3.6] ~ V ~ = e + f for some pair of minimal orthogonal projections e, f of M.
Thus, if M is real {(~ V ~)M(~ V ~)} ~M~(R)~ and so, counting dimensions, we find dim B(~, o ) = dim M2(R)~- 1 = 2. The complex and quaternionic cases are treated similarly.
Next, assume M~=M~, and let e~EM~ be the matrix units (i, j = l , 2, 3). Since M is of type I3, 1 - e - f ~ e33 via a symmetry, so we m a y as well assume e + f = e l l + e22. Thus
{(~+/)M(e+/)} ~= ~ , so dim B(~, o) = dim M S - 1 = 9 .
Finally, if M is a spin factor, N is a Hilbert ball, so (iv) follows trivially. []
Definition. The state space K of a JB-algebra is said to have the 3-ball property if, for any pair ~, a of distinct extreme points of K, the ball B(~, o) has dimension 1 or 3.
(By Proposition 2.3 dim B(~, o) = 3 iff ~ and a are equivalent.)
Definition. A JB-algebra A is said to be of complex type if all its dense representations are into a type I factor isomorphic to B(H)~ a where H is a complex Hilbert space. Similarly, we m a y define JB-algebras of real, quaternionic, spin, and purely exceptional types.
COROLLARY 3.3. A JB-algebra is o] complex type iff its state space has the 3-ball property.
Pro@ The JB-algebra .4 is of complex type iff .4e is complex for ~11 pure states ~.
278 E . M . A L F S E N E T AL.
Since the normal state space of A 0 is isomorphic to Fo, the corollary follows directly from
Theorem 3.1 and Proposition 3.2. []
The relevance of the above discussions for our purpose stems from the following lemma. As will be seen later on, its converse is false.
L E ~ M A 3.4. The sel]-ad]oint part o / a C*-algebra is a JB-algebra o/ complex type.
Proo/. L e t A be the self-adjoint p a r t of a C*-algebra, and let ~: A-->M be a dense representation. As in the proof of Proposition 2.2 we can find a central idempotent c EA**
such t h a t &: coA**-)M is a surjective isomorphism. Since A** is the self-adjoint p a r t of a v o n N e u m a n n algebra, M m u s t be isomorphic to the self-ad]oint p a r t of a t y p e I y o n N e u m a n n factor, i.e. to B(H)s ~ for some complex Hilbert space H. []
P R O P O S I T I O ~ 3.5. A J B-algebra is o/complex type i// it is special and all its irreducible concrete representations are dense.
Proo/. Assume t h a t A is of complex type. B y [7, w 9] A is special. Now let ~: A-> B(H)s a be an irreducible concrete representation. Then ~ ( A ) - is an irreducible JW-algebra, hence [39; Th. 4.1] it is a t y p e I JBW-factor, and therefore is a complex factor. The proof of Theorem 3.1 shows t h a t actually ~ ( A ) - = B(H)~a, i.e. z is a dense representation.
Conversely, assume A is a special J B - a l g e b r a all of whose irreducible representations are dense. L e t ~: A--*M be a dense representation. Since A is special, M is n o t isomorphic to M s. Then M can be represented as an irreducible J W - a l g e b r a [37; Th. 51], say M ~ B(H)s ~. (That this is also true when M is a spin factor, is seen by first representing M on a H i l b e r t space, and then choosing an irreducible representation of the C*-algebra generated b y M.) B u t then ~, viewed as a m a p into B(H)~a, is an irreducible representation, and hence it is dense. Thus M = ~ ( A ) = B ( H ) s ~ , and A is of complex type. []
De/inition. L e t A be a J B - a l g e b r a of complex type. We say an irreducible concrete representation ~: A-~ B(H)sa is associated with Q E~eK if there exists a (unit) vector ~ E H such t h a t
<a, e> = (7t(a)~]~:) for all a e A . (3.3) Note t h a t the unit vector ~ of (3.3) is uniquely determined up to scalar multiples (of modulus one) b y virtue of the density of g(A) in B(H)s a. We will say t h a t this vector ~ represents Q
w . r . t o ~ .
PRO~OSIT~O~ 3.6. Let A be a JB-algebra o/complex type. Then/or each pure state
~ E ~ K there is associated at least one irreducible concrete representation. Two irreducible
STATE SPACES OF C*-ALGEBRAS 279 concrete representations associated with the same ~ E OeK are either unitarily equivalent or conjugate; both happen i[/ the representations are one-dimensional. Furthermore, i/ an ir- reducible concrete representation ~ o / A is associated with ~ EO~K, then the set o / p u r e states with which ~ is associated, is precisely ~e Fo = Fo N ~ K .
Proo]. B y Proposition 3.5, an irreducible concrete representation of A is the same as a dense concrete representation. B y Proposition 2.2, such a representation ~: A ~ B(H)s a is J o r d a n equivalent to ~0 iff ~* m a p s the normal state space of B ( H ) bijeetively onto F o.
Since the pure normal states of B ( H ) are the vector states, this is equivalent to 7e being associated with ~. This proves the final s t a t e m e n t of the proposition, and also the first since to each ~ EOe K is associated the dense representation ~0: A ~ A q and A o is isomorphic to B(H)~a b y assumption.
Finally, the second s t a t e m e n t is a direct consequence of Proposition 2.4, since two irreducible representations associated with ~ b o t h are J o r d a n equivalent to ~o, and there-
fore to each other. []
w 4. Reversibility
Following Stormer [36; p. 439] we will say t h a t a JC-algebra A is reversible if
ala2 ... an § as an_l ... al E A (4.1)
whenever a I .... , a~EA. Note t h a t the left hand side of (4.1) is the J o r d a n triple product for n = 3 . Thus, (4.1) always holds for n = 3 , b u t it is worth noting t h a t it can fail already for n = 4 . (In fact, n = 4 is the critical value; if (4.1) holds for n = 4 , then it holds for all n > 4 as shown b y P. M. Cohn [13].)
For a given JC-algebra A ~_ B(H)sa we denote b y ~o(A) the real subalgebra of B ( H ) generated b y A, and we denote b y ~(A) the norm closure of ~0(A). We observe t h a t
~0(A) is closed under the *-operation since (ala 2 ... a~)*=a~a~_ 1 ... a 1 for a 1 ... a~EA.
F r o m this it follows t h a t ~(A) is a norm closed real *-algebra of operators on H. (Such an algebra is sometimes called a "real C*-algebra".) I f A is reversible and b = a l a 2 ... an where al, ..., a~EA, then the self-adjoint p a r t b~= 89247 will be in A. F r o m this it fol- lows t h a t A is reversible iff ~0(A)sa=A.
Assume now t h a t A is reversible and consider an element bE ~(A)sa, say b =b* and b =limn bn where bnE R0(A) for n = 1, 2, ... (norm limit). Then b =limn (b~)saeA since A is closed. F r o m this it follows t h a t A is reversible iff ~(A)s a = A .
B y definition, reversibility is a spatial notion involving the n o n - c o m m u t a t i v e mul- tiplication of tIilbert space operators. I n general it is not an isomorphism invariant;
it is possible for a reversible and a non-reversible JC-algebra to be isomorphic. This situa-
2 8 0 E . M . A L F S E N E T A L .
t i o n is i l l u s t r a t e d b y t h e s p i n - f a c t o r s . A s p i n f a c t o r A_~ B(H)s a is a l w a y s r e v e r s i b l e w h e n d i m A = 3 or 4, n o n - r e v e r s i b l e w h e n d i m A # 3 , 4 or 6, a n d i t c a n be e i t h e r r e v e r s i b l e or n o n - r e v e r s i b l e w h e n d i m A = 6, e v e n t h o u g h all s p i n f a c t o r s of t h e s a m e d i m e n s i o n a r e i s o m o r p h i c . Of t h e s e r e s u l t s we will p r o v e o n l y t h e one w i t h d i m A = 4, since we shall n o t n e e d t h e others.
R e c a l l t h a t t h e t t i l b e r t n o r m of a s p i n f a c t o r is e q u i v a l e n t w i t h t h e J B - a l g e b r a n o r m , a n d t h a t t h e t w o coincide on N = ( 1 ) " (ef. [41]). I t follows t h a t e v e r y spin f a c t o r is a B a n a c h d u a l space, hence a JBW-algebra. I t is e a s i l y verified t h a t t h e c e n t e r of a n y s p i n f a c t o r is t r i v i a l , h e n c e i t is a f a c t o r (which justifies t h e t e r m i n o l o g y ) . I n fact, t h e s p i n f a c t o r s a r e p r e c i s e l y t h e JBW-factors of t y p e I S (see [7; w 7] for d e f i n i t i o n a n d proof).
I f S is a s p i n f a c t o r , t h e n t h e h y p e r p l a n e N = {1)" consists of all e l e m e n t s 4s w h e r e 4 E R, s :4= • 1, a n d s is a symmetry, i.e. s ~ - 1. N o t e also t h a t t w o e l e m e n t s of N are o r t h o g o n a l iff t h e i r J o r d a n p r o d u c t is zero. Thus, if {s~} is a n o r t h o n o r m a l basis in S such t h a t s:0 = 1 for s o m e i n d e x :r t h e n all t h e o t h e r b a s i s - e l e m e n t s a r e s y m m e t r i e s s a t i s f y i n g s : o s~ = ~:. Z 1.
F o r l a t e r references we o b s e r v e t h a t t h e o r t h o g o n a l c o m p o n e n t s of a n e l e m e n t a E S w i t h r e s p e c t t o such a basis, can be e x p r e s s e d in t e r m s of t h e J o r d a n p r o d u c t . I n fact, if a = 4o + ~*~o 4~s~, t h e n for e a c h ~ # ~ o :
(aosa) os a = (40s ~ +4~ 1)osa = 401 +4~s~;
h e n c e for a n y i n d e x fl:4=~0 d i s t i n c t f r o m a:
m o r e o v e r :
( ( (aos~)o%)os p)os z = 401;
( a - 4 0 1 ) o s ~ = 4 a l .
(4.2) (4.3)
S i m p l e e x a m p l e s of s p i n f a c t o r s a r e t h e J o r d a n a l g e b r a M2(R)s of all s y m m e t r i c 2 • 2- m a t r i c e s o v e r R a n d t h e J o r d a n a l g e b r a M~(C)sa of all s e l f - a d j o i n t 2 • 2 - m a t r i c e s o v e r C.
F o r t h e s e algebras, o r t h o n o r m a l bases a r e r e s p e c t i v e l y (So, Sl, s2) a n d (So, Sl, s 2, sa), w h e r e s o is t h e u n i t m a t r i x a n d Sl, s~, s 3 a r e t h e e l e m e n t a r y s p i n m a t r i c e s (cf. (3.2)).
I t follows f r o m t h e a b o v e discussion t h a t t w o s p i n f a c t o r s of t h e s a m e d i m e n s i o n m u s t be i s o m o r p h i c . I n p a r t i c u l a r , e v e r y s p i n f a c t o r of d i m e n s i o n t h r e e is i s o m o r p h i c t o M~(R)s, a n d e v e r y spin f a c t o r of d i m e n s i o n f o u r is i s o m o r p h i c t o M~(C)sa.
LEMMA 4.1. The /our-dimensional spin /actor M2(C)s a i8 reversible in every concrete representation.
Pros/. L e t M ~ B(H)s a be a concrete s p i n f a c t o r of d i m e n s i o n four. L e t 1, s 1, s 2, s 3 b e a basis for M , w h e r e s~ = 1 a n d s~osj = 0 for i=4=].
STATE SPACES OF C*-ALGEBRAS 281 B y multilinearity, it suffices to prove t h a t x = a 1 ... a n + a n ... a l E M , whenever t h e a / s belong t o t h e a b o v e basis. Using t h e relations s~ = 1, s~sj = - s j s ~ when i:4=], we m a y per- m u t e the aj's (possibly reversing a sign in t h e expression for x) a n d cancel terms until we find x = • 1 ... b ~ + b m ... bl), where m < 3 . T h u s x E M . (If m = 3 , this expression is the J o r d a n triple p r o d u c t {b x b 2 ba} = (b x o b 2 ) o b a + (b 2 o ba) o 51 - (b x o ba) o b 2.) [ ] W e will reduce t h e p r o b l e m of reversibility for a given J C - a l g e b r a t o t h e same p r o b l e m for its weak closure in an a p p r o p r i a t e representation. T h e n we are in a setting where t h e s t r u c t u r e t h e o r y for J W - a l g e b r a s applies. Recall in this connection t h a t a n y given J W - algebra A ~ B(H)s a can be written as
A = A I @ A 2 q ) ... | (4.4)
where A x is an abelian J W - a l g e b r a , Ai is of t y p e I j for j = 2 , 3 ... oo, a n d B is t h e n o n t y p e I s u m m a n d . (See [41; Theorems 5 & 16] for precise definitions a n d proofs, b u t note in particular t h a t t h e direct sum (4.4) is given b y o r t h o g o n a l central i d e m p o t e n t s zx, z 2 .. . . . zoo, w E A such t h a t z j A = A j for j = l , 2 ... oo a n d w A = B . )
W e will see later t h a t t h e I ~ - s u m m a n d is t h e k e y to reversibility. Therefore we will n o w s t u d y J W - a l g c b r a s of t y p e 12. W e begin b y t w o technical lemmas.
L E M MA 4.2. F o r each integer n >~ 1 there exists a J o r d a n p o l y n o m i a l P ~ i n n + 2 variables such t h a t / o r a n y s p i n / a c t o r S a n d a n arbitrary p a i r s, t o / o r t h o g o n a l s y m m e t r i e s i n S we have Pn(s, t, a x . . . an) = 0 i / / a 1 .. . . . a~ E S are linearly dependent.
Proo/. B y t h e well k n o w n G r a m criterion for spaces with an inner product, n elements a I ... a~ of a spin f a c t o r S will be linearly d e p e n d e n t iff d e t ((a~Iaj)}~.j=~ = 0 . Since t h e J o r d a n multiplication in S reduces to scalar multiplication in R1 ___ S, we can rewrite this condition as
Q~((a~ lax) 1, (a 11a2) 1 ... (an l a~) 1) = 0, (4.5) where Qn is an a p p r o p r i a t e J o r d a n polynomial in n ~ variables.
Assume n o w t h a t s, t are t w o a r b i t r a r y (but fixed) o r t h o g o n a l s y m m e t r i e s in S. F o r a n y set (a 1 .... , a n) of n elements of S we decompose each aj as aj = ~j 1 + nj where n~ E N = {1 }~.
F o r given i, ] t h e multiplication rules for spin factors give:
(a~[aj) 1 = ~ g j l + ( n i ] n j ) 1 = ( ~ l ) o ( e j l ) + n ~ o n j = ( a ~ l ) o ( a j l ) + ( a ~ - - ~ l ) o ( a s - a ~ l ) . I t follows from (4.2) t h a t (a t [aj)1 can be expressed as a J o r d a n polynomial in s, t, a~, aj for i, ] = 1, 2 ... n. S u b s t i t u t i n g these polynomials into Q,, we o b t a i n a J o r d a n p o l y n o m i a l Pn in t h e n + 2 variables s, t, a 1 .. . . . an, which will h a v e t h e desired p r o p e r t y . Clearly, P n is i n d e p e n d e n t of t h e spin f a c t o r S a n d t h e choice of s a n d t. [ ]
19 - 792902 Acta mathematica 144. Imprim6 le 8 Septembre 1980.
282 E. ~. ALFSEN ET AL.
Observe for later applications t h a t if A is a J W - a l g e b r a of t y p e I s a n d if ~: A - > M is a dense representation, t h e n M m u s t be a spin factor. I n fact, if p a n d q are e x c h a n g e a b l e abelian projections in A with s u m 1, t h e n ~(p) a n d ~(q) are exchangeable abelian projec- tions in M with s u m 1, so M is a n I2-factor, i.e. a spin factor.
F o r t h e n e x t l e m m a we also need some new terminology: T w o elements a, b of a J B - algebra are said t o be J-orthogonal if aob = 0 . Clearly this generalizes t h e o r t h o g o n a l i t y of symmetries in a spin factor. N o t e also t h a t if A is concretely represented as a JC-algebra, t h e n a, b are J - o r t h o g o n a l iff t h e operator ab is skew. F o r a given i d e m p o t e n t p in a J B - algebra A we s a y t h a t an element s E A is a p-symmetry if s ~ = p .
LEMMA 4.3. I / a projection p in a JW-algebra A o / t y p e 12 admits two J-orthogonal p-symmetries, then p is central.
Proo/. L e t s, t be t w o J - o r t h o g o n a l p - s y m m e t r i e s in A , a n d define q= 89247 r = 8 9 T h e n q + r = p , a n d q, r are exchangeable projections; in fact t h e s y m m e t r y u = (1 - p ) + t satisfies uqu =r, so it exchanges q a n d r.
N o t e t h a t t h e central covers c(p), c(q), c(r) are all equal. W e assume for contradiction t h a t p # c ( p ) . T h e n t h e central covers of q a n d of c ( p ) - p will n o t be orthogonal, so b y [41; L e m m a 18] there will exist exchangeable non-zero projections x<~q, y ~ c ( p ) - p . Defining z = uxu, we get z ~< uqu = r.
N o w x, y, z are non-zero o r t h o g o n a l projections with x, y exchangeable a n d x, y ex- changeable. T h e n a n y h o m o m o r p h i s m which annihilates one of t h e projections x, y, z, will annihilate t h e other two. T h u s there exists a dense representation ~: A ~ M which does n o t annihilate a n y of t h e three projections x, y, z (cf. [7; Cor. 5.7]). B y t h e r e m a r k preceding this lemma, M m u s t be a spin factor. B u t a spin f a c t o r c a n n o t contain a set of three non- zero o r t h o g o n a l projections. This c o n t r a d i c t i o n completes t h e proof. [ ]
T h e n e x t l e m m a is crucial.
L E M~A 4.4. I / A ~ B(H)s ~ is a JC-algebra o/complex type, then every dense representa- tion o/ the I2-summand o/ A is onto a spin/actor o/ dimension at most/our.
Proo]. L e t z be t h e central projection in A such t h a t t h e I ~ - s u m m a n d of 3 is equal t o zA, a n d let ~: z ~ M be a dense representation. As r e m a r k e d earlier, M m u s t be a spin factor.
N o t e t h a t M 0 = ~ ( z A ) will be a n o r m closed J o r d a n subalgebra of M containing t h e identity. I t is n o t difficult to verify t h a t such a subalgebra is itself a spin f a c t o r unless it
S T A T E SPACES OF C*-ALG]~BRAS 283 is of dimension less than three. I n the latter case M 0 will be associative (in fact M ~ R or M =~ R | I n the former case the spin factor M 0 satisfies M 0 -~ B(H0),a for some finite or infinite I-Iilbert space H0; but B(H0)sa is a spin factor only ff H 0 is of (complex) dimension 2, in which case B(H0)s~ is of (real) dimension 4. Hence dim M 0 = 1, 2 or 4.
We will next show t h a t dim M~<4. Let p, q be exchangeable abelian projections in z ~ with p + q = z . Then there exists a z-symmetry sEz.4 such t h a t s p s = q . Now p s = s q , so s ( p - q) = ( q - p ) s . Thus the elements s and t = p - q are symmetries in the J o r d a n algebra z-~ satisfying s o t = O . Consider now an arbitrary dense representation ~0 of zA. B y the above argument (with F in place of ~), F is a spin factor representation and dim F(zA) <~4.
B y Lemma 4.2 we have
~p(Ps(s, t, za 1 .... , za~) ) = Ps(y~(s), ~f(t), ~0(Z~tl) , ..., ~)(za5) ) = 0
for a n y set of five elements a I ... as E A . Since the dense representations separate points [7; Cor. 5.7 and Prop. 8.7J, it follows t h a t
Ps(s, t, za 1 ... zas) : 0 , all a I . . . a s E A . (4.6) B y the Kaplansky density theorem for JC-algebras [18; p. 314], the unit ball of zA is strongly dense in the unit ball of z-4. Hence it follows from (4.6) t h a t P6(s, t, x 1 ... xs) = 0 for all x 1 ... x 5 EzA. Applying ~, we get
P5(~9(8), 9~(t), ~(Xl) ... ~(X5) ) = 0 all x 1 ... x s e z . ~ .
B y L e m m a 4.2 ~(xl) .... , ~(xs) is a linearly dependent set of elements of M for a n y set of five elements x 1 ... x 5 E z ~ . Hence dim ~(zX) ~< 4, and by a-weak density, dim M ~< 4. [ ] I t follows from the next result t h a t the dense spin factor representations of L e m m a 4.4 have dimension precisely four.
LwMlgA 4.5. Let A g B(H)s a be a JC-algebra o / c o m p l e x type and let s o be the central projection in .~ such that so A is the I~-summand o/ .4. T h e n so.4 contains a subalgebra M = l i n R (so, sl, s~, sa) which is a / o u r dimensional s p i n / a c t o r with sl, s2, sa J-orthogonal %- symmetries. Moreover each b Eso.~ can be uniquely expressed as:
3
b = ~. /js~, (4.7)
J-O
w h e r e / j is in the center Z o / s o , 4 / o r j = O , 1, 2, 3.
Proo/. Let (Pa} be a maximal orthogonal set of central projections in s 0 ~ with the property t h a t each Pa admits three J-orthogonal p~-symmetries, say sla, s~a, sa~ and let p =
284 ~ . M. A L F S E N E T A L .
~ p~. A priori, there m a y n o t exist a n y such p~, in which case the s u m m a t i o n over the e m p t y set of indices would give p = 0. However, we shall see t h a t this eventuality cannot occur; in fact we will prove t h a t p = s 0.
Assume t h a t p ~ s 0. Now we will first show t h a t every dense representation of (s o - p ) is of dimension at most three, then we will see t h a t this leads to a contradiction. B y L e m m a 4.4 all dense representations of (s o - p ) A are onto spin factors of dimension at most four (since each extends to the I e - s u m m a n d of A). Now if ~: (s 0 - p ) A - ~ M is a four-dimensional spin factor representation, t h e n b y [34; L e m m a 3.6] we can find orthogonal symmetries 81, s3, s 8 in M, an idempotent q E (s o - p ) . ~ , and J-orthogonal q-symmetries tl, t3, t 3 m a p p i n g onto s 1, %, %, respectively. Note t h a t (s o - p ) A is of t y p e I S, and so b y L e m m a 4.3, q is a central idempotent. This contradicts the m a x i m a l i t y of {pa}, so we conclude t h a t all dense representations of (s o - p ) A are onto spin factors of dimension three. Now, as in the proof of L e m m a 4.4, all such representations restricted to (s o - p ) A have associative range. Thus (s o - p ) - 4 m u s t be associative (i.e. abelian). B u t this is impossible, so p = s o as claimed.
Define sj=~.~ si~ for ] = 1 , 2, 3. Then sl, s3, sa are J-orthogonal so-symmetries , and M =lin R (so, sl, s3, ss) is a spin factor of dimension four.
I t remains to establish the decomposition (4.7). F o r a given b ~ s o A we define
]o = ( ( ( b ~ 1 7 6 1 7 6 (4.8)
/j = ( b - / o ) O S j for i = 1, 2, 3. (4.9)
Consider now a dense representation y~ of s 0.~. Since y) is a dense spin factor represen- tation of dimension a t most four (by L e m m a 4.4), and since ~(sl) , ~f(s3) , yJ(sa) are orthogonal symmetries, we have a decomposition
3 3
~(b) = ~ ~j~(sj)= ~ (~j ~)o~(s~),
./~0 i - 0
where the coefficients ~ are given as in the formulas (4.2) a n d (4.3). Comparing these formulas with (4.8) and (4.9) (with a =yJ(b)), we conclude t h a t
211 =~([t), for i = 0 , 1, 2, 3, (4.10) and therefore
( ( )
v?(b) =y~ lsosj . (4.11)
J
B y (4.10) and (4.11), ~ will m a p the elements ]0, ]1, ]3, ]a onto central elements and the element b - ~ = 0 / i o s j onto zero. Since the dense representations separate points, it fol- lows t h a t /0,/1,/3,/a E Z and t h a t b - ~ = 0 / i o s j = 0 . The uniqueness follows from (4.8)
and (4.9). []
STATE SPACES OF C*-ALGEBRAS 2 8 5
R e m a r k . Note t h a t L e m m a 4.5, equation (4.7), implies t h a t the I~-summand of is isomorphic to C ( X , M2(C)~) where X is a hyperstonean space such t h a t C ( X ) is iso- morphic to the center of s oA.
The n e x t theorem is the main result of this section.
T H E OREM 4.6. A n y JC.algebra A o / c o m p l e x type is reversible.
Proo/. 1. L e t 9~ be the C*-algebra generated b y A _~ B(H)s ~ and let z: 9 ~ - ~ B ( H ' ) be t h e universal representation of 9~. Since reversibility of A only depends on the embedding of A in 0/, we can, and shall, identify 9~ and zr(9~). First we will show t h a t A is reversible in this representation.
B y [37; Th. 6.4 & Th. 6.6] it suffices to show t h a t the I~-summand of A is reversible.
Let this s u m m a n d be s 0 ~ where s o is a central projection in .4, and let 81, s2, s 3 be J - o r t h o - gonal s0-symmetries with the properties explained in L e m m a 4.5. Consider now an a r b i t r a r y finite set of elements
3
bi = ~ /~jsjESo-4, i = 1, . . . , n, t=0
where the coefficients /~j are in the center of s0.4. B y L e m m a 4.1 the spin factor M = linR (so, sl, s2, sa) is reversible. Hence
b l . . . b ~ + b , . . . b l = ~ / l j , . . . f ~ j ~ ( s j l . . . s j ~ + s j , . . . s j , ) E S o A .
(J~, ...,in)
This shows t h a t s o J is reversible, and thus A is reversible.
2. We now show t h a t A is reversible. Suppose a 1 ... a , EA; b y reversibility of x = a l a 2 ... a~ +a~a=_ 1 ... a x E_~.
B u t x is also in 9~, and so it lies in 9~ N A. We are done if we show 9~ fl A = A.
Recall t h a t ~ can be identified with 9~**. The weak and a-weak closures of A will coincide [39; L e m m a 4.2], so .4 is also the a-weak closure of A (i.e. the closure in w(9~**, 9~*)).
Now _~ N 9~ is obtained b y intersecting 9~ with the intersection of all w(~**, 9~*)-closed hyperplanes containing A; b u t these hyperplanes are of the form ~-1(0) where q0 Eg~*, and thus A N 9~ = A since A is norm closed. This completes the proof. []
w 5. The enveloping C*-algebra
Two JC-algebras, even if t h e y are isomorphic, m a y act on their respective Hilbert spaces in quite different ways; in fact, even the C*-algebras t h e y generate m a y be non- isomorphic. I n this section we prove the existence, for a n y special JB-algebra, of a "largest"
286 E . M . A L F S E N E T A L .
C*-algebra generated b y it, such t h a t in a n y concrete representation, the C*-algebra generated b y the given J B - a l g e b r a is a quotient of the "largest" one. Then we specialize to J B - a l g e b r a s of complex type.
T ~ E O ~ M 5.1. Let A be a JB.algebra. There exists a C*-algebra ~ and a Jordan homo- morphism ~p: A-+9~ such that ~ is generated by ~p(A) and such that for any Jordan homomor- phism 0: A->~sa where B is a C*.algebra, there exists a *-homomorphism O: ~ ~ B satisfying 0 = ~ o w.
Proof. L e t A C = A | be the complexification of A. I t is a J o r d a n *-algebra, i.e. a complex J o r d a n algebra with an involution satisfying (aob)* =a*ob*. (A c can be normed to become a "JJB*-algebra" or " J o r d a n C*-algebra" [44], but we will not need this.)
L e t u: A c-> ~ be the "unital universal associative specialization" of A c [21; p. 65].
Here ~ is a unital complex associative algebra and u: A c-~ ~ is a J o r d a n homomorphism with roughly the universal p r o p e r t y stated in the theorem, only with the C*-algebras replaced b y associative algebras.
We briefly indicate how ~ is constructed: ~ is the tensor algebra of A c factored b y the ideal generated b y all elements of the form
a o b - 8 9 1 7 4 1 7 4 a, b e A c,
and the difference between the unit of the tensor algebra and t h a t of A s . The details are found in [21; p. 65].
N e x t we note the existence of a unique involution on "U such t h a t u is a *-map. This is done b y defining an involution on the tensor algebra b y
(al| ... | a*| ... | a I . . . a~fiA c, and noting t h a t this involution preserves the above-mentioned ideal.
I t is easily seen t h a t u [ A: A-~ ~ satisfies the p r o p e r t y of the theorem, with C*-algebras replaced b y *-algebras.
Next, we define a seminorm on ~ b y
Ilxl[ = s u p B(H) is a *-representation}. (5.1)
We h a v e to prove t h a t ]]x]] < oo for x e ~/. Since u(A) generate ~ as an algebra and I[" ]l is clearly sub-additive, it is enough to prove this for x of form x = u ( a l ) . . , u(an), where
a l , . . . , a n E A .
But, whenever xe: ~ ~ B(H) is a *-representation, ~ o u [ a is a J o r d a n representation of A, and is consequently of n o r m 1. Thus
STATE SPACES OF C*-ALGEBRAS 287 HTe(x)H = ]lzou(al) ... 7~ou(an) H ~ H~ou(al)ll ... ][~ou(an)ll ~ lla~l] ... Ilanll ,
so llx]l ~ Ilal]l ... Ilanll < c~.
Letting
N = (xE 'U: Ilxll = o ) , (5.2)
we obtain a C*-norm on '~/N. 9~ is defined to be the completion of 'U/N. L e t
y J ( a ) = u ( a ) + N , a E A . (5.3) Obviously, 9~ is generated as a C*-algebra b y ~o(A). To complete the proof, let B be a C*-algebra and 0: A--->Bsa a J o r d a n homomorphism. Then 0 factors through ~ , i.e. there exists a *-homomorphism ~: ~ - > B with 0 = ~ o u . Assume B is faithfully represented on a Hilbert space. Then ~ is a *-representation of ~ , and therefore it annihilates N, and b y definition of the n o r m induces a *-representation of ~ / N of norm 1. I t s continuous exten- sion ~ to ~ is easily seen to satisfy the conditions of the theorem. []
Remark. Note t h a t the *-homomorphism 0 in Theorem 5.1 is necessarily unique. The theorem states t h a t the following diagram commutes:
6
A---~ ~sa I
B y abstract nonsense, the pair yJ, ~I is uniquely determined (in the obvious sense).
I f A is special, t h e n b y definition A can be faithfully represented on a ttilbert space.
Factoring such a representation through ~I, we conclude that, in this case, yJ is injective.
Then we identify A with its image ~o(A) in 9~, and call 9~ the enveloping C*-algebra of A.
We can then rephrase the above results as follows: A is a JB-subalgebra o/~s~, and generates 9~ as a C*-algebra. A n y Jordan homomorphism O: A ~ Bs a where B is a C*-algebra extends uniquely to a *-homomorphism ~: 9..[---> B.
I n the general case, the kernel of ~ is easily seen to be the "exceptional ideal" of A defined in [7; w 9]. Then 9~ is the enveloping C*-algebra of A/ker ~.
The fact t h a t " J o r d a n multiplication knows no difference between left and r i g h t "
is reflected in the following:
COROLLARY 5.2. I / A is special, there exists a unique *-anti-automorphism 6P o/ 9~
leaving A pointwise invariant. Also, r I.
288 E.M. ALFSEN ET AL.
Proo]. L e t 9 ~ denote the opposite C*-algebra of 9)[. Then 9 is the *-homomorphism
~ - > ~ ~ extending the Jordan homomorphism ~0: A - ~ [ ~ Also, O 2 is a *-automorphism of 9~
leaving A pointwise invariant, and is therefore the indentity. []
Throughout the rest of this chapter, A will be a JB-algebra of complex type, and K its state space. 9~ is its enveloping C*-algebra, with state space Jr. The restriction map from 3( onto K will be denoted by r. (It is the dual of the embedding ~: A ~ f . ) Obviously, toO* = r .
P R O r O S I T I O ~ 5.3. Let A be a JB-algebra o/ complex type, with state space K. Let 9A be the enveloping C*-algebra o/ A , with state space ~ . I[ ~ E~e~, then r(~)E~eK, and the restriction map r maps FQ bi]ectively onto F~(~).
Proo]. Consider the GNS representation ~0: ~-+B(Ho). Since A generates ~, then zQ] A is irreducible. B y Proposition 3.5, z~Q[ A is dense, and so by Proposition 2.5, (~0 [~)*
maps the normal state space N of B(H) bijectively onto a split face of K. Consider the commutative diagram:
Since :~* maps N bijectively onto _F0, it follows t h a t r maps F o bijectively onto a split
face of K, and the result follows. []
L ] ~ A 5.4. With assumptions as in Proposition 5.3, the restrictions to A o/the represen.
rations ze, resp. ~rr are conjugate irreducible representations o[ A associated with r(~ ).
Proo]. L e t ~0EH 0 denote a representing vector for o. Choose an involution ] of H e fixing ~=0, and define the transpose map a~-->a ~ on B(H~) b y at=]a*]. Then xb-->xr~(O(x)) ~ is an irreducible representation of 9.I, and ~:e is seen to represent the state O*p under this representation, which can therefore be identified with oz~.~.
For x EA, we have r = ?':zQ(x) ?', which proves t h a t the restriction of this represen- tation is conjugate to :zQ I a.
T h a t ~rel A is associated with r(~) follows from the following, if x E A:
( ( % [ 2 (x)$o [$o) = <x, q> = <x, r(q)>. []
P R 0 ~ 0 S I ~z I 0 ~ 5.5. With assumptions as in Proposition 5.3 the/ollowing are equivalent:
(i) O* e =e, (ii) .Fr = {~}, (iii) F o fl O * ( F o ) # 0 .
STATE SPACES OF C * - A L G E B R A S 289 Moreover, the inverse image r-l(r(o)) o/ r(~) in ~ equals the line segment [if, qb*~] which degenerates to a point i / t h e above requirements are/ul/ilted.~
Proof. W e prove (i) ~ (iii) ~ (i!) ~ (i). Of these implications, t h e first is i m m e d i a t e since O*(F~) = F r e.
Assume (iii). Since minimal split faces are either disjoint or equal, we h a v e (I)*# E Fe, so ~, (I)*Q are equivalent. F r o m this it follows t h a t z0, ~ * e are u n i t a r i l y equivalent. T h e n the same holds, of course, for their restrictions to A. But, b y L e m m a 5.4, these restrictions are conjugate, so b y P r o p o s i t i o n 2.4 w e h a v e dim •Q = 1, or F e = (~}.
Assume n e x t t h a t F e = {~}, i.e. dim ze = 1. T h e n we h a v e zQ(x)~ = (x, ~}~, a n d similarly for (I)*~, t h e r e b y p r o v i n g (since ~ a n d (I)*# have t h e same restrictions to A) t h a t ~ ] A is unitarily equivalent to ~*e[A" A u n i t a r y equivalence of these two representations is also an equivalence of ~e a n d ~r T h u s (I)* e E F e --(e}, so (I)* e = e .
Finally, assume t h a t ~ E ~e K a n d r(~) = r(~). T h e n 7~r I A is an irreducible representation of A associated with r(#), a n d is therefore either u n i t a r i l y equivalent or conjugate to
~e I A, i.e. unitarily equivalent to either ~e I A or z r e I A ( L e m m a 5.4). T h u s z~ is equivalent to either zQ or 7~r i.e. a E F e or a E F ~ , e. Since FQ, F r e are m a p p e d injectively into K (Prop. 5.3), a = e o r a=(I)* e follows. T h u s r-i(r(Q))N ~e~={e, OP*e}. Since r - l ( r ( e ) ) i s a closed face of :K, t h e Krein-Milman t h e o r e m shows t h a t r-i(r(o~)) = [~, (I)*#], a n d t h e proof
is complete. [ ]
De/initions. W e divide t h e pure state space of A in t w o parts as follows:
~e,,~K = {o E ~ K : F~ = {~)}} (5.5)
~. ~ K = ~ e K ~ O e , o K . (5.6)
Thus ~e.o K correspond to one-dimensional representations. ~. 0K is easily seen to be closed in t h e facial t o p o l o g y [1; w 6], b u t we shall n o t use this fact.
C o ~ o L L A ~ Y 5.6. The restriction map r: ~I~-+ K maps Oe. o ~ one-to-one onto ~. 1K and
~e.l~ two-to-one onto ~e.i K.
Proo]. Since r ( ~ ) = K , t h e Krein-Milman t h e o r e m proves t h a t r(~e:~ ) ~ e K . Proposi- tion 5.3 contains the opposite inclusion a n d proves t h a t ~,. 0 ~ (resp. ~e. 1 ~ ) is m a p p e d into
~,0 K (resp. OelK). B y Proposition 5.5, t h e proof is completed. [ ] Our n e x t l e m m a is essential. I t i s our only use of t h e results o f w 4, so i t is n o t as innocent as it looks. W e repeat our standing hypothesis t h a t A is Of complex type.
LEMZ~A 5,7. The/ixed point set in 9~ o] (I) is A + i A . 1 9 t - 792902 Acta mathematica 144. Imprim6 le 8 Septembre 1980.
