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9 B ~ M 2 (B), for B equal to the relative commutant of the set of matrix units {ei]} in M (Lemma 4.27). If M is represented as a weakly closed *-subalgebra of B (H), then the same will be true of B, so B is a von Neumann algebra. Now let * be a second Jordan compatible product on M wch that zr * s * t = 1. We will show * coincides with the given product. We first sketch the idea. The Jordan product on M so. extends uniquely to a complex linear Jordan product on M. As seen above, we can represent elements of M as 2 x 2 matrices. Then the associative product is determined by the Jordan product in the following sense: (4.25) (Note that a similar calculation (3.7) was behind the Jordan coordinatization theorem (Theorem 3.27). In that case, lacking M = Msa+zMsa, the Jordan product on the self-adjoint part of the algebra did not determine the Jordan product on the whole algebra, and so we needed 3 x 3 matrix units instead of 2 x 2.) Now we give the details of the proof. For each associative product on M, we can create 2 x 2 matrix units from linear combinations of 1, r, s, t, as shown above, and so can represent M as the algebra of 2 x 2 matrices over an algebra B, where B is the relative commutant of r, s, t in M. Since r, s, t are symmetries, any element x in M element commutes with each of r, s, t iff UrX = UsX = UtX = x. Since these are Jordan expressions, the relative commutants for the two products coincide. Thus we can represent M with each product as 2 x 2 matrices over a fixed subset B. The two products on M will then coincide iff the two products coincide when restricted to B. Since the two products must induce the same Jordan product on M, the equality of the two products then follows from (4.25). 0 By the bicommutant theorem, an irreducible von Neumann algebra B(H) must be equal to B(H). For JW-algebras, this is not necessarily the case, since for example l\1,,(R) acts irreducibly on c n . However, if we add the assumption that NI is isomorphic to the self-adjoint part of B (H) the analogous conclusion holds, as we will show below (cf. Proposition 4.59). The remaining results will also be useful in our characterization of normal state spaces of von Neumann algebras.
M
C
4.57. Lemma. Let M be a reversible lW-subalgebra of B(H)sa. Suppose that there exists a self-adjoint system {eij} of n x n matrix umts
CARTESIAN TRIPLES
135
in B (H) (wzth 1 < n < 00) such that ei) + eli E M for all z, J. Suppose further that there zs a partwl symmetr·y t such that ell - e22, e12 + e21, t form a Carteswn trzple of partwl symmetrzes zn M. Then there is an element J zn the center of the real algebra Ro(Jv1) generated by M such that J* = -J and J2 = -1. Proof. Note that ei) = eii (ei] + e);) E Ro (lIn for all z i= J, so {ei)} is a self-adjoint system of matrix units for Ro(M). By Lemma 4.27 we can identify Ro(M) with Mn(B) where B is the relative commutant of these matrix units in Ro(M). By assumption Ai is reversible, so M = Ro(M)sa = Hn(B). Define r = ell - e22 and s = e12 + e21. Then by assumption there is an (ell + e22)-symmetry t such that T, s, t form a Cartesian triple of (ell + e22)-symmetries. Define Jo = Tst, and
By Lemma 4.54 applied to M 12 , Jo satisfies J5 = -(ell + e22) and Jo = -jo, and Jo is central in Ro(M12). Since Jo is in Mn(B) and Jo = JO(ell + e22) = (ell + e22)Jo, then Jo admits a unique representation 2
(4.26)
Jo =
L
bijei)
i.j=l with each bi ) in B. We will show that there is a central element z in B such that Jo = z(ell +e22), and that J = L:~l zeii will have the properties specified in the lemma. Since Jo is central in R o (M 12 ), it commutes with ell, so
By the uniqueness of the representation (4.26), b21 = b12 = O. Since jo commutes with e12 + e21, a similar calculation shows bll = b22 . Thus there is an element z E B such that
Since jo = -jo and j5 = -(ell +e22), then the element z satisfies z* = -z and z2 = -1. We are going to show z is central in B. Let b E B, and let x = be12+b*e21. Then x E Ro(M)sa = M, and (ell +e22)x(ell +e22) = x, so x E (ell + e22)M(ell + e22). In particular, Jo must commute with x. Then
136
4.
REPRESENTATIONS OF JB-ALGEBRAS
which gives
This implies zb = bz, so z is central in B as claimed. Finally, define J = zeii· Then J is clearly central in Mn(B) Ro(M), and satisfies J2 = -1, J* = -J. 0
L
4.58. Lemma. Let M be a lEW-algebra zsomorphzc to B (H)sa. If e is a projectIOn m M and rand S are lordan orthogonal e-symmetrzes, then there exzsts an e-symmetry t m M such that (r, s, t) is a Carteszan triple. Proof. It suffices to prove the result for M = B(H)sa. Define t = zrs. Since rand s anticommute, one finds that t is an e-symmetry that is Jordan orthogonal to rand s. Furthermore, rst = -ze and then tsr = (rst)* = (-ze)* = ze. Therefore for x E eB(H)e,
so (r, s, t) is a Cartesian triple of e-symmetries in M. (See also (A 192)). 0 We say a concrete JW-algebra Me B(H)sa is zrreduczble if there are no proper closed subspaces of H invariant under lVI.
4.59. Proposition. If M zs a lEW-algebra zsomorphzc to B(K)sa for a complex Hilbert space K, and lVI zs represented as an zrreduczble concrete lW-algebra m B(H)sa, then M = B(H)sa. Proof. The case where K is one dimensional is obvious, and so we assume that dim K > 1. We first show that there is an element J in the center of the real algebra Ro(M) generated by lVI such that J* = -J and J2 = -1. We are going to apply Lemma 4.57, so we show that M ~ B(K)sa satisfies the hypotheses of that lemma. (Note that there is a subtlety here: associative computations are taking place in B (H) and not B (K).) We can write 1 = Pl + ... + Pn where Pl,P2, ... ,Pn are projections exchangeable by symmetries in Ai and 2 F p" Now Fp = {O" E K I O"(p') = O} and (5.1) imply that Fp n Fq = Fpl\q. Thus the greatest lower bound of projective faces in :F is their intersection. 0
Dual maps of Jordan homomorphisms 5.11. Lemma. Let M be a lEW-algebra, and let p be a proJectwn m M wdh assocwted face F = Fp. Then p lS the umque posltwe bounded affine functwnal on the normal state space K of M wdh value 1 on F and value 0 on F' = Fp' .
Proof. Recall that we can identify M with the set of bounded affine functions on K (Corollary 2.60). Let b E M+ satisfy b = 1 on F and b = 0 on F'. Then Up,b = 0, so by the complementarity of Up and Up' we have Upb = b. We next show b :S 1. If 0" E K, since 11U; I :S 1 (Proposition 1.41), then U;O" = AW for some w E F = im U; n K and A E R with 0 :S A :S 1. Then for all 0" E K, O"(b)
= O"(Upb) = (U;O")(b) = Aw(b) = A :S 1,
since b = 1 on F. Thus b :S 1 follows. Then
Since 1 - b is 0 on Fp and 1 on Fp" the same argument shows 1 - b :S p' = 1 - p, so p :S b. Thus b = p. 0 5.12. Lemma. Let A be alE-algebra wdh state space K. Then the map a I-> alK lS an lsomorphlsm from the order umt space (A, 1) onto the order umt space A(K) of w*-contmuous affine functwns on K wlth the pomtwise ordermg and norm, wlth the constant functwn 1 as dlstmgmshed order unit.
Proof. The fact that (A, 1) is a complete order unit space was established in Theorem 1.11. The rest of the lemma follows from the corresponding statement for complete order unit spaces: (A 20). 0 5.13. Lemma. If M I , M2 aTe lEW-algebms wdh nOTmal state spaces K I ,K2 , then T I-> T*IK2 lS a 1-1 cOTTespondence of positive unital 0"weakly contmuous maps T : MI ---> M2 and affine maps from K2 mto K I . Similarly, if A I ,A 2 aTe lB-algebms with state spaces K I ,K2 , then
DUAL MAPS OF JORDAN HOI\IOI\IORPHISMS
145
T f---> T* IK2 zs a 1-1 correspondence of posztwe unital maps T : Al and affine w*-contmuous maps from K2 mto K 1 .
->
A2
Proo j. Assume first that K 1, K 2 are the normal state spaces of the JBW-algebras lvh, M2 respectively. If w E K 2, then T*w = w 0 T is aweakly continuous, and thus is normal (Corollary 2.6). Thus T* maps K2 into K 1 . If Tl and T2 are bounded linear maps from Ml into M2 such that Ti and T2 agree on K 2, then Tl a and T 2a agree on all normal states of !l12 , so we must have Tl = T 2. Thus T -> T*IK2 is 1~1. Now let ¢ : K2 -> Kl be an affine map. Then ¢ extends uniquely to a linear map from the linear span of K 2 into the linear span of K 1. By Corollary 2.60, the unit ball of the preduals (M2 )*, (Md* are co(K2 U (-K2)) and co(K 1 u( -Kd) respectively, so the extension of ¢ has norm at most one. We also denote this extension by ¢, and let T = ¢* : l'vh -> M 2 . Then T is a positive unital a-weakly continuous map, and T* restricted to K2 equals ¢. This shows that T -> T* IK2 is surjective, which completes the proof of the first statement of the lemma. Assume next that K l , K2 are the state spaces of JB-algebras AI, A2 respectively. If T : Al -> A2 is a unital positive map, then T* is a w*continuous affine map from K 2 into K 1. Since states separate elements of a JB-algebra, then T -> T* IK2 is injective. To see that this map is surjective, let ¢ be a w*-continuous affine map from K2 into K l . Identify A; with A(K,) for z = 1,2 (cf. Lemma 5.12). Define T: Al -> A2 to be the map given by Ta = a 0 ¢. Then T* restricted to K2 coincides with ¢. Thus T -> T*IK2 is bijective. D
5.14. Lemma. Let Ail, !lh be lEW-algebras wzth normal state spaces K 1 , K 2 . Let ¢ : K2 -> Kl be an affine map, and T : !vh -> M2 the posztwe umtal a-weakly contmuous map such that ¢ = T*IK2' Let Fl and F2 be pTOJectwe faces of Kl and K2 respectwely, wzth assocwted pTOjectwns PI and P2. Then these are equwalent:
(i) TPI = P2, (ii) ¢~l(Fd = F2 and ¢~l(F{)
=
F~.
If ¢ zs surJectwe, then (2) and (u) are also equwalent to (iii) d;(F2)
= Fl and
¢(F~)
= F{.
PTOOj. (i) (ii) Viewing elements of Ml and !viz as bounded affine functions on their normal state spaces, using Tpl = PI 0 (T*) = PI 0 ¢, and applying Lemma 5.11, we have TPI
= p2
(Tpl)~l(l)
-¢=?
= F2 and q;-l(pll(l)) = F2 and
-¢=?
d;~l(Fl)
-¢=?
=
F2
and
(Tpd~l(O)
=
d;~l(pll(O))
d;~l(F{)
=
F~.
F~
=
F~
146
5.
STATE SPACES OF JORDAN ALGEBRAS
(iii) =} (i) Assume (iii) holds, and let b = TPI' Then b ?: 0 and b = PI 0 ¢ on K 2, so by (iii) b is a positive element of M2 with value 1 on F2 and value 0 on F~. By Lemma 5.11, b = P2, so TPI = P2. (ii) =} (iii) Assume ¢ is surjective, and that (ii) holds. Then applying ¢ to each side of the equations in (ii) gives (iii). D
5.15. Lemma. Let MI and Nh be lEW-algebras wlth normal state spaces KI and K 2 , and let ¢ : K2 ---> KI be an affine map. Le~ T : MI ---> M2 be the unital posltwe a-weakly contmuous map such that T* IK2 = ¢. Then the followmg are equwalent:
(i) T lS a umtal lordan homomorphlsm from Nfl mto lvf2, (ii) ¢-l preserves complements of proJectwe faces, (iii) ¢-l as a map from the lattlce of proJectwe faces of KI mto the lattlce of projective faces of K2 preserves lattlce operatwns and complements. Proof. Note first that if p is a projection in M I , and q is the range projection of Tp', then
where the penultimate equality follows from the fact that positive elements of M2 and their range projections have the same annihilators in K2 (d. (2.6)). Thus the inverse image of a projective face is projective. (i) =} (ii) Let FI be any projective face of KI and let PI E MI be the associated projection. If (i) holds, let P2 = TPI E lvh, and let F2 be the corresponding projective face of K 2. By Lemma 5.14, ¢-I(Fd = F 2, and ¢-I(F{) = F~, so ¢-I(F{) = (¢-I(FI ))'. Thus ¢-l preserves complements. (ii) =} (i) We first show T maps projections to projections, preserving orthogonality of projections. Let PI E MI be a projection with associated face FI C K I . Let F2 = ¢-I(Fd; by the remarks in the first paragraph of this proof, F2 is a projective face of K 2. Let P2 E AI2 be the associated projection. By (ii), ¢-I(F{) = F~, so by Lemma 5.14, TPI = P2. Thus T maps projections to projections and preserves complements of projections (since T1 = 1). Now suppose P and q are orthogonal projections. Since T is a positive map, then p:S; q' implies Tp:S; T(q') = (Tq)', so Tp...L Tq. Thus T also preserves orthogonality of projections. It follows that T preserves squares of elements with finite spectral decompositions. Since T is a positive unital map and M has the order unit norm, T is norm continuous (A 15). By continuity and the spectral theorem (Theorem 2.20), T preserves squares of all elements. It follows that T is a Jordan homomorphism. (ii) ¢? (iii) The map ¢-l preserves intersections and thus lattice greatest lower bounds (d. Proposition 5.10). Since complementation is order re-
TRACES ON JB-ALGEBRAS
147
versing, if ¢-l preserves complements, it will then necessarily also preserve least upper bounds. The reverse implication (iii) implies (ii) is trivial. 0 5.16. Proposition. If M l , M2 are lEW-algebras wdh normal state spaces K l , K 2, then T f-7 T*IK2 zs a 1-1 correspondence of lordan zsomorphzsms from !vfl onto A12 and affine zsomorphzsms from K2 onto K l . Similarly, zf Al and A2 are lE-algebras wzth state spaces K l , K 2, then the map T f-7 T*IK2 zs a 1-1 correspondence of lordan zsomorphisms from Al onto A2 and affine homeomorphzsms (for the w*-topologies) from K2 onto K l .
Proof. Assume that M l , !vf2 are JBW-algebras with normal state spaces K l , K 2. Then T f-7 T*IK2 is a 1-1 correspondence of unital order isomorphisms from Ml onto M2 and affine isomorphisms from K2 onto Kl (d. Lemma 5.13). Note that T is a unital order isomorphism iff T is a Jordan isomorphism. In fact, unital order isomorphisms are Jordan isomorphisms (Theorem 2.80), and conversely, each Jordan isomorphism preserves squares, so is an order isomorphism. This completes the proof of the first statement of the proposition. The proof of the second statement is similar. 0 5.17. Corollary. Let AI be a lEW-algebra wzth normal state space K. Let F and C be pro)ectwe faces with assocwted pro)ectwns p, q respectwely. Let T be a lordan automorphzsm of M. Then T*(F) = C iff Tq=p.
Proof. If Tq = p, then T*(F) = C follows from Lemma 5.14((i) =} (iii)). Conversely, suppose T*(F) = C. Since T* is an affine automorphism of K (Proposition 5.16), it preserves complements of projective faces (see the remarks following (5.6)). Thus T*(F') = C'. Now by Lemma 5.14 ((iii)=} (i)), Tq = p. 0 Traces on JB-algebras
5.18. Lemma. Let M be a lEW-algebra, and let T be a normal state on M. The following four propertzes are equwalent.
(i) T((a 0 b) 0 c) = T(a 0 (b 0 c)) for all a, b, c EM, (ii) T(6Zoa) = T({bab}) for all a,b E M, (iii) + U;,)T = T Jar all pro)ectwns p E M, (iv) U;T = T Jar all symmetrzes s E M.
(U;
Proof. (i) =} (ii) Let a, bE M. By the definition of {bab} (cf. (1.13)),
(5.8)
T (
{bab} =
T
(2b
0
(b
0
a) - b2
0
a).
5.
148
STATE SPACES OF JORDAN ALGEBRAS
If (i) holds, then
T(2b
0
(b
0
a)) = T((2b
0
b)
0
a) = T(2b 2
0
a)
which when substituted into (5.8) gives (ii). (ii) =? (iii). Let a E M. For a projection p, taking b b = pi in (ii) gives
T((Up + Up' )(a)) = T(p 0 a)
+ T(pl 0
=
p and then
a) = T(a),
which proves (iii). (iii) ¢} (iv) By spectral theory, the symmetries s in M are the elements of the form s = 2p - 1 = p - pi, for p a projection. Now the equivalence of (iii) and (iv) follows from Us = 2(Up + Up') - I (cf. (2.25)). (iii) =? (i) Let a, b, c E M. From the definition (1.12) of the triple product we have
(5.9)
(a
0
b)
0
c - a 0 (b
0
c) = ~ ( { bac} - {acb}).
Thus to prove (i) it suffices to show (5.10)
T ( {bac} - {acb})
=
o.
By linearity of the triple product in each factor, the spectral theorem, and norm continuity of Jordan multiplication, it suffices to prove (5.10) for the case where b = p is a projection. By (iii)
T({pac} - {acp}) = T(Up + Up,)({pac} - {acp}), so we will be done if we can show (5.11) for all projections p in M and all elements a, c E M. To prove (5.11), we begin with the identity (5.12)
2{bac}
0
b = {b(a
0
c)b}
+ {b 2 ac}.
(This is easily verified for any special Jordan algebra, and then holds in all Jordan algebras by Macdonald's theorem (Theorem 1.13).) Applying (5.12) with b = p gives (5.13)
2{pac}
0
p = {p(a
0
c)p}
+ {pac}.
149
TRACES ON JB-ALGEBRAS
Interchanging the roles of a and e and subtracting gives
2p
(5.14) Let x ~(I
+ Up
0 (
{pac} - {pea})
=
{pac} - {pea}.
{pac} - {pea}. Then by (5.14), x = 2p - Up') (cf. (1.47)), then x = (I
(5.15)
+ Up
0
x.
Since pox
- Up' )x.
Applying Up to both sides of (5.15) gives Upx = 2Upx so Upx = O. Applying U; to both sides of (5.15) gives Up'x = O. This proves (5.11), which completes the proof of (i). D
5.19. Definition. Let M be a JB-algebra. A state T on A is a tracial state if T((a 0 b) 0 c) = T(a 0 (b 0 c)) for all a, b, e E A, or (when M is a JBW-algebra) if T satisfies any of the equivalent conditions of Lemma 5.1S. A tr'acial state T is fadhful if whenever a ;::: 0 and T(a) = 0, then a = O. Remark. Recall that a tracial state on a von Neumann algebra M is a state T satisfying T (ab) = T (ba) for all a, b EM. If s is a symmetry in Msa and T is a tracial state on M, then T(sas) = T(as 2 ) = T(a) so T is a tracial state on the JBW-algebra M sa' Conversely, let T be a tracial state on M = M sa and also denote by T its complex linear extension to a state on M. Then for all symmetries s, t, we have T(st) = T(t(st)t) = T(tS). Then T(pq) = T(qp) for all projections p, q EM. By linearity and the spectral theorem, T(ab) = T(ba) for all a and b in M. Thus T is a tracial state on M. 5.20. Corollary. Every tracwl state T on a lEW-algebra M satisfies
a,b;:::O
=?
T(aob);:::O.
Proof. By Lemma 5.18(ii)
and {a 1 / 2 ba 1 / 2 }
;:::
0 by Theorem 1.25. D
5.21. Lemma. A spin factor admzts a umque tracwl state. Thzs state is faithful and normal.
Proof. Write M = Rl EB N as in Definition 3.33, where N is a real Hilbert space and x 0 y = (xIY)l for x, yEN. Recall that N consists of
150
5
STATE SPACES OF JORDAN ALGEBRAS
the scalar multiples of non-zero symmetries other than ±l (Lemma 3.34). Define T : A ---7 Rl by
T(A1
+ a) =
A
for A E R and a E N.
If s is any symmetry, then Us is a Jordan automorphism (Proposition 2.34), and so Us takes multiples of symmetries to multiples of symmetries, and therefore maps N onto itself. Thus
T(Us (Al T
+ a)) =
Al
+ T(Us(a)) =
Al = T(AI
+ a).
is positive since
Thus T is a tracial state on lvI. Furthermore, T((AI + a)2) = 0 evidently implies A = 0 and a = 0, so T is faithful. We now prove uniqueness. The smallest possible dimension of N is 2, so if s is a symmetry in N there is at least one unit vector t in N orthogonal to s, and then sot = O. By the definition of the triple product (5.16)
Uts
= 2t 0 (t 0 s) - t 2 0
S
= -so
Thus for any tracial state T1,
so T1(S) = O. This holds for all symmetries in N. Since every element of N is a multiple of a symmetry, then T1 = 0 on N. Then T = T1 follows. Finally, by Proposition 3.38 every state on a spin factor is normal. 0
5.22. Proposition. Every lEW-factor M of type In (1 ::; n < (0) admds a umque traczal state. Thzs state takes the value l/n on every mimmal pTOJectwn. PTOOf. If M is of type 1 2 , then 1\1 is a spin factor by Proposition 3.37, and so there is a unique tracial state on Jvl (Lemma 5.21). Now consider the case where M is of type In with n i- 2. Then M is finite dimensional (Theorem 3.32). Let G be the group of affine automorphisms of the state space K of M. Since the unit ball of M* is co(K U -K), then G extends to a group of isometries of M*. Thus G forms an equicontinuous group of affine automorphisms of the compact convex set K. By the Kakutani fixed point theorem [108, Thm. 5.11], there is a fixed point T of G. Since for each symmetry s the dual map U; is in G, then U;T = T, so T is a tracial state.
TRACES ON JB-ALGEBRAS
151
Now we prove uniqueness. Let M be of type In with 1 ::; n < =, and let T be any tracial state on !vI. Let P and q be any two minimal projections. Recall that minimal projections and abelian projections coincide in a JBW-factor (Lemma 3.30), so P and q arc abelian. Since P and q have the same central cover (namely 1), then P and q are exchangeable by a symmetry (Lemma 3.19). It follows that T(p) = T(q). Since M is a factor of type In, by definition there is a set S of n abelian (therefore minimal) projections with sum 1. Since T takes the same value on each projection in S, it must have the value lin on each. Since all minimal projections in M are exchangeable, T takes the value lin on all minimal projections. By Lemma 3.22 any set of orthogonal non-zero projections has cardinality at most n, so every projection in !vI is a finite sum of minimal projections. By the spectral theorem T is determined by its value on minimal projections. Thus T is unique. D
5.23. Definition. Let lvi be a lBW-algebra. A positive linear map
T from !vI onto the center Z of !vI is a center-valued trace if (i) T is the identity map on Z, and (ii) T(Usx) = Tx for all x E M and all symmetries s. We say T is fmthful if x :::: 0 and Tx = 0 imply x if it is O"-weakly continuous (Proposition 2.64). Note that if A is a JBW-factor, then Z is of the form x f--+ T (x) 1 for a tracial state
= O. Recall T is normal
= R1, so a center valued trace T.
5.24. Lemma. If T zs a center-valued trace or a traczal state on a JB W-algebra .M and PI, . .. ,Pn are orthogonal pmJectzons wzth P = PI + '" + Pn, then for x E AI
Proof. We will prove the lemma for center valued traces; the proof for tracial states is essentially the same. It suffices to prove the lemma for the case n = 2. (The general case follows by induction.) Let P be any projection and let s = P - p'. Since Us = 2(Up + Up' -1) (cf. (2.25)), by the invariance of center-valued traces under the map Us, for each x E !vI We have
which implies
(5.17)
Tx = T((Up + Up' )x).
152
5
STATE SPACES OF JORDAN ALGEBRAS
Let Pi and P2 be orthogonal projections in M. Applying (5.17) with Pi in place of P and UP1 +P2 x in place of x gives (5.18) Since Pi 1\ (Pi +P2) =Pi and P~I\(Pi+P2) =Pi+P2-Pi =P2 (d. (2.17)), then by Proposition 2.28, (5.19) Combining (5.18) and (5.19) gives
This completes the proof. D
5.25. Proposition. Every lEW-algebra M of type In (1 T:;T is continuous from the a-weak topology on lvI to the weak topology on /1.1* (i.e., the weak topology for the duality of lvI* and M). Since the closed unit ball lvh of M is a-weakly compact (Corollary 2.56), then the set Ko = {T:; T I h E Md is the continuous image of a a-weakly compact convex set, and so is a weakly compact convex subset of M* containing O. We are going to show that a E Ko. Let b be an arbitrary element of M such that p(b) :s; 1 for all p E Ko. We will show that a(b) :s; 1 in order to apply the bipolar theorem. Let b = b+ - b- be the orthogonal decomposition of b into the difference of orthogonal positive clements (Proposition 1.28). Let p be the range projection of b+. Then by definition of the range projection, b+ E Mp. Since b+ and b- are orthogonal, by (2.9) b- E M p" We also have Mp 0 M p' = {O} (Lemma 1.45), so
NORl\I CLOSED FACES OF JBW NORMAL STATE SPACES
157
Thus (p - p')
(5.27)
lip - p'li =
Since
0
b = b+
+ b~.
1, and by hypothesis a :S T, then T;~p'T E K a, so
By the bipolar theorem (or a direct application of the Hahn~Banach theorem), we conclude that a E K a. It remains to show that we can choose 0 :S h :S 1. Let h = h+ - h~ be the orthogonal decompositon of H, and let p = r(h+), and s = p - p'. As in the calculation above, soh = h + + h ~. Then
so T(h~)
(h~)2 :S Ilh~11 h~, then T((h~)2) = O. Now by the inequality, T(h~ 0 a) = 0 for all a E M. Thus a(a) = T(h+ 0 a), and Ilh+ II :S Ilhll :S 1, so O:S h+ :S 1. 0
= O. Since
Cauchy~Schwarz
T(h+
0
a)
=
Norm closed faces of JBW normal state spaces The main result of this section is Theorem 5.32, which shows that every norm closed face of the normal state space K of a lBW-algebra M is projective, i.e., of the form Fp = {a E K I a(p) = I} for a projection p in K. We then establish various properties of the norm closed faces of K.
5.28. Definition. If X is a subset of the normal state space of a lBW-algebra, the intersection of the faces containing X is called the face generated by X, and is denoted face(X). Similarly, the intersection of the norm closed faces containing X is called the norm closed face generated by X. If X = {a}, then we write face( a) instead of face ( {a} ), and similarly We write face (a, T) instead of face ( {a, T} ). 5.29. Lemma. Let !vI be a lEW-algebra wdh normal state space K and T a normal tracial state on !vI. Let h E M+, wzth range proJectzon satisfying r(h) = l. Define w on M by w(a) = T(h 0 a). Then T zs zn the norm closure of the face of K generated by w. Proof. Let {e.>.} be the spectral resolution for h (cf. Definition 2.21). By the definition of a spectral resolution and Theorem 2.20, for each A E R
5
158
STATE SPACES OF JORDAN ALGEBRAS
On the other hand, each e~ operator commutes with h, so by (1.62)
0< Ue ,A h < h. Combining these gives
o :::;
Ae~
:::; h.
Now for each A E R define a functional
T),(a) = T(e~ By Corollary 5.20,
T),
by
T),
0
a).
is a positive functional, and for a E M+,
AT),(a) = T(Ae~
0
a) :::; T(h
0
a) = w(a).
Thus for any A > 0, we conclude that T), is in the face of !v!: generated by w, and so IIT)'II- 1 T), is in the face of K generated by w. Finally, we show that {IIT)'II- 1 T),} converges to T in norm as A ~ O. By the definition of a spectral resolution, as A ~ 0, e~ / eS. Since eS = r(h) (cf. (2.15)), and by hypothesis, r(h) = 1, then e~ /
(5.28)
1.
By the Cauchy-Schwarz inequality (1.53), if a E M with
IT(a) - T),(a)1
=
IT(a) - T(e~
=
IT ((1 -
:::; T(l -
e~)
0
Iiall :::;
1, then
a)1 a) I
0
e~)1/2T(a2)1/2
:::; T(l - e~)1/2T(1)1/2
= T(l -
e~)1/2.
Then
and the last term goes to zero by (5.28) and normality of
T.
D
5.30. Lemma. Let M be a lEW-algebra wzth normal state space K. If T zs a fazthful normal traczal state on M, then the face of K generated by T zs norm dense m K.
Tb
E
Proof. Let F be the face of K generated by T. For b E M define M. by Tb(a) = T(b 0 a). Recall that if a, b :0:: 0, then T(a 0 b) :0:: 0
NORM CLOSED FACES OF JEW NORMAL STATE SPACES
159
(Corollary 5.20). Thus if b ~ 0, then Tb ~ 0. Note also that if b E M+, then b Ilbll1, so Tb Tllblll = IlbiIT. Thus
s:
s:
(5.29)
If b = b+ - b- is the orthogonal decomposition of b, we will prove that the orthogonal decomposition of Tb is given by Tb = Tb+ - Tb-' Let p be the range projection of b+ and pi = 1 - p. Then as in the proof of Lemma 5.27 (cf. (5.27)), we have (p - pi) 0 b = b+ + b-. Thus
The opposite inequality is clear, so it follows that Tb = Tb+ - Tb- is the orthogonal decomposition of Tb. Now let X be the linear space of all functionals Tb for b E M. Note that by the previous paragraph
Furthermore, since (Tb)+ = Tb+, then by (5.29) we have
(5.30) as long as ()+ # 0. We next show X is norm dense in M*. If a E M annihilates X, then Tb(a) = T(b 0 a) = for all b E M. In particular, T(a 2 ) = 0. Since T is faithful, then a 2 = 0, and so a = 0. Thus no non-zero element of M annihilates X, so X is norm dense in 1'v1*. Now fix w E K and choose a sequence (}n in X such that (}n --> w (in norm). Then II(}nll --> Ilwll = 1, so
°
(5.31 ) On the other hand, (}n(l)
II(}~ I -->
+ II(}~ I
-->
1.
w(l) = 1, so
(5.32) Combining (5.31) and (5.32) shows II(};;:-II --> 0, so ()~ --> W in norm. Let = 11(};t"11- 1 (};;. Then by (5.30), Wn is a sequence in F converging to w, Which completes the proof that F is norm dense in K. 0
Wn
5.31. Lemma. Let M be a lEW-algebra wdh normal state space E K, and let p be the carner proJectwn of (). Then the face generated by () zs norm dense in Fp = {w E K I w(p) = I}.
K. Let ()
Proof. We first reduce to the case where p = 1. Let Mp = {pMp}. Recall from Proposition 2.62 that we can identify the normal state space of
5
160
STATE SPACES OF JORDAN ALGEBRAS
Mp with Fp. Note that since Fp is a face of K, then face ( a) is the same whether calculated as a subset of Fp or of K. Similarly by Proposition 2.62 the norm topology on F p , viewed as the normal state space of M p , is just the norm topology inherited as a subset of K. Finally, the carrier projection of a viewed as a normal state on Mp will still be p. Therefore it suffices to prove the lemma with M replaced by M p , and we may assume hereafter that p = 1. We then have carrier( a) = 1. We now show this implies that a is faithful. If 0 ::; a E M and a(a) = 0, then a(r(a)) = 0 by equation (2.6). Then a(l - r(a)) = 1, so by the definition of the carrier projection of a, we must have 1 ::; 1 - r(a). Therefore r(a) = 0 and so a = o. Thus a is faithful as claimed. It remains to show that the face generated by a is norm dense in K. (i) First consider the case where AI has no part of type 12 or 1 3 . By Theorem 4.23, M is a JW-algebra. Let M be the universal von Neumann algebra for M (cf. Definition 4.43); we may view M as a JW-subalgebra of Msa. M acts reversibly in M (Corollary 4.30), so there is a *-antiautomorphism c1i of period 2 of M such that AI coincides with the set of self-adjoint elements of M fixed by c1i (Proposition 4.45). Observe that ~(I + c1i) will be a positive idempotent map from Msa onto M. Note that c1i preserves the Jordan product on M sa , and so is a Jordan isomorphism. Thus it is a-weakly continuous. For each normal state w on M, define = ~w 0 (I + c1i). Then is an extension of w to a normal state on M sa. We also denote by the extension of to a complex linear normal functional on M, which will then be a normal state on M. As shown above, a is faithful on M. We now show that is faithful on M. Let a E M+ and suppose a(a) = o. Then by definition a(~(a + c1i(a)) = o. Since ~(a+c1i(a)) isin M, by faithfulness of a on M, a+c1i(a) = O. Since c1i is a *-anti-isomorphism of M, it preserves order, so c1i(a) :::: o. It follows that a = c1i(a) = o. Thus is faithful on M. Let K be the normal state space of M. Since is faithful on M, its carrier projection is the identity, so Fcarrier(;) = K. Thus the norm closure of the face generated by equals K, d. (A 107). As shown above, each w E K has an extension w E K. Therefore the restriction map is an affine map from K onto K. The restriction map is norm continuous and sends face ( a) into face( a), so it follows that face( a) is dense in K. (ii) Now consider the case where M is either of type 12 or 1 3 . Let a E K be faithful. By Lemma 5.26 there is a faithful normal tracial state T on !v! such that a ::; nT where n = 2 or n = 3. By the JBW RadonNikodym type theorem (Lemma 5.27), there is a positive element h Eo M such that a(a) = T(hoa) for all a E M. We will show the range projection r(h) equals 1. Suppose q is a projection with Uqh = h. Then by (1.47)
w
w
w
w
a
a
a
a
q'
0
h
= ~(I
+ Uq, -
Uq)h
=
o.
NORM CLOSED FACES OF JBW NORMAL STATE SPACES
161
Thus a(q') = T(h 0 q') = 0. By faithfulness of a, q' = 0, so q = 1. Thus the range projection of h is 1. By Lemma 5.29, T is in the norm closure offace( a). Since a ::; nT, then a is in face( T), so face (a) = face (T). Since a is faithful and T :::: (l/n)a, then T is also faithful. By Lemma 5.30, face(a) = face(T) = K. (iii) Finally, let M be an arbitrary JBW-algebra. Write M = M2 EB M3EBMO where A12 isoftypeI2' A13 isoftypeI3, and !vIa has no summand of types I2 or I3 (cf. Lemma 3.16 and Theorem 3.23). Let a be a normal faithful state on M. Then we can write a as a convex combination of normal faithful states a2, a3, ao on !vI2, !vh, and Mo respectively. To verify this, let C2, C3, Co = 1 - C2 - C3 be the central projections such that Mi = CiM for l = 0,2,3. Let Ai = a(ci) and ai = X;lU;;a for l = 0,2,3. Then a = Aoao + A2a2 + A3a3 is the desired convex combination. If W is any normal state on M, we can similarly decompose W into a convex combination of normal states W2, W3, Wo on M 2, M 3, and Mo. By (i) and (ii), Wi E face(ai) for l = 0,2,3. Since a = Aoao + A2a2 + A3a3, each of a2, a3, ao is in face(a), so it follows that wE face(a). D Recall that a face F of K is norm exposed if there is a positive bounded affine functional a on K whose zero set equals F, cf. (A 1).
5.32. Theorem. Let M be a lEW-algebra wdh normal state space K, and let F be a norm closed face of K. If p lS the carner pro]ectzon of F, then F = Fp = {a E K I a(p) = I}. Thus every norm closed face of K is norm exposed and pro]ectwe. Proof. By the definition of the carrier projection, F C Fp. To prove the opposite inclusion, let a E Fp. For Wl and W2 in F, the carrier projection of ~(Wl +W2) dominates the carrier projections of Wl and W2. Thus the set of carrier projections of states in F is directed upward. The least upper bound of this set of projections is carrier(F) (cf. (5.2)). Therefore there is a net {we>} in F such that {carrier( we»} is an increasing net with least upper bound carrier(F). Then a(carrier(we») ----; a(carrier(F)) = 1. Thus we can choose a sequence {w;} in F such that a(carrier(wi)) ----; 1. Define an element W of F by W = 2::i 2-iwi, and observe that carrier(w) :::: carrier(wi) for each L Thus a(carrier(w)) :::: a(carrier(wi)) ----; 1. Then a(carrier(w)) = 1, so a E Fcarrier(w)' By Lemma 5.31, this implies that a is in the norm closed face generated by w, and thus is in F. Thus We have shown Fp C F, which completes the proof that F is projective. Viewing p as an affine function on K, then 1-p takes the value precisely on F, so F is norm exposed. D
°
162
5.
STATE SPACES OF JORDAN ALGEBRAS
5.33. Corollary. Let M be a lBW-algebra wIth normal state space K. Then p f-7 Fp IS an IsomorphIsm from the lattice of proJectwns of M onto the lattIce of norm closed faces of K (where Aa Fa = na Fo CJb is an affine isomorphism from the closed unit ball of the Hilbert space N onto the state space of M. Let bEN with Ilbll ::; 1. Note first that CJb(l) = 1. Furthermore, IICJbl1 = 1, because for each a E N we have
ICJb(a +
Al)1 ::;
I(a
I b)1 + IAI ::; Ilallllbll + IAI ::; Iiall + IAI = Iia + Alii·
To show CJb is positive, let 0 ::; x E M with 111- xii::; 1, so because Ihll ::; 1 we have
Ilxll ::;
1. By spectral theory,
Thus CJb (x) ;::: 0 as claimed, so CJb is a state. (Alternatively, this follows from (A 16).) If CJb, = CJb2, then for all a EN we have (a I b 1 - b2 ) = 0, so b 1 = b2 . Thus the map b f---> CJb is 1-1. Finally, we show this map is surjective. If CJ is any state, then CJ restricts to a functional on N of norm at most 1. Since the JBW and Hilbert norms coincide on N, then there exists b in the closed unit ball of N such that CJ(a) = (a I b) for all a E N. Since CJ(I) = 1, then CJ = CJb. 0 We illustrate Proposition 5.51 with a particular example. Recall that
M = (M2 (C))sa is a spin factor, cf. Example 3.36. In this case, the state space will be a 3-ball. States are given by the maps a f---> tr(da) where d is a positive matrix of trace 1. According to (A 119), an affine isomorphism from the state space of lvI onto the Euclidean ball B3 is given by the map W f---> (;31, ;32, ;33), where
For later use, suppose that p is an atom in a spin factor lvI. Then, s = p - pi is a symmetry in N, since N includes all symmetries not equal to ±1. Thus the state CJb defined in (5.36) for b = s satisfies
170
5
STATE SPACES OF JORDAN ALGEBRAS
It follows that (J"s is the unique (normal) state with value 1 at p, or in the notation of Definition 5.40, for each atom p in a spin factor, (5.37)
p
=
(J"s
where s
=p
- p'.
We are now going tQ describe the circumstances under which the face generated by a pair of extreme points in the normal state space of a JBWalgebra degenerates to a line segment. 5.52. Definition. Let (J" and T be points in a convex set K. We say and T are separated by a spld face if there is a split face F of K such that (J" E F and TEF'. (J"
Note that for any split face F of a convex set K, since the convex hull of F and F' is all of K, then each extreme point must lie either in F or in F'. Thus two extreme points not separated by a split face always lie in the same split faces. 5.53. Lemma. Let M be a JEW-algebra, and p, q dzstmct atoms zn M. Then ezther c(p) = c( q) or c(p) ~ c( q). In the former case, Mpvq zs a spm factor, and m the latter case, d zs zsomorphzc to Rl EB Rl. Proof. Since c(p) and c(q) are minimal projections in the center of M (Lemma 3.43), then either c(p) = c( q) or c(p) ~ c( q). We first consider the case c(p) = c(q). Note that by minimality of c(p) in the center of M, c(p)M is a JBW-factor. Since Mpvq
=
{(p V q)(c(p)M)(p V q)},
then Mpvq is a JBW-factor (Proposition 3.13). Define l' = P V q - p. By the covering property (Lemma 3.50), l' is an atom, and p + l' = P V q. Minimal projections are abelian, and in a factor have the same central cover (namely, the identity), so are exchangeable (Corollary 3.20). Thus p V q = p + T is the sum of two abelian exchangeable projections, so lvlpvq is by definition a type 12 factor, and therefore is a spin factor. Now consider the case c(p) ~ c( q). Then p and q are orthogonal, and qc(p) ::; c(q)c(p) = 0 implies qc(p) = O. Thus p = (p + q)c(p) is central in Mpvq (Proposition 3.13), as is q (by the same argument). Therefore by Proposition 2.41 lvlpvq is the direct sum of the algebras Mp = Up(lvlpvq) and AIq = Uq (lvlpvq). Since p and q are minimal projections, then Aip = Rp and Mq = Rq. Thus M = Rp EB Rq. 0 5.54. Lemma. Let M be a lEW-algebra wzth normal state space K. Let (J" and T be dzstmct extreme pomts of K. Then face ( (J", T) coinczdes with the line segment [(J", T] iff (J" and T aTe separated by a split face.
THE HILBERT BALL PROPERTY
171
Proof. Let F = face (cr, T). Let p and q be the carrier projections of cr T respectively. Suppose F = [cr, T]. Then F is norm closed, and so equals the minimal norm closed face containing cr and T. Then by Corollary 5.33 the carrier projection of F is p V q and F = Fpvq. Since the linear span of F is two dimensional, and F can be identified with the normal state space of Mpvq (Proposition 2.62), then Ivlpvq has dimension 2. Since spin factors have dimension at least 3, then Ivlpvq is not a spin factor. By Lemma 5.53, c(p) ..1 c(q). Then c(cr) ..1 C(T), so Fa C F~ (cf. (5.33) and Corollary 5.43). Thus cr and T are separated by the split face FT' Suppose now that F =J [cr, T]. Then the norm closed face generated by cr and T has affine dimension 2 or more, so Mpvq has dimension 3 or more. By Lemma 5.53, c(p) = c(q). Then c(cr) = C(T), so Fa = FT' Thus cr and T generate the same split face, so are not separated by any split face. (Alternatively, tllls follows from the fact that in any convex set, the face generated by two extreme points cr, T separated by a split face is the line segment [cr, T], cf. (A 29).) 0 and
5.55. Proposition. Let Ivi be a lEW-algebra, and cr, T extreme points of the normal state space K. Then face( cr, T) lS norm exposed and is affinely lsomorphlc to a Hllbert ball.
Proof. Assume first that cr =J T. Let G be the norm closed face generated by cr and T. p = carrier(cr) and q = carrier(T). Then the c:arrier of G is p V q. By Proposition 2.62 the normal state space of Alpvq is affinely isomorphic to Fpvq. By Lemma 5.53, lvlpvq is either a spin factor or isomorphic to R1 EEl Rl. In the first case, by Proposition 5.51 its normal state space is affinely isomorphic to a Hilbert ball. In the second case, the dual space of lvlpvq is two dimensional and so its state space is a line segment, which is again a Hilbert ball. Thus G is a Hilbert ball, and face(cr, T) is a face of G. Since the only proper faces of a Hilbert ball are extreme points, we must have face(cr, T) = G. Thus face(cr, T) is a Hilbert ball, and is norm exposed since G = Fpvq. Finally, if cr = T, then face(cr,T) = {cr}, which is a zero dimensional Hilbert ball. If p is the carrier projection of the norm closed face {cr}, then {cr} = Fp , so {cr} is norm exposed. 0 5.56. Corollary. Let A be a lE-algebra, and cr, T extreme pmnts of the state space K. Then face( cr, T) lS norm exposed and lS affinely isomorphlc to a Hllbert ball. In partlcular, extreme pomts of K are norm exposed.
Proof. By Corollary 2.61 we can identify the state space of A with the normal state space of the JBW-algebra A **, so the result follows from Proposition 5.55. 0
172
5
STATE SPACES OF JORDAN ALGEBRAS
In the last part of this book, we will abstract the property of state spaces described in Corollary 5.56, and call it the Hzlbert ball property. This property of JB state spaces will turn out to be a key tool in our characterization of these state spaces. Recall that if p is an atom in a JBW-algebra M, then unique normal state with value 1 at p (Definition 5.40).
p denotes
the
5.57. Corollary. (Symmetry of transition probabilities) Let p and q be atoms in a JBW-algebra M. Then
p(q) = q(p). Proo]. The result is clear if p = q, so we may assume p I q. Since in Fpvq , we can replace !v! by !v!pvq, and thus arrange that p V q = 1. Then by Lemma 5.53, M is either the direct sum of two one dimensional algebras or else is a spin factor. In the first case, there are only two distinct atoms p, q and one easily sees p(q) = 0 = q(p). Suppose instead that !v! is a spin factor, and define s = p - p' and t = q - q'. By equation (5.37)
p and q are
p( q)
=
oA ~ (t + 1)) =
~
+ ~ (t Is) =
q(p) ,
which completes the proof. D Remark. The property described in Corollary 5.57 has an interesting physical interpretation. If (Y and T are pure states, and p, q the atoms such that p = (Y and q = T, then the probability p(q) can be thought of as the "transition probability" for a system prepared in the pure state (Y = P to be found in the pure state T = q. Then Corollary 5.57 is a statement about symmetry of transition probabilities. \iVe will return to this concept in the axiomatic portion of this book. \iVe also observe that a purely geometric proof can be given for Corollary 5.57, based on the geometry of Hilbert balls. We will give such a proof in the axiomatic context later, cf. Proposition 9.14. The (Y-convex hull of extreme points
In this section we will show that every normal state on an atomic JBWalgebra can be written as a countable convex combination of an orthogonal set of extreme points, and that more generally the set of such countable convex combinations forms a split face of the normal state space of a JBWalgebra M. Recall that oeK denotes the set of extreme points of a convex set K. Recall also that a JBW-algebra is atomic if every non-zero projection
THE u-CONVEX HULL OF EXTREME POINTS
173
dominates a minimal projection, i.e., an atom, and that this is equivalent to every projection being the least upper bound of an orthogonal collection of atoms. (See the remark after Definition 3.40.) 5.58. Lemma. If Iv! ~s an atom~c lEW-algebra, then lis normal state space K ~s the norm closed convex hull of ds extreme pomts.
Proof. It suffices to show the cone generated by oeK is norm dense in the positive cone of Iv!*. By the Hahn-Banach theorem, this will follow if we show oeK determines the order on Iv!, i.e., that for a E M, (5.38)
u(a)
~
0
for all
u E oeK
=}
a
~
O.
Let a E M satisfy the left side of (5.38). Let a = a+ -a- be the orthogonal decomposition of a (Proposition l.28), and let p = r(a-). Then by (2.9) Upa+ = 0, so a- = -Upa. For each u E oeK, U;u is a multiple of an element of oeK (Corollary 5.49), so (5.39) Since a- ~ 0, by (5.39) we must have u(a-) = 0 for all u E oeK. Since M is atomic, then 1 is the least upper bound of atoms. By Corollary 5.33, the lattices of projections and norm closed faces are isomorphic, with atoms corresponding to extreme points (Proposition 5.39), so the fact that 1 is the least upper bound of atoms implies that the norm closed face of K generated by the extreme points of K is K itself. Since the set of normal states annihilated by a- is a norm closed face of K, we conclude that ais zero on K, and thus a- = O. We've then shown a = a+ - a- = a+ ~ O. This completes the proof of (5.38), and thus of the lemma itself. D Recall that a lBW-algebra A1 has a unique predual M*, i.e., a Banach space such that M is isometrically isomorphic to (M*) *, with Iv!* consisting of the normal linear functionals on Iv! (Theorem 2.55). If {Ui} is a sequence in A1* with L, Iluill < 00, then by completeness of Iv!*, Li Ui converges in norm to an element of M*. 5.59. Lemma. Let M be a lEW-algebra, P1,P2,'" orthogonal atoms, and A1, A2, ... scalars such that Li IAil < 00. Define u = Li A;j;, E M*. Then Ilull = Li IAil and the unique orthogonal decompos~tzon u = u+ -uis given by (5.40)
Proof. Define
U1
and
U2
by
174
5
Clearly Define
(5
=
(51 -
STATE SPACES OF JORDAN ALGEBRAS
(52,
and each of
(51
and
(52
is a positive linear functional
Note that for distinct 2, J, by orthogonality of p, and Pj, Pi (PJ) ::; p, (p;) = 0, so Pi(PJ) = 0. Then (5(s) = L, IA,I, and by spectral theo:y Ilsll = 1. Thus
which implies that
11(511 = Li IA,I.
which completes the proof that of (5. 0
(5
=
Furthermore,
(51 -(52
is the orthogonal decomposition
5.60. Lemma. Let AI be an atormc .fEW-algebra AI wIth normal state space K. There eXIsts a umque lmear order IsomorphIsm W from A1. mto M, such that Ilwll ::; 1 and w(p) = P for all atoms p.
Proof. Define W on lin (8eI 0 the projection (1- e a ) has finite dimension in the lattice of projections (cf. (A 42) and Proposition 3.51). Suppose that 0' > 0 and that PI, ... ,Pn are orthogonal atoms under 1 - en. Then
Since W is bipositive, then
Applying both sides to the identity in M, we conclude that n::; 0'~1. By atomicity, l-e a is the least upper bound of an orthogonal set of atoms, and by the result just established this set of orthogonal atoms has cardinality ::; 0'~1. Thus we conclude that 1 - e a has finite dimension for each 0' > O. This implies that there is a strictly decreasing sequence of positive scalars bd (which may be a finite sequence), such that each e~; is finite dimensional, and such that e,\ is constant for A > 0 except for jumps at the Ii. For simplicity of notation we will consider the case where the sequence {Ad is infinite, and leave to the reader the minor changes in wording needed for the case where the sequence terminates. Let I = lim; Ii. Then I> 0 would contradict finite dimensionality of e~ for I > 0, so limi Ii = O. Let qi be the jump in the spectral resolution e,\ at A = Ii, i.e., qi = e-Yi - e-Yi~E for any E with 0 < E < Ii - li+l. The
THE a-CONVEX HULL OF EXTREME POINTS
177
spectral theorem (Theorem 2.20) now gives \{t(a) = Li riqi. Each qi has finite lattice dimension (since qi :s: e-y, - e-Yi+l :s: e~'+l)' Replacing each qi by a finite sum of atoms leads to an expression \{t (a) = Li AiPi for orthogonal atoms Pi and a sequence of positive scalars Ai. Since \{t is bipositive, for each n, n
L
n
AiPi
:s: \{t (a)
=?
L AiP,
:s: a.
i=l
Applying both sides of the inequality on the right above to the identity element 1, we conclude that L~=l Ai :s: 1 for each n, and so L~l Ai :s: 1. Thus L~l AiPi converges in norm to an element of M*. Since
\{t
and
\{t
(~ AiPi) =
\{t ( a),
is bijective, we conclude that 00
a=LAiPi. i=l
Finally, applying both sides of this equation to the identity element 1 gives L~l Ai = 1. 0 5.62. Definition. The a-convex hull of a bounded set F of elements in a Banach space is the set of all sums Li A,ai where AI, A2, ... are positive scalars with sum 1 and a1,a2,'" E F. 5.63. Corollary. If "Al lS a lEW-algebra wdh normal state space K, then the a-convex hull of the extreme powts of K lS a spilt face of K. Proof. Let z be the least upper bound of the atoms in M. Recall that z is a central projection, zfvl is atomic, and z'"Al contains no atoms (Lemma 3.42). Then since every atom is under z, by the correspondence of atoms and extreme points (Proposition 5.39), every extreme point of K is in the split face F z . By Proposition 2.62, Fz is affinely isomorphic to the normal state space of zfvl, and thus equals the a-convex hull of its extreme points by Theorem 5.61. Every extreme point of a face of K is also an extreme point of K, so Fz is the a-convex hull of the extreme points of
K.D 5.64. Corollary. If A lS a iE-algebra wlth state space K, then the a-convex hull of the extreme points of K is a spilt face of K. Proof. The state space of A can be identified with the normal state space of the lBW-algebra A** by Corollary 2.61. 0
178
5.
STATE SPACES OF JORDAN ALGEBRAS
Trace class elements of atomic JBW-algebras 5.65. Definition. Let IvI be an atomic JBW-algebra. An element a E M is trace class if there are orthogonal atoms Pl,P2,'" and )'1, A2,'" E R such that
(5.44)
a = L AiPi
L
and
exp(tTa)* restricts to give a one-parameter group of order automorphisms of M*. Orbits of normal states then stay inside the positive cone of M*, and if normalized the orbit will lie in K. We are going to show, for the special case where a = p - p' for a projection p, that these normalized orbits trace out half of an ellipse. Since T p_ p' = Up - Up' (cf. (1.47)), for later applications we will express our results in terms of the maps Up - Up" The following is a simple technical lemma that we will use both noW and later, and so we state it separately for easy reference.
185
SYl\IMETRY AND ELLIPTICITY
5.73. Lemma. Let a, 7 and p be lmearly mdependent elements of a vector space, let a be a real number wzth < a < 1, and set p = ~(a(1 - a))-1/2 p. For each t E R the pomt
°
(5.62)
hes on the ellzpse conszstmg of all pomts (5.63)
Thzs ellzpse has the dzameter [a,7]. When t vanes from -= to =, the parametenzed curve Wt traces out, m the dzrectzon from 7 to a, the halfellzpse between these poznts whzch contazns the poznt aa + (1 - a)7 + p (attazned for t = 0). The other half of the ellzpse zs traced out zn the same fashzon by the parameterized curve obtamed by replacmg p by -p zn
(5.62). Proof. It is easily verified that when t varies from (Xt, Yt) E R2 defined by
-= to =, the point
2(a(1 - a))1/2 Yt = aet + (1 - a)e- t
(5.64)
traces out the upper half of the circle x2 + y2 = 1 in the direction from left to right. The affine map cI> : R2 f---* M. defined by (5.65)
cI>(X, y) = ~ (a
+ 7) + x (~( a -
7))
+YP
carries this circle to an ellipse. Rewriting (5.65) in the form
cI>(X,y)
=
~(:r;+ l)a+ ~(I-X)7+YP
and substituting x = Xt, Y = Yt, we find that cI>(Xt, Yt) = Wt. Thus Wt lies on the specified ellipse for all t E R. Since cI>(I, 0) = a and cI>(-1,0) = 7, the line segment [a,7] is a diameter of this ellipse. By (5.64), (Xt, Yt) --> (±l, 0) when t --> ±=, so Wt --> 7 when t --> -= and Wt --> a when t --> =. Thus Wt traces out, in the direction from 7 to a, the half-ellipse between these points which contains the point Wo = aa + (1 - a)7 + p. In the same fashion, the parameterized curve obtained by changing the sign of p in (5.62) traces out the half-ellipse containing the "opposite" point aa + (1 - a)7 - p (i.e., the image of Wo under reflection about the diameter [a, 7]). This proves the last statement of the lemma. 0
5
186
5.74.
STATE SPACES OF JORDAN ALGEBRAS
Lemma.
Let M be a lEW-algebra wzth normal state space
K, let p be a pro]ectwn m M wzth assoczated face F = Fp and refiectwn R = U;~p' (cf. Proposztwn 5.72), and let w E K. Assume w f/c F U F ' , and define IJ = IIU;wll~lU;w, T = 11U;,wII~lUl~,w. Conszder first the orbzt {WdtER of w under the one-parameter group of order automorphzsms of M. gwen by t f-+ exp( t(U; - U;,)). Thzs orbzt zs one half of a hyperbola. Conszder also the normalzzed orbzt {WdtER of w, gwen by Wt = IlWt II ~l Wt. If w f/c co( F U F ' ), then thzs orbzt zs the (unzque) halJ-ellzpse wzth dzameter [IJ, T] whzch contams the pomt w, traced out m the dzrectwn from T to IJ, and the other half of thzs ellzpse zs the normalzzed OT'bzt of Rw. If w E co(F U F ' ), (but stzll w f/c F U F ' ), then the normalzzed orbzt of w degenerates to the line segment from T to IJ, traced out m the same dZT'ectwn.
Since w f/c F U F ' , then IIU;wll = (U;w)(l) = w(p) oF 0, and likewise 11U;,wll = W(p') oF O. Therefore the two points IJ and T of the lemma are well defined. Now we can rewrite the equation above in the form (5.66)
~ Wt = ae t IJ
+ (1 -
a ) e ~t T
+p
where a = IIU;wll and p = (1 - Up - Up' )*w. This equation shows that the orbit {wdtER is one half of a hyperbola. For each t E R, IIWtl1 = Wt(1) = ae t + (1 - a)e~t, so the normalized orbit of w is given by (5.62). Assume now w f/c co(F U F'). Then IJ, T, P are linearly independent; otherwise p would be a linear combination of the two (linearly independent) elements IJ and T, and then w = Wo would also be a linear combination of IJ E F and TEF', so w would be in im (U; + U;,) n K = co(FuF ' ). Thus we can apply Lemma 5.73, from which the description of the normalized orbit of w follows. Since U)J~p' = 2Up + 2Up' - I, the reflection R = U;~p' sends p = (1 - Up - Up' )*w to -p, and fixes IJ and T. Since U; and U;, commute with R, the orbit of Rw equals R applied to the orbit of w. Therefore the description of the normalized orbit of Rw also follows from Lemma 5.73. Finally, if w E co(F U F ' ), then p = 0 and it follows from (5.62) that Wt can be written as a convex combination Wt = AtIJ + (1 - AdT where At ----> 0 when t ----> -= and At ----> 1 when t ----> =. Thus Wt traces out the line segment from T to IJ in this case. 0 Note that the fact that the normalized orbit of the point w of Lemma 5.74 is part of an ellipse follows easily from the fact that the non-normalized
SYl\U.,1ETRY AND ELLIPTICITY
187
orbit is half of a hyperbola. Actually, the passage from the non-normalized orbit to the normalized orbit is just the projection via the origin onto the plane through p, 0', T, and since the image of the hyperbola under this projection is a conic section in the bounded set K, it must be an ellipse. Note also that if the point W of Lemma 5.74 lies in either F or F' , then it is easily verified that w is a fixed point under the one-parameter group t f-+ exp(t(U; - U;*)).
5.75. Proposition. Let !vI be a lEW-algebra, and F a norm closed face of the normal state space K. Let W = U; + U;, be the canonzcal affine proJectwn of K onto co(F U F ' ), and 0' E F, TEF'. Then w- 1 ([0', T]) has ellzptzcal cross-sectwns, z. e., the zntersectwn of thzs set wzth every plane through the lzne segment [0', T] zs an ellzptzcal dzsk (whzch may degenerate to a lzne segment). Proof. Let p be the carrier projection of F. As observed in the remarks preceding Lemma 5.73, for each w E K the normalized elliptical orbit Wt described in Lemma 5.74 stays inside K. Let Lo be the intersection of w-I([O',T]) with a plane L through [O',T]. If Lo is a line segment, then Lo is a degenerate ellipse, so we are done. Hereafter, we assume this is not the case. Since K is closed and bounded for the base norm, and all norms on the finite dimensional subspace L are equivalent (cf., e.g., [108, Thm. 1.21]), Lo = K n L is closed and bounded and thus is compact. Let W be a point on the boundary of Lo in L. Then since w(w) E [0', T], there is a scalar A with 0 < A < 1 such that u;w
+ U;,w =
AO'
+ (1
- A)T.
Applying U; to both sides, we find that 0' = IIU;wll- 1 U;w. Similarly = II U;,w 11-1 U;,w, so 0' and T have the same meaning as in Lemma 5.74. Let E(w) denote the ellipse with diameter [0', T] passing through w, together with its interior. Then each half of this ellipse is the normalized orbit of either w or the reflected point U;w (where s = p - pi) by Lemma 5.74, and so this ellipse is contained in K. Thus E(w) is contained in L o . We will show that E(w) = Lo. If WI were another element of Lo not in E(w), then the ellipse through 0', T, WI would not meet the ellipse through 0', T, W except at the points 0' and T. Therefore E(WI) would be contained in Lo and would contain w in its interior. But w is a boundary point, so this is a contradiction. We conclude that Lo = E(w) as claimed. D T
We will refer to the elliptical cross-section property described in Proposition 5.75 as ellzptzcityof K. We will define this in an abstract context later, and show that the combination of admitting a suitable spectral theory and ellipticity characterize normal state spaces of JEW-algebras. Both of the properties symmetry and ellipticity can be viewed as illustrations that there are "many" affine automorphisms of the normal state
188
5
STATE SPACES OF JORDAN ALGEBRAS
space of a JBW-algebra (or of the state space of a JB-algebra). Dually, the positive cone of every JB-algebra admits "many" Jordan automorphisms. One way to make this precise is to observe that the affine automorphisms of the interior of the positive cone of a JB-algebra A are transitive. For a more complete discussion of such homogenelty, see the discussion in the notes to Chapter 6. As discussed there, in the context of self-dual cones, another notion of homogeneity (sometimes called "facial homogeneity"), is due to Connes [36]. The definition is that exp( t( P F - PVl )) ~ 0 for all t and F, where his maps PF are projections onto faces of the associated cone, and thus are closely related to our maps Up. In the JBIV-context, positivity of the analogous maps exp( t(Up - Up ')*) is what we used above to establish ellipticity, and we will see in an abstract context that positivity of such maps is equivalent to ellipticity. Notes The introductory results of this chapter on the lattice of projective faces can be found in [10]. The characterization of Jordan homomorphisms via their dual maps on the state space (Lemma 5.15) is from [117]. The result describing alternative characterizations of traces on JB-algebras (Lemma 5.18) is due to Pedersen and St0nner [101]. The result that type In JBWalgebras have a center-valued trace (Proposition 5.25) can be found in [62]; our proof is modeled on the similar proof for type I n von Neumann algebras in [80]. For a more detailed treatment of traces on Jordan operator algebras, see [23]. The Radon~Nikodym type theorem for JBW-algebras in Lemma 5.27 generalizes a well known result for von Neumann algebras [112], and the proof is essentially the same. The result that norm closed faces of the normal state space of a JRW-algebra are projective (Theorem 5.32) is due to Iochum [68]. After quite a bit of work in the lemmas preceding Theorem 5.32, the proof eventually reduces to using the analogous result for von Neumann algebras (A 107), which is due to Effros [46] and to Prosser
[105]. In the sequel, we will only use a special case of this 1 esult: namely, that extreme points of the state space of a JB-algebra are norm exposed. This is used to show that the factor representations associated with pure states are type I (Corollary 5.42), and will play an important role in our characterization of JB-algebra and C*-algebra state spaces. In [8] and in [10] the result that extreme points are norm exposed is based on the characterization of pure states on JC-algebras due to St0rmer [126]. The general correspondence of ideals and split faces (Proposition 5.36 and Corollary 5.37) is quite analogous to the similar results for C* -algebras and von Neumann algebras, cf. (A 112) and (A 114). For C*-algebras, this result is due to Segal [115]. For JB-algebras, this result is in [65]. The Hilbert ball property for JB-algebra state spaces (Corollary 5.56) was first discussed in [10] as part of the characterization of JB state spaces
NOTES
189
(Theorem 9.38), as were the related facts that the state space of a spin factor is a Hilbert ball (Proposition 5.51), and the symmetry of transition probabilities (Corollary 5.57). The representation of normal states on an atomic JBvV-factor as a countable convex combination of orthogonal pure states (Theorem 5.61) also is in [10], and generalizes the analogous result for B(H), which can be found in [100]. The use of trace class elements to describe the normal state space of atomic JBW-algebras (Theorem 5.70) parallels the corresponding result for B (H) (A 87). The symmetry property (Proposition 5.72) first appears in [10]. As discussed at the end of this chapter, ellipticity as described in Proposition 5.75 is formally similar to the notion of Jaczal homogenedy due to Connes [36]. The geometric interpretation of ellipticity given in Proposition 5.75 appeared in [71].
6
Dynamical Correspondences
In this chapter we will discuss the notion of a "dynamical correspondence" , which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms ill quantum mechanics. It is closely related to Connes' concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones of Tomita-Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure. In the first section we introduce the general notion of a "Connes orientation", which carries over to JBW-algebras the concept Connes introduced. In the second section we introduce the notion of a "dynamical correspondence" (first defined in [12, Def. 17]). We show that a JB-algebra is the self-adjoint part of a C*-algebra iff it admits a dynamical correspondence, with a similar result for JBW-algebras, and that the dynamical correspondences are in both cases in 1-1 correspondence with "Jordan compatible" associative products (Definition 6.14). We also show that for JBW-algebras there is a 1-1 correspondence of Connes orientations and dynamical correspondences. The last section sketches a geometric interpretation of these concepts, and relates them to the concepts of orientation for C*-algebra state spaces and von Neumann algebra normal state spaces discussed in
pJ
[AS]. Connes orientations Both Connes' concept of orientation and our notion of a dynamical correspondence refer to order derivations. If A is an ordered Banach space, a bounded linear map 0 : A ---+ A is an order der'lVatwn if exp(to) :::0: 0 for all t E R, i.e., if 0 is the generator of a one-parameter group of order automorphisms of A. A bounded linear operator 0 acting on a C*-algebra A is said to be an order del'lvatwn if it leaves Asa invariant and restricts to an order derivation on Asa (A 181).
6.1. Definition. If A is a C*-algebra A, and a E A, then we define ---+ A by
(Sa : A (6.1)
DaX
= ~(ax
+ xa·).
We recall the following results from [AS].
192
6.
DYNAMICAL CORRESPONDENCES
6.2. Theorem. If A zs a C*-algebra and a E A, then 6a zs an order derzvation on A. If A zs a von Neumann algebra, then every order derzvatwn on A has the form 6a for some a EA. Proof. The first statement is (A 182), and the second is (A 183). D
6.3. Proposition. Let A be a C*-algebra. one-parameter group at = exp(t6a ) zs gwen by
If a E .4., then the
(6.2) In partzcular, zf h E Asa and a
=
h (the self-adjomt case), then
(6.3) and zf a
= zh
(the skew case), then
(6.4) Proof. Consider the left and right multiplication operators L : x f--t ax and R: x f--t xa* defined on A for a E A. Since Land R commute, exp(2t6a )(x) = exp(tL
+ tR)(x)
= exp(tL)exp(tR)(x) = etaxe ta *.
This proves (6.2), from which (6.3) and (6.4) both follow. D The orbits of the one-parameter group a; = exp(6a )* can be easily visualized in the simplest non-commutative case where A = lvI2(C). Here the state space is a Euclidean 3-ball, and the pure state space is the surface of the ball, i.e., a Euclidean 2-sphere. If a E A is self-adjoint and has two distinct eigenvalues >'1 < A2 corresponding to (unit) eigenvectors ~1,6, then the vector states W~" w6 are antipodal points on the sphere (South Pole and North Pole in Fig. 6.1). If a = zh where h E Asa (the skew case), then it can be seen from (6.4) that a; is a rotation of the ball by an angle t(A2 - Ad/2 about the diameter [W~1,W~2], d. (A 129). Thus the one-parameter group a; represents a rotational motion with rotational velocity (A2 - Ad/2 about this diameter, and the orbits on the sphere are the "parallel circles" (in planes orthogonal to [w~" w~2l). If a = h where hE Asa (the self-adjoint case), then the orbits will take us out of the state space. But this can be remedied by a normalization, i.e., by considering the parametric curves t f--t Ila;(0')11- 1 a;(0') instead of t f--t a;(O'). Now it can be seen from (6.3) that the (normalized) orbits are the "longitudinal semi-circles" on the sphere (in planes through [w~ll W~2])' cf. Lemma 5.74.
CONNES ORIENTATIONS
193
Fig. 6.1
In the example above we can easily see how the one-parameter group is determined by the geometry in the self-adjoint case, and we can also see what indeterminacy there is in the skew case. The element a E Asa determines a real valued affine function W f-7 w(a) on the state space. This function attains its minimum A1 at WI;! and its maximum A2 at w1;2. In the self-adjoint case the orbits are the longitudinal semi-circles traced out in the direction from wl;! to WE2. In the skew case the orbits are the parallel circles, but they can be traced out in two possible directions, "eastbound" and "westbound". The mere knowledge of the affine function a does not tell us which direction to choose. This would require a specific orientation of the ball (right-handed or left-handed around wl;~t:;2).
a:
If a, b are elements of a C*-algebra, then we will often make use of the identity (6.5) We are going to transfer Connes' concept of orientation [36, Definition 4.11]' originally defined in the context of certain self-dual cones, to the context of lBW-algebras. For motivation, note that if M is a von Neumann algebra and a E M sa, then
so that
Oia
essentially gives the Lie product. Thus we can express the
194
6
DYNA1VIICAL CORRESPONDENCES
ordinary left multiplication by a as
Oid from the set D( M ",) Now the idea is to axiomatize the Illap Od of order derivations into itself. Note that as it stands this map is not well defined on the set of all order derivations, since if d is not self-adjoint, then Od does not determine d. In fact, for any central skew-adjoint d the map Od will be zero. Thus we will define the Illap instead on D( M ,a) modulo the center of the set of order derivatiolls, which we now defillc. f--)
6.4. Definition. Let A be a JB-algebra. The center of the Lie algebra D(A) consists of all the order derivations 0 in D(A) that commute with all members of D(A). We will denote the center of D(A) by Z(D(A)). We review some basic facts about order derivations on JB-algebras. If
a is an element of a JB-algebra A, we let oa denote Jordan multiplication by a, i.e., oax = aox for x E A. (Note that this agrees with (6.1) when A is the self-adjoint part of a C* -algebra A.) Each such map 0(1 is an order derivation on A (Lemma 1.56). The center of a .IB-algebra A consists of those z E A such that ozoa = 0(10Z for all a E A (Definition 1.51). The space D(A) of order derivations of A is a Lie algebra with respect to the usual Lie bracket given by commutators: [01,02] = 0] 02 - 0201 (Proposition 1.59). An order derivation on A is self-adJomt if it has the form oa for some a E A. An order derivation 0 on A is skew if 0(1) = o. Every order derivation admits a unique decomposition as the sum of a selfadjoint derivation and a skew derivation (Proposition 1.60). A skew order derivation is a Jordan derivation, i.e., satisfies the Leibniz rule (6.6)
o(a
0
b)
=
o(a)
0
b + a oo(b),
and the skew order derivations are precisely the boullded lillear maps that generate one-parameter groups of Jordan automorphisms (Lemma 2.81). 6.5. Lemma. If A
(6.7)
lS
a iE-algebra, then
Z(D(A)) = {oz I Z
E
Z(A)}.
Proof. Assume first that 0 E Z(D(A)). Then [o,oal a E A. Let Z = 0(1). Then for every a E A,
o(a) = 5O a(l) = oao(l) = O(1(z) = a 0
Z
=
Z 0
o for
every
a.
Hence 0 = Oz. Also ozoa = oaoz, so z E Z(A). Assume next that Z E Z(A). By the definition of Z(A), Oz commutes with every self-adjoint order derivation oa. Therefore we only have to show
CONNES ORIENTATIONS
195
that 6z commutes with every skew derivation 6. But such a derivation is a Jordan derivation, so it follows from the Leibniz rule (6.6) that
(6.8)
66 z (x) = 6(z
0
x) = (6z)
0
x
+Z 0
(6x)
for every x E A. Since 6 is skew, 6z = 0 (Proposition 2.82), so (6.8) shows that (66z)(x) = (6 z 6)(x). Thus 66 z = 6z 6 as desired. D
6.6. Definition. Let A be a JB-algebra and 6 E D(A). Write 6 = 61 + 62 where 61 is self-adjoint and 62 is skew. Then we define 6t =61- 62. Note that if A is a JB-algebra and 6 E D(A), then (6 t )t = 6, so 6 --) 6t is linear and is an involutive map, i.e., has period 2. Note also that 6 is self-adjoint iff 6t = 6, and skew iff 6t = -6.
6.7. Definition. Let M be a lBW-algebra. We write D(M) in place of D(M)/Z(D(M)), and denote the equivalence class of an element 6 of D(M) modulo Z(D(M)) by b. Note that the involution (b)t = (bt) is well defined on D(M), for if 61 = 62, then 61 - 62 = 6z for some Z E Z(AI) (Lemma 6.5). Hence
-
-
6i - 6~ = 61 = 6z , so (6i) = (6~). 6.8. Definition. A Connes orzentatwn on a lRW-algebra IvI is a complex structure on D(M), which is compatible with Lie brackets and involution, i.e., a linear operator I on D(NI) which satisfies the requirements: (i) 12 = -1 (where 1 is the identity map), (ii) [I'll, b2 ] = [b1 , Ib2 ] = I[b1 , b2 ], and (iii) I(bt ) = -(Ib)t.
6.9. Proposition. Let M
-
be a von Neumann algebra. Then
-
I(6 x ) = 6u
for all x E M
defines a Connes onentatwn on Mba. Proof. Let IvI = M sa. We first verify that I is well defined. For this purpose, it suffices to show that if x E M and bx = 0, then bix = o. We first verify
(6.9)
ox=o
=}
xEZ(M),
196
6
DYNAMICAL CORRESPONDENCES
where Z( M) is the center of M. Write x = a + Ib with a, b E M. If Ox = 0, then Ox E Z(D(M)), so Ox = Oz for some z in the center of the JBW-algebra M (Lemma 6.5). Then z is also central in the von Neumann algebra M (Corollary 1.53). Note that the self-adjoint and skew parts of the order derivation Ox are oa and Oib respectively. Equating the selfadjoint and skew parts of the order derivations Ox and oz, we conclude that oa = Oz (so a = z), and Oib = 0 (so b is central). Thus x = a + Ib is central in M, so (6.9) is verified. Now let x E M with Sx = O. Then x is central in M, so we can write x = a + Ib with a, b self-adjoint elements of the center of M. Then Oix = Oia-b = Oia - Ob· Thus Oia = 0 and Ob E Z(D(M)), so Six = 0, which completes the proof that I is well defined. Since every order derivation on M has th~Jorm Ox for some x E M (Theorem 6.2), then the domain of I is all of D(M). Now we verify the three properties of Definition 6.8. The property (i) follows at once from the definition of I, and (ii) is an immediate consequence of the identity (6.5). To verify (iii), note that if a, b E M sa, then oa is self-adjoint and Otb is skew, so
Thus
O!
= Ox'
for all x EM.
Now it follows that I satisfies Definition 6.8(iii). D We will see later (in Corollary 6.19) that a JBW-algebra M is the selfadjoint part of a von Neumann algebra iff M admits a Connes orientation, and that such orientations are in 1-1 correspondence with W*-products on M+IM.
Dynamical correspondences in JB-algebras When a C*-algebra or a von Neumann algebra is used as an algebraic model of quantum mechanics, then it is only the self-adjoint part of the algebra that represents observables. However, the self-adjoint part of such an algebra is not closed under the given associative product, but only under the Jordan product. Therefore it has been proposed to model quantum mechanics on Jordan algebras rather than associative algebras [75], [98]. This approach is corroborated by the fact that many physically relevant properties of observables are adequately described by Jordan constructs. However, it is an important feature of quantum mechanics that the physical variables play a dual role, as observables and as generators of
DYNAMICAL CORRESPONDENCES IN JB-ALGEBRAS
197
transformation groups. The observables are random variables with a specified probability law in each state of the quantum system, while the generators determine one-parameter groups of transformations of observables (Heisenberg picture) or states (Schrodinger picture). Both aspects can be adequately dealt with in a C*-algebra (or a von Neumann algebra) A. But unlike the observables whose probability distribution is determined by the functional calculus in Asa and thus by Jordan structure, the generators cannot be expressed in terms of the Jordan product. Each self-adjoint element a = d~ of a C*-algebra defines a derivation Jih, and these derivations generate one-parameter groups as in (6.4) which describe the time development of the physical system. But generators are determined not by the symmetrized product a 0 b = ~ (ab + ba) in A sa, but instead by the antisymmetrized product a * b := ~(ab - ba) on Asa. Therefore both the Jordan product and the Lie product of a C*-algebra are needed for physics, and the decomposition
(6.10)
ab = a
0
b-
2 (a
* b)
separates the two aspects of a physical variable. It is of interest to find appropriate constructs, defined in terms of the Jordan structure of a JB-algebra (or the geometry of its state space), which makes it the self-adjoint part of a C*-algebra. One such construct is the concept of a dynam2cal correspondence (defined below), which axiomatizes the transition h f---+ bih from the self-adjoint part of a C*-algebra to the set of skew order derivations on the algebra.
6.10. Definition. A dynamzcal correspondence on a JB-algebra A is a linear map 1/J : a f---+ 1/Ja from A into the set of skew order derivations on A which satisfies the requirements (i) [1/Ju,1/Jb] = -[b a , bb] for a, bE A, (ii) 1/J"a = 0 for all a E A. A dynamical correspondence on a JB-algebra A will be called complete if it maps A onto the set of all skew order derivations on A.
6.11. Proposition. If A zs the self-adJomt part of a C*-algebra or a von Neumann algebra, then the map a f---+ 1/Ja = bia ZS a dynamzcal correspondence. In the von Neumann algebra case, thzs dynamzcal correspondence zs complete.
Proof. In both cases, property (i) above follows from the identity (6.5), and (ii) is immediate, so 1/J is a dynamical correspondence. A skew order derivation on a von Neumann algebra has the form bia for some self-adjoint element a (A 184). Thus in a von Neumann algebra the map a f---+ 1/Ja = bia is a complete dynamical correspondence. 0
6
198
DYNAMICAL CORRESPONDENCES
The skew order derivations on a JB-algebra A determine oneparameter groups of Jordan automorphisms of A (Lemma 2.81). Therefore the dual maps determine affine automorphisms of the state space of A (and also of the normal state space in the JBW case), cf. Proposition 5.16. Thus a dynamical correspondence gives the elements of A a double identity, which reflects the dual role of physical variables as observables and as generators of a one-parameter group of motions. Hencp the name "dynamical correspondence" . Since the Jordan product is commutative, there is no useful concept of "commutator" for elements in a JB-algebra, but the commutators of the associated Jordan multipliers can be used as a substitute in view of the identity [Oa,Ob] = ~O[a,bJ for elements a, b in a C*-algebra. Thus the condition (i) above is a kind of quantization requirement, relating commutators of elements to the commutators of the associated generators. Note also that the equation 1/Ja(a) = 0 is equivalent to exp(t1/Ja)(a) = a for all t E R. Thus condition (ii) says that the time evolution associated with an observable fixes that observable We will also state the definition of a dynamical correspondence in another form. In this connection we shall need the following lemma. 6.12. Lemma. Let A be a fE-algebra and let a f-7 1/Ja be a map from A mto the set of all skew order derivatIOns on A. Then for all pall'S a,b E A, (6.11)
Proof. Since 1/Ja is skew, it is a Jordan derivation. Hence for all c E A,
which can be rewritten
This gives (6.11). D Note that linearity of 1/J is not listed among the requirements in the proposition below, since it follows from the other requirements. 6.13. Proposition. Let A be a fE-algebra and let 1/J : a f-7 1/Ja be a map from A mto the set of skew order derwations of A. Then 1/J lS a dynamical correspondence lff the followmg reqmrements are satlsfied for a,b E A:
(i) [1/Ja,1/Jb] = -loa, bb], and (ii) [1fJa, bb] = [ba, 1fJb].
DYNAl\IICAL CORRESPONDENCES IN JB-ALGEBRAS
199
Proof. Assume first that 'lj; is a dynamical correspondence. Condition (i) above is satisfied as it is the same as condition (i) of the definition of a dynamical correspondence (Definition 6.10). By condition (ii) of Definition 6.10 and Lemma 6.12, for all a E A
where the last equality follows from linearity of the map a again by linearity of 'lj;, for all a, b E A,
f--->
'lj;a. Therefore,
which gives
This proves condition (ii) above. Assume next that 'lj; satisfies conditions (i) and (ii) of the proposition. Condition (i) of Definition 6.10 is trivially satisfied. It follows from condition (ii) above that for all a E A,
so ['lj;a, <Sal = 0 for all a E A. By Lemma 6.12, <S",,,a = ['lj;a, <Sa], so <S",,,a = O. Hence 'lj;aa = <S",,,a1 = 0, so condition (ii) of Definition 6.10 is also satisfied. It remains to show that 'lj; is linear. By Lemma 6.12 and condition (ii) above, for all a, b E A
Hence 'lj;ab = -'lj;ba. Since a f---> 'lj;ba is linear, it follows that a linear map from A into D(A). D
f--->
'lj;a is a
In the last paragraph of the proof above we have shown that if 'lj; is a dynamical correspondence on A, then (6.12) We shall make more use of this equation later. 6.14. Definition. Let A be a lB-algebra. A bilinear associative product * on the complexified space A + zA is a Jordan compatzble product if
200
6
(ii) (a y*
+
DYNAMICAL CORRESPONDENCES
Ib)* = a - Ib is an involution for (A for all x, yEA + LA.
* x*
+ LA, *),
i.e., (x * y)*
=
Such a product will also be called a C*-product OIl A + IA. If A is a JBWalgebra, then we call such a product a W*-product or a von Neumann product.
Remark. If A is a JB-algebra and * is a Jordan compatible product on A + LA, then we can define a norm on A + l.A by Ilxll = 11.1:*:1':111/2. With this norm, A + IA will become a C*-algebra (A 59), justifying our use of the term "C*-product". In principle this norm could depend on the associative product as well as on the norm on A, but in fact it is uniquely determined by the JB-algebra A (A 160). (This follows from Kadisoll's theorem that a Jordan isomorphism between C*-algebras is an isometry (A 159).) If A is a JBW-algebra and A + 'LA is equipped with a Jordan compatible product, then A + LA will be a C*-algebra whose self-adjoint part A is monotone complete with a separating set of normal states, so will be a von Neumann algebra (A 95), justifying the term \V*-product (or von Neumann product). We are now ready to state and prove our main theorem, which relates dynamical correspondences to associative products. 6.15. Theorem. A JE-algebra A LS (Jordan IsomorphIc to) the selfadJomt part of a C*-algebra Iff there eXIsts a dynarmcal correspondence on A, In thIS case there IS a 1-1 correspondence of C*-products on A + IA and dynamzcal cor7'espondences on A. The dynarmcal correspondence on A assoczated WIth a C* -product (a, b) f---> ab IS ( 6.13) and the C*-product on A + IA assoczated wl,th a dynaTnlc(11 correspondence IS the complex bz/mear extenS1.On of the product defined on A by
1/; on A (6.14)
ab
=
a
0
b - 11/;ab.
Proof. Assume first that A admits a dynamical correspondence 1/). By equation (6.12) we can define an anti-symmetric bilinear product (a, b) 1-7 a x b on A by writing
(6.15) Next define a bilinear map (a, b) f---> ab from the Cartesian product A x A into A + zA (considered as a real linear space) by writing (6.16)
ab=aob-z(axb).
DYNMIICAL CORRESPONDENCES IN JB-ALGEBRAS
201
This map can be uniquely extended to a bilinear product on A + ~A (considered as a complex linear space). We will show that this product IS associative. By linearity, it suffices to prove the associative law
a( cb) = (ac)b
(6.17)
for a, b, c E A. Writing out (6.17) by means of (6.16), we get
a 0 (c 0 b) - ~(a x (c 0 b)) - ~(a
0
(c x b)) - a x (c x b)
= (a 0 c) 0 b - z((a 0 c) x b) - z((a x c) 0 b) - (a x c) x b. Separating real and imaginary terms (and using the anti-symmetry of the x -product), we get two equations. The first one can be written as (6.18)
a x (b x c) - b x (a x c)
=
-a 0 (b
0
c)
+ b0
(a
0
c),
and the second one as (6.19)
a x (b
0
c) - b 0 (a x c)
= a 0 (b x c) - b x (a 0 c).
The left-hand side of (6.18) is just [1f!a,1f!b](C) and the right-hand side of (6.18) is -[b a, bb](C). Similarly the left-hand side of (6.19) is [1f!a,6b](C) and the right-hand side of (6.19) is [b a,1f!b](C). Thus these two equations follow directly from the characterization of a dynamical correspondence in Proposition 6.13. We must also show that the bilinear product on A + zA is compatible with the natural involution (a+zb)* = a-lb on A+1A, i.e., that (xy)* = y*x* for x, yEA + ~A. By linearity it suffices to show that (ab)* = ba for a, b E A. But this follows directly from the anti-symmetry of the xproduct, as
(ab) * = (a
0
b - 1 (a x b)) * = a 0 b + 1 (a x b) = boa -
1
(b x a) = ba.
We have now shown that A + ~A is an associative *-algebra. By the definition of the involution, the self-adjoint part of A A. Thus by (6.16) and the anti-symmetry of the x -product, ~ (ab
+ ba) = a 0
+ lA
is
b
for all a, b E A. Therefore the associative product in A + ~A induces the given Jordan product on A, and thus is a C*-product. Assume next that A is Jordan isomorphic to the self-adjoint part of a C*-algebra A. Extending the isomorphism from A onto Asa to a complex linear map from A + iA onto A and pulling back the associative product
6.
202
DYNAMICAL CORRESPONDENCES
in A, we get a C*-product (a,b) f--+ ab in A+2A. Defining a map 1/J from A into D(A) by equation (6.13), we get a dynamical correspondence on A (cf. the remark following Definition 6.14, and Proposition 6.11). Then we define a new product * on A + 2A to be the complex bilinear extension of (6.14). By the definition of 1/J, we have a * b = a 0 b - 21/Ja(b) = ab for a, b E A, so the new product is the same as the original. It remains to prove that if we start out from a dynamical correspondence 1/J on A and construct first the C*-product in A+iA via (6.14) as in the first part of the proof, and then from this product the dynamical correspondence given by (6.13) as above, then we will end with the dynamical correspondence with which we began. Let a, bE A. By (6.15), (6.16) and the anti-symmetry of the x -product, we find that ~i(ab - ba)
=
~(a x b - b x a)
=
a x b = 1/Ja(b),
which completes the proof. 0 Note that it follows from equation (6.13) of Theorem 6.15 that a dynamical correspondence 1/J on a JB-algebra A can be recovered from the associated C*-product on A + 2A through the equation
(6.20)
1/Ja
= Ow for all
a E A.
6.16. Corollary. A lEW-algebra M zs (lordan zsomorphzc to) the self-adJomt part of a von Neumann algebra zfJ there eX2sts a dynam2cal correspondence on M. In thzs case the constructzon m Theorem 6.15 promdes a 1-1 correspondence of W*-products on !vI + 2M and dynam2cal correspondences on M. Proof. This follows from Theorem 6.15, together with the observation in the remarks preceding Theorem 6.15 showing that if ~M is a JBWalgebra, then a Jordan compatible product on !vI + 2!vI makes M + zM into a von Neumann algebra. 0
We will now explain the relationship between Connes orientations and dynamical correspondences, and we begin with the following: 6.17. Lemma. Let I be a Cannes orientatzon on a lEW-algebra Ai. If a E !vI, then there exzsts a umque skew order denvation 6 zn I (5a ). Proof. Choose 61 E I(5a ). By Definition 6.8 (iii), since (6 a )t = Oa,
DYNAMICAL CORRESPONDENCES IN JB-ALCEBRAS
oi
Hence 5i + 51 = 0, so + 01 define 0 = 01 - ~oz = ~(01 -
203
= Oz for some z E Z(M) (Lemma 6.5). Now Then ot = -0, so 0 is skew. Also
oi).
If 0' is an arbitrary skew order derivation such that 0' E I(5a ), then 0 - 0' is both central and skew. By Lemma 6.5, a central order derivation is selfadjoint. Thus 5 - 5' is both skew and se!.f-adjoint, so 5 - 5' = o. Hence 5 is the unique skew order derivation in I(5 a ). 0 The concept of a Connes orientation is defined for JBW-algebras, while the concept of a dynamical correspondence is defined for all JB-algebras. However, in the context of JBW-algebras, they are equivalent, as we now show. In this connection we shall need the Kleinecke-Shirokov theorem, which says that if P and Q are bounded operators on a Banach space and their commutator C = [P, QJ commutes with P (or Q), then C is a quasi-nilpotent, i.e., Ilcnlll/n ---+ 0 when n ---+ 00 (cf. [63, p. 128]). (In this reference the theorem is stated for Hilbert space operators, but the proof works equally well for operators on a general Banach space.)
6.18. Theorem. If /1,1 zs a JEW-algebra, then there zs a 1-1 correspondence between Connes orzentatwns on /1,1 and dynamzcal correspondences on lvI, and any dynamzcal correspondence on M is complete. If I is a Connes onentatwn on iIi[, then the assocwted dynamzcal correspondence zs the map a f--+ 1/Ja E D( M), where Wa zs the unique skew adjomt order denvation such that (6.21)
t/Ja
E
I(5 a) for all a
E
M.
For each dynamzcal correspondence 1/J, the assocwted Connes onentatwn I is gwen by (6.22)
where 6x and Oix are defined wzth respect to the W*-pTOduct correspondmg to 1/J (as m (6.14)). Proof. Let I be a Connes orientation on !v1. Let a E M. Denote the unique skew order derivation in I(5u) by 1/Ja for each a E M. Clearly 'IjJ : a f--+ 1/Ja is a linear map from Minto D(lvI). Let a, b E M. By Definition 6.8 (i),(ii),
204
6.
DYNAMICAL CORRESPONDENCES
Thus for some z E Z(M),
Since 1/;a and 1/;b are skew, then [1/;a, 1/;b](l) = O. Also [6 a, 6b](1) = a 0 b boa = O. Hence z = 6z (1) = O. Thus [1/;a,1/;b] = -[6 a, 6b], so 1/; satisfies condition (i) of Definition 6.10. To show that 1/; also satisfies condition (ii) of Definition 6.10, we first observe that for all a E M, by Definition 6.8 (ii),
Therefore there exists z E Z(M) such that [1/;a,6 a] = 6z . Thus 6z is a commutator of two bounded linear operators on /\1, and it commutes with each of them, so it follows from the Kleinecke-Shirokov Theorem that 6z is quasi-nilpotent, i.e., lim
116:'
But 6~(1) = zn for all n. Since IlzI1 2", so
IIa 21
n-+(X)
Il
lln
=
= O.
IIal1 2 for
all a E M, then
Ilz2" I
=
hence z = O. Thus [1/;a,6 a] = O. Now it follows from Lemma 6.12 that 61jJ"a = 0, and hence also 1/;aa = 0, so we have verified condition (ii) of Definition 6.10. Thus 1/; is a dynamical correspondence. Now let 1/; be any dynamical correspondence on !'vI, and let (a, b) f--+ ab be the associated W*-product on M + ~M (Corollary 6.16). By Proposition 6.9, there is a Connes orientation I on !vI given by (6.22). With this we have from each given Connes orientation constructed a dynamical correspondence, and from each given dynamical correspondence constructed a Connes orientation. To show that these two constructions are inverses, start with a Connes orientation I on lvI. Let 1/; be the dynamical correspondence constructed from l, so that l(la) = ~a holds for all a E M. Note also that this equation implies l(~a) = l2(la) = -Ja for all a E M. From 1/;, construct the corresponding W*-product (a, b) f--+ ab on lvI +~lv[ as in Theorem 6.15, so that 1/;a = 6ia for all a E M (cf. (6.20)). Then for a, b EM,
Thus we have I(Jx ) = "SiX for all x E M + lM. By (6.22) this shows that I is the same as the Connes orientation constructed from 1/;.
COl\lPARING NOTIONS OF ORIENTATION
205
Now let 'Ij; be any dynamical correspondence on M, construct the corresponding \,y*-product (a, b) f---> ab on lYl + ~M as in Theorem 6.15, so that 'Ij;" = 6ia for all a EM. Let I be the Connes orientation defined by (6.22). Then for a E M, we have ;j;a = 5ia = 1(5a ), so 'Ij; satisfies (6.21). Thus 'Ij; coincides with the dynamical orientation constructed from the Connes orientation I. Finally, we prove that every dynamical correspondence 'Ij; on M is complete. For the W*-product on lYl + ~M associated with 'Ij;, we have 'lj;a = 6ia for all a E 1Y1 + ~M, i.e., 'Ij; is the dynamical correspondence associated with this product. Then 'Ij; is complete (cf. Proposition 6.11). 0 6.19. Corollary. Let M be a JEW-algebra. Then lYl ~s Jordan isoto the self-adJomt part of a von Neumann algebra ~ff M admzts a Connes onentatwn. There ~s a 1-1 correspondence of Connes onentatwns on M and W*-products on lvl + ~M. The Cannes onentatwn correspondmg to the W*-product (a, b) f---> ab zs gwen by morph~c
1(6 x ) = 6ix
for all x E M
+ ~M.
Proof. This follows from Theorem 6.18 and Corollary 6.16. 0 Comparing notions of orientation In [AS], two notions of orientation were defined: one for C*-algebra state spaces (A 146), and the other for von Neumann algebras and their normal state spaces, cf. (A 200) and (A 201). Here we will sketch how these are related to our current notion of a dynamical correspondence (and thus also to Connes orientations.) The relationships are easiest to see for the special example of the C*-algebra M of 2 x 2 matrices over C. In this case, the state space is affinely isomorphic to a Euclidean 3-ball (A 119). An orientation in the C* sense consists of an equivalence class of parameterizations, i.e., affine isomorphisms from the standard Euclidean 3-ball onto the state space, with two parameterizations being equivalent if they differ by composition with a rotation. An orientation in the von Neumann sense corresponds to an equivalence class of Cartesian frames for the state space, with two frames being equivalent if one can be rotated into the other. Note in each case there are two equivalence classes; these correspond to the standard multiplication on M, and to the opposite multiplication (a,b) f---> ba. Now let s be a symmetry in M. Then s can be viewed as an affine function on the state space, and will take its maximum and minimum values at antipodal points on the boundary of the 3-ba11. Let 'Ij; be the dynamical correspondence associated with the standard multiplication on M. The dual of the map 'lj;s = ,sis generates a rotation about the axis
206
G
DYNMlIICAL CORRESPONDENCES
determined by these two antipodal points. (See Figure G.l and the aCCOlllpanying remarks.) As s varies, the axis ofrotation varies, and each map 1/J, determines an orientation of the 3-ball. However, as s varies continuously, these rotations will vary continuously, and thus give the same orientation. If we replace the multiplication on M by the opposite one, the effect is to replace 1/Js by -1/Js, and so the associated rotation moves in the opposite direction. Thus in all cases, one can think of the orientatiOli as determming a direction of rotation about diameters of this 3-ball, with consistency conditions (e.g., continuity) as the axis varies. The rotations 1/Js are determined by (and determine) the dynamical correspondence a f--7 'Pa' Now we sketch how this extends to arbitrary C* and von Neumann algebras. In the case of C* -algebra orientations, each pair of equivalent pure states determine a face of the state space which is a 3-ball, and a requirement of continuity is imposed, so that the orientation of these 3- balls is required to vary continuously with the ball (A 146). In the von Neumann algebra case, there may be no pure normal states, and thus no facial 3-balls. Instead one works with "blown up 3-balls" (A 194), which also will appear later in our characterization of von Neumann algebra normal state spaces. One can describe an orientation of such blown up 3-balls, cf. (A 206), in terms of pairs (s, 1/Js), where s is a partial symmetry, andl/;s = Gis is the generator of a "generalized rotation". Thus in all contexts, the notion of orientation can be expressed in terms of maps S f--7 1/Js that are related to (generalized) rotations of (blown up) 3-balls, and then one needs to impose consistency conditions, as specified in the various definitions of orientation. Piecing these orientations together consistently is equivalent to choosing the maps 1/Js so that they can be extended to a dynamical correspondence a f--7 <Pa.
Notes The description of the order derivations on a von Neumann algebra (Theorem 6.2) is in [36]' where Connes introduced the notion of an order derivation. The proof of Theorem 6.2 quickly reduces to the result that derivations on a von Neumann algebra are inller, due to Kadison [79] and Sakai [111]. Part of the history of nonned Jordan algebras is tied to homogeneous self-d ual cones. A cone A + in a vector space A is self-dual if there exists an inner product on A such that a ;::: 0 iff (a I b) ;::: 0 for all b ;::: O. It is homogeneous if the order automorphisms are transitive on the interior of the positive cone. Note that for any lB-algebra A, if a E A is positive and invertible, then Ua l/2 is an automorphism of the interior of A + taking 1 to a, so the group of affine automorphisms of the interior of A + is transitive. Koecher [86] and Vinberg [134]' by rather different methods, showed that the finite dimensional homogeneous self-dual cones are exactly the positive cones of formally real Jordan algebras. As we remarked in the
NOTES
207
notes to Chapter 1, these are exactly the finite dimensional JB-algebras. The book of Faraut and Koninyi [52] discusses in detail finite dimensional homogeneous self-dual cones and their connections with Jordan algebras and with symmetric tube domains. The results in this chapter are related to the general problem of characterizing von I'\eumann algebras as ordered linear spaces. Sakai [109] characterized the L2 -completion (with respect to a trace) of a finite von Neumann algebra by means of self-duality of its positive cone and existence of an "absolute value", which models the left absolute value a f---t (a*a)1/2. Connes, in his order-theoretic characterization of (T-finite von Neumann algebras [36], showed that these algebras are in 1-1 correspondence with cones that are self-dual, "homogeneous", and have an "orientation". (The latter two are new concepts defined ill Connes' paper.) Later on, Bellissard and Iochum showed that the (T-finite JRW-algebras are in 1-1 correspondence with the cones that have the first two of Connes' three properties, i.e., are self-dual and homogeneous [27]. Such cones were studied in greater detail by Iochum [68], who introduced the name "facial homogeneity" for Connes' concept. As remarked at the end of Chapter 5, facial homogeneity is closely related to ellipticity. Homogeneity (in the sense of transitivity of the automorphism group on the interior of the positive cone) is equivalent to facial homogeneity in finite dimensions as shown in the paper [26] by Bellissard, Iochum, and Lima. Once one has Jordan structure, something must be added to get the C*-structure, since the Jordan product does not determine the associative multiplication. Connes' definition of orientation is one way to proceed; dynamical correspondences are another approach, closely related to Connes', and in Chapter 11 we will discuss yet a third (more geometric) approach. The results in this chapter are taken from [12] and [13]; the latter also discusses the relationship between the concepts of orientation in Connes' paper [36] and that of the authors in [10]. The notion of a "Connes orientation" (Definition 6.8) carries Connes' definition of an orientation from the natural cone of a JBW-algebra to the algebra itself. For JBW-algebras, the concept of a dynamical correspondence is closely related to that of a "Connes orientation", as illustrated by Theorem 6.18. However, the notion of a dynamical correspondence also makes sense in the broader context of a general JB-algebra. Crgin and Petersen [58] axiomatize an observable-generator duality in terms of what they call a Hamilton algebra, which is a linear space H equipped with two bilinear operations: a product T and a Lie product a, satisfying certain conditions. Emch [49] discussed "Jordan-Lie" algebras, in which both a Jordan product and a Lie product are given, satisfying certain conditions relating the two products. There is also a discussion of such algebras in the book of Landsman [91]. In each case the focus is on deformations of the algebraic structure that approach a kind of classical limit as a parameter approaches zero.
PART II Convexity and Spectral Theory
7
General Compressions
Vie will now approach the theory of state spaces from a new angle, starting with geometric axioms which are at first very general, but which will later be strengthened to characterize the state spaces of Jordan and C*-algebras. The first part of this program is to establish a satisfactory spectral theory and functional calculus. Here the guiding idea is to replace projections p by "projective units" p = PI determined by "general compressions" P defined by properties similar to those characterizing the compressions Up in (A 116) and Theorem 2.83. The first section of this chapter presents basic results on projections in cones. The second section defines general compressions. The third section contains results on projective units and projective faces. The fourth and last section gives a geometric characterization of projective faces together with some concrete examples in low dimension.
Projections in cones We will first work in a very general setting, assuming only that we have a pair X, Y of two positively generated ordered vector spaces in separating ordered duality under a bilinear form (-,.) (A 3). (An ordered vector space X is said to be positively generated if X = X+ - X+.) Unless otherwise stated we will use words like "continuous", "closed", etc. with reference to the weak topologies defined by the given duality.
7.1. Definition. If F is a subset of a convex set C C X, then the intersection of all closed supporting hyperplanes of C which contain F will be called the tangent space of C at F, and it will be denoted by TancF, or simply by Tan F when there is no need to specify C. If there is no closed supporting hyperplane of C which contains F, then TancF = X by convention. Clearly a tangent space of C is a supporting (affine) subspace, but a supporting subspace is not a tangent space in general. In Fig. 7.1 we have shown one supporting subspace which is a tangent space and one which is not.
212
7
GENERAL COlvIPRESSIONS
Fig. 7.1
By (A 1) a face F of a convex set C C X is semz-exposed if it is the intersection of C and a collection of closed supporting hyperplanes containing F. Thus F is semi-exposed iff
F = CnTancF.
(7.1)
We will use the symbol BO to denote the anmhzlator of a subset B of X in the space Y. We will now also use the symbol Be to denote the posztwe anmhzlator of B. Thus
(7.2)
Be = {y E y+ I (x, y) = 0 for all x E B}.
Similarly we use the same notation with BeY and X, Y interchanged. For each x E B C X+, the set x-1(0) = {y E Y I (x,y) = O} is a closed supporting subspace of Y+, so the positive annihilator
Be = y+ n
n
x-1(0)
xEB
is a semi-exposed face of Y+. We also make the following observation, which we state as a proposition for later reference.
7.2. Proposition. If X, Yare ordered vector spaces m separatmg ordered duaZzty and B c X+, then Bee zs the semz-exposed face generated by B m the cone X+ and Beo zs the tangent space to X+ at B. Proof The closed supporting hyperplanes of X+ are precisely the sets of the form y-l (0) for y E Y+. Thus if B c X+, then
Bee = X+ n (
n
yEBO
y-l(O))
and
B eo =
n
y-l(O).
yEBO
The intersections above are extended over all supporting hyperplanes of X+ which contain B, so the proposition follows from the definition of a semi-exposed face and of a tangent space. 0
213
PROJECTIONS IN CONES
If P : X -> X is a posdwe pro)ectwn, i.e., if P is a linear map such that P(X+) C X+ and p2 = P, then we will use the notation ker+ P = X+ n ker P and im+ P = X+n im P. Now it follows from the simple characterization (A 2) of faces in cones, and from the assumption that X is positively generated, that if P : X -> X is a positive projection, then ker+ P is a face of X+ and im P is a positively generated linear subspace of X. We will now study the tangent spaces of the cone X+ at ker+ P and im+ P, and for simplicity we will omit the subscript X+ and just write Tan (ker+ P) and Tan(im+ P). If P is a continuous positive projection on X, then we denote by P* the (continuous) dual pro)ectwn on Y, defined by (7.3)
(Px, y)
= (x, P*y) when x
E X and y E
Y.
Clearly X and Y may be interchanged in all the statements above.
7.3. Proposition. Let X, Y be posdwely generated ordered vector spaces m separatmg ordered dualdy. If P lS a contmuous positwe projection on X, then Tan(ker+ P)
(7.4)
= (ker+ pta c ker P.
Proof. By Proposition 7.2,
Since (ker P)O = imP* and im P* is positively generated, (ker Pt Hence (ker P)·o
= im+ P* - im+ P* = (ker P)· - (ker Pt.
= (ker P)OO = ker P, which completes the proof.
D
7.4. Corollary. Let X, Y be posltwely generated ordered vector spaces in separating ordered duahty. If P lS a contmuous posltwe pro)ectwn on X, then ker+ P lS a seml-exposed face of X+. Proof. Clearly (7.4) implies X+ n Tan(ker+ P) C ker+ P. The opposite inclusion is trivial, so (7.1) holds with ker+ P in place of F. Thus ker+ P is a semi-exposed face of X+. D The inclusion opposite to the one in (7.4) is not valid in general, but projections for which this is the case form an important class containing the compressions Up in the case of C*-algebras and JB-algebras (as we will show in Theorem 7.12).
7
214
GENERAL COMPRESSIONS
7.5. Definition. Let X, Y be positively generated ordered vector spaces in separating ordered duality. A continuous positive projection P on X will be called smooth if ker P is equal to the tangent space of X+ at ker+ P, i.e., if Tan(ker+ P)
(7.5)
= ker P.
Note that the non-trivial part of the equality (7.5) is the inclusion ker Pc Tan(ker+ P), which by Proposition 7.2 is equivalent to ker P
(7.6)
c
(ker+ p)eo.
We will now show that a smooth projection P is determined by ker+ P and im+ P. (This is not the case for general positive projections. A simple example of a non-smooth projection is the following. Let X = Y = R3 with the standard ordering, and let P : X ----> X be the orthogonal projection onto the (horizontal) line consisting of all points (Xl, X2, X3) such that Xl = X2 and X3 = O. Then ker P is the (vertical) plane consisting of all points such that Xl = ~X2, but the tangent space to ker+ P is only the (vertical) line consisting of all points such that Xl = X2 = 0.) 7.6. Proposition. Let X, Y be posltwely generated ordered vector spaces m separatmg ordered dualzty. If P zs a smooth projectwn on X and R zs another contmuous posztwe plOJectwn on X such that im+ R = im+ P and ker+ R = ker+ P, then R = P.
Proof.
By (7.5) and (7.4) (the latter with R in place of P), ker P
= Tan (ker+ P) = Tan(ker+ R)
C ker R.
Since im+ P = im+ R, and the sets im P and im R are both positively generated, im P = im R. From the two relations ker P C ker Rand im P = im R, we conclude that P = R. D Smoothness of P will also imply that ker+ P and im+ P dualize properly under positive annihilators. The precise meaning of this statement is explained in our next proposition. 7.7. Proposition. Let X, Y be positwely generated ordered vector spaces m separating ordered dualzty. Let P be a contmuous posztive proJectwn on X and conszder the equatwns im+ Pt
= ker+ P*,
(7.7) (7.8)
(ker+ Pt =:> im+ P*,
(7.9)
(ker+ Pt = im+ P*.
215
PROJECTIONS IN CONES
Of these, (7. 7) and (7.8) hold generally, and (7. 9) holds zff P zs smooth. Proof. Since im P is positively generated, (im+ P)O ker P*, from which (7.7) follows. Generally im P*
= (ker Pt
= (im P)O =
C (ker+ Pt,
so im+ P* C (ker+ P)·, which proves (7.8). To prove the last statement of the proposition, we assume that P is smooth. By (7.5), (ker+ P)·o = ker P. The set (ker+ Pl· is trivially contained in its bi-annihilator, so (ker+
pr
C
(ker Pt
= im P*.
Hence (ker+ Pl· C im+ P*. The opposite relation is (7.8), so the equation (7.9) is satisfied. Finally, (7.9) implies (7.5), so the equation (7.9) is only satisfied if P is smooth. 0 We saw in Corollary 7.4 that ker+ P was a semi-exposed face of X+ for every continuous positive projection P on X. The corresponding statement for im+ P is false. In fact, im+ P need not be a face of X+, and if it is a face, it need not be semi-exposed. Examples in which im+ P is not a face of X+ are easily found already in R 2 , where it suffices to consider the orthogonal projection P onto a line through the origin and some interior point of the positive cone X+. An example in which im+ P is a non-semiexposed face of the cone X+, can be obtained by ordering R 3 by a positive cone X+ generated by a convex base with a non-exposed extreme point. A standard example of a convex set with non-exposed extreme points is a square with two circular half-disks attached at opposite edges. (Such a set may be thought of as a sports stadium with a soccer field inside the running tracks. Then the non-exposed extreme points will be located at the four corner flags of the soccer field). Now the orthogonal projection P onto the line through the origin and such a non-exposed extreme point of the base will give the desired example. It turns out that smoothness of P* is a necessary and sufficient condition for im+ P to be a semi-exposed face of X+, as we will now show. 7.8. Corollary. Let X, Y be posztively generated ordered vector spaces in separatzng ordered dualzty, and let P be a contznuous positive pT'OJectwn on X. The dual prOjection P* zs smooth zff the cone im + P is a semiexposed face of X+.
Proof. By Proposition 7.2, im+ P is a semi-exposed face of X+ iff
216
7.
GENERAL COMPRESSIONS
and by (7.7) this is equivalent to im+ P
= (ker+ P*t.
But this equality is (7.9) with P* in place of P, which is satisfied iff P* is smooth. D Recall from (A 4) that two continuous positive projections P, Q on X (or on Y) are said to be complementary (and Q is said to be a complement of P and vice versa) if
(7.10) Note that since X is positively generated, (7.10) implies that PQ = QP = O. Recall also that the two projections P, Q are said to be stTOngly complementary if (7.10) holds with the plus signs removed, whIch is equivalent to
(7.11)
PQ = QP = 0,
P+Q
=
I.
Thus P admits a strong complement iff I - P ;::: O. A continuous positive projection P on X (or on Y) is to be complemented if there exists a continuous positive projection Q on X such that P, Q are complementary, in which case Q is said to be a complement of P. Also P is said to be b~complemented if there exists a continuous positive projection Q on X such that P, Q are complementary and P*, Q* are also com plementary.
Fig. 7.2
PROJECTIONS IN CONES
217
A pair of projections which are complementary, but not strongly complementary, will satisfy the first, but not the second, of the two equalities in (7.11). An example of such a pair is shown in Fig. 7.2 where X = R3 and X+ is a circular cone.
7.9. Proposition. Let X, Y be posltwely generated ordered vector spaces m separating ordered dualdy, and let P be a contmuous posltwe proJectwn on X (or on Y). If P lS complemented, then the dual projectwn P* is smooth, and If P has a smooth complement Q, then Q lS the unique complement of P.
Proof. If P has a complement Q, then the cone im+ P = ker+Q is a semi-exposed face of X+ (Corollary 7.4), so the dual projection P* must be smooth (Corollary 7.8). If P has a smooth complement Q, then an arbitrary complement R of P must satisfy ker+ R = im+ P = ker+Q and im+ R = ker+ P = im+Q. By Proposition 7.6 this implies R = Q. Thus, Q is the unique complement of
P.
D
Complementary projections are not unique in general. One can obtain an example of a positive projection with infinitely many complements by replacing the circular base of the cone in Fig. 7.2 by the convex set in Fig 7.3 which has a vertex at im+ P = ker+Q and also at im+Q = ker+ P. Then ker+Q will be a sharp edge of the cone X+, and by tilting the plane ker+Q about this edge we obtain infinitely many positive projections which are all complements of P. Similarly with P and Q interchanged. (We leave it as an exercise to picture the dual cone {y E X* I (x, y) ;::: O} with the sub cones ker+ P*, im+ P*, ker+Q*, im+Q*, and to show that in this example P, Q are complementary but not bicomplementary.)
im+Q
im+P
Fig. 7.3
The theorem below is a geometric characterization of bicomplementarity in terms of smoothness.
218
7.
GENERAL COMPRESSIONS
7.10. Theorem. Let X, Y be posztwely generated ordered vector spaces m separatmg ordered dualIty. If P and Q are two continuous POSItwe projectIOns on X, then the followmg are equwalent:
(i) P, Q are complementary smooth proJectIOns, (ii) P*, Q* are complementary smooth proJectIOns, (iii) P, Q are bIcomplementary prOJectIOns. Proof. It suffices to prove (i) ¢? (iii) since P*, Q* are bicomplementary iff P, Q are bicomplementary. (i) =} (iii) Assuming (i) and using Proposition 7.7, \ve get
Interchanging P and Q, we also get ker+Q* = im+ P*. Hence we have (iii) . (iii) =} (i) Astmming (iii) we have im+ P = ker+Q. By Corollary 7.4 ker+Q* is semi-exposed, therefore im+ P* is semi-exposed. Now it follows from Corollary 7.8 (with the roles of P and P* interchanged) that P is smooth. Similarly, Q is smooth. Hence we have (i). D
7.11. Corollary. Let X, Y be posztwely generated ordered vector spaces m separatmg ordered duahty. If P IS a blcomplemented contmuous posltwe projectIOn on X, then P has a umque complement. Proof. By Theorem 7.10, P has a smooth complement Q, and by Proposition 7.9, Q is the unique complement of P. D Our next result, concerning C*-algebras and JB-algebras, can be found in (A 116) and Theorem 2.83. But it is actually only the easy argument in the first part of the proofs of these results that is needed, so for the reader's convenience we will give this argument.
7.12. Theorem. Let X be the self-adJomt part of a C*-algebra (or von Neumann algebra) and let Y be the self-adjOInt part of ds dual (respectwely Its predual). Then the compreSSIOn Up determmed by a projectIOn p E X IS a smooth projectIOn on X, the dual map U1: IS a smooth projectIOn on Y, and the two maps Up and Uq where q = pi (= 1 - p) are blcomplementary prOjectIOns on X. The same IS true when X IS a lBalgebra (or lEW-algebra) and Y IS zts dual (respectively ItS predual). Thus m all these cases Up and Uq are blcomplementary projectwns. Proof. Let p be a projection in the C*-algebra X. We will first show that Up is a smooth projection on X by proving (7.6). Thus we must shoW that if a E ker Up, then w(a) = 0 for each wE (ker+Up)-.
PROJECTIONS IN CONES
219
Let a E ker Up, i.e., pap = 0, and W E (ker+Up)·, i.e., W ;::;. 0 and = 0 on ker+Up. Clearly the projection q = pi is in ker+Up, so w(q) = o. By (A 74) ((iii) =? (vi)), w = p. W· p. Thus w(a) = w(pap) = 0 as desired.
W
Next we observe that by (A 73) ((iii).;=;.(iv)), ker+Uq = im+Up , and likewise with p and q interchanged. Thus Up and Uq are complementary smooth projections. Thus the statement (i) of Theorem 7.10 holds when P = Up and Q = Uq . Then the statement (ii) also holds, so U; and U; are complementary smooth projections. This completes the proof in the C*-algebra case. (The same proof works in the von Neumann algebra case, now with w a normal linear functional). In the JB-algebra (and JBW-algebra) cases we can use the same argument with reference to Proposition 1.41 instead of (A 74), and Proposition 1.38 instead of (A 73). 0 We will now establish a direct sum decomposition which generalizes the Pierce decomposition of Jordan algebras [67, §2.6]' and we begin by a lemma which will also be needed later. Here we shall need the concept of a direct sum of two wedges in a linear space. Recall that a wedge W is a (not necessarily proper) cone, so it is defined by the relations W + ~V c W and AW c W for all A ;::;. 0 (cf. [AS, p. 2]). We will say a wedge W is the direct sum of two subwedges WI and W 2 iff each element of W can be uniquely decomposed as a sum of an element of WI and an element of W 2 • We will use the notation EBw for direct sums of wedges (reserving EB for direct sums of subspaces). 7.13. Lemma. Let X, Y be posdwely generated ordered vector spaces in separatmg ordered duallty. If F and G are subcones of X+ such that
(7.12)
X+
c F EBw TanG,
X+
c G EBw TanF,
and N := Tan F n Tan G, then (7.13)
Tan F
= lin F EB N, Tan G = lin G EB N, X = lin FEB lin G EB N.
Proof. Clearly lin F
+N c
Tan F
+N c
Tan F.
To prove the converse relation, we consider an element x E Tan F. Since X is positively generated, x = Xl - X2 where XI,X2 E X+. By (7.12) we also have decompositions Xi = ai + bi where ai E F and bi E Tan G for i = 1,2. Now al - a2 E lin F and b l - b2 E Tan G. But we also have
7.
220
GENERAL COMPRESSIONS
Thus Tan F = lin F + N. Similarly Tan G = lin G + N. Observe now that lin F n Tan(G) = {O}. In fact, if x E lin F n Tan(G), then x = Xl - X2 where Xl, X2 E F, and the equation X2 + X = Xl + 0 provides two decompositions of the element Xl of X+ as a sum of an element of F and an element of Tan(G), so X2 = Xl and X = O. Similarly lin G n Tan(F) = {O}. N ow also lin F n N = {O} and lin G n N = {O}, so Tan F = lin F EEl N and Tan G = lin G EEl N. Since X is positively generated, it follows from (7.12) that X = linF +Tan(G). In fact, X = lin F EEl Tan G, and then also X = lin F EEl lin G EEl N. D
7.14. Theorem. Let X, Y be posztively generated ordered vector spaces m separatmg ordered dualdy, let P, Q be a pazr of bicomplementary contmuous posdwe proJectwns on X, and set F = im+ P, G = im+Q and N = Tan F n Tan G. Then (7.12) holds, and P and Q are the proJectwns onto lin F and lin G determmed by the dzrect sum (7.14)
X = lin F EEl lin G EEl N.
Thus P + Q zs a projection wzth kernel N onto the subspace lin F EEl lin G. Proof. By Theorem 7.10, P is a smooth projection. Hence (7.15)
ker P = Tan(ker+ P) = Tan(im+Q).
Let X be an element of X+. Then we can decompose X as the sum of the element Px in the cone im+ P and the element X - Px in the subspace ker P = Tan(im+Q). If X = Xl + X2 is an arbitrary decomposition with Xl E im+ P and X2 E Tan(im+Q), then X2 E ker P, so Px = Xl. Thus the decomposition is unique. With this we have shown the first inclusion in (7.12). The second inclusion follows by interchanging P and Q. Since im P is positively generated, im P = F - F = lin F, and by (7.15) and (7.13) ker P = Tan G = lin G EEl N. Thus P is the projection onto lin F determined by the direct sum in (7.14). Similarly Q is the projection onto lin G determined by this direct sum. This completes the proof. D We will now generalize the uniqueness theorem for the conditional expectations E = Up + Up' in a von Neumann algebra (A 117) to the geometric context of bicomplementary projections in cones, which by Theorem 7.12 includes compressions in general C*-algebras, von Neumann algebras, JB-algebras and JBW-algebras.
221
PROJECTIONS IN CONES
7.15. Theorem. If X, Yare posdwely generated ordered vector spaces m separating ordered dualzty and P, Q are bzcomplementary contmuous posztwe projections on X! then the map E = P + Q zs the unzque contmuous posztive projectwn onto the subspace im (P + Q) = im PEEl im Q. Proof. Note first that since PQ = QP = 0, E is a positive projection onto im (P + Q) = im P EEl im Q and ker E = ker P n ker Q. To prove the uniqueness, we consider an arbitrary continuous positive projection Tonto im (P + Q). Thus imE = imT, so ET = T. We will prove that we also have ker EckerT, which will imply ker E = kerT and complete the proof. Observe that it suffices to prove (7.16)
ker P
c
ker PT
and
ker Q
c
ker QT,
since this will imply ker E = ker P n ker Q c ker PT n ker QT
= ker (PT + QT) = ker ET = ker T. Assume x E ker P. To show that x E ker PT, it suffices to show that (PTx, y) = 0 for all y E Y+ (since Y is positively generated). Let y E Y+ be arbitrary. Note that T* P*y E (ker+ P)·. In fact, for each a E ker+ P = im+Q C im T,
(a, T* P*y)
= (Ta, P*y) = (a, P*y) = (Pa, y) = o.
But since P is smooth, x E ker P implies that x E Tan (ker+ P) (ker+ P)·o, so we get
(PTx, y)
= (x, T* P*y) = 0
as desired. Thus ker Pc ker PT. Similarly ker Q C ker QT. With this we have proved (7.16) and have completed the proof of the theorem. D We will briefly discuss the geometric idea behind the above proof. The key to the argument is smoothness. Since P and Q are smooth projections, the cones ker+ P and ker+Q determine the subspaces ker P = Tan (ker+ P) and kerQ = Tan(ker+Q), and then also their intersection, which is shown to be the kernel not only of the projection E = P + Q but of any positive projection onto the subspace M = im (P + Q). The fact that the uniqueness of E depends on the smoothness of P and Q, can be easily seen in the example given in Fig. 7.2. Si~ce E is the projection onto M with kernel equal to the intersection of the tangent planes to the cone X* at the two opposite rays ker+ P and ker+Q, E must
222
7
GENERAL COMPRESSIONS
be the orthogonal projection onto lvI, and this is the only projection onto M which maps X+ into itself. The situation is different in the example obtained by replacing the circular cone in Fig. 7.2 by a cone with a base as in Fig. 7.3, having two antipodal vertices. Now we can find infinitely many pairs of (non-smooth) positive projections P, Q with ker+ P and ker+Q equal to the rays through these vertices (passing from one such pair to another by tilting the planes ker P and ker Q about the rays ker+ P and ker+ Q). Two distinct pairs will determine distinct intersections ker Pnker Q, and then distinct positive projections of X onto AI. Thus there is no uniqueness in this case. Definition of general compressions
We will now specialize the discussion to a pair of an order unit space
A and a base norm space V in separating order and norm duality (A 21). Note that order unit spaces and base norm spaces are positively generated, cf. (A 13) and (A 26), and so the previous results in this chapter are applicable. As in the previous section, we will study positive projections that are continuous in the weak topologies defined on A or V by the given duality. But we will no longer omit the words "weak" or "weakly" when referring to these topologies since we now also have to deal with the norm topologies. We will say that a positive projection on A or V is normalzzed if it is either zero or has norm 1, or which is equivalent, if it has norm ::::; 1. Note that if P is a normalized positive projection on A, then the dual projection P* is a normalized positive projection on V, and vice versa. Recall from (A 22) that if P is a weakly continuous positive projection on A, then (7.17)
IIP* pil
= (PI, p) for all p E V+.
We also make two observations, which we state as lemmas for later reference. 7.16. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty. A posdwe pTOJcctwn P on A zs normalzzed zJJ PI ::::; 1.
Proof. To prove the "if" part of the lemma, we assume PI ::::; 1 and consider a E A with Iiall ::::; 1, i.e., -1 ::::; a ::::; 1, d. (A 13). Since P is positive, -PI::::; Pa ::::; PI; hence -1 a (where a(p) = (a, p) for p E V) is an order and norm preserving isomorphism from A onto a point-separating subspace of the space V* = Ab(K) of all bounded affine functions on K, equipped with the pointwise ordering and the uniform norm (A 11). For simplicity We will identify elements a of A with their representing affine functions on K. Thus if a, b E K and F c K, then we will write a ::::: b on F instead of (a,p) ::::: (b,p) for all p E K, etc. As before we wili. use the notation for the positive part of the norm closed unit ball of A.
a
Ai
7.25. Lemma. Let A, V be an order umt space and a base norm space m separating order and norm duality. If P lS a compresswn on A, then PI lS the greatest element of whzch vamshes on ker+ p', and
Ai
(7.21)
im+ P'
= {p
E
V+ I (PI, p)
= (1, p) }.
Proof. Let P be a compression on A. If p E ker+P', then (Pl,p) = (1, P' p) = 0, so PI vanishes on ker+ P*. Consider now an arbitrary element a of Ai which vanishes on ker+ P*. Thus 0 ::; a ::; 1 and a E (ker+ P*)·. By equation (7.9) (with P' in place of P), a E im+ P. Hence a = Pa ::; PI. Thus PI is the greatest element of Ai which vanishes on ker+ p'. The equation (PI, p) = (1, p) is equivalent to IIP* pil = Ilpll, and since P is neutral, this gives (7.21). D 7.26. Proposition. Let A, V be an order unit space and a base norm space m separating order and norm dualzty, and let P be a compresswn on A. Then the two cones im + P, ker+ Pare semz-exposed faces of A +, the two cones im+ P*, ker+ P* are seml-exposed faces of V+, and each one of these four cones determmes the compresswn P. Proof. The cones in question are semi-exposed by Corollary 7.4 and the existence of a complementary positive projection. By Lemma 7.25, ker+ P* determines PI, and through (7.21) also im+ P*. By Proposition 7.6, ker+ P* and im+ P* together determine P*, and hence also P. With this, we have shown that ker+ P* determines P. When im+ P' is given, we also know the cone ker+Q* = im+ P* where Q is the complement of P. Thus it follows from what we have just proved that im+ P* determines Q, and then also its complement P (Corollary 7.11). It follows from Proposition 7.7 that im+ P determines ker+ P* and ker+ P determines im+ P*. Thus it follows from what we have shown above that each of the two cones im+ P and ker+ P determines P. D Remark. Proposition 7.26 fails if complementary compressions are replaced by bicomplementary weakly continuous positive projections. Thus
PROJECTIVE UNITS AND PROJECTIVE FACES
227
the requirement that the projections be normalized is crucial. To see this, observe that if V+ is a circular cone, for each pair of distinct faces F, G of V+ there are bicomplementary projections P, Q with im+ P* = F and im+Q* = G.
Projective units and projective faces We will continue the study of compressions defined in the context of a separating order and norm duality of an order unit space A and a base norm space V with distinguished base K. In the motivating example of a von Neumann algebra and its predual, each compression is of the form P = Up where p is a projection in the algebra, uniquely determined by the equation p = PI, cf. Proposition 7.23. Starting out from this equation, we will define a class of elements pEA which will play the same role as the projections p in the algebra, and a class of faces F c K which will play the same role as their associated norm closed faces Fp = K n im P* of the normal state space (cf. (A 110)).
7.27. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality. If P is a compression on A, then the element p = PI of is called its associated pmjectwe unit, and the face F = K n im P* is called its associated pro]ectwe face.
At
In the case of von Neumann algebras and lBW-algebras, every norm closed face of the normal state space is projective, cf. (A 109) and Theorem 5.32. This is certainly not true in general. In fact, most convex sets K encountered in elementary geometry have no projective faces at all (other than K and 0). For simplexes and smooth strictly convex sets, all faces are projective. (They are split faces in the former case, and just the extreme points in the latter.) Other examples of 3-dimensional convex sets with projective faces, are not so easy to come by. But we will soon give a geometric characterization of projective faces (Theorem 7.52), which we will use to give more examples. (Examples are pictured in Fig. 7.6 and Fig. 7.7, where F and G are complementary projective faces, and in Fig. 8.1, where all faces are projective.) We will first prove some elementary properties of projective units and projective faces.
7.28. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm duaizty, and let a E A. Each projectwe unit p zs assoczated with a umque compresswn P. Also each pro]ectwe face F is associated with a unique compresswn P, and then Pa ;:::: 0 iff a ;:::: 0 on F. Proof. Assume first p = PI where P is a compression on A. By (7.21) im+ P* is determined by p, and thus by Proposition 7.26 the compression P is itself determined by p.
228
7
GENERAL COl\lPRESSIONS
Assume next F = ]{ n im P* where P is a compression on A. Then im+ P* is the cone generated by F (i.e., im+ P* consists of all AW where A E R + and w E F). Thus im + P* is determined by F, and then P is determined by F. If a ;::: 0 on F, then a;::: 0 on the cone im+ P* generated by F, hence (Pa, w) = (a, P*w) ;::: 0 for all w E K Thus a ;::: 0 on F implies Pa ;::: O. Conversely if Pa ;::: 0, then (a, w) = (a, P*w) ;::: 0 for all w E F C im+ P*. Thus Pa ;::: 0 implies a ;::: 0 on F. D
If P is a compression with associated projective unit p and associated projective face F, then it follows from the proposition above that each one of P, p, F determines the others, and we will say that they are mutually assocwted with each other. We will also denote the (unique) compression which is complementary to P by pi, and we will denote the projective unit and the projective face associated with pi by pi and F' respectively Also we will say that pi, pi, F' are the complements of P, p, F respectively. Note that by equation (i) of Lemma 7.17, pi = 1 - p, so this notation is compatible with our previous use of the prime to denote the (orthogonal) complement of projections. Note also that by (7.21), the projective faces F and F' are determined by p through the explicit formula (7.22)
F = {w E]{ I (p,w) = I},
F' = {w E]{ I (p,w) = O},
and conversely, by Lemma 7.25, that the projective unit p is determined by F or F' through the formula (7.23)
p=
V{a E At
I
a = 0 on F'} =
1\ {a E At
I
a = Ion F},
where the second equality follows from the first by replacing p by pi, F by F ' , and a by 1 - a. Thus p is the least element of At which takes the value 1 on F, and p is the greatest element of At which takes the value o on F'. Hence p is the unique element of At which takes the value 1 on F and also takes the value 0 on F'. By Definition 7.27, we have the following formula for the projective face associated with a compression: (7.24)
F = {w E ]{ I P* w = w},
F' = {w E ]{ I P* w = O}.
From this it follows that for each a E A we have (Pa, w) = 0 when wE F'. Thus
=
(a, w) when
wE F and (Pa, w)
(7.25)
Pa = a on F,
Pa = 0 on p'.
Recall from (A 2) that if a E A +, then the face generated by a in A +, denoted by face (a), consists of all b E A + such that b ::::; Aa for some
PROJECTIVE UNITS AND PROJECTIVE FACES
229
>. E R +, and that the semi-exposed face generated by a in A + is equal to {a}·· (Proposition 7.2).
7.29. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualdy and conszder a compresswn P wzth
assocwted projectzve umt p. If a E A +, then the followmg are equzvalent: (i) a E im+ P, (ii) a::: Iiall, (iii) a E face(p), (iv) aE{p}··.
Proof. (i) =} (ii) If a E im+ P, then a::: IIal11 and a = Pa ::: IiallP1 = Iialip. (ii)=}(iii) and (iii)=}(iv) are trivial. (iv)=}(i) Assume (iv). By Proposition 7.26, im + P is a semi-exposed face of A +. Hence a E {p} •• C (im+ P)·· = im+ P. 0 By (A 75) and Proposition 1.40, an element of a von Neumann algebra or a JBW-algebra is an extreme point of the closed unit ball iff it is a projection. Our next proposition generalizes the "if' -part of this result. 7.30. Proposition. Let A, V be an order umt space and a base norm space m separating order and norm dualdy. If p zs a projective umt, then p is an extreme pmnt of At.
Proof. Let p be the projective unit associated with a compression P. Clearly p = PI E At. Assume (7.26)
p
= >.a + (1 - >.)b
with a, bEAt and 0 < >. < 1. Then a::: >.-lp and b::: (1- >.)-lp. Hence a, b E face(p) = im+ P, so a = Pa ::: PI = p and b = Pb ::: PI = p. By (7.26), a = b = p, so p is an extreme point of At. 0 In the proposition below (and in the sequel) we will use the standard square bracket notation for closed order intervals, so [a, b] will denote the set of all x E A such that a ::: x ::: b. 7.31. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty. If P zs a compresswn on A wzth associated proJectzve umt p and assocwted proJectzve face F, then
(i) [-p,p]=A1nimP, (ii) co(F u -F) = VI n imP*. Moreover, P( A) = im P is an order umt space wzth distinguzshed order umt p and P* (V) = im P* is a base norm space wzth dzstinguzshed base
7
230
GENERAL COMPRESSIONS
F. These two spaces are zn sepamtmg order and norm dualzty under the ordenng, norm and bzlmear form (".) relatwzzed from A and V. Proof. (i) If a E [-p,p], then it follows from Lemma 7.29 ((iii)=? (i)) that a + p E im P. Hence also a E im P, and since Iiall ::; Ilpll ::; 1, then a E At n im P. Conversely, if a E At n imP, then -1 ::; a ::; 1, so -p ::; Pa ::; p, and also Pa = a. (ii) The set at the left-hand side of (ii) is trivially contained in the set at the right. To prove the reverse containment, we consider an element w in VI n im P*. Without loss of generality, we may assume Ilwll = 1. By the definition of a base norm space (d. Definition 2.45), VI = conv(K U -K). Thus there exists (J E K, T E -K and A E [0,1] such that w = A(J + (1 A)T. Then w
= P*w = AP*(J + (1 - A)P*T.
Hence
1 = Ilwll ::; AIIP*(JII
+ (1- '\)IIP*TII,
which implies IIP*(JII = IIP*TII = 1. Thus p*(J E im+ P* n K = F, and similarly -P*T E F, so w E co(F U -F). The rest of the proposition follows from (i) and (ii) and the fact that P is a positive projection with IIPII ::; 1. 0 7.32. Proposition. Let A, V be an order umt space and a base norm space zn sepamtzng order and norm duaZzty. Let P, Q be compresswns wdh assocwted proJectwe umts p, q and assocwted proJectwe faces F, G, respectwely. Then the followzng statements are equwalent: (i) (ii) (iii) (iv)
imP C imQ, QP = P, p::; q,
Fe G, (v) PQ = P,
(vi) ker Q c ker P, (vii) im Q' C im P'. Proof. (i) =? (ii) Trivial. (ii) =? (iii) If QP = P, then p = PI = QPl ::; Ql = q. (iii) =? (iv) Clear from (7.22). (iv) =? (v) If F c G, then im+ P* C im+Q* and then also im P* C im Q*, so Q* P* = P*. Dualizing, we get PQ = P. (v) =? (vi) Trivial. (vi) =? (vii) If ker Q C ker P, then trivially ker+Q C ker+ P, so im+Q' C im+ P', and then also imQ' C imP'. (vii) =? (i) Use the implication (i) =? (vii) with Q' in place of P and P' in place of Q. 0
PROJECTIVE UNITS AND PROJECTIVE FACES
231
7.33. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality and let P, Q be compressions with associated projective units p, q and associated projective faces F, G respectively. If the condition im P C im Q (together with the equivalent conditions of Proposition 7.32) are satisfied, then we will write P ::< Q and F ::< G. Note that if P ::< Q and Q ::< P, then im P = im Q and ker P = ker Q, so P = Q. Thus the relation ::< is a partial ordering. By the equivalence of (i) and (vii) in Proposition 7.32, P ::< Q iff Q' ::< P', so P f--> P' is an order reversing map for this relation, and similarly for p f--> p' and F f--> F'.
7.34. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty, and let P, Q be compressions on A wzth assocwted proJectwe umts p, q and assocwted pTOJectwe faces F, G, respectwely. Then the followmg are equwalent: (i) QP = 0, (ii) P::< Q', (iii) p::; q', (iv) Fe G', (v) PQ = O.
Proof. (i) =} (ii) If QP = 0, then im P C ker Q. Hence im+ P C ker+Q = im+Q', so P::< Q'. (ii)¢? (iii) ¢? (iv) Follows directly from Proposition 7.32. (iv)=}(v) If F c G', then it follows from the implication (iv) =} (v) of Proposition 7.32 that PQ' = P. Multiplying from the right by Q gives 0 = PQ. (v) =} (i) Use the implication (i) =} (v) with P and Q interchanged. 0 7.35. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality, and let P, Q be compressions with associated projective faces F, G and projective units p, q. If the condition PQ = 0 (or any of the equivalent conditions in Proposition 7.34) is satisfied, then we will say that P zs or·thogonal to Q and we will write P ..l Q, and we will also say that p zs orthogonal to q and write p ..l q, and that F zs orthogonal to G and write F ..l G. The notion of orthogonality of projective faces F, G depends on the duality of A and V (since G' is defined in terms of this duality). We will now show that in the important special case where A = V*, this notion corresponds to the geometric notion of antipodality defined in (A 23). Recall that two subsets F, G of a convex set are antzpodal if there is a function a E Ab(K) with 0 :s: a :s: 1 such that (7.27)
a
= 1 on
P,
a
=0
on G.
232
7
GENERAL COMPRESSIONS
7.36. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualdy, and assume A = V*. Let F and G be projectwe faces of K. Then F, G are orthogonal iff F, G are anttpodal. In parttcular, two extreme pomts p, 0' E K are orthogonal tff IIp-O'II = 2.
Proof. Assume first that F 1.. G, and let p, q be the projective units associated with F,G. Then p pi as well as the lattice operations P, Q f-> P AQ and P, Q f-> P V Q.
8
258
SPECTRAL THEORY
8.16. Theorem. Assume the standmg hypothesis of thzs chapter. A set L of compresswns zs a Boolean algebra (under the operatwns mduced from the complemented lattzce C of all compresswns) zff zt zs a complemented sublattzce of C and each pazr P, Q E L zs compatzble. Proof. Consider first a complemented sublattice Lee which is a Boolean algebra. If P, Q E L, then by the distributive law,
P = P
A
(Q
V
Q') = (P
A
Q)
V
(P
A
Q'),
so the compatibility criterion (8.8) is satisfied. Assume next that L is a a complemented sublattice of C for which each pair P, Q E L is compatible. To prove the distributive law (8.14), we consider an arbitrary triple P, Q, R of compressions in L with associated projective units p, q, r. By Corollary 8.14 (together with Theorem 8.3 and Proposition 8.7), PA (Q V R) 1
= P(Q-tRQ') 1 = PQ1 -tPRQ'l.
Hence
s: (p A q) V (p A r). In fact, the equality sign must hold in this relation, since p A q s: P A (q V r) and also pAr s: p A (q V r). This gives (8.14) and completes the proof. 0 P A (q V r)
= (p A q)
V (p A r A q')
Recall from Proposition 7.31 that if A, V is a pair of an order unit space and a base norm space in separating order and norm duality and if Po is a compression with associated projective unit Po and associated projective face Fo, then Po(A) = im Po is an order unit space with distinguished order unit Po, Po (V) = im Po is a base norm space with distinguished base Fo, and Po(A), Po(V) are in separating order and norm duality.
8.17. Proposition. Assume the standmg hypotheszs of thzs chapter and let Po be a compresswn wzth assocwted proJectwe unit Po and associated proJectwe face Fo. Then the pmr Po(A), Po(V) wzll also satzsfy the standmg hypotheszs. For thzs pmr the compresswns are the compresswns P of A such that P .:::S Po (restncted to Po(A)}, the proJectwe umts are the proJectwe units p of A such that p Po, and the proJectwe faces arc the proJectwe faces F of K such that F C Fo. For thzs pmr, the complementary proJectwn of P .:::S Po zs P' (restrzcted to Po(A)).
s:
Proof. Let F be an exposed face of the distinguished base Fo of Po (V). This means that there exists an element of Po (A) + which vanishes
precisely on F. Otherwise stated, there exists an element a of Po(A)+ such that a = 0 on F and a > 0 on Fo \ F.
THE LATTICE OF COMPRESSIONS
259
Now define b = a + P61. We know that PM = 0 on Fa, and also that P61 = pS > 0 on K \ Fa (d. (7.25) and (7.22)). Hence b = a = 0 on F, b = a > 0 on Fa \ F, and also b > 0 on K \ Fa. Thus F is an exposed, hence projective, face of K. Let P be the compression and p be the projective unit associated with F. Since F c Fa, then P ~ Po, so P and Po, and pI and Po, are compatible, and therefore commute (Lemma 7.42). It follows that P and pI leave Po(A) invariant. The restrictions PIPo(A) and PllPo(A) are seen to be complementary compressions (with duals P* IPO' (V) and p l*IPO' (V)) whose associated projective units are p and p' 1\ Po, and whose associated projective faces are F and Fa 1\ F'. This shows that the pair Po (A), PO' (V) satisfies the standing hypothesis, and also that the compressions, projective units and projective faces for this pair of spaces are as stated in the proposition. 0
In Proposition 8.17, we observe that if A that the dual of im PO' is im Po.
= V*, then it is easy to check
8.18. Corollary. Assume the standmg hypothesIs of thIs chapter and let Po be a central compresswn. Then a compresswn P ~ Po is central for Po(A) Iff It IS central for A.
Proof. By Proposition 7.48, P is central for A (or Po(A)) iff Pa ~ a E A+ (respectively, all a E Po(A)+). is central for A, then trivially Pa ~ a for all a E Po(A)+ C A+, central for Po(A). Conversely, assume P is central for Po(A). ~ Po, then P = PPo, so for each a E A+,
for all a If P so P is Since P
Pa = P Poa
~
Poa
~
a.
Thus P is central for A. 0
Remark. We will continue the discussion of the order theoretic approach to quantum mechanical measurement theory initiated in the Remark after Proposition 7.49 in the preceding chapter, but now under the standing hypothesis of the present chapter. This hypothesis says that each exposed face, i.e., each set of the form F = {w E K I (a, w) = O} with a E A+, is a projective face. Interpreted physically, this means that for each positive observable a E A+ the question: "does a have the value 0 7" is a quantum mechanical proposition, i.e., a question which can be answered by a quantum mechanical measuring device, say a filter for a beam of particles. This is true in standard quantum mechanics, and it is natural to assume it as an axiom in the general order-theoretic approach. As explained before, the expected value of an observable a E A measured on a system in a state w E K, is (a, w). For a proposition pEP, the
8
260
SPECTRAL THEORY
expectation (p, w) is the same as the probability of the value l. Therefore the face F associated with p is the set of all states for which the particle will certainly (i.e., with probability 1) appear with the value l. Now it follows from Theorem 8.10 that when the standing hypothesis is satisfied, then the propositions form an orthomodular lattice with the same physical interpretation as in standard quantum mechanics. Thus the proposition lattice has the properties of "quantum logic", which are often used as the basic assumption in axiomatic quantum theory. (See, for example, Piron's paper [87].)
The lattice of compressions when A
= V*
In this section we will continue to study a pair of spaces A, V satisfYlllg the standing hypothesis of this chapter, but we will now also assume that A = V*. Clearly, this condition is satisfied in the motivating examples of von Neumann algebras and JBW-algebras in duality with their preduals. We know that if A = V*, then we can identify A with the space A b(]() of all bounded affine functions on K (by the map a f--+ of (A 11)). Then the ordering in A will be the same as the pointwise ordering in A b (](), and the weak topology in A (defined by the duality with V) will be the same as the topology of pointwise convergence in Ab(K). Since A = V*, the ordered vector space A is monotone complete. Thus each increasing net {aoJ bounded above has a least upper bound a E A, which is also the weak (= pointwise) limit of {aa}, and in this case we will write a", / a. Similarly each decreasing net {a",} bounded below has a greatest lower bound a E A, which is also the weak (= pointwise) limit of {an}, and in this case we will write a", ~ a. Note that if aD: / a (or a", ~ a) and P is a compression, then (by weak continuity) Pac> / Pa (respectively
a
Pa",
~
Pa).
8.19. Lemma. Assume the standmg hypothesls of thls chapteT and A = V*. If {Pc>} lS a deCTeasmg net of pTO]ectwe umts, and Pc> ~ a E A, then a lS a pTO]ectwe umt, so d lS also the gTeatest 10weT bound of {p,,} m P. SlmllaTiy, If {Pa} lS an mCTeasmg net of pTo]ective umts, and Pa / a E A, then a lS a pTO]ectwe umt, so d lS also the least uppeT bound of {Pa} m P. PTOOj. Assume P'" ~ a. Let Fa be the projective face associated with P'" for each ex, and set G = {w E K I (a,w) = O}. Now G is an exposed, hence projective, face. Let P be the compression, and p the projective unit, associated with G'. For each ex,
G => {w E K I (p""w) = O} = F~. Hence G' c Fa, so p ::; Pc" and then p ::; a. But by (7.23), p is the greatest element of At which vanishes on G, so p = a. Thus a is a
THE LATTICE OF COMPRESSIONS WHEN A
= V*
261
projective unit. The remaining statement can be proven in similar fashion, or can be derived from the first by considering the net {I - Pa}. 0
8.20. Proposition. IJ the standing hypothesis oj this chapter is satisfied and A = V*, then C, P, F are complete lattzces, and the lattzce operations are gwen by the Jollowmg equatwns Jor each Jamzly { Fa} c F: (8.15)
Moreover, zJ {PaJ zs an mcreasmg net oj compresswns compatzble wzth an element a E A, then the compresswn P = Va: P a zs also compatzble with a and Paa / Pa. Szmzlarly, zJ {Pa} is a decreasing net oj compressions compatzble wzth a E A, then P = Aa Po. is compatzble wzth a and P",a "" Pa. Proof. By Proposition 8.1, C, P, F are isomorphic lattices, so it suffices to show that F is a complete lattice with lattice operations as in (8.15). In fact, by (8.1) it suffices to show that the first equality in (8.15) is satisfied for each decreasing net {F",} in :F. Let {F",} be a decreasing net in F and let Po. be the projective unit associated with Fa for each oo. By Lemma 8.19, Pa "" P where p is the greatest lower bound of {Po.} in P. Therefore the projective face F associated with p is the greatest lower bound of {Fa} in F. For each a, F", is the set of points in K where Po. takes the value 1. Also F is the set of points in K where p takes the value 1. If w E Fa, then (Pa, w) = 1 for each a, hence also (p, w) = 1, so w E F. Conversely, if w E F, then 1 = (p,w)::; (Pa,w) for each a, so wE Fa. Thus F = naFa, and we are done. It suffices to prove the last statement of the proposition for an increasing net {Pa}. Without loss of generality we assume 0 ::; a ::; 1. We have Paa ::; a for each oo. If a < (3, then P a ::S P{3 so Paa = P{3Pa a ::; P{3a. Thus {Paa} is an increasing net bounded above by a. Since Paa ::; a, applying P to both sides gives Paa ::; Pa for each oo. Set P = Va Po. and also p = PI and Po. = Po. 1 for each oo. By Lemma 8.19, Pa / p, so p - Pa "" O. For each a, let Ra = P 1\ P~. By Lemma 8.9,
nO.
nO.
Hence
Thus Paa /
Pa.
262
8
SPECTRAL THEORY
By compatibility, Paa ::: a for each oo. Thus Pa ::: a, so P compatible with a. 0
IS
also
Recall from (A 1) that a subset F of K is a semi-exposed face if there is a family {aa} in A + such that (8.16)
F = {w E K I (aCt, w) = 0 for all oo}.
By our standing hypothesis, each exposed face of K is projective. With the additional assumption A = V*, each semi-exposed face is also projective, (and therefore is exposed, cf. (7.22)), as we will now show.
8.21. Corollary. If the standmg hypothesis of th~s chapter and A = V*, then each semi-exposed face of K zs pro]ectwe.
~s sat~sfied
Proof. Let F be a semi-exposed face of K, given as in (8.16). For each index 00, we denote by Fa the set of points in K where a", takes the value zero. Then each Fa: is an exposed, hence projective, face. By (8.16), F = F",. Now it follows from Proposition 8.20 that F is a projective face. 0
n",
8.22. Definition. Assume the standing hypothesis of this chapter and A = V*. If a E A, then the set of all compressions compatible with a and with all compressions compatible with a, will be called the C-bzcommutant of a. We will also refer to the corresponding sets of projective faces and projective units as the F-bzcommutant and P-bzcommutant respectively. The use of the term "bicommutant" is of course motivated by its use in operator algebras. It is also partly justified by the fact that in our present context, compatibility of compressions is the same as commutation.
8.23. Proposition. Assume the standmg hypothes~s of this chapter and A = V*. The C-bzcommutant of an element a E A is a complete Boolean algebra. More speczfically, it ~s closed under arbztrary (fimte or mfinite) latt~ce operatwns m C and zs a Boolean algebra under these operatwns. Proof. Let B be the C-bicommutant of a. Consider two compressions PI, P 2 E B. Each of them is compatible with a and with all compressions compatible with a. Hence they are compatible with each other. Then by Lemma 8.4, PI A P 2 and PI V P 2 are also compatible with a. Moreover, if Q is a compression compatible with a, then PI A P 2 and PI V P 2 are also compatible with Q (again by Lemma 8.4, this time with the projective unit q = Q1 in place of a). Thus B is closed under finite lattice operations induced from C. It follows by an easy application of Proposition 8.20 that B is also closed under infinite lattice operations.
THE LATTICE OF COMPRESSIONS WHEN A
= V*
263
Clearly B contains 0 and 1, and P E B implies pi E B. Thus B is a complemented sublattice of C. Since each pair of compressions in B is compatible, B is a Boolean algebra (Theorem 8.16). 0
8.24. Proposition. Assume the standzng hypothes~s of th~s chapter and A = V*. For each a E A + there ~s a least pro]ectwe unzt p such that a E face(p), and p is the unzque element of P such that
(8.17)
{w E K
I
(a,w;
= O} =
{w E K
I
(p,w;
= O}.
Proof. Let G be the set of points in K where a takes the value zero. Then G is an exposed, hence projective, face. Let p be the projective unit associated with G'. Thus G' is the set of points where p takes the value 1, and G is the set of points where p takes the value O. Now (8.17) is satisfied, and this equation determines p uniquely (by the 1-1 correspondence of projective faces and projective units). Leaving aside the trivial case a = 0, we define al = lIall-1a E By (7.23), p is the greatest element of At which vanishes on G, so al ::; p. Now a::; Iiall p, so a E face(p). Let q be a projective unit such that a E face(q). Then a ::; )..q for some ).. ::::: 0, so the projective face F of points where q takes the value zero is contained in the projective face G of points where a takes the value zero. Hence G' c F', so q' ::; pi, and therefore p::; q. Thus p is the least projective unit such that a E face(q). 0
At.
8.25. Definition. Assume the standing hypothesis of this chapter and A = V*. For each a E A+ we denote by r(a) the least projective unit p such that a E face(p). Note that by (8.17), r(a) is characterized among projective units by the following equivalence where w E K:
(8.18)
(a,w;
=0
{==}
(r(a),w;
= O.
Clearly r(a) is the customary range projection in the von Neumann algebra case, d. (A 99).
8.26. Proposition. Assume the standzng hypothes~s of th~s chapter and A = V*. If a E A+, then r(a) ~s characterized by each of the followzng statements.
(i) The compression assocwted w~th r(a) ~s the least compresswn P such that Pa = a. (ii) The complement of the compresswn associated with r(a) is the greatest compression P such that Pa = O. (iii) r(a) ~s the least projective unzt p such that a::; Iialip.
8
264
SPECTRAL THEORY
(iv) r(a) zs the greatest pro]ectwe umt contazned zn the semi-exposed face of A + generated by a. Proof. (i) and (iii) follow from Lemma 7.29, and (ii) is equivalent to f--> pi reverses order and ker+ pi = im+ P. For the proof of (iv), we recall that the semi-exposed face generated by a is equal to {a}-(Proposition 7.2). Thus a projective unit p is in this face iff;t is annihilated by each w E K that annihilates a, i.e., iff
(i) since P
{w E K I (a,w)
= O}
C {w E
K I (p,w) = O}.
By (8.17) and (7.22), the left side IS the projective face associated wIth r(a)/, and the right with pi, so this is equivalent to r(a)' ::; p'. Thus p ::; r( a), so r( a) is the greatest projective unit in the semi-exposed face generated by a. 0 By statement (iii) of the proposition above, we generally have a Note in particular that
A2 > ... > 0 and limn An = O. By repeated use of Lemma 8.51, we construct a decreasing sequence {Qn} of compressions compatible with a such that for each n = 1,2, ... the compression Qn and its associated projective unit qn satisfy the inequalities (8.43) Consider the compression Q = An Qn with associated projective unit q. Let G n be the projective face associated with Qn, and G the projective face associated with Q. By (8.43), for each n we have a::; An on G n , and therefore a ::; An on G = Am G m C G n . Then a ::; limn An = 0 on G. Since a:::: 0, then a = 0 on G. By (8.42), G C F. Conversely if w E F, then by (8.43) and the compatibility of Q;, and a,
so (q~,w) = 0 for n = 1,2, .... By Lemma 8.50, qn '" q, so q;, / q'. Hence (q',W) = 0, which shows that w E G. Thus G = F, which proves that F is projective. 0 Spectral duality can be defined in various other equivalent ways (among those the ones in [7] and [9]), as we proceed to show. In the next lemma (and in the sequel) we will write "a > 0 on F" when an element a E A is strictly greater than zero everywhere on a set Fe K, i.e., if (a,w) > 0 for all wE F.
8.53. Lemma. Assume the standmg hypotheszs of this chapter and A = V*. Let a E A, let P be a compression with assocwted proJectzve face F, and assume a zs compatzble wzth F and a :::: 0 on F and a ::; 0 on F'. Then G = {w E K I (Pa, w) = O} zs a proJectzve face whose complement P = G' zs compatzble wzth a and satzsfi,es the conditzons P C F, a > 0 on P and a ::; 0 on pi . Moreover, Fa '= Pa where F is the compresszo n assocwted wIth F, Proof. Observe first that Pa = 0 on F' and pia = 0 on F by (7,25), Furthermore, Pa :::: 0 and pi a ::; 0 follow from Proposition 7.28. Clearly
SPACES IN SPECTRAL DUALITY
277
G is an exposed, hence projective, face of K. Since Pa = 0 on FI, we have FI C G, and then G I c F. If wE G, then (a,w)
If wE G I
C
= (Pa,w) + (pla,w) = (pla,w)
F, then (a, w)
O.
Here the inequality must be valid, otherwise w E G. Thus a 0 on G I • Let Q be the compression associated with G. We have Pa = 0 on G, so QPa = 0 and QI Pa = Pa (Lemma 7.43). Since G I C F, we have pI a = 0 on G I. Therefore also QI pI a = 0 and Q pI a = pI a. Hence Qa
+ Qla =
Q(Pa
+ pIa) + QI(Pa + pIa) =
pIa
+ Pa =
a.
Thus Q, and then also G and G I , are compatible with a. Now has all the properties announced in the lemma. Finally, the compression F = QI associated with F satisfies so the equality QI Pa = Pa gives Fa = F Pa = Pa. D
F=
GI
F ::S
P,
8.54. Lemma. Assume that A, V are a pair of an order unit and base norm space m separatmg order and norm dualzty, wzth A = V*. Then the followmg are equwalent:
(i) A and V are m spectral dualzty. (ii) For each a E A there exzsts a umque pTOJectwe face F compatzble wzth a such that a > 0 on F and a 0 on F and a 0 ~n F and a 0 on F. By minimality, F = F. Thus a> 0 on F. To prove uniqueness, we consider an arbitrary projective face FI compatible with a such that a > 0 on FI and a 0 since w E Fl and (a, w; ::; 0 since w E F ' , a contradiction. Thus Fl = F as desired. (ii) =} (i) We first show that exposed faces of K are projective. Fix a E A+, and let G = {w I (a,w; = O}. Let F be as in (ii); we will show
G=F'. Since a 2 0 on K and a ::; 0 on F ' , then a = 0 on F ' , so F' c G. Let w E G, and let P be the compression associated with F. Then P and a are compatible, so (8.44)
0= (a, w; = (Pa
+ pi a, w; =
(a, P*w;
+ (a, P'*w;.
Note that P*w and P'*W are multiples of elements of F and F' respectively. Since a = 0 on F ' , then (a, pl*w; = o. Thus by (8.44), 0= (a, P*w;. Since a > 0 on F, we must have P*w = 0, and so w E F'. Thus G = F ' , and G is projective. Now let a E A be arbitrary, and let F be the projective face with the properties in (ii). Let Fl be an arbitrary projective face compatible with a such that a 2 0 on Fl and a ::; 0 on F 1 . Consider once more the projective face F C Fl which satisfies the same requirements as F l , but now with the strict inequality a > 0 on F (cf. Lemm~ 8.53). Since F is unique under these requirements, we must have F = F C Fl. Thus F is the least projective face compatible with a such that a 2 0 on F and a ::; 0 on F'. By Lemma 8.44, the proof is complete. D 8.55. Theorem. Assume the standmg hypothesis of thlS chapter and A = V*. A and V aTe m spectral duallty lff each a E A has a umque orthogonal decomposltwn.
Proof. Assume A and V are in spectral duality. Let a E A and let
F be the unique projective face compatible with a such that a > 0 on F and a::; 0 on F' (Lemma 8.54). Then the compression P associated with F is compatible with a, and satisfies the inequalities Pa :;,. 0 and pi a ::; 0 (Lemma 8.44). Thus we have the orthogonal decomposition a = b - c where b = Pa and c = -pia. To prove uniqueness, we consider an arbitrary orthogonal decomposition a = b1 - Cl with b1 = PI a and Cl = - P{ a for a compression Pr compatible with a. Let Fl be the projective face associated with Pr. Since PI a = bl :;,. 0 and P{ a = -Cl ::; 0, we have a :;,. 0 on Fl and a ::; 0 on Fl (Lemma 8.44). By Lemma 8.53, there is a projective face F c F{ compatible with a such that a > 0 on F and a ::; 0 on F'. Moreover, Fa = PIa where F is the compression associated with F. But F was the unique projective face compatible with a such that a > 0 on F and a ::; 0 on F ' , so F = F and F = P. Thus
br
=
PIa = Fa = Pa = b,
Cl
= bi
-
a = b - a = c,
SPACES IN SPECTRAL DUALITY
279
so a = b - c is the unique orthogonal decomposition of a. Assume next that a is an element of A with a unique orthogonal decomposition, say b = Pa and c = -pia for a compression P compatible with a, and let F be the projective face associated with P. Again we consider the projective face F of Lemma 8.53. Thus a is compatible with F, and a > 0 on F and a ::::: 0 on Fl. Recall also that F = G I where G = {w E K I (Pa,w) = O}. We will prove that A and V are in spectral duality by showing that F is contained in an arbitrary projective face FI compatible with a such that a :;:, 0 on FI and a ::::: 0 on FI (Lemma 8.44). If PI is the compression associated with F I , then we have PIa:;:' 0 and P{a ::::: 0 (Lemma 8.44), so a = PI a - (- P{ a) is an orthogonal decomposition. By uniqueness, PIa = band -P{a = c. We have b = 0 on F{ (7.25), so F{ c G. Hence F = G I C F I , which completes the proof. 0
8.56. Definition. Assume A and V are in spectral duality. For each a E A we will write a+ = band a- = c where a = b - c is the unique orthogonal decomposition of a. 8.57. Proposition. Assume A and V are m spectral dualzty, and let a EA. Then the projective face F assoczated wzth the pro]ectzve umt r(a+) is the umque projective face F compatzble wzth a such that a > 0 on F and a ::::: 0 on F I , and zt zs also the least pro]ectzve face F compatzble with a such that a :;:, 0 on F and a ::::: 0 on Fl. The compresszon P associated with r(a+) zs the least compresszon such that Pa :;:, a and Pa :;:, O. Proof. Let H be the projective face associated with r(a+). By (8.18) (and (7.22)), HI = {w E K I (a+,w) = O}. Therefore we will show that H = F where F is the unique projective face compatible with a such that a> 0 on F and a::::: 0 on Fl. Let P be the compression associated with F. Then Pa :;:, 0, pia::::: 0 and a = Pa - (_pia) (Lemma 8.44). Thus a+ = Pa and a- = -pia (by uniqueness of orthogonal decompositions). Since a+ = Pa, the set G of Lemma 8.53 is equal to HI, so H = G I = F where F has the same properties as those that determine F. Thus H = F. With this we have proved the first characterization of r(a+) in terms of its associated projective face F. The second characterization of F follows from the first by Lemma 8.54. Then the characterization of r(a+) in terms of its associated compression P follows by the equivalence of (iii) and (i) in Lemma 8.44. 0 In a von Neumann algebra the spectral projections e).. of a self-adjoint element a are explicitly given as the complements of the range projections r((a - .>..1)+) (and similarly for a JBW-algebra, d. (A 99) and Theorem 2.20. This will now be generalized to the context of an order unit space
280
8.
SPECTRAL THEORY
A = V* in spectral duality with a base norm space V. For simplicity of notation we will write r)..(a) := r((a - A1)+).
(8.45 )
Note that by Proposition 8.57, T)..(a) is compatible with a - Al and then also with a, and the projective face F).. associated with T).. (a) is the unique projective face compatible with a such that (8.46)
a
> A on F)..,
a::;
A on
F~.
It also follows from Proposition 8.57 that the compression R).. associated with r)..(a) is the least compression compatible with a such that (8.47) Clearly r)..(a) = r(a) if a 2': 0 and A = O. In this case r)..(a) is compatible, not only with a, but also with all compressions compatible with a (Proposition 8.41). It turns out that the same is true for arbitrary a E A and A E R. This is an important technical result, which we now proceed to prove (Proposition 8.62).
8.58. Lemma. Assume the standmg hypotheszs of this chapter and A = V*. Let F, C be a pair of projectwe faces both compatzble with a E A and let A E R. If F ..1 C and a > A on F U C, then a > A on F V C.
Proof. Let p, q be the projective units and P, Q be the compressions associated with F, C. Let wE F V C. By Proposition 8.8 (and (7.22)), (p,w)
(8.48)
+ (q,w) =
(pVq,w) = 1.
Set ~ = (p,w), and note that by (8.48), (q,w) = 1 -~. Note also that (p, w) = 1 implies w E F and that (q, w) = 1 implies w E C. In these two cases there is nothing to prove, so we will assume 0 < ~ < 1. Since (1, P*w) = (p, w) = ~ and (1, Q*w) = (q, w) = 1 - ~, the two positive linear functionals p = ~-lp*w and (J = (1 - ~)-lQ*w are states. Moreover, p E F and (J E C (7.24). Since w E FvC, then (PVQ)*w = w. Hence by Proposition 8.5,
(a,w) = ((PVQ)a,w) = (Pa,w) + (Qa,w) = (a, P*w) + (a, Q*w) = ~ (a, p) + (1 -~) (a, We are done. 0
(J)
> A.
SPACES IN SPECTRAL DUALITY
281
8.59. Lemma. Assume A and V are m spectral duallty. If a and A > 0, then r.da) :s: r(a).
E
A+
Proof. We know that r(a) is compatible with all compressions compatible with a (Proposition 8.41). Therefore r(a) is compatible with the compression associated with r).,(a), or which is equivalent, the projective face F associated with r(a) is compatible with the projective face F).,(a) associated with r).,(a). Hence F' is compatible with F)." so by Lemma 8.12,
F'
=
(F'
A
F).,)
V
(F'
A
F{).
We will show that F' A F)., = 0. If w E F' A F)., = F' n F)." then (a,w) = 0 since wE F, and (a,w) > A > 0 since w E F)., (cf. (8.46)), a contradiction. Thus F' = F' A F{ c F{, so F)., c F. D
8.60. Lemma. Assume A and V are m spectral dualzty. Let a, b E A+ and let A> O. If a -.l b, then r).,(a) -.l r).,(b) and (8.49)
Proof. Since a -.l b, then also r(a) -.l r(b) (Proposition 8.39). By Lemma 8.59, r).,(a) :s: T(a) and r).,(b) :s: T(b), so we have T).,(a) -.l r).,(b). Note that by Proposition 8.8, this implies (8.50) Let F and G be the projective faces associated with r(a) and r(b) respectively. Also let F)., and G)., be the projective faces associated with T).,(a) and r).,(b) respectively. Then F)., is compatible with a and G)., is compatible with b (Proposition 8.57). Since r(a) and r(b) are orthogonal and T).,(a) :s: r(a), then T).,(a) -.l r(b). If R)., is the compression associated with r).,(a), then 0 :s: R).,b :s: IlbIIR).,r(b) = 0, so R)., is compatible with b (Lemma 7.37). Thus F)., is compatible with b, and similarly G)., is compatible with a. Hence F)., and G)., are compatible with both a and b, and therefore also with a + b. Since F)., -.l G)." then F)., V G)., is also compatible with a + b (Proposition 8.5). By (8.46) a > A on F)., and b > A on G)." so a + b > A on F)., u G).,. By Lemma 8.58, also a + b > A on F)., V G).,. We next prove (8.51 )
a:::; Ar(a) on F{
and
b:::; AT(b) on
G~.
To prove the first ofthese inequalities, let R)., be the compression associated with r).,(a), and Ro the compression associated with r(a). Since r).,(a) :::;
8.
282
SPECTRAL THEORY
r(a) (Lemma 8.59), then R>.. and Ro are compatible, and then also and Ro. Then by (8.47)
R~
Thus R~(Ar(a) ~ a) 2' 0, which gives the first inequality in (8.51); the second follows in a similar manner. Since l' (a) -.l 1'( b), then l' (a) + l' (b) ::; 1. Then
a
+ b ::;
A(r(a)
+ r(b))
::; A on
F~
n G~ = (F>..
V
G>..)'.
With this we have shown that (8.46) is satisfied with a + b in place of a and F>.. V G>.. in place of F>... Thus F>.. V G>.. is the projective face associated with 1'>.. (a + b). Hence
1'>.. (a
+ b) = 1'>.. (a)
V
r>..(b).
By (8.50) this gives (8.49). 0
8.61. Lemma. Assume A and V are m spectral dualzty. If a and Q lS a compresswn compatzble wzth a, then for each A > 0,
E
A+
r>..(Qa) = Q(r>..(a)).
(8.52)
Proof. We have Qa -.l Q'a, so it follows from Lemma 8.60 (and compatibility) that (8.53)
r>..(Qa) + 1'>.. (Q'a) = r>..(Qa + Q'a) = T>..(a).
By Lemma 8.59, T>..(Qa) ::; T(Qa) ::; q. Hence r>..(Qa) E face (q) = im+Q. Similarly r>..(Q'a) E im+Q' = ker+Q. Now applying Q to (8.53) gives (8.52). 0
8.62. Proposition. Assume A and V are m spectral dualdy. a E A, then each r>..(a) lS m the P-bzcommutant of a.
If
Proof. Let Q be a compression compatible with a. Assume first a 2' 0. If A < 0, then T>..(a) = 1, so r>..(a) is compatible with Q. If A = 0, then r>..(a) = r(a), so it follows from Proposition 8.41 that Q and T>..(a) are compatible in this case also. Assume now A > 0. By Lemmas 8.61 and 8.60, Q(r>..(a))
+ Q'(T)..(a))
=
1').. (Qa)
+ TA(Q'a)
=
Thus Q and r)..(a) are compatible in this case.
r)..(Qa
+ Q'a)
=
1').. (a).
283
SPACES IN SPECTRAL DUALITY
In the general case we define b = a + Ilalll and 11 = A + Iiali. Then bE A+ and Q is compatible with b. By the above, Q is also compatible with rll(b). Let F).. be the projective face associated with r)..(a) and let Gil be the projective face associated with rll(b). Now Gil is compatible with b, and then also with a. For W E K, (b, w) > 11 iff (a, w) > A, so it follows from the criterion (8.46) that Gil = F)... Thus rll(b) = r)..(a), so Q and r)..(a) are compatible. 0 8.63. Corollary. Assume A and V are m spectral dualzty. Let a E A and A E R, and let F).. be the pro)ectwe face assoczated with r).. (a). If F is a pro)ectwe face compatzble wzth a such that a ;::: 11 on F where 11 > A, then Fe F)...
Proof. By Proposition 8.62, F is compatible with F)... Hence (by Lemma 8.12) F = (F 1\ F)..) V (F 1\ F~). But F 1\ F~ = F n F~ = 0 since a ;::: 11 on F and a A on F~. Thus F = (F 1\ F)..) c F)... 0
:s:
We are now ready to prove our spectral theorem which generalizes the corresponding theorem for von Neumann algebras (A 99) and JBWalgebras (Theorem 2.20). As in these cases, we define the norm 111'11 of a finite increasing sequence 1= {Ao, AI, ... , An} of real numbers by
(8.54) 8.64. Theorem. Assume A and V are m spectral dualzty, and let a EA. Then there zs a umque famzly {e).. hER of projective umts wzth assoczated compresswns U).. such that
(i) e).. zs compatzble wzth a for each A E R, (ii) U)..a:S: Ae).. and U~a ;::: Ae~ for each A E R, (iii) e).. = 0 for A < -ilall, and e).. = 1 for A> (iv) e)..:S: ell for A < 11, (v) 1\11 >).. ell = e).. for each A E R.
Iiall,
The famzly {e)..} zs gwen bye).. = 1 - r((a - Al)+), each e).. zs m the P-bzcommutant of a, and the Rzemann sums n
(8.55)
S-y
=
L Ai(e)..; -
e)..i_l)
,=1
converge
m norm
to a when
iii I
---->
o.
Proof. (i)-(v) Define e).. = 1 - r((a - A1)+) and let U).. be the compression associated with e)... By Proposition 8.62 each e).. is in the Pbicommutant of a. With the notation from (8.45) and (8.47) we have e).. = r)..(a)' and U).. = R~, which gives (i) and (ii).
284
8
SPECTRAL THEORY
If A < -ilall, then a e,\ = T,\(a)' = O. Similarly,
Al > 0 on K, so T,\(a) = T(a - AI) = 1 alld A> Iiall implies e,\ = 1. This proves (iii). Let F,\ be the projective face associated with T,\ (a) for each A E R. Then assume A < IL. By Corollary 8.63, F I , C F,\. Hence T I , :::: T,\ which gives e,\ :::: ell and proves (iv). We will prove (v) by establishing the corresponding equality for the projective faces F~ associated with the projective units e,\ = T,\ (a)" i.e., Ap >,\ F~ = F~. Choose an arbitrary A E R and consider the decreasillg net {F~} p>'\. By Proposition 8.20, the projective face G
=
1\ F~ = p>'\
n
F:l
11 >'\
is compatible with a. Since F(, :J F~ for each IL > A, then G :J F~. "Ve must show that G = F~. Since G' C F,\, then a> A on G' by (8.4G). Assume now w E G. For each IL> A we have w E F~, so (a, w) :::: 1". Hence (a, w) :::: A. Thus a:::: A on G. With this we have shown that the criterion (8.4G) is satisfied with G' in place of F,\. Thus G' = F,\, so G = F~ as desired. To prove uniqueness, we consider an arbitrary family {e,\ hER of projective units with associated compressions U,\ such that (i)-(v) are satisfied. We will show that for a given A E R the projective face H,\ associated with e,\ is equal to the projective face F~ associated with T,\(a)'. We know from (8.47) that the compression R,\ associated with T,\(a) is the least compression compatible with a such that
By (ii), this condition is satisfied with U~ in place of R,\. Therefore R,\ ::S U~. Hence H,\ C F~. By (i) each Hil is compatible with a, and by (ii) Hil :J Hv when IL> 1/. Then by (v), (8.56)
H,\ =
1\ Hll = 11>'\
n
HI"
Il>'\
For each IL, we have U;la = a on H;l' cf. (7.25). By (ii), U;,a ~ lLe~, so a ~ IL on H;l' If IL > A, we can apply Corollary 8.63 with the projective face H~ in place of F. This gives H;1. C F,\ for each IL > A, so F~ c HJ1' Thus by (8.56), F~ c H,\, so H,\ = F~ as desired. To prove convergence of the Riemann sums, we choose for given E > o a finite increasing sequence {AO, AI,"" An} of real numbers such that AO < -ilall, An> Iiall and Ai - Ai-1 < E for z= 1, ... , n. By (iv), e" ::; e p when A < IL, so in this case e" -1 e~ and e p - e" = e p 1\ e~ (Lemma 8.9). If A < IL, we also have (Up 1\ U~) a = Upa - U"a
SPACES IN SPECTRAL DUALITY
285
(Lemma 8.9). By (ii), U/1a::; /-LeI' and U~a:::: Ae~, so applying U~ to the first inequality and UJ1 to the second gives
Therefore
Now define Pi = e;." ~ e;",_l for l = 1, ... , n. Pi = U;.,; 1\ U~'_l associated with Pi satisfies
Note that 2:.,Pi gives
Thus
= 1 by (iii). Since
2:.:'=1 Pia = a
n
n
i=l
i=l
Ils'Y ~ all < E
Then the compression
(Proposition 8.7), this
and the proof is complete. 0
8.65. Corollary. If A and V are m spectral duality, then each a E A can be approxlmated m norm by lmear comlnnatwns AiP, of mutually orthogonal proJectwns Pi m the P-blcommutant of a.
2:.:'=1
Proof. Clear from Theorem 8.64. 0 Note that if A and V are in spectral duality, then the first of the two requirements for the C-bicommutant (compatibility with a) is redundant in Definition 8.22. In fact, if a compression is compatible with all compressions compatible with a, then it is compatible with the projective units e;., of a, and by an easy application of the spectral theorem (Theorem 8.64) also with a itself. (Similarly for the P-bicommutant and the F-bicommutant.)
8.66. Definition. If A and V are in spectral duality and a E A, then the family {e;.,} ;"ER of Theorem 8.64 will be called the spectral resolutwn of a, and the projective units e;., in this family will be called the spectral units of a. If we define Riemann-Stieltjes integrals with respect to {e;.,} ;"ER as the norm limit of approximating Riemann sums in the usual way, then we can restate the last result in Theorem 8.64 as
(8.57)
a= JAde;.,.
8.
286
SPECTRAL THEORY
Similarly the integral with respect to {e A } AER of an arbitrary continuous function f defined on the interval [-Ilall, Iiall] is given by (8.58) where the (norm) limit exists by the same argument as in the last part of the proof of Theorem 8.64. More generally we can define the integral with respect to {eAhER of a bounded Borel function f defined on this interval as the weak mtegral in A = V', determined by the equation
(J
(8.59)
f(>-.) de A , ¢ ) =
J
f(>-.) d(e A , ¢)
for all ¢ E V+,
where d(e A , ¢) denotes the measure associated with the increasing, right continuous function >-. f---+ (e A , ¢). Using this notion of integral, we define the spectral functzonal calculus for an element a E A. This is the map which assigns to each bounded Borel function f on [-Ilall, Iiall] the element f(a) E A given by (8.60)
f(a)
=
J
f(>-.) de A ,
where {e A hER is the spectral resolution of a.
8.67. Proposition. If A and V are m spectral duahty and a E A, then the spectral functional calculus for a sat7sfies the followmg reqwrements (where f,g and all fn are bounded Borel functzons on [-Ilall, I all] and Q, f3 are real numbers):
(i)
Ilf(a)ll::; Ilflloo.
(ii) (Qf
+
(iii) f::; g
f3g)
=
=}
f (a) ::; g (a) .
Qf(a)
+
f3g( a).
(iv) If Un} 7S a bounded sequence and fn f(a) m the w*-topology of A = V'.
f---+
f pomtw7se, then fn(a)
f---+
Proof. (i) We know from (A 11) that
Ilf(a)11
=
sup IU(a),w)1 ::; sup wEK
wEK
JIf('\)1
d(eA,w).
For each state w, the function ,\ f---+ (e A, w) is increasing and right continuous with values ranging from 0 to 1, so d(eA'w) denotes a probability measure in the integral above. TherEfore Ilf(a)11 ::; Ilflloo. (ii), (iii) Trivial. (iv) Easy application of the monotone convergence theorem. 0
287
SPACES IN SPECTRAL DUALITY
Remark. The functional calculus described in Proposition 8.67 has a natural physical interpretation. If a is an observable and f is a bounded Borel function, then f(a) is the observable determined by measuring a and then applying the fUllction f to the result. Now define pos('\) = max(,\,O) and neg('\) = - min('\, 0), and let a+ = pos(a) and a- = neg(a). Then a = a+ -a-, and a+ and a- are orthogonal since a+ Iialip and aIlall(l-p), where p = X(o.oo)(a). Thus the existence of orthogonal decompositions, which will be a key assumption in our characterization of state spaces of C*-algebras (Theorem 1l.59), is well justified on physical grounds. Uniqueness of such decompositions is perhaps less evident, but we will see that uniqueness is automatic in the context in which we will be using it. By virtue of Theorem 8.55, this also provides another physical justification for the assumption of spectral duality.
s:
s:
We will now give a characterization of spectral duality which involves the concept of the F-bicommutant.
8.68. Lemma. Assume the standmg hypothesIs of thIS chapter and A = V*. For each a E A there IS a greatest compression Q m the Cbicommutant of a such that Qa 0; equwalently there IS a largest proJective face C m the F-bIcommutant of a such that as: 0 on C.
s:
Proof. Let Q be the set of compressions Q in the C-bicommutant of a such that Qa O. Recall that the C-bicommutant of a is closed under the lattice operations of C, and is a Boolean algebra (Proposition 8.23). Thus if Ql, Q2 E Q, then Q1 V Q2 is in the C-bicommutant of a, and by the distributive law for Boolean algebras, (Q1 VQ2) = (Q1 +Q~ /\Q2)' By Proposition 8.5
s:
Hence Ql V Q2 E Q. Thus Q is directed upwards, so we can organize Q to an increasing net {Q",} such that Q", / Q := Va Qa. By Proposition 8.20, Qaa / Qa, so Qa O. By Proposition 8.23, Q is in the Q-bicommutant of a. Thus Q is in Q, and in fact is the greatest element of Q. This proves the result for compressions. The result for projective faces follows by Proposition 7.28. 0
s:
A
8.69. Theorem. Assume the standmg hypotheszs of thzs chapter and Then the following are equwalent:
= V*.
(i) A and V are in spectral duality. (ii) For each a E A there is a proJectwe face C m the F-bIcommutant of a such that a ::; 0 on C and a > 0 on C/. (iii) For each a E A the largest proJectwe face C m the F-bIcommutant of a such that a ::; 0 on C satIsfies the mequality a > 0 on C'.
288
8
SPECTRAL THEORY
Proof. (i) =} (ii) If A and V are in spectral duality and a E A, then the projective face G associated with the spectral unit eo = r(a+)' has the properties in (ii) by virtue of Proposition 8.62 and equation (8.46). (ii) =} (iii) Let Go be a projective face in the F-bicommutant of a such that a:"::: 0 on Go and a > 0 on GS. If G is the largest projective face in the F-bicommutant of A such that a :"::: 0 on G (cf. Lemma 8.68), then Go c G. Hence G' c GS, so we also have a> 0 on G' . (iii) =} (ii) Trivial. (ii) =} (i) Suppose F is a projective face compatible with a such that a> 0 on F and a:"::: 0 on F'. Then FlnG' = 0 and GnF = 0. Since G is in the F -bicommutant of a, then G and F are compatible. By (8.8)
G = (G /\ F) so G
C
V
(G /\ F') = G /\ F',
V
(F' /\ G' ) = F' /\ G,
F'. We also have
F' = (F' /\ G)
so F' c G. Therefore F' = G, and so F = G' , which proves condition (ii) of Lemma 8.54. Thus A and V are in spectral duality. 0 Note that statement (iii) in Theorem 8.69 differs from all the preceding characterizations of spectral duality in that it imposes conditions on a specific face G, whereas the other ones just demand the existence of a projective face or compression with certain properties without telling how it can be found. Therefore it is easier to apply statement (iii) when proving spectral duality in special cases, as we will do in Theorem 8.72. In this theorem V is supposed to have a weakly compact base, and we shall need the following elementary result on compact convex sets. 8.70. Lemma. Let K be a compact convex set (m some locally convex space), and let A(K) be the order umt space of all contmuous affine functwns on K. If ¢ ~s a state on A(K), (~.e., ¢ E A(K)*, ¢ ~ 0 and ¢(1) = I), then there zs a pomt w E K such that ¢(a) = a(w) for all a E A(K).
Proof. Recall that a linear functional1/; on the order unit space A(K) is positive iff 1/;(1) = 111/;11, d. (A 16). Therefore the linear functional ¢ on A(K) can be extended with preservation of the properties ¢ ~ 0 and ¢(1) = 1 to a linear functional on C(K) by the Hahn-Banach theorem. This functional on C(K) can be represented by a probability measure !L on K (Riesz representation theorem), and this measure has a barycenter
SPACES IN SPECTRAL DUALITY
289
wE K (cf. [6, Prop. 1.2.1]) such that
cjJ(a)
=
J
adp.
=
a(w)
for all a E A(K). 0 A subset of a Banach space is weakly compact if it is compact in the weak topology determined by the dual space. Thus the distinguished base K of a base norm space V is weakly compact iff it is compact in the weak topology determined by the order unit space V*. By elementary linear algebra, V* is isomorphic to the order unit space Ab(K) of all bounded affine functions on K, cf. (A 11). Hence K is weakly compact iff it is compact in the weak topology determined by Ab(K), i.e., in the weakest topology for which all bounded affine functions are continuous. Thus if K is weakly compact, then A(K) = Ab(K). Recall also that a Banach space is rejiexzve if it is canonically isomorphic to its bidual. Thus a base norm space V with distinguished base K is reflexive iff it is isomorphic to the space V** ~ Ab(K)* under the map w f---> w (where w(a) = a(w) for all a E Ab(K)), or equivalently, iff the map w f---> W from V into V** is surjective. 8.71. Lemma. A base norm space V IS rejiexzve Iff Its dIstmgUIshed base K IS weakly compact. Proof. Assume K is weakly compact. By Lemma 8.70, each state cjJ on A(K) = AdK) is of the form cjJ = w for some w E K. The dual of the order unit space Ab(K) is a base norm space (A 19), so the state space of Ab (K) spans Ab (K) * = V**. Therefore the map w f---> W maps V onto V**, so V is reflexive. Conversely, if V is reflexive, then the closed unit ball V1 of V is weakly compact (Banach-Alaoglu), so the weakly closed subset K of V1 is also weakly compact. 0
We will now specialize the discussion to dual pairs (A, V) for which A and V are reflexive spaces with A = V*. Clearly this is true in all finite dimensional cases. 8.72. Theorem. If the dual paIr (A, V) satIsfies the standmg hypothesis of thIS chapter with A = V*, and A and V are rejiexzve, then A and V are m spectral dualIty. Proof. For given a E A we will show that the largest projective face G in the F-bicommutant of a such that a ::::; 0 on G satisfies the inequality
8.
290
SPECTRAL THEORY
a > 0 on C'. Assume for contradiction that (a, w) :::: 0 for some w
E
C'
and define
f3= inf(a,w)::::O.
(8.61)
wEG'
a
a
(8.62)
H
Let = a - f31. Then ~ 0 on C'. Let Q be the compression associated with C and observe that Q'a ~ 0 (Proposition 7.28). Then define
= {w
E
K
I
(Q'a,w)
= O}.
Since V is reflexive, the distinguished base K, and then also the set C' c K, is weakly compact. Hence there is an Wo E C' such that (a, wo) = f3. Therefore
(Q'a,wo)
= (a,Q'*wo) = (a,wo) = (a,wo) - f3 = o.
Thus Wo E H, so H n C' -I- 0. We will show that H n C' is in the F-bicommutant of a. Observe first that H' is the projective face associated with r(Q'a) (8.18). It follows from Proposition 8.41 that H' is in the F-bicommutant of Q'a. Therefore the compression R associated with H is in the C-bicommutant of Q'a. Clearly Q is compatible with Q'a, so R is compatible with Q and then also with Q'. Since Rand Q' are mutually compatible and both are compatible with Q'a, then R 1\ Q' is compatible with Q'a (Lemma 8.4). Since Q'a = Q'a - f3Q'l, and R 1\ Q' ::::: Q' is compatible with Q'l, then R 1\ Q' is compatible with Q' a. Since Rand Q' are compatible, R 1\ Q' = RQ', so (R 1\ Q')Qa = O. Observe now that Qa :::: 0 since a :::: 0 on C (Proposition 7.28), and note that a compression which annihilates a positive or negative element of A is compatible with this element (Lemma 7.37). Therefore R 1\ Q' is compatible with Qa. Thus R 1\ Q' is compatible with both Q' a and Qa, so is compatible with a = Qa + Q' a. Let P be a compression compatible with a. Then P and also P' are compatible with the compression Q' in the C-bicommutant of a, so
P(Q'a)
+ P'(Q'a)
=
Q'(Pa
+ P'a)
=
Q'a.
Thus P is compatible with Q' a, and then is also compatible with Q'a = Q' a - f3Q'l. Therefore P is compatible with the compression R in the C-bicommutant of Q'a. Hence
RQ'P
= RPQ' = PRQ'.
Thus the compression R 1\ Q' = RQ' is compatible with P. With this we have shown that R 1\ Q' is in the C-bicommutant of a. Now H 1\ C' is in the F-bicommutant of a, as claimed.
SPECTRAL CONVEX SETS
291
Consider the projective face G := G V (H 1\ G') and note that G properly contains G since H 1\ G' = H n G' -=I- 0. The projective faces G and H 1\ G' are both in the F-bicommutant of a, and therefore G is also in the F-bicommutant of a (Proposition 8.23). We have a::; 0 on G, so Qa ::; 0 (Proposition 7.28). Since a ::; ii, then Q'a::; Q'ii. Since Q'ii = 0 on H, then Q'a ::; 0 on H, and so we also have RQ'a;:; o. Now by Proposition 8.5 the compression Q RQ' associated with G satisfies the inequality
+
(Q
+RQ') a
=
Qa
+ RQ'a::; o.
Hence a ::; 0 on G. This contradicts the maximality of G. By Theorem 8.69, the proof is complete. D
Remark. Theorem 8.72 is not true without the condition that V be reflexive, by which K is weakly compact. A counterexample is given in [7, Prop. 6.11]. We will just sketch the idea leaving the details to the interested reader. In the example, K is a smooth and strictly convex set, and the space A = V* has elements not attaining their maximum and minimum on K. (Actually, K is affinely isomorphic to the set of sequences that are eventually zero in the closed unit ball of the Banach space l2 of square summable sequences.) The standing hypothesis of this chapter is trivially satisfied, as the only faces are the extreme points and each of them is a projective face (whose complement is the unique antipodal point). But no omch face is compatible with an element a E A that does not attain its maximum and minimum, so the only projective faces compatible with this element are K and 0. Thus by Lemma 8.44 (iii), A and V are not in spectral duality. In this example K is not norm complete. But spectral duality can not be rescued just by demanding norm completeness. In fact, one can construct a similar example where K is a norm complete (but not weakly compact) smooth and strictly convex set, as explained in [9, §2]. Spectral convex sets In our definition of spectral duality for a pair of spaces A, V we assumed A = V*. Then the pair A, V is completely determined by the distinguished base K of V, as we will show in Proposition 8.73 below. In this section we will shift the focus from the pair of ordered normed spaces A, V to the geometry of K, and we will give various examples of convex sets which in this way determine spaces in spectral duality. The proof of Proposition 8.73 involves elementary properties of order unit and base norm spaces, presented in [AS, Chpt. 1]. For the readers convenience we will briefly review what is needed.
292
8
SPECTRAL THEORY
If K is a convex set for which the bounded affine functions separate points, then the set Ab(K) of all such functions is a norm complete order unit space (whose distinguished order unit is the constant unit function), the dual space Ab(K)* is a norm complete base norm space (whose distinguished base is the "state space" consisting of all linear functionals ¢ such that ¢ ;::: 0, or equivalently II¢II = ¢(1)), and these two spaces are in separating order and norm duality. Observe that Ab(K) is also in separating duality with the subspace of Ab(K)* which consists of all alscrete linear functionals, i.e., functionals of the form L::'=l AiWi with Wl, ... ,Wn E K. (Here, and in the sequel, W denotes the evaluation functional at a point w, i.e., w(a) = a(w) for all a E Ab(K).) Recall also that an lsomorphlsm of order unit spaces, or base norm spaces, is a bijective map that preserves all relevant structure (i.e., linear structure, ordering and norm). 8.73. Proposition. Let K be the dlstmgmshed base of a base norm space V, and set A = V* (so A lS a norm complete order umt space m separatmg order and norm duality wlth V). Then the dual pmr A, V lS canomcally lsomorphlc to the pmr conslstmg of the space Ab(K) and the space of all dlscrete lmear functwnals on Ab(K).
Proof. Each element (8.63)
W
W=Ct(T~(3T
in the base norm space V can be written as where CT,TEKanda,(3ER+.
This representation is of course not unique, but by elementary linear algebra each ao E Ab(K) has a well defined extension to a linear functional a E V* given by the equation
(a,w)
=
aao(CT)
~
(3ao(T),
d. (A 11). Since V* = A and the spaces A and V are in separating order and norm duality, the extension map 1> : ao f---> a is an isomorphism of the order unit space Ab(K) onto the order unit space A. By linearity, the dual map 1>* : V --+ Ab(K)* is given by 1>*w = aCT ~ (37 when W is represented as in (8.63). Thus 1>*w is a discrete linear functional on Ab(K). Clearly each discrete linear functional ¢ = L:~=l AiWi with Wi E K and Ai E R for l = 1, ... , n can be expressed as a difference ¢ = aCT ~ (37 where CT, T E K and a, (3 E R +. Therefore 1>* is an isomorphism of the base norm space V onto the base norm space of all discrete linear functionals on Ab(K). 0 8.74. Definition. A convex set K will be called spectral if it is affinely isomorphic to the distinguished base of a complete base norm space V in spectral duality with the order unit space A = V*.
SPECTRAL CONVEX SETS
293
By Proposition 8.73, this definition is intrinsic in that the spaces A and V are determined by the convex set K. Note however, that A and V will be given in all of our concrete examples, so the identification of these spaces in Proposition 8.73 will not be needed for this purpose. The requirement that K be affinely isomorphic to the distinguished base of a base norm space is equivalent to the assumption that the bounded affine functions on K separate points of K, together with a weak kind of compactness requirement, namely, that co(K U -K) be radially compact when K sits on a hyperplane missing the origin, cf. [6]. In particular, any compact convex subset of a locally convex space will satisfy this requirement. We begin with the motivating examples already mentioned in connection with the definition of spectral duality.
8.75. Proposition. If /-L zs a measure on a O"-field
X, and K zs the set of all 1
E £1(1 O?" is a quantum mechanical proposition, i.e., a question that can be answered by a quantum mechanical measuring device. By Theorem 8.52, the spectral axiom implies the standing hypothesis of this chapter. Thus if A and V are in spectral duality, then the propositions
APPENDIX PROOF OF THEOREM 8 87
303
form a complete orthomodular lattice as in standard quantum mechanics (Theorem 8.10 and Proposition 8.20). But the standing hypothesis is not enough to give all aspects of quantum mechanical measurement theory. An important ingredient that is missing is a mathematical procedure to determine the entire probability distribution (not only the expectation) of an observable a E A. Thus what we would like to have is a method to compute the probability that the value of an observable is in a given Borel set E c R when the observable is measured on a system in a given state w E K. We will show that this can be achieved if we assume that A and V are in spectral duality. Spectral duality of A and V provides a spectral resolution {e)..} )..ER of each a E A (Theorem 8.64 and Definition 8.66). This spectral resolution provides a functional calculus for a, i.e., a map f f-+ f(a) from the space B(R) of bounded Borel functions into A satisfying the conditions (i)-(iv) of Proposition 8.67. As explained in the remark after Proposition 8.67, the element f(a) of A represents the observable operationally defined by a measurement of the given observable a followed by evaluation of the function f on the recorded result. Thus we can for each Borel set B c R compute the observable XE(a) E A, which has the value 1 when the value of a is in E and the value 0 when the value of a is in R \ E. This observable is a proposition, more specifically, it is the question that asks if the value of a is in E. The expected value (XE(a), w) is the same as the probability that the proposition XE(a) has the value 1, and therdore that the observable a has a value in E, when the system is in the state w. In this way, the functional calculus gives the desired probability distribution of the observable in every state of the quantum system. Note, however, that although all aspects of quantum mechanical measurement theory which depend on spectral theory and functional calculus can be adequately described in the context of spectral duality, there are also features which require additional assumptions. This is the case, for example, for the pure state properties (among those the symmetry of transition probabilities) which will be introduced and studied in the next chapter.
Appendix. Proof of Theorem 8.87 In this appendix S will denote a finite dimensional real linear space, i.e., S ~ R k for a natural number k. If N is an affine subspace of S, then the vertzcal subspace over N is the affine subspace N x R of S x R ~ R k+1 which consists of all pairs (w, A) with wEN and A E R. An affine subspace M of S x R is said to be vertlcal if it is the vertical subspace OVer the subspace N = AI n S of S. Note that in order to prove that an affine subspace !vI of S x R is vertical, it suffices to show that M contains a single vertical line {w} x R. (This observation is based on the simple fact that if an affine subspace contains three of the vertices of a parallelogram, then it also contains the fourth vertex.)
8.
304
SPECTRAL THEORY
8.88. Lem~a. Let K be a compact convex set m a fimte d!menswnal space S and let K be a compact convex set m S x R such that K n S = K. Assume that FI and F2 are complementary proJectwe faces of K and that the followmg condztzons are satzsfied:
(i) (w, t)
K
zmplzes wE K. (ii) FI and F2 are faces of K as well as of K. (iii) The affine tangent spaces of K at FI and F2 are the vertzcal subspaces over the affine tangent spaces of K at FI and F2 respectwely. E
Then FI and F2 are complementary proJectwe faces of K. Proof. Let NI and N2 be the affine tangent spaces of FI and F2 respectively. By Theorem 7.52, (8.75) We must show that also
K (8.76)
C
FI EBc (N2 x R),
Kc
F2 EBc (NI x R),
(NI x R) n (N2 x R)
= 0.
To prove the first inclusion in (8.76), we consider an arbitrary point E K and we will show that it can be uniquely expressed as a convex combination of a point in FI and a point in N2 x R. By condition (i), W E K, and by (8.75) the point w can be uniquely expressed as a convex combination
(w, t)
(8.77) where WI E F I , W2 E N 2 , and 0 ::; A ::; 1. Here we can assume A < 1; otherwise the point (w, 0) is in the face FI of K and there is nothing to prove. Assuming A < 1, we define t2 = (1 ~ A)~lt. Then
and by the uniqueness of the convex combination (8.77), this expression of (w, t) as a convex combination of a point in FI and a point in N2 x R is also unique. The second inclusion in (8.76) is proved in the same way, and the last equality in (8.76) follows trivially from the last equality in (8.75). 0 8.89. Lemma. Let K be a compact convex set in a finite dimenswnal R be a compact convex set in S x R such that R n S = K.
space S and let
APPENDIX PROOF OF THEOREM 8 87
305
Assume that FI and F2 are complementar'Y proJectwe faces of K and that the followmg conddwns are satzsjied:
(i) (w, t) (ii) (w, t)
K
lmphes wE K. and w E F) or w E F2 zmplzes t = O. (iii) There zs a lme L through two pomts Wl E Fl and W2 E F2 such that the plane set D = K n (L x R) has vertzcal tangents at F) and F 2 . E
E K
Then F) and F2 are complementary pT'OJectwe faces of K.
wE
Proof. Vie will first prove statement (ii) of Lemma 8.88. Fl and
Assume
K for z = 1,2 and 0 < A < l. By condition (i) above, Since W = AUl + (1 - A)U2 E F l , then Ul, U2 E Fl. By condition (ii) above, t; = 0 so that (u;, t;) E F, x {O} for z = 1,2. Thus Fl is a face of K. Similarly F2 is a face of K. We will now prove statement (iii) of Lemma 8.88. Let AI be any supporting hyperplane of K at Fl' Either the hyperplane AI contains the plane L x R or it intersects it in an affine subspace which will be a supporting affine subspace of the convex set D in L x R, so it must contain the vertical tangent {Wl} x R of D at Wl. In either case, by the observation preceding Lemma 8.88, AI will be a vertical subspace of S x R. Observe that the vertical hyperplane IvI in S x R supports K at Fl iff it is of the form IvI = H x R where H is a hyperplane in S that supports K at Fl. (If ¢ is a defining function for H, then (w, t) f-> ¢( w) is a defining function for H x R.) The intersection of all such hyperplanes H x R is equal to N) x R (since Nl is the intersection of all hyperplanes H in S that supports K at F)). Therefore the tangent space for K at Fl is equal to N) x R. By the same argument, the tangent space for K at F2 is equal to N2 x R. This completes the proof. 0 where
(11;, t i ) E
Ul, U2 E K.
Consider a direct convex sum K = B EElc C of two Euclidean balls, say that Band C are the closed unit balls of Rm and Rn respectively, and admit also the case where morn is zero so that B or C is an "improper ball" consisting of a single point. Let X = R, Y = Rm, Z = Rn and consider the Euclidean space S = X x Y x Z ~ Rm+n+l. Assume (without loss) that K is imbedded in S by the affine isomorphism which identifies the point b E B with the point (1, b, 0) E S and the point c E C with the point (0,0, c) E S. Then the point A b EEl (1 - A) c E K with b E B, C E C and A ::; 1 will be identified with the point (x, y, z) E S where x = A E X, Y = AbE Y, Z = (1 - A) c.
° ::;
8
306
SPECTRAL THEORY
Thus we can represent K as the set of all points
w = (x,y,z) E 5,
(8.78) where 5 (8.79)
= X x Y x Z with X = R, Y = Rm, Z = Rn and
0:S x :S
1,
IIYII:S x,
Ilzll:S 1 -
x.
8.90. Lemma. Let K be as above. Then all pTOper faces of KaTe proJectwe, and they are of the followzng types where 'U and v are boundar'y pmnts of Band C respectwely:
(i) B or C (balls), (ii) [u,v] = {u} EBc {v} (lzne segments), (iii) B EBc {v} or {u} EBc C (caps of cones), (iv) {u} or {v} (szngletons). Band C are complementary spht faces. If u ' zs the antzpodal poznt of u on the boundary of the ball B and Vi zs the antipodal poznt of v on the boundary of the ball C, then the lzne segments [u, v] and [U', Vi] aTe complementary pTOJectwe faces, the czrcular cone B EBc {v} and the szngleton {Vi} are complementary pTOJectwe faces, szmzlarly the czrcular cone {u} EBc C and the szngleton {u ' } are complementary pTOJectwe faces. Proof. Clear from Proposition 8.86. 0
Note that in Lemma 8.90, if B is a single point, then only (i) and the first part of (iii) need be considered, and similarly if C is a single point, then only (i) and the second part of (iii) apply. The theorem below is Theorem 8.87 restated in the terminology of this appendix. 8.91. Theorem. If K zs the dzrect convex sum of two Euclzdean balls Band C represented as a subset of the Euclzdean space 5 = X x Y x Z where X = R, Y = RTn, Z = Rn as in (8.78) and (8.79), then the set K C 5 x R conszstzng of all poznts (w, t) with w = (x, y, z) E K, t E R and
(8.80) zs a non-decomposable spectral convex set. Proof. Let f be the positive function defined on K by the equation f(w) = t where w = (x, y, z) and (8.81 )
APPENDIX. PROOF OF THEOREM 8.87
307
Now K consists of all points (w, t) E S x R such that wE K, t E Rand
It
(8.82)
I
S;
f(w).
Clearly f is a continuous function, so K i~ compact. We will show that is a concave function, which implies that K is convex. Consider the two functions
(8.83)
g(x, y) = /x 2 -
Ily112,
h(x, z) = /(1 - x)2
f
-llzI12,
where 9 is defined on the set of all points (x, y) E X x Y such that o S; x S; 1 and IIYII S; x, and where h is defined on the set of all points (x,z) E XxZ such that 0 S; x S; 1 and Ilzll S; I-x. Observe that for u 20 the equation u = g(x, y) is equivalent to u 2 + IIyl12 = x 2, which defines the boundary of a convex cone in X x Y x R (with top-point (0,0,0) and with base {(1,y,u) I u 2 + IIyl12 S; I} that is a Euclidean ball). The subgraph of 9 (i.e., the set of all points (x, y, t) such that 0 S; t S; g(x, y)) is the "upper half' of this cone (i.e., the intersection with the upper half-space consisting of all points (x, y, t) E X x Y x R such that t 2 0). Thus the subgraph of 9 is convex. Therefore 9 is concave. Similarly, the subgraph of h is the upper half of a convex cone in X x Y x R) (with top-point (1,0,0) ), from which it follows that h is concave. Note that f(x,y,z) = /g(x,y)h(y,z). From this and the concavity of 9 and h it follows that f is concave, as we will now show. Let W1 = (Xl, Y1, zd and W2 = (X2' Y2, Z2) be two distinct points of K, let 0 < ), < 1, and define Wo = (xo, Yo, zo) by
(8.84) For simplicity of notation, set fi = f(Wi), gi = g(Xi' Yi) and hi = h(yi' Zi) for 2 = 0,1,2. By the Cauchy~Schwarz inequality and the concavity of 9 and h, we have
)'h + (1 - )')12 =yf):i; J):h; + /(1 (8.85)
S; /),gl S;
+ (1 -
,;go ~ =
),)g2 /(1 - )')h2
),)g2 /)'h 1 + (1 - )')h2 fo.
Thus f is concave on K, so K is convex. We will now show that f is strictly concave in the interior of the set K c S. By the definition of K and f (i.e., by (8.79) and (8.81)), the boundary of K consists of all points W E K such that f(w) = o. Thus f(w) > 0 for w in the interior of K. Assume for contradiction that Wo is an interior point of K for which f is not strictly convex. Then Wo can be written as a convex combination
8
308
SPECTRAL THEORY
(8.84) with 0 < A < 1 and WI =f. W2 for which the equality signs hold in (8.85). Moreover, we can and will choose this convex combination such that WI and W2 are interior points of K. Then for z = 0,1,2 the variables fi, gi, hi are all non-zero. Since the last ::; relation in (8.85) holds with equality, (8.86) From the first of these equalities, it follows that the line segment between the points ((xl,yd,gl) and ((X2,Y2),g2) in (X x Y) x R lies on the boundary of the (conical) subgraph of g, hence on a ray from the point (0,0,0). By the second equality in (8.86), the line segment between the points ((Xl, zd, hd and ((X2' Z2), h2) in (X x Z) x R lies on the boundary of the (conical) subgraph of h, hence on a ray from the point (1,0,0). Thus there exist two scalar factors a, {3 such that
(8.87)
X2= OOX l, 1 - X2
Y2=OOYl,
= {3(1 - xd,
Z2
g2=oogl,
= {3zl'
h2 = {3h l .
Since the first::; relation in (8.85) holds with equality and this relation follows from Cauchy-Schwarz, the two R 2 -vectors (JAgl, J(l - A)g2) and (JAh l , J(l- A)h2) are proportional. Thus hl/g l = h2/g2, so g2/g) = h2/ hI. Therefore there exists a scalar factor " such that g2 = "gl, h2 = "hI. Combining with (8.87), we find that a = {3 = " and that
Adding these two equations, we conclude that a = {3 = " = 1 so that the two points WI = (Xl, Yl, zd and W2 = (X2' Y2, Z2) coincide, contrary to assumption. The interior of the set K consists of all points (w, t) E K x R for which the inequality (8.80) is strict, that is, the ones for which It I < f(w). Therefore the boundary of K consists of all (w, t) E K x R for which f(w) = Itl, and the boundary points with w in the interior of K are those for which It I = f(w) > O. Since f is strictly concave in the_interior of K, all such boundary points are seen to be extreme points of K. We will now determine the faces of K. By the above, they include all singletons {(w,t)} on the boundary of K for which w is an interior point of K. We will show that the only other faces of K are the subsets that are proper faces of K. Let F be a proper face of K (i.e., F =f. K). To prove F is a face of K, let (w,O) be an arbitrary point of F and assume
APPENDIX PROOF OF THEOREM 8 87
309
where 0 < A < 1 and (wi,td E K for l = 1,2. Then W = AWl +(1-A)w2' so wI and W2 are in F. Since F is a proper face, it contains no interior point of K. Hence Iti I -s: f(wd = 0 for l = 1,2. Thus Itll = It21 = 0, so that (WI, td and (W2' t 2 ) are in F. This shows F is a face of i 0, so that W would~be an interior point of K and then (w, t) would be an extreme point of K, contrary to the fact that all these points are located on an open line segment in K. ~ ~With this we have shown that the faces of K are those extreme points of K that are not in K together with the faces of K. We will show K is spectral by proving that all these faces are projective (Theorem 8.72). The non-trivial faces of K are the ones described in Lemma 8.90. We will show they are all projective in K by verifying the assumptions in Lemma 8.89. Conditions (i) and (ii) are easily verified, so we only have to construct lines L with the property demanded in condition (iii) for pairs F I , F2 of projective faces of K. First assume FI = Band F2 = C. Let L be the line through the centers WI = (1,0,0) and W2 = (0,0,0) of the two balls Band C. Thus L consists of the points w).. = (A, 0, 0) with A E R, and Lx R consists of the points (w)..,t) with (A,t) ER2. By (8.80), the convex set D=Kn(LxR) consists of all points (w).., t) for which t 4 -s: A2(1 - A)2. Thus D is the circular disk defilled by the inequality (8.88) Clearly, condition (iii) of Lemma 8.89 is satisfied in this case. Next assume that FI = [u, v] and F2 = [u' , Vi] where u and v are boundary points of Band C respectively, and where u ' and Vi are the antipodal boundary points. Now let L be the line through the mid-points WI = ~(u + v) and W2 = ~(U' + Vi) of the line segments [u, v] and [u' , Vi]. Therefore u = (I, b, 0) and u ' = (I, -b, 0) where b E Band Ilbll = l. Similarly v = (0,0, c) and Vi = (0,0, -e) where eE C and Ilell = l. Hence WI = (~, ~b, ~c) and W2 = (~, -~b, -~e). L consists of all points (8.89)
W)..
=
AWl + (1 - A)W2
and with the above values for W)..
=
WI
with A E R,
and W2,
(~, (A - ~)b, (A - ~)e).
Substituting x = ~, Y = (A - ~)b, Z = (A - ~)e into (8.80), we get the same inequality (8.88) as in the preceding case. Thus D is a circular disk
8
310
SPECTRAL THEORY
and condition (iii) of Lemma 8.89 is satisfied also in this case. Finally assume Fi = B EBc {v} and F2 = {Vi} where v is a boundary point of C and Vi is the antipodal boundary point. (The case where Fi = {u} EBc C and F2 = {u' } is similar.) Let 11 be the center of Band let L be the line through the points Wi = ~ (u + v) and W2 = Vi. Now u = (1,0,0), v = (0,0, e) and Vi = (0,0, -e) where e E C and Ilell = 1. Hence Wi = (~, 0, ~e) and W2 = (0,0, -e). The points of L are expressed in terms of Wi and W2 as in (8.89). Therefore we find that W).=(~A,O, ~A-1).
Substituting x
=
~ A, Y
= 0,
Z
=
~ A -1 into (8.80), we get the inequality
(8.90) This inequality defines an oval disk (of degree 4) with vertical tangents for A = 0,1. Thus D has vertical tangents at Wi and W2, so condition (iii) of Lemma 8.89 is satisfied also in this case. Now we consider extreme points of X. Clearly f is cOEtinuously differentiable in the interior of X, from which it follows that X has a unique tangent hyperplane at each boundary point (w, t) with W in the interior of X, and this hyperplane is equal to Tan {( w, t)} (as defined in Definition 7.1). Clearly (w, t) has a unique antipodal point (Wi, t') (also with Wi in the interior of X) defined by the property that the tangent space of K at (Wi, t') is parallel to the tangent space at (w, t). (By the results of the previous paragraph, this antipodal point is not in X.) Now it follows by an elementary argument in plane geometry that each point in X can be expressed in a unique way as a convex combination of the point (w, t) and a point in Tan{(w ' , t')} (cf. the proof of Proposition 8.84). Thus the conditions (i) and (ii) of Theorem 7.52 are satisfied by the pair o~ faces F = {(w, t)} and G = {(Wi, t')}. Th~refore all extreme points of X that are not in X are projective faces of X. The proof is complete. D It can be shown that the set D = K n (L x R) defined by the line L between arbitrary interior points Wi and W2 of Band C is an elliptical disk (which degenerates to a line segment if Wi or W2 are boundary points.) Thus K is obtained by "blowing up" the line segments between points in Band C to elliptical disks in an extra dimension. If m = n = 0, then K is a circular disk, so it is the state space of the spin factor Nh(R)sa. In all other cases K is different from the state space of a lB-algebra. (The inequality (8.90) violates the elliptical cross-section property of Proposition 5.75.) The convex sets K in Theorem 8.91 are mutually non-isomorphic except for the trivial isomorphism between sets with transposed pairs of parameters m, nand n, m (under the map x ........ 1 - x, Y ........ z, z ........ Y).
NOTES
311
Notes The results in this chapter on the lattice of compressions, and on spectral duality, were established either in [7] or [9], with a few exceptions noted below. Loosely speaking, spectral duality requires that there be "enough" projective faces or compressions. The concept of spectral duality in [7] was defined in terms of projective faces by statement (ii) of our present Theorem 8.69, which involves projective faces in the bicommutant. The alternate characterization of spectral duality in statement (ii) of Lemma 8.54 was given in [7, Thm. 7.5], and was used to prove the existence of unique orthogonal decompositions (Theorem 8.55) in [9, Thm. 2.2]. Our present definition of spectral duality in terms of compressions (Definition 8.42) does not involve the bicommutant, but we have to work harder to prove the key result that norm exposed faces are projective (Theorem 8.52). An important first step in the argument that leads up to this theorem is Lemma 8.46, which was originally proved by Riedel [106] in a slightly different context (with his "fundamental units" in place of our projective units). Note that in [7] there is also a concept of "weak spectral duality", which is shown to imply existence, but not, a prwn, uniqueness, of spectral resolutions. It is still an open question if weak spectral duality implies spectral duality (or, which can be seen to be equivalent, if the minimality requirement is redundant in Definition 8.42). The proof given here that Choquet simplexes are spectral (Proposition 8.80) reduces to the fact that normal state spaces of a JBW-algebra or a von Neumann algebra are spectral. A more direct lattice-theoretic proof is given in [7]. The "exotic" examples in Theorem 8.87 are presented here for the first time. What was known about the relationship of split faces to Choquet simplexes evolved over several years. It was shown by Alfsen [5] that every w*-closed face of a Choquet simplex is a split face (where for a regularly imbedded compact convex set we refer to the given compact topology as the w*-topology). Then Asimow and Ellis [19] showed that every norm closed face of a Choquet simplex is split. The fact that a compact convex set is a Choquet simplex iff every norm closed face is split (Proposition 8.83) is proved in [9], and the result that a compact convex set is a Choquet simplex iff every w*-closed face is split was shown by Ellis [51]. The fact that the traces of a C*-algebra form a Choquet simplex is due to Thoma [127]. The converse result by which every metrizable Choquet simplex is the trace simplex of a C*-algebra, was found by Blackadar [30] and Goodearl [57], who independently proved that for every metrizable Choquet simplex K there is a simple, unital C*-algebra whose trace simplex is affinely homeomorphic to K. This result can also be obtained from the characterization of dimension groups among all ordered abelian groups, which was found slightly later by Effros, Handelman and Shen [47]. Note that the trace simplex has recently found application as an impor-
312
8
SPECTRAL THEORY
tant invariant for C*-algebras, and that a series of interesting new results on the interplay between the theory of infinite dimensional simplexes and C*-algebras have been found in connection with the Elliott program for a K-theoretic classification of various kinds of C*-algebras. (See [107]' and in particular the references to R0rdam, Thomsen and Villadsen in that book, and the book [92].) Many authors have studied spectral theory from an order-theoretic point of view, but we will just mention a few papers that relate fairly closely to the approach in this book. Some of those are the papers by Abbati and Mania [1], and by Bonnet [31]' which are based on axiom systems similar to (but not identical with) that in the paper by Riedel [106] mentioned above. Another relevant reference is the paper by C. M. Edwards [41], which extends spectral duality from order unit spaces to the wider class of G M -spaces (not necessarily with unit). There is also a rich literature on quantum mechanics modeled on ordered vector spaces, and again we will only give a few references that are directly related to the material in this book. The "operational approach" to axiomatic quantum mechanics (consistent with the point of view discussed in the remark after Proposition 7.49) goes back at least as far as Davies and Lewis [39], who proposed analyzing the space of observables and the convex set of states, expressed as an order unit and a base norm space in duality. In this context, it is natural to axiomatize filters, and numerous authors have taken that approach. We will just point out the papers of Mielnik [97]' Gunson [59]' Araki [17], and Guz [60]. The attempt to axiomatize quantum mechanics by assuming a linear space of observables equipped with a suitable functional calculus was the basis of Segal's paper [114], and is also present in von Neumann's paper [98]. In both cases one is led to define a candidate for a Jordan product given by squares: a 0 b = -j-((a + b)2 - (a - b)2). However, distributivity of such a product is not automatic, so additional axioms must be assumed to guarantee it. This is exactly what is needed to make the transition from spectral theory to Jordan algebras, and is the topic of the next chapter.
PART III State Space Characterizations
9
Characterization of Jordan Algebra State Spaces
In this chapter we will give a characterization of the state spaces of JB-algebras and of normal state spaces of JBW-algebras. As a preliminary step, we will characterize the normal state spaces of JBW-factors of type I. (This includes as a special case a characterization of the normal state space of B(H).) Our standing hypothesis for this chapter will be that A, V is a pair of an order unit space and a complete base norm space in separating order and norm duality with A = V*, with the additional assumption that each exposed face of the distinguished base K of V is projective. (This is the same as the standing hypothesis of the last chapter, except for the additional requirements that V be complete and that A = V*.) By Proposition 8.76, the normal state space of a JBW-algebra A is spectral, i.e., A is in spectral duality with its predual V = A*. Thus by Theorem 8.52, such a pair A, V satisfies the standing hypothesis of this chapter. If instead B is a JB-algebra with state space K, then K can be identified with the normal state space of the JBW-algebra B** (Corollary 2.61), so again K is spectral and thus satisfies the standing hypothesis for the duality of V = B* and A = B**. We will add to the standing hypothesis two equivalent sets of axioms. One set (the "pure state properties") is of particular physical interest, while the other set is geometric in nature. We will use these axioms to characterize the normal state spaces of JBW-factors of type I, and then use this result to achieve the characterization of state spaces of JB-algebras. The axioms just discussed involve the extreme points of the state space, and so are not relevant in the more general situation of normal state spaces of arbitrary JBW-algebras, where there may be no extreme points. The ellipticity property (cf. Proposition 5.75 and the remarks following) will be used to characterize these normal state spaces, and we will also give a characterization in which ellipticity is replaced by a more physically relevant assumption. The pure state properties 9.1. Definition. If V is a base norm space with distinguished base ----+ V a positive map, we say T preserves extreme rays if T maps each extreme point of K to a multiple of an extreme point.
K and T : V
316
9
CHARACTERIZATION OF JORDAN STATE SPACES
Recall that under the standing hypothesis of this chapter, the sets C,
P, :F of compressions, projective units, and projective faces form isomorphic lattices. Each is a complete orthomodular lattice (Theorem 8.10 and Proposition 8.20). We now briefly review some lattice terminology. The least element of a complete lattice is called zero. A projective unit that is minimal among the non-zero projective units is called an atom. The same term is used for any element of a complete lattice that is minimal among the non-zero elements of the lattice, cf. (A 41). A complete lattice is atomIC if each non-zero element is the least upper bound of atoms. In this section, we will sayan element in a complete lattice is fimte if it is the least upper bound of a finite set of atoms. Note that this lattice-theoretic concept may differ from the notion of finiteness for projections in von Neumann algebras (A 166), as defined in terms of Murray-von Neumann equivalence, if the projection lattice is not atomic. Recall that under the standing hypothesis, for each atom u in A there is a unique extreme point u of K such that U(u) = 1 (Proposition 8.36).
9.2. Definition. Assume the standing hypothesis of this chapter. Then K has the pure state properiles if (i) every extreme point of K is norm exposed, (ii) for every compression P, the map P* preserves extreme rays, and (iii) (11, v) = (v, u) for all atoms u, v in A. 9.3. Proposition. The normal state space of a lEW-algebra and the state space of a lE-algebra have the pure state propertzes.
Proof. Since the state space of a lB-algebra can be identified with the normal state space of its bid ual (Corollary 2.61), it suffices to establish the pure state properties for normal state spaces of lBW-algebras. By Theorem 5.32 every norm closed face is projective, so (i) holds. By Corollary 5.49, each compression on a lBW-algebra preserves extreme rays of the normal state space. Finally, (iii) follows from Corollary 5.57. 0
Remark. If we interpret K as the set of states of a physical system, we can view each rJ E V+ as representing a beam of particles with intensity IlrJll. We further interpret compressions P as representing physical filters. (See the remark after Proposition 7.49.) If each norm exposed face is projective, then (i) says that for each pure state there is a filter that prepares that state. The property (ii) says that filters transform pure states to pure states. Finally, (iii) is a statement of "symmetry of transition probabilities". See the remarks after Corollary 5.57, and [AS, remarks following Lemma 4.10 and Cor. 6.43], for further discussion of this interpretation.
THE PURE STATE PROPERTIES
317
In a general finite dimensional compact convex set, there may be faces that are not exposed. However, we have the following result, due to Minkowksi, cf. [123, Thm. 3.4.11]. 9.4. Theorem. If F zs a proper face of a non-empty convex set K m R n, then there zs a proper exposed face G of K contaznzng F. Proof. By working in the affine span of K, we may assume without loss of generality that the affine span of K is Rn. The interior of K is non-empty (e.g., it contains the arithmetic mean of any finite number of points in K whose affine span equals the affine span of K), and is convex and dense in K. Since F is a proper face, then F does not meet the interior of K. By Hahn-Banach separation, there is an affine function a with a > 0 on the interior of K and a :s; 0 on F. Since the interior of K is dense in K, then a = 0 on F. Then G = a- 1 (0) n K is the desired proper exposed face. 0
In the current context, we can say more. The following result shows that the first pure state property is automatic in finite dimensions. (This result will not be needed in the sequel.) 9.5. Proposition. Let K be a jinzte dzmenszonal compact convex set m whzch every exposed face zs pro]ectzve. Then every face F of K is exposed, and therefore pro]ectzve. Proof. Let G be the intersection of the exposed faces of K that contain
F. By finite dimensionality, G is itself an exposed face of K. (An intersection of any family of hyperplanes will equal the intersection of some finite subfamily, and if a1, ... ,am are positive affine functions on K such that i a;I(O) = F, then F is exposed by al + ... +a m .) Suppose F i- G. By Minkowksi's theorem (Theorem 9.4) applied to F c G, there is an affine function a on G, not the zero function, such that a = 0 on F, and a :::- 0 on G. Let P be the compression associated with G, so that G = im P* nK. Note that G is a base for the cone im+ P*, and so we can extend a to a linear function on lin G = im+ P*. Let b = a 0 P*. Then b = 0 on F, and b :::- 0 on K. Thus H = {O" E K I (b,O") = D} is an exposed face of K containing F and properly contained in G, contrary to the definition of G. We conclude that F = G. 0
n
The second of the pure state properties is closely related to the covering property for lattices, cf. (A 43), which we now review for the reader's convenience. 9.6. Definition. Let p and q be elements in a complete lattice L. We say q covers P if p < q and there is no element strictly between p
318
9
CHARACTERIZATION OF JORDAN STATE SPACES
and q. We say L has the covermg property if for all pEL and all atoms 'U
E
L,
(9.1)
p Vu
= p or p Vu covers p.
We say L has the fimte covermg property if (9.1) holds for all finite p in L and all atoms u E L. Recall that by Lemma 8.9, (p V u) 1\ pi = (p Vu) - p for each p, u E P. Thus P has the covering property iff for each atom u E P and each pE P,
(9.2)
(p V u)
1\
pi is an atom or zero.
We will use the following result to relate the covering property to the second of the pure state properties.
9.7. Proposition. Assume the standmg hypotheszs of thzs chapter, and let P be a compresswn wzth assocwted proJectwe umt p. Assume also that each extreme pomt of K zs norm exposed. The followmg are equwalent:
(i) P* preserves extreme rays. (ii) For each atom u E P, (u V pi) 1\ P zs ezther an atom or zero. (iii) P maps atoms to multzples of atoms. Proof. Let P be the projective face associated with p. Let u be an atom, and ()' the associated extreme point of K. By Theorem 8.32, r(Pu)
(9.3)
= (pi
V u) 1\ p,
and by Corollary 8.33, (9.4)
ProjFace(P*(),)
= (Pi
V {()'}) 1\ P,
where ProjFace(P*(),) denotes the least projective face containing IIP*()'11- 1 P*()', or the empty set if P*()' = O. (i) =? (ii) By (i), P*(), is a multiple of an extreme point, and thus of an exposed point of K. Therefore ProjFace(P*(),) is an atom in the lattice :F, or else the empty set. Now (ii) follows from (9.4) and the isomorphism of the lattices P and :F. (ii) =? (iii) By (ii) and (9.3), r(Pu) is zero or an atom. If Pu = 0, then Pu is certainly a multiple of an atom. Otherwise q = r(Pu) is an atom. Let Q be the associated compression. By Proposition 8.36, im Q = Rq. By Lemma 7.29, Pu E face(q) = im+Q c Rq, which proves (iii). (iii) =? (i) By (iii), Pu is a multiple of an atom, so r(Pu) is an atom or zero. Thus by (9.3), (pi V u) I\p is an atom or zero. By the isomorphism
THE HILBERT BALL PROPERTY
319
of 'P and F, (F'v {(j}) /\ F is an atom in F or equals the empty set. By (9.4) the same is true of ProjFace(p*(j). Since minimal projective faces are sets consisting of a single extreme point (Proposition 8.36), then p*(j is a multiple of an extreme point. D 9.8. Corollary. Assume the standmg hypothesIs of tlm chapter', and assume that eveTY extTeme pomt of K lS nOTm exposed. Then the second of the pUTe state pmpertles lS equwalent to the coveTmg property JOT the lattlces C, 'P, F. Pmoj. Let 71 be an atom in 'P and let p E 'P. By (9.2), the covering property for 'P is equivalent to (ii) in Proposition 9.7 holding for all p E 'P and all atoms 71. The corollary now follows from the equivalence of (i) and (ii) in Proposition 9.7. D
The Hilbert ball property Recall that a HdbeTt ball is a convex set affinely isomorphic to the closed unit ball of a finite or infinite dimensional real Hilbert space. 9.9. Definition. A convex set K has the HllbeTt ball propeTty if the face generated by each pair of extreme points of K is a norm exposed Hilbert ball. Note that in Definition 9.9 we allow the pair of extreme points to coincide, so the Hilbert ball property implies that each extreme point of K is norm exposed. Note also that by the standing hypothesis of this chapter, the Hilbert ball property implies that the face generated by each pair of extreme points of K is projective.
9.10. Proposition. State spaces of lE-algebras and C*-algebras, and normal state spaces of lEW and von Neumann algebras, have the Hllbert ball propeTty. Prooj. By Proposition 5.55 and Corollary 5.56, normal state spaces of JBW-algebras and state spaces of JB-algebras have the Hilbert ball property. This implies as a special case the stated results for C*-algebra state spaces and von Neumann algebra normal state spaces, but the fact that the face generated by two distinct pure states of a C*-algebra is a Hilbert ball (of dimension 1 or 3) is also stated directly in (A 143). Since norm closed faces of the normal state space of a von Neumann algebra are norm exposed (A 108), and the state space of a C*-algebra can be identified with the normal state space of the enveloping von Neumann algebra (A 101), it follows that the face generated by a pair of pure states in a C*-algebra state space is norm exposed. This completes the proof that C*-algebra state spaces have the Hilbert ball property. 0
320
9.
CHARACTERIZATION OF JORDAN STATE SPACES
We will now derive some consequences of the Hilbert ball property. As remarked above, the first of the pure state properties follows from the definition of the Hilbert ball property. Recall that the second pure state property is equivalent to the covering property (Corollary 9.8). We now show that the Hilbert ball property implies the finite covering property.
9.11. Proposition. Assume the standmg hypotheszs of thzs chapter. If K has the Hzlber-t ball pmper-ty, then the latt?ces C, P, F have the fimte covenng pmper-ty.
Pmof. Assume K has the Hilbert ball property. We first establish the implication for atoms 1L, v and projective units q, q
q implies
(p V u) - p < (p V u) - q. By (9.5), the only elements under an element of dimension 2 are atoms or zero, so this finishes the proof that (p V u) - p is an atom or zero. Thus we have shown that the finite covering property follows from the Hilbert ball property. D Remark. In the proof above, the only property of the Hilbert ball that is used is that it has no proper faces other than singletons. Thus the finite covering property would be implied by the weaker assumption that the face generated by each pair of extreme points is norm exposed and strictly convex.
If an additional geometric assumption is added to the Hilbert ball property, we will see in Proposition 9.13 below that the second pure state property follows. Recall from Definition 5.71 that a convex set K is symmetllc wzth respect to a convex subset Ko if there exists a refiectwn of K whose set of fixed points is precisely Ko. 9.12. Lemma. Assume the standmg hypotheszs of thzs chapter. If F is a pro)ectwe face of K and P zs the correspondmg compression on A, then the followmg are eqwvalent:
(i) K zs symmetllc wzth respect to co(F U F ' ), (ii) 2P + 2P' - I :::: o. If these eqwvalent condztwns hold, then (2P + 2P' - 1)* zs the umque refiectwn of K wzth co(F U F') as zts set of fixed pomts. Pmoj. Let T = 2P+2P' -I. If (ii) holds, then T is a positive map such that T2 = I, and T1 = 1. It follows that T* is an affine automorphism of K of period 2. For IJ E K we have T*IJ
= IJ
¢===}
(P
+ PI)*IJ = IJ
¢===}
IJ E co(F U F').
(The last equivalence follows from Theorem 7.46). Thus T* is a reflection of K with fixed point set co(F U F ' ), so by definition (i) holds. Conversely, suppose (i) holds, and let R be a reflection of K with fixed point set co(F U F'). Then !(R + I) is an affine projection of K onto co(F U F'). By Theorem 7.46, there is just one affine projection of K onto co(FUF'), namely (p+p/)*, so we must have !(R+1) = (p+pl)*. Thus R = (2P+2P' -I)*, and (ii) follows. This also establishes uniqueness of the reflection that has co(F U F') as its set of fixed points. 0
322
9
CHARACTERIZATION OF JORDAN STATE SPACES
Remark. By Proposition 5.72, the normal state space K of a JBWalgebra (and the state space of a JB-algebra) is symmetric with respect to co( F u F') for every norm closed face F. The same holds for von Neumann algebra normal state spaces, since the self-adjoint part of a von Neumann algebra is also a JBW-algebra. In the von Neumann context, let p = PI be the projection associated with the face F, let s be the symmetry 2p - 1, and let Us be the map a f----o> sas. Then Us = 2Up + 2Up ' - I, so from Lemma 9.12 it follows that U; is the reflection whose fixed point set is co(F U F'). The following result gives one circumstance in which the Hilbert ball property implies the second of the pure state properties, but will not be needed in the sequel.
9.13. Proposition. Assume the standmg hypotheszs of thzs chapter. If K has the Hzlbert ball property and zs symmetrzc wzth respect to co(F U F') for every proJectwe face F, then P* preserves extreme rays for every compresswn P. Proof. Let 0' be an extreme point of K, and let P be a compression. We will show that P*O' is a multiple of an extreme point of K. If P*O' = 0' or P*O' = 0 this is trivial, and so we assume neither occurs. By bicomplementarity of P and P', it follows that neither P'*O' = 0' nor P'*O' = 0 occurs. Let A = IIP*O'II. Note that
IIP'*O'II
= (1, P'*O') = (1 - PI, 0') = 1 - IIP*O'II = 1 - A.
By neutrality of P* (Proposition 7.21) and our assumptions, 0 Define 0'1
= A-I P*O' and
0'2
< A
2a - 1 is an affine homeomorphism of [0,1] onto [-1,1]' then {2p - 1 I pEP} is weakly dense in oe[-I,I]. As in the proof of Lemma 9.20, for each pEP we can choose a net Po: of finite projective units converging weakly to p and a net qo: of finite projective units cOllverging to 1 - p. Then 1 ::;> (Po: - qo:, 0') for each a, and (Po: - q"" 0') ----> (2p - 1,0'). Hence (2p - 1, 0') ~ 1. By weak compactness of [-1,1] and the Krein-Milman theorem, we conclude that 110'11 ~ 1. 0
9.23. Lemma. Assume the workzng hypotheses of th~s sectzon. Then there ex~sts a umque posdive lznear map ¢ : Af ----> V such that ¢( u) = U fOT all atoms u. The equatzon (alb) = (a, ¢(b)) defines a symmetnc bzlznear form on Af such that (9.11 )
(ulv)
=
(u, v)
for all pairs of atoms u, v. Furthermore, each fimte or cofinite compresszon P maps Af znto itself and sat~sfies the symmetry conditzon
(9.12)
(Palb)
=
(aIPb)
for all a, bE A f . Proof. First let
Ul,""
Un
be atoms and
)'1, ... ,
An
scalars such that
~~=l Aiui ::;> 0. By (v) of the working hypotheses,
for each atom v. Hence ~~=l AJii 2: 0 by Lemma 9.20. In particular, if L~=l AiUi = 0, then L~=l AiUi = O. Thus there is a well defined positive
THE TYPE I FACTOR CASE
329
linear map ¢ : Ar --) V given by (9.13)
Bilinearity of the form (alb) = (a,¢(b» is trivial. So is (9.11), and symmetry of the bilinear form follows from (9.13) and (v) of the working hypotheses. Now assume that P is a finite or cofinite compression. By Lemma 9.19, P maps atoms to multiples of atoms, hence P maps Ar into A r . Let u and v be atoms. In order to prove that P is symmetric with respect to the bilinear form on A r , we start by showing that (9.14)
((1-P)uIPv) =0.
Write Pv = AW, where W is an atom. We may assume A # 0, since otherwise (9.14) is clear. Since wE imP, then by (9.10), iii E imP*. This together with the definition of the bilinear form on Ar gives ((I - P)u I Pv) = A((1- P)u I w) = A((I - P)u, iii) = A(U, (1- P)*iii) = 0,
which proves (9.14). From (9.14) we conclude that (u I Pv) = (Pu I Pv) for all atoms u, v. Exchanging the roles of u and v, and using (Pu I Pv) = (Pv I Pu), we get (Pulv)
= (PuIPu) = (uIPv)
for all atoms u, v. Now (9.12) follows by linearity, since by definition Ar is the linear span of atoms. D 9.24. Proposition. Assume the workmg hypotheses of thls sectwn. Let P be a compresswn, and let ¢ be the map defined m Lemma 9.23. If P is fimte or cofimte , then (9.15)
¢(Pa)
= P*(¢(a») for all a
EAr.
If P lS fimte and Q zs any compr-esswn such that Q ~ P, then
(9.16)
(Q
+ Q')*¢(P1)
=
¢(P1).
Proof. Assume that P is finite or cofinite. By (9.12) and the definition of the bilinear form (-I,), for all a, bEAr, (b, ¢(Pa) = (b I Pal = (Pb I a) = (Pb, ¢(a) = (b, P*¢(a).
330
9.
CHARACTERIZATION OF JORDAN STATE SPACES
Since Af and V are in separating duality (Lemma 9.22), then ¢(Pa) P*¢(a) follows. Assume now that P is finite and Q is any compression such that Q ~ P. By Lemma 7.42, P and Q are compatible. Since P is finite and Q ~ P, then Q is finite (Proposition 9.18). Applying (9.15) and compatibility of Q with PI, we get
(Q
+ Q')*¢(P1) = ¢((Q + Q')(P1)) = ¢(P1),
which proves (9.16). 0 Note that (9.16) says that ¢(P1) is a tracial state of imP for the duality of im P and im P*, cf. Definition 8.81. Recall that (J, T E K are defined to be orthogonal if II(J - Til = 2 (cf. (8.20)).
9.25. Lemma. Assume the workzng hypotheses of tiLlS sectwn. Let u, v be atoms. Then u, v are orthogonal lff U, v are orthogonal.
PTOOj. If u ..1 v, then (v, u) :s: (1 - u, u) = 0, so (v, u) = 0, and similarly (u, v) = 0. Then (u - v, U - v) = 2. Since Ilu - vii :s: 1, then Ilu - vii = 2, so U, are orthogonal. Conversely, suppose that ..1 Define w = then this must coincide with the unique orthogonal decomposition of w (cf. Theorem 8.27). Thus by the same result, there exists a compression P such that P*U = and p'*v = v. Then by (9.10), Pu = u and P'v = v, so u and v are orthogonal. 0
v
u v.
u - v;
u
9.26. Lemma. Assume the workzng hypotheses of thls sectwn. Let P be a fimte compression. Then every (J E im P* can be WTltten as a fimte lznear combmatwn (J = 2.:~=1 Ai(Ji of orthogonal poznts (JI, ... ,Un E (OeK) n imP*.
PTOOj. By working in the duality of im P and im P* (cf. Proposition 8.17), we may assume that PI is equal to the order unit 1, (i.e., that P is the identity map), so that 1 is finite. We will use induction on the lattice-theoretic dimension of P. If dim(P) = 1, then P is a minimal non-zero compression, so by Proposition 8.36 the corresponding projective face consists of a single extreme point, and thus the proposition holds in this case. Now assume that the result is true for dim(P) < n. Let dim(P) = 71, and let (J E im P* be arbitrary. By assumption 1 is finite, and thus is a finite sum of orthogonal atoms (Proposition 9.18). Define T = ¢(1). By Lemma 9.25, T is a finite sum of orthogonal extreme points of K. By Proposition 9.24, for all compressions Q we also have (9.17)
(Q
+ Q')*T
= T.
THE TYPE I FACTOR CASE
331
If a IS a multiple of T, then we are done. If not, then there is a scalar A E R such that neither a :::: AT nor a ;::: AT. Thus for that scalar A, if we define w = a - AT, then w is neither positive nor negative. By Theorem 8.27, a can be written as a difference of positive elements w = w+ - w- in such a way that there is a compression Q such that w+ = Q*w, w- = -Q'*w. Then (Q + Q')*w = w. Thus by equation (9.17) (Q
+ Q')*a =
(Q
+ Q')*(w + AT) =
(w
+ AT) = a.
Now let al = Q*a and a2 = Q'*a; note that a = al +a2. Since w L- 0 and Q*w = w+ ;::: 0, then we cannot have Q = P (since P is the identity map), and similarly Q' I:- P. By Proposition 9.18, the lattice dimensions of Q and of Q' are strictly smaller than that of P. By our induction hypothesis, al is a finite linear combination of orthogonal extreme points of K in im Q*, which we can write in the form Ul, ... ,Uk for atoms U1, ... , Uk in imQ (cf. (9.10)). Similarly we can write a2 as a finite linear combination of orthogonal elements ih, ... , vm in im Q'* n oeK, where VI, ... , Vm are atoms in im Q'. Then each Ui is dominated by Q1 and each Vj is dominated by Q'l, so Ui 1.. Vj for 1 :::: z :::: k and 1 :::: J :::: m. By Lemma 9.25, the corresponding extreme points of K are orthogonal. Substituting these decompositions into a = al + a2 gives the desired decomposition of a. 0
9.27. Corollary. Assume the workmg hypotheses of thls sectwn. Then every a E Ar can be wrztten as a finde lmear combmatwn a = 2::7=1 Aiui of orthogonal atoms Ul,··· ,Un·
a = ¢(a)
Proof. We first show that the map ¢ : Ar -> V is one-to-one. Let 2::7=1 Aiui be any finite linear combination of atoms 'Ul, ... , Un. If = 0, then for all atoms v in A,
Let P be the finite compression corresponding to Ul V ... V Un. Then a annihilates every E oeK in im P*, so by Lemma 9.26, a annihilates all of im P*. Since a E im P, the separating duality of im P and im P* implies that a = o. Finally, applying Lemma 9.26 to ¢(a) shows that we can write ¢(a) = 2:::: 1 f3Wi for orthogonal points al, ... , am E oeK. If we choose atoms vI, ... , Vm such that Vi = ai for each l, then ¢(a) = ¢(L::: 1 f3iVi) and so a = L:::l f3iVi is our desired representation. 0
v
332
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CHARACTERIZATION OF JORDAN STATE SPACES
9.28. Definition. Assume the working hypotheses of this section. For each atom u = Pe (with P a compression), and each bE A we define (9.18) Note that by (1.47), u
*b=
u
0
b when A is a JBW-algebra.
9.29. Lemma. Assume the workzng hypotheses of thl.s secb'Jn. Let u, v be atoms zn A. Then'
(i) u * v = v * u. (ii) If u -.l v and bE A, then u (iii) If u -.l v, then u (iv) U H l = U.
* v = O.
* (v * b) = v * (u * b).
Proof. (i) Let P, Q be the compressions such that PI = u and Q1 = u V v. On im Q the complement of P is the restriction of P' (Proposition 8.17), and thus u * v is the same whether calculated in A or in im Q. Hence without loss of generality we may assume that uVv = 1. If u = v then (i) is obvious, so we may also assume that u # v. Then dim(l) = dim(uVv) = 2. Now dim(l - u) = 1, so 1 - u is also an atom. Since u is an atom, then the image of P is one dimensional (Proposi.. tion 8.36), and thus im P consists of all multiples of u and im P* consists of all multiples of (cf. (9.10)). Therefore for each atom w,
u
PW=AU for some scalar A. Applying u to both sides and using P*u (w, u) = A, and so (9.19) Since u'
Pw
= (w,u)u = (ulw)u.
= 1 - u is also an atom, we have P'w
= (w, ;;')u' = (u'lw)u'.
By (9.18) we can compute u
u
*v =
*v
as follows.
+ (ulv)u - (u'lv)u') = ~ (v + (ulv)u - (u'lv)(l - u)) = ~ (v + (ulv)u + (u'lv)u - (u'lv)l) = ~ (v + (llv)u - (u'lv)l). ~(v
= u gives
THE TYPE I FACTOR CASE
Since (1Iv)
333
= (1, v) = 1, then u *v
= ~(v
+ u-
(1 - (ulv))I).
Since this last expression is symmetric in u and v, (i) follows. (ii), (iii) Now assume u ~ v, and let P, Q be the compressions corresponding to u, v. Then P, Q, pi, Q' commute (Lemma 7.42), so (ii) follows. Also, v :"::: u' implies that Pv = 0, and then P'v = v, so (iii) follows from the definition (9.18). (iv) follows at once from (9.18). 0 In the proof of several results below we shall also need the following elementary result on order unit spaces: If Ao is a linear subspace of an order unit space A and 0 is commutative bilinear product on Ao with values in A such that
(9.20)
-1:":::a:":::l
=}
O:":::aoa:":::1
for a E A o , then
Iia 0 bll :": : Ilallllbll
(9.21)
for all pairs a, bE Ao (A 50). This can be readily seen from (9.20) and the identity
9.30. Proposition. Assume the workmg hypotheses of thls sectwn. Let Ao be the norm closure of Af + Rl. Then there lS a umque product a 0 b on Ao such that Ao lS alB-algebra wlth ldentlty 1 and such that
uov=u*v. Proof. We first define the product on Af by
(9.22)
a 0
b=
L
cxi{3jUi
*vJ
i,j
where a = 2.:i CXiUi and b = 2.: j {3jVj with Ul, ... , Un and Vi, ... , Vm atoms. From Definition 9.28 it follows that the right side of (9.22) is independent of the representation of b as a linear combination of atoms. From Lemma 9.29 (i) it follows that it is also independent of the representation of a, and so the product is well defined on A f . There is clearly a unique extension of this product to a commutative bilinear product on Af + R1 such that 1 acts as the identity. (If it happens that 1 is in A f , then from the definition (9.18) it is easy to check that 1 already acts as the identity on Ad
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9
CHARACTERIZATION OF JORDAN STATE SPACES
Next we show that this product on Af
+ Rl
satisfies
(9.23) By Corollary 9.27, each element b in Af is a finite linear combination of orthogonal projective units, say VI, .. . , V". Then the same is true of each element b + Al with A E R, since b + Al is a linear combination of the orthogonal projective units VI, ... , V"' q where q = 1 - U1 - ... - Un. It is straightforward to check that q2 = q, and that q * Uj = 0 fo~ J = 1, ... , n. Thus for a E Af + Rl we can find real scalars AI, . .. ,An and A such that
Now by applying the compressions associated with U1, ... , Un, q to an element a E Af + Rl such that -1 S; a S; 1, we conclude that AI, . .. , An and A are between -1 and 1, and thus that (9.23) holds. A similar calculation shows that all groupings a * (a * ... * (a * a) * ... ) of n factors of a give the same product, namely
and thus the product is power associative. Now by the remarks preceding this proposition, (9.23) implies that (9.21) holds in Af + Rl. Hence there is a unique extension of the product o to the norm closure Ao of Af + Rl. By continuity, the resulting product on Ao is commutative, power associative, squares are positive, the order unit 1 acts as an identity, and (9.21) holds. Then by definition Ao is a commutative order unit algebra (cf. Definition l.9), and thus Ao is a JB-algebra by Theorem 2.49. D 9.31. Lemma. Assume the workmg hypotheses of thIs sectIOn. For each a E Ao and each a E V there IS a umque element a 0 a E V such that (9.24)
(b, a 0 a) = (a
0
b, a)
for all b E Ao.
Proof. Suppose first that a is an atom, and let P be the compression such that a = Pl. Then by definition of the Jordan product on A o , for bE A o , a
0
b = ~(I
+P
- P')b.
Then
(aob,a) = (~(I+P-P')b,a) = (b,~(I+P-P')*a),
335
THE TYPE I FACTOR CASE
and so (9.24) holds with a 00"= ~(I + P - P')'O". We can take 100" to be 0", and so by linearity an element a 0 0" satisfying (9.24) exists for all a in Af + Rl. For an arbitrary element a of A o , choose a sequence {an} in Af + Rl converging in norm to a. By the norm property (9.21), {an 0 O"} is a Cauchy sequence in V. By completeness of V, this sequence converges in norm to an element of V; we define that element to be a 0 0". It is then evident that (9.24) holds. Since Af and V are in separating duality, the element a 0 0" satisfying (9.24) is unique. D
9.32. Lemma. Assume the workmg hypotheses of thzs sectwn. Then A can be equzpped wzth a product that makes zt an atomic ]B W-algebra wlth predual V (wzth the gwen order and norm on A and V ). Proof. By Lemma 9.22, Af and V are in separating norm duality, as are A and V by hypothesis. Thus Ao and V are also in separating norm duality, so we can imbed V isometrically into AD. We therefore identify V with a subspace of AD. By Corollary 2.50, AD' is a lBW-algebra, with a separately w*-continuous product that extends the product on Ao. By w*-density of Ao in AD', together with (9.24), we conclude that for each 0" E V, (9.25)
(b, a
0 0")
= (a 0 b, 0") for all a E Ao and all bEAD'.
Now let] be the annihilator of V in AD'. By (9.25), a 0] c ] for all (= O"-weakly closed), it follows that ao] C ] for all a E AD', i.e., ] is a O"-weakly closed ideal of AD'. By Proposition 2.39 there is a central projection c E Ao' such that ] = im Uc = cAD*. Since V is complete, it is a norm closed subspace of AD' so by the bipolar theorem, V is the annihilator of ] = im Uc in AD. The annihilator of im Uc is ker U;, which coincides with im Ui-c. By Propositions 2.9 and 2.62, Ui_c(A D) is the predual of the lBW-algebra im U 1 - c , and thus we have the isometric isomorphism a E Ao. Since] is w*-closed
Hence A can be equipped with a product making it a lBW-algebra with predual V. The order on an order unit space is determined by the norm and the order unit (since for an element a with norm 1, a 2: iff Ill-all :s; 1), so the order on A as a lBW-algebra coincides with that inherited as the dual of the base norm space V, and thus matches the given order on A. By uniqueness of the predual of a lBW-algebra (Theorem 2.55), K will be affinely isomorphic to the normal state space of A. By assumption the lattice F of norm exposed faces of K is atomic, and therefore so is the projection lattice of A, i.e., A is an atomic lBW-algebra. D
°
336
9
CHARACTERIZATION OF JORDAN STATE SPACES
Remark. For the interested reader, we sketch the structure of the proof above in a familiar context, without providing details. In the case where A = B (H),," then Af consists of the finite rank operators, and Ao will be ]{ + R1 where ]{ is the space of (self-adjoint) compact operators Then Ao = V EEl Rwo, where Wo is the unique state on ]{ + R1 that annihilates ]{, and Ao* ~ B(H)"" + Rco, where Co is a central projection and an atom. Thus essentially we have defined the Jordan product on the finite rank operators and then recovered the Jordan product on B(H)s" as the bidual. 9.33. Theorem. Let ]{ be the base of a complete base nOTm space. Then ]{ zs affinely zsomoTphzc to the nOTmal state space of an atormc lEW-algebm ziJ all of the followmg hold.
(i) EveTY nOTm exposed face of ]{ zs pTO)ectzve. (ii) EveTY non-empty spilt face of ]{ contams an extTeme pomt. (iii) ]{ has the pUTe state pTOpeTtzes (cf. Defimtwn 9.2). FUTtheTmoTe, ]{ wzll be affinely zsomoTphzc to the nOTmal state space of a lEW-factoT of type I zff ]{ satzsfies the thTee condztwns above and m addztwn]{ has no pTOpeT splztface (Le., one otheT than]{ OT 0). PTOOj. Assume first that ]{ is affinely isomorphic to the normal state space of an atomic JBW-factor lvI. By Theorem 5.32 every norm closed face of ]{ is projective. so (i) holds. By definition of an atomic JBVv-algebra, the lattice of projections of AI is atomic. By Proposition 5.39, atoms in the projection lattice correspond to projective faces consisting of a single extreme point, so every projective face of ]{ contains an extreme point. By Proposition 7.49 every split face of ]{ is a projective face. (Alternatively, every split face is norm closed (A 28), and therefore projective (Theorem 5.32)). Thus (ii) holds. By Proposition 9.3, the normal state space of AI has the pure state properties, so (iii) holds. Now suppose that ]{ is the distinguished base of a complete base norm space V, and satisfies (i), (ii), and (iii). Let A = V*. We are going to show that the working hypotheses of this section hold. The pair A, V are in separating order and norm duality (A 27). Then (i) implies that the standing hypothesis for this chapter holds. The properties (iv) and (v) of the working hypotheses of this section are part of the pure state properties. The covering property follows from the second of the pure state properties (Corollary 9.8). It remains to show that the lattices C, P, :F are atomic. Let z be the least upper bound of the atoms in P. By the covering property and Proposition 9.16, z is central, every projection under z dominates an atom, and 1 - z dominates no atom. We will show z = 1, which will then imply that the lattices P, C, :F are atomic.
THE TYPE I FACTOR CASE
337
The projective face F corresponding to z is a split face by Proposition 7.49. Let F' be the complementary split face. If F' is not empty, then by (ii) it contains an extreme point a, which is norm exposed by the first of the pure state properties. Then z' = 1 ~ z must dominate the atom corresponding to a (cf. Proposition 8.36), which contradicts the fact that z' dominates no atom. Thus F' must be the empty set, and F must be all of K, so z = 1, which completes the proof of atomicity. Therefore the working hypotheses of this section are satisfied. By Lemma 9.32, A = V* can be equipped with a product making it a JBWalgebra. By the uniqueness of the predual (Theorem 2.55), K is affinely isomorphic to the normal state space of A. Finally, suppose that K satisfies (i), (ii), (iii), and has no proper split faces. Then as established in the first part of this proof, K is the normal state space of a JBW-algebra IvI. Since K has no proper split faces, then M is a JBW-factor (Corollary 5.35). Since K contains extreme points, then M is a JBW-factor of type 1 (Corollary 5.41). Conversely, if K is the normal state space of a JBW-factor AI of type I, then M is atomic, so (i), (ii), (iii) hold, and since M is a factor, then K has no proper split face (Corollary 5.35). This completes the proof of the last statement of the theorem. D
9.34. Theorem. Let K be the base of a complete base norm space. Then K zs affinely zsomorphzc to the normal state space of an atomzc lEW-algebm iff all of the followmg hold.
(i) Every norm exposed face of K zs proJectwe. (ii) The a-convex hull of the extreme pomts of K equals K. (iii) K has the HIlbert ball property. Furthermore, K wzll be affinely zsomorphzc to the normal state space of a lEW-factor of type I zff K satIsfies the three condItIOns above and m addition K has no proper spilt face. Proof. Assume first that K is affinely isomorphic to the normal state space of an atomic JBW-factor M. By Theorem 5.32 every norm closed face of K is projective, so (i) holds. By Theorem 5.61, (ii) holds. By Proposition 9.10, (iii) holds. Now suppose that K is the distinguished base of a complete base norm space V, and satisfies (i), (ii), and (iii). Let A = V*. We are going to show that the working hypotheses of this section hold. The pair A, V are in separating order and norm duality (A 27). Every extreme point of K is norm exposed by the Hilbert ball property. Property (ii) implies that the lattices C, P, :F are atomic (Proposition 9.15). The Hilbert ball property implies the finite covering property (Proposition 9.11). The symmetry of transition probabilities, i.e., (v) of the working hypotheses of this section, also follows from the Hilbert ball property (Proposition 9.14).
338
9
CHARACTERIZATION OF JORDAN STATE SPACES
Thus the working hypotheses of this section are satisfied. The remainder of the proof of the current theorem is the same as the last two paragraphs of the proof of Theorem 9.33. D
Remark. Instead of assuming in Theorems 9.33 and 9.34 that every norm exposed face of K is projective, we could instead assume the stronger property that K is a spectral convex set, d. Definition 8.74 and Theorem 8.52. Recall that if C5, T are extreme points of the normal state space of a JBW-algebra, the face they generate is a norm exposed Hilbert ball (Proposition 5.55). \Ve will see that the affine dimension of this ball distinguishes the normal state spaces of type I JBW-factors. Recall from Theorem 3.39 that type I JBW-factors are either R, spin factors, H 3 (O), or the bounded self-adjoint operators on a real, complex, or quaterniollic Hilbert space.
9.35. Definition. A type I JBW-factor is said to be roeal (resp. complex resp. quaternwmc) if it is isomorphic to B(H)sa for some real (resp. complex resp. quaternionic) Hilbert space H. 9.36. Proposition. Let C5, T be dzstmct extreme pamts of the normal state space K of a lEW-factor M of type 1.
(i) If If If If If
(ii) (iii) (iv) (v)
M M M M M
real, then dzm face(C5, T) = 2. complex, then dzm face(C5, T) = 3. quaternwmc, then dzm face(C5, T) = 5. a spm factor, then dIm face(C5, T) = dzm M = H 3 (O), then dzm face(C5,T) = 9. zs zs zs zs
~
1.
Proof. Let p and T be the carrier projections of C5 and T. Then by the isomorphism of the lattice of norm closed faces and the lattice of projections (Corollary 5.33), p and T are atoms (i.e., minimal projections), and the carrier of face (C5, T) is pVT. Let q = p V T ~ p. Then p and q are orthogonal atoms (Lemma 3.50), with p V q = pVT. Thus face(C5, T) can be identified with the normal state space of lvIp +q • The affine dimension of face(C5, T) is then one less than the linear dimension of lvIp +q . By the classification of type I JB\V-factors, lvI is isomorphic to a spin factor, to H 3 (O), or to B(H)sa for a real, complex, or quaternionic Hilbert space. If lvI is a spin factor, then !vI is of type 12 , so any pair of orthogonal projections must add to 1 (Lemma 3.22). Thus in this case lvIp +q = lvI, aud hence (iv) holds. So it remains to consider the case where !vI is H 3 (O), or B (H)sa for a real, complex, or quaternionic Hilbert space. In each case, there are orthogonal minimal projections p and q such that !vI~+~ p q is isomorphic to 1\,f2(R)sa, lvI2 (Ck" M 2 (H)sa, or M 2 (O)sa' The results stated in the proposition will follow if we show that p + q is equivalent to p + q, so that M p + q is isomorphic to M;+q'
CHARACTERIZATION OF STATE SPACES OF JB-ALCEBRAS
339
Since p and p are minimal projections in a JBW-factor, they are exchangeable by a symmetry s (Lemma 3.19). Then Usq is orthogonal to UsP = p, as is (j, so both are minimal projections in M 1 _ p' which is a JBW-factor (Proposition 3.13). Thus there is a (1 ~ p)-symmetry to that exchanges Usq and (j. Then t = P+ to is a symmetry that also exchanges Usq and (j (since Jordan multiplication by p on M 1 _ p is the zero operator), while Ut fixes p. Thus UtUs takes p to p and q to (j, so takes p + q to p + (j. This completes the proof. D
Characterization of state spaces of JB-algebras We will continue to assume the standing hypothesis of this chapter, namely, that A is an order unit space and V a complete base norm space with distinguished base K, such that every norm exposed face of K is projective. Recall from Proposition 8.34 and Definition 8.35 that for each a E K there is a smallest central projective unit c(a) (called the central carT"'ler of a) such that (c( a), a) = 1. The corresponding face FeCa) is the smallest split face containing a. We will write Fa in place of FeCa).
9.37. Lemma. Assume the standmg hypothes7s of th7s chapter. If is an extreme pomt of K, then Fa contams no proper spht face.
(J
Proof. Suppose that G is a non-empty split face of K properly contained in Fa. Since K is the direct convex sum of G and the complementary split face G' , and a is an extreme point of K, then (J must be in either G or G' . By minimality of Fa, a cannot be contained in G, so a E G' . Then by (A 7), G' n Fa is a split face of K containing a and properly contained in Fa (since it contains no point of G). This violates the fact that Fa is the smallest split face containing (J. Thus we conclude that no such split face G of K exists. Finally, a split face of Fa would also be a split face of K, as can be verified directly from the definition of a split face (A 5), or from Corollary 8.18. Thus Fa contains no proper split face. D Recall that positive elements al, a2 of A are orthogonal if there exists a projective face F of K such that al = 0 on F and a2 = 0 on F' (Definition 8.38). Recall that under the standing hypotheses of this chapter, every element a E A admits a unique decomposition as a difference of orthogonal positive elements iff A, V are in spectral duality (Theorem 8.55). In the rest of this section K will be a compact convex set. Without loss of generality, we may assume that K is regularly imbedded as the base of the base norm space V = A(K)*, where A(K) denotes the continuous affine functions on K. (See the description of this imbedding following
340
9
CHARACTERIZATION OF JORDAN STATE SPACES
Proposition 8.76.) We let A = Ab(K) = V* be the space of bounded affine functions on K (A 11). Then A and V are in separating order and norm duality (A 27). Note that A(K) C Ab(K), and so notions such as orthogonality of positive elements of A(K) are to be interpreted with respect to the duality of A = Ab(K) and V. One of the key properties that we will use to characterize JB-state spaces is that each a E A(K) admits a decomposition as a difference of positive orthogonal elements of A(K). (For a discussion of a physical interpretation of this property, see the remarks after Proposition 8.6:.) Note that we will not require uniqueness of this decomposition, but we will require that the orthogonal elements be in A(K) and not just in A = Ab(K). Combining this property with the properties previously discussed, we can now characterize the state spaces of JB-algebras. 9.38. Theorem. A compact convex set K zs affinely homeomorphzc to the state space of a lB-algebm (wdh the w*-topology) ztf K satzsjies the conditwns:
(i) Every a E A(K) admds a decomposdwn a A(K)+ and b.l c.
=
b - c wzth b, c E
(ii) Every norm exposed face of K zs pTOJectwe. (iii) K has the pure state propertzes.
Here the condition (m) can be replaced by the alternatwe conddwn (iii ') The a-convex hull of the extreme pmnts of K zs a spld face of K and K has the Hilbert ball pTOperty.
Proof. Let B be a JB-algebra with state space K, and let V = B*. We will show K satisfies the properties above. Recall that we can identify K with the normal state space of the JBW-algebra B** (Corollary 2.61), and so we work in the duality of V and A = V* = B**. (i) By Proposition 1.28, each element of A admits a unique decomposition as the difference of orthogonal positive elements. Those elements will also be orthogonal as elements of the JBW-algebra A **. By Proposition 2.16, positive elements a, b of the JBW-algebra B** arc orthogonal iff their range projections are orthogonal. Viewing a, b as elements of A = Ab(K), this is the same as the notion of orthogonality with respect to the duality of A and V, d. Definition 8.38 and Proposition 8.39. Thus (i) holds. (ii) Norm closed faces of the normal state space of a JBW-algebra are projective (Theorem 5.32), so (ii) follows. This also follows more simply from the fact that positive elements of the JBW-algebra B** and their range projections have the same annihilators in the normal state space, d. (2.6). (iii) follows from Proposition 9.3. (iii ') follows from Corollary 5.63 and Corollary 5.56.
CHARACTERIZATION OF STATE SPACES OF JB-ALCEBRAS
341
Conversely, assume that (i), (ii), and either (iii) or (iii ') hold. For each extreme point (J" of K, let Pe(a) denote the compression associated with C((J") , and let Aa = im Pe(a) and Va = im Pe'(a)" By Proposition 8.17 and Proposition 8.34, we can identify Aa with the space of all bounded affine functions on Fa (the minimal split face generated by (J"). Fix an extreme point (J" of K. By Lemma 9.37, Fa contains no proper split faces. We are going to show that Fa satisfies the conditions (i), (ii), and (iii) of Theorem 9.33 or of Theorem 9.34, and thus that Aa can be equipped with a product making it a type I JBW-factor. By (ii) of the current theorem, A, V satisfy the standing hypothesis of this chapter. Thus by Proposition 8.17, we can identify Aa with the dual of the complete base norm space Va (with distinguished base Fa), and every norm exposed face of Fa is projective for the duality of Aa and Va. The projective faces for this restricted duality will be precisely the projective faces of K that are contained in Fer. Assume that the pure state properties hold for K. Then by the remarks following Proposition 9.17, the first and third of the pure state properties will also hold for Fu' By the second pure state property and Corollary 9.8, the covering property holds for the lattice of projective faces of K, and therefore also for the lattice of projective faces of Fa. Therefore again by Corollary 9.8, the second of the pure state properties holds for Fa. Since Fu contains the extreme point (J", and has no proper split faces, then we have shown that Fu satisfies the hypotheses of Theorem 9.33, and so we conclude that Aa admits a product making it a JBW-factor of type I, with the given order and norm. Now assume alternatively that (iii ') holds. If w, T are extreme points of Fa, then the face G generated by w, T in K is the same as in Fa, so the fact that G is norm exposed and a Hilbert ball in K implies the same in Fa. Thus the Hilbert ball property holds in Fa. Now let G be the split face of K that is the (J"-convex hull of the extreme points of K. Then G n Fa is a split face of K that contains (J", so by minimality of Fa, the split face G must contain Fu. Thus if W is any element of Fa, then W is a (J"-convex combination of extreme points of K. Each of those extreme points of K must in fact be in the face Fu (cf. (9.8)), so must be extreme points of Fa. Thus Fa is the (J"-convex hull of its extreme points. Hence (iii ') holds for Fu. Now by Theorem 9.34, Aer can be equipped with a product making it a JBW-factor of type I, with Fu as its normal state space. Thus we've shown that (i), (ii), and (iii) or (iii ') imply that each Fu is the normal state space of a type I JBW-factor. For each extreme point (J" of K, define Tru to be the restriction map which sends a E A(K) to alFu E Ab(Fu). We have identified Aa with Ab(Fu ), so Trer maps A into Au. Define B = EBerE&,K Au (cf. Definition 2.42) where oeK denotes the set of extreme points of K. Then B is a JB-algebra with pointwise operations and the supremum norm. By the Krein~Milman theorem, the map Tr that takes a to EBuE&e K TrO"(a) is an isometric order isomorphism of A(K) into B.
342
9
CHARACTERIZATION OF JORDAN STATE SPACES
Next, we are going to show that 7r(A(K)) is a lB-subalgebra of B. We first will show that 7r(A(K)) is closed under the map x I-> x+ where x = x+ - x- is the unique orthogonal decomposition in the lB-algebra B (cf. Proposition 1.28). For that purpose, let a E A(K) and let a = b-c be a decomposition of the type described in (i). By the definition of orthogonality, there is a projective face C such that b = 0 on C and c = 0 on C'. For each (J' E 8 e K, by (A 6) the faces F(TnC and F(TnC' are complementary split faces of Fa, and thus are complementary projective faces of Fa by Proposition 7.49. Thus band c restricted to Fa are orthJgonal elements of the order unit space AT' As discussed at the start of this proof, b and c restricted to Fa will also be orthogonal as elements of Aa viewed as a lB'-V-algebra. Thus by the uniqueness of the orthogonal decomposition 7r(a) = 7r(a)+ - 7r(a)- in the lB-algebra A a , we have 7r a (b) = 7r a (a)+ for all (J' E 8 e K. Hence
and in particular 1T(A) is closed in B under the map x I-> x+. Now let X be the spectrum of 1T( a) (cf. Definition 1.18). We will show f (1T( a)) E 1T(A) for each f E CR(X). Let Co denote the set of all f E CR(X) such that f(1T(a)) E 1T(A), where f acts on 1T(a) by the continuous functional calculus (Definition 1.20). Since A(K) is complete, and 1T is an isometry, then 1T(A(K)) is complete. Therefore Co is a linear subspace of CR(X) closed under the map f I-> f+, and so is a vector sublattice of CR(X). Since f I-> f(1T(a)) is an isometry on CR(X) (Corollary 1.19), then Co is a norm closed vector sublattice of CR(X) containing the constants and the identity function. Hence by the lattice version of the Stone-Weierstrass theorem (A 36), Co = CR(X). In particular, Co contains the squaring map, so 1T(A(K)) is a lB-subalgebra of B. Thus 1T is a unital order isomorphism from A(K) onto a lB-algebra, and so its dual gives an affine homeomorphism of the state space of a lB-algebra onto K. 0 Remark. Instead of assuming (i) and (ii) in Theorem 9.38, we could assume the stronger property that K is a strongly spectral convex set. (See the remark after Proposition 8.80.) Recall that a lC-algebra is a lB-algebra which is isomorphic to a norm closed 10rdan subalgebra of B (H)sa (Definition 1.7). 9.39. Corollary. A compact convex set K lS affinely homeomorphlc to the w*-compact state space of a le-algebra iff it satisfies, not only the conditlOns of Theorem 9.38, but also the additional requirement that every mmimal Spllt face of dimenslOn 26 lS a Hllbert ball.
CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS
343
Proof. By Lemma 4.17 and Corollary 4.20, a JB-algebra A is a JCalgebra iff there is no closed Jordan ideal J in A such that AI J is isomorphic to the exceptional Jordan factor H3(O). Recall from Corollary 5.38 that for every closed ideal J of A there is a natural affine isomorphism of the state space of AI J onto the split face F = JO n K corresponding to J. Thus it remains to show that H3(O) is the only JB-algebra whose state space is a 26-dimensional convex set which has no proper split face and is not a Hilbert ball. Counting (real) parameters in self-adjoint 3 x 3-matrices over the octonions, we find that H3(O) is a 27-dimensional vector space (over R). Therefore the state space of H3(O) is a 26-dimensional convex set. Since H3(O) is a factor, it contains no non-trivial central projection. Therefore its state space contains no proper split face. The state space of H3(O) is not a Hilbert ball since it has proper faces that are not singletons, e.g., the projective faces associated with projections p such that pH3(O)P is isomorphic to H2(O). If B is any JB-algebra whose state space is a 26-dimensional convex set which has no proper split face and is not a Hilbert ball, then B is a JBWfactor (Corollary 5.35), and is of type I (since it is finite dimensional). We conclude from Theorem 3.39 that the only type I factors with 26dimensional state space are the 27-dimensional spin factor and H3(O). The state space of a spin factor is a Hilbert ball (Proposition 5.51), so B = H3(O). We are done. 0 The state spaces of JC-algebras can also be characterized among JBstate spaces by a condition on the dimensions of the Hilbert balls generated by pairs of pure states: cf. [10, Prop. 7.6].
Characterization of normal state spaces of JEW algebras In this section we will always be working with a spectral convex set K which is then the distinguished base of a base norm space V, in spectral duality with A = V* (cf. Definition 8.74). Note then that A = Ab(K) (Proposition 8.73). By Theorem 8.52, every norm exposed face of K will be projective, so the standing hypothesis of the previous chapter is satisfied.
9.40. Lemma. Let K be a spectral convex set, and A = Ab(K). Let {e>.} be a spectral resolutwn for a E A, and let E = {e>'2 - e>., I A1 < A2}. Then:
(i) Each pazr e, f in E is compatzble. (ii) E zs closed under the map (e, 1) f---> e !\ f. (iii) Each element b E lin E can be expressed as a linear combination of orthogonal elements of E.
344
9.
CHARACTERIZATION OF JORDAN STATE SPACES
Proof. Let Eo be a finite subset of E, say Eo
=
{e/3, - e",; II'S
2
'S n},
where ai < Pi for each L Let Al, ... , A2n be the set {al' ... ,an, Pl, ... ,Pn} arranged in increasing order. Let F = {e'\,+l - e,\, 11 'S 2 'S 2n - I}. The projective units in F are orthogonal, and each member of Eo is a sum of elements of F. The items (i), (ii), (iii) follow. 0
9.41. Lemma. Let K be a spectr'al convex set, and A = Ab(K). Let {e,\} be a spectral resolutwn for a E A, let E = {e'\2 - e,\, I Al < A2}, and let M be the norm closure of the lmear span of E. Then there 2S a compact Hausdorff space X and an 2sometnc order 2somorph2sm from CR(X) onto M such that the mduced product * on M sat2sfies e * f = e 1\ f for all e,f E P. Proof. Let Mo
= lin E. Define * on Mo by a*b=
L aJ3 ei J
1\
fj
t,J
where a = 2:i aiei and b = 2: j pjfj with el,···, en, il,··., fm E E. We will show this product is well defined. By Lemma 9.40 all elements of E are compatible. Thus if Pe ; is the compression associated with ei for 2 = 1, ... ,n, then by Theorem 8.3,
for 1 'S
2,
J 'S n. Hence
Thus the definition of a * b is independent of the representation of b, and similarly of the representation of a. Since the operation 1\ is commutative and associative, the same is true of * on 1\110 . Clearly * is bilinear. Next we show that for a E 1\110, (9.26) By Lemma 9.40, we can write a as a linear combination of orthogonal elements el, ... , en of E, say a = Li Aiei. Since -1 ::; a ::; 1, then IAil ::; 1 for all i (as can be seen by applying suitable compressions to both sides of the equation a = Li Aiei)' Then a*a = Li Arei, so 0::; a*a ::; 1, which proves (9.26).
CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS
Thus (9.20) holds for a E M o, and holds, i.e., for a, bEMa,
*
345
is commutative, so (9.21) also
Iia * bll ::; Iiall Ilbll·
(9.27)
(See also the argument after (9.21).) Now we can extend * to 11'1 (the norm closure of Mo) by continuity. The resulting product on M still satisfies (9.27) and is commutative and associative. By norm continuity of the map * and the fact that A + is norm closed, it follows that a*a 2: 0 for all a E M. It is clear from the definition of the product that the order unit acts as an identity for M. Thus M is a commutative, associative order unit algebra, and so AI ~ CR(X) follows from (A 48). D
9.42. Definition. Let K be a spectral convex set, and A = Ab(K). Let e E P and let P e be the corresponding compression. Then we define Te : A ---> A by
I,
Note that if e and so
I
are compatible projective units, then (Pe+P:)1
=
Thus for compatible e, I we have (9.28) Recall from (1.47) that if A is a JB-algebra, then Te is the Jordan multiplication operator a I--> e 0 a. Thus in this case Tel = Tie = eo I for every pair of projections e, I. For a general spectral convex set K, the equation Tel = Tie may fail for a pair of projective units e, f. But we will now show that if it holds, then the map (e, f) I--> Tel = Tie from P x P into A extends to a bilinear map from A x A into A which organizes A to a JBW-algebra with normal state space affinely isomorphic to K. This result provides our first characterization of normal state spaces of JBW-algebras. Note that this construction of a bilinear product in A is similar to that in Proposition 9.30 with atoms replaced by general projective units. However, in the proof of Proposition 9.30 we made use of the fact that every linear combination of atoms can be written as a linear combination of orthogonal atoms (Corollary 9.27), which is not true if atoms are replaced by arbitrary projective units. Thus we take a different tack, using the spectral theorem (Theorem 8.64) and Lemma 9.41.
346
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CHARACTERIZATION OF JORDAN STATE SPACES
9.43. Theorem. A spectral convex set K is the normal state space of a lEW-algebra iff (9.29) for all paIrs of proJectwe umts e, f E A = Ab(K). More specl.ficaliy, If thIs condztwn IS satIsfied, then there IS a umque bIlmear product 0 on A whIch satIsfies the norm conditwn (9.21) for a, bE A and the equatwn eo e = e for e E P, and thIs product organizes A to the umque lEW-algebra whose normal state space IS affinely IsomorphIc to K. Proof. Clearly (9.29) is a necessary condition that K be the normal state space JBW-algebra since Te is the Jordan multiplication operator a f-> e 0 b in this case. To prove sufficiency, we assume (9.29). Define (e, J) = Tef = Tfe for each pair e, f E P. Let a = Li ociei for ei E P, OCi E R where 1 ::; i ::; 71, and b = LJ3 j f) for fj E P, Pj E R where 1 ::; J ::; m. Then
(2:OCiTei) (b) = i
2: OC,p) (ei' f)) = (2: PjTfJ ) (a). i,]
J
Thus there is a commutative bilinear product 0 on the space Ao := lin P with values in A given by the following definition, which is independent of the representation of a and b:
(9.30) Clearly, this product is norm continuous in each variable separately. We next show that the implication (9.20) holds in Ao. Fix a E Ao. Define E, lvI, and the product * as in Lemma 9.41. By Lemma 9.40, each pair e, fEE is compatible, so by (9.28) e0 f
= Te f = e /\ f = e * f.
Thus 0 and the product * agree on linE. Let {an} be a sequence in linE converging to a. (Such a sequence exists by the spectral theorem (Theorem 8.64)). By separate norm continuity of 0 on Ao and norm continuity of * on M ~ CR(X), (9.31 )
a
0
a
= lim lim an n Tn
0
am
= lim lim an * am = a * a. n Tn
Now in M ~ CR(X) the implication (9.20) holds, so it follows from (9.31) that it also holds in Ao. Thus (9.32)
CHARACTERIZING NORMAL STATE SPACES OF .JBW ALGEBRAS
347
for a E Ao. Now (9.21) also holds, i.e.,
Iia 0 bll :::: Ilallllbll
(9.33)
for all pairs a, bE Ao. By (9.33) and the spectral theorem (Theorem 8.64), there is a unique continuous extension of 0 to a product on A. By continuity, the extended product is still bilinear and commutative, and satisfies (9.32) and (9.33). To verify that A is a JB-algebra for this product, by Theorem 1.11 we need only show that A satisfies the Jordan identity
a2
(9.34)
0
(b
0
a) = (a 2
0
b)
0
a.
So suppose that el, ... , en are orthogonal projective units and AI, ... , An scalars such that a = Li Aiei' Then for each ~,J, the compressions Pei , P:" Pe), P:) associated with ei, ej commute (Lemma 7.42), so Tei and Tej commute. From the definition (9.30) together with commutativity, we have (ei 0 b) 0 ej = ei 0 (b 0 ej), so
',]
i,j
We have then shown that the Jordan identity (9.34) holds for a E Ao with finite spectral decomposition and all b E Ao. By continuity and the spectral theorem (Theorem 8.64), the Jordan identity holds for all a, bE A, By Theorem 1.11, the order from the JB-algebra structure (given by the cone of squares) coincides with the given order from the order unit space
A. The spectral convex set K is the distinguished base of a complete base norm space V (Definition 8.74) whose dual space is the order unit space A = Ab(K) (Proposition 8.73). Thus A is a dual space, hence a JBW-algebra. The JBW-algebra A has a unique predual space A., which consists of all normal linear functionals on A (Theorem 2.55). Therefore the Banach space V is isometrically isomorphic to the space of all normal linear functionals on A. Since V as well as A. are in separating order duality with A, these two base norm spaces are also order isomorphic. Therefore the distinguished bases of these two base norm spaces are affinely isomorphic. Thus K is affinely isomorphic to the normal state space of A. The uniqueness of the product 0 follows from the fact that two bilinear products on A which satisfy the norm condition (9.21) and the equation eo e = e for e E P, must agree on P and by linearity on Ao, and then by norm continuity and the spectral theorem also on A. If A is any J~W -algebra whose normal state space is affinely isomorphic to K, then A is isomorphic to A by Proposition 5.16. D
348
9
CHARACTERIZATION OF JORDAN STATE SPACES
The corollary below gives another characterization of normal state spaces of JBW-algebras. It is of interest mainly in showing that the functional calculus for affine functions on a spectral convex set provides a natural candidate for a Jordan product, which will give a JBvV-algebra exactly under the simple but unappealing requirement that this product be bilinear. Let a E A and let E, lvI, and * be defined as in Lemma 9.41. vVe observe for use below that the proof above showed that the products * and o agree on lin E, and thus by norm continuity of each ~roduct, agree 011
M. 9.44. Corollary. Let K be a spectral convex set, and A = Ab(K). Define a product
0
on A by
(9.35) where the squares are gwen by the functwnal calculus on A defined by equatwn (8.60). A wzth thzs product and the order umt norm zs a JBWalgebra zff thzs product zs bzlznear. Proof. Suppose that this product is bilinear, and let a E A. Then squares for this product by definition coincide with those given by the functional calculus, so by Proposition 8.67,
(9.36) Furthermore, with the notation of Lemma 9.41, a*a = aoa (see the remarks preceding this corollary), so in particular a 0 a is again in AI. Then we have (a 0 a) 0 a = (a * a) * a, and similarly for any product of factors of a grouped in any fashion. Since AI ~ CR(X) is power associative, the same follows for the product o. Thus A is a complete order unit space with a commutative, power associative product for which 1 is an identity, and for which (9.36) holds. By (A 51), A is a commutative order unit algebra with positive cone A2 (A 47), and thus by Theorem 2.49, A is a lB-algebra. Since A is monotone complete and admits a separating set of normal states, then A is a lBW-algebra. 0 The corollary below shows that the condition Tef = Tje in Theorem 9.43 can be restated in terms of the general version of the "closeness operator" (9.37)
c( e, 1) = e f e
+ e' J' e'
associated with a pair of projections e, f in a von Neumann algebra. This operator, which was first defined in B(H) by Chandler Davis [40], is a useful tool in the theory of algebras generated by a pair of two non-commuting
CHARACTERIZING NORl\lAL STATE SPACES OF JBW ALGEBRAS
349
projections, sometimes referred to as "non-commutative trigonometry". (If e and f are projections onto two lines in C 2 with an angle a, then c(e, 1) = (cos 2 a)I. The element c(e, 1) has an interesting physical intepretation, discussed in the remark after Corollary 9.46. For more information on this topic, see the discussion in [AS, Chpt. 6] and also the references in the notes to that chapter.) The compression Pe associated with a projection e in a von Neumann algebra is the map a f-+ eae, and the compression P~ associated with the complementary projection e' = 1 - e is the map a f-+ e' ae'. Therefore we define c(e, 1) for a pair e, f of two projective units in an order unit space A in spectral duality with a base norm space V by
(9.38)
c(e,1)
= Pef +
P~f'.
9.45. Corollary. A spectr'al convex set K of a lEW-algebra zJJ
(9.39)
c(e,1)
IS
the normal state space
= c(f, e)
for all pazrs of pro)ectwe umts e, f E A := Ab(K). Proof. Let e, f be a pair of projective units in A. Using the equality P~f'
=
P~(1 -
1) = 1 - e -
P~f,
we find that Tef
= ~(f + Pef - P~1) = ~(e + f -1 + Pef + P~f').
Subtracting the corresponding equation with e and find that
Thus the equation Tef
f
interchanged, we
= Tje is equivalent to the equation
(9.40)
which is the same as c(e,1) Theorem 9.43. 0
=
c(f, e).
Now the corollary follows from
The above characterization of the normal state space of a JBW-algebra Was first proved in [9, Cor. 3.7] under the condition (9.40), written as the commutator relation [Pe, PI] I = [Pj, P:]l, and with the Jordan product defined as in (9.35).
350
9
CHARACTERIZATION OF JORDAN STATE SPACES
The symmetry condition c( e, 1) = c(j, e) for projective units in a spectral convex set is related to symmetry of transition probabilities (the third of the pure state properties). In fact, if e and f are minimal projections in a von Neumann algebra or JBW-algebra M, computing in the subalgebra Mevf gives
c(e, 1)
= (e,l) (e V 1) = (j,e) (e V 1).
Of course, by Corollary 9.45 and Corollary 5.57, the symmetry condition c(e,1) = c(j, e) for projective units in a spectral convex set K implies that Ab(K) can be equipped with a product making it a JBW-algebra with normal state space K, and so in particular that K has the property of symmetry of transition probabilities.
9.46. Corollary. A compact spectral convex set K zs the state space of a iE-algebra zJJ zt satzsfies the conddwns
(i) c(e, 1) = c(j, e) for all pairs e, f E P, (ii) a 2 E A(K) for all a E A(K). Proof. Clearly these are necessary conditions that K be the state space of a JB-algebra (which must be isomorphic to A(K) equipped with the product 0 of (9.35)). They are also sufficient, for if (i) is satisfied, then Ab(K) is a JBW-algebra with the product 0 of (9.35); and if (ii) is also satisfied, then A(K) is closed under this product which will organize A(K) to a JB-algebra with state space affinely isomorphic and homeomorphic to
K.o Remark. The "closeness operator" c( e, 1) associated with projections in a von Neumann algebra has a physical interpretation discussed in the remark following [AS, Cor. 6.43]. We will show that the element c(e,1) of A+ defined by (9.38) for a pair of projective units in the space A = Ab(K) of a spectral convex set K can be interpreted in a similar way. Then we assume that e and f represent propositions (or questions) and that the compressions Pe and Pf represent measuring devices for these propositions, as explained in the remark after Proposition 7.49. Thus each of these measuring devices records if particles in a beam appear with the value 1 or the value 0 (the two possible values for the proposition). What happens if the beam from one apparatus is sent through the other one? Clearly, there are four possible outcomes with probabilities p(l,l), p(l, 0), p(O, 1), p(O, 0) respectively. (Here the first variable of the function p(., .) refers to the first apparatus and the second variable refers to the second apparatus). We will now determine these probabilities. If a beam of particles in a state w E K is sent into the apparatus Pe , then the particles that are recorded with the value 1 will leave the apparatus in a new state represented by P;w E V+. More specifically,
CHARACTERIZING NORMAL STATE SPACES OF .JBW ALGEBRAS
351
they will be in the state IlPe*wll- 1 Pe*w E K. Here IIP;wll is the relative intensity of the partial beam consisting of all particles that emerge with the value 1. Otherwise stated, IIP*wll is the probability that a particle in the state w is recorded with the value 1 by the apparatus Pe . Assume now that the beam from the apparatus Pe is sent into the apparatus Pf . Then the states are transformed once more in the same fashion as in the first apparatus. Therefore the particles that are recorded with the value 1 by both apparata will emerge in a state represented by Pj Pe*w, and the partial beam of such particles will have the relative intensity IIPjPe*wll. Thus the probability that a particle is recorded with the value 1 by both apparata is
In the same way we find that p(l,O) = (Pe!',W),
p(O, 1) = (P:f,w),
p(O,O) = (p:!"w).
Note that in general these probabilities will depend on the order in which the measurements are performed. However, it follows from the above that
p(l,l)+p(O,O) =
(Pef+P~!',w)
= (c(e,f),w).
Thus (c( e, f), w) is the probability that two consecutive measurements on particles in the state w by the apparata Pe and Pf in that order coincide (both with the value 1 or both with the value 0). Therefore the equation c(e, f) = c(j, e) says that for all w this probability is the same when the order of the measurements is reversed. (Thus we might refer to this equation as "the symmetry of coincidence probabilities".) We now give an alternate characterization of normal state spaces that is more geometric in nature, replacing the condition (9.29) by ellipticity. We have previously discussed this concept in the context of normal state spaces of lBW-algebras (d. Proposition 5.75). We now define the corresponding concept for spectral convex sets and show that it characterizes normal state spaces of lBW-algebras.
9.47. Definition. Let K be a spectral convex set. Let P and pi be complementary projective faces of K, with corresponding compressions P and P'. Let I}i = (P + PI)* be the canonical projection of K onto co(P U Pi), let a E F, T E pi, and let [a, T] be the line segment with endpoints a, T. Then we say I}i-l ([a, T]) has ellzptzcal cross-sectwns if the intersection of this set with every plane through the line segment [a, T] is an ellipse together with its interior. We say K is elhptzc if such cross-sections are elliptical for all F, F ' , a E F, T E P'.
352
9.
CHARACTERIZATION OF JORDAN STATE SPACES
By Proposition 5.75, the normal state spaces of JBW-algebras are elliptic. Now let K be a spectral convex set, represented as the distinguished base of a complete base norm space V. Let F and FI be complementary projective faces of K, with corresponding compressions P and pi, and fix (J E F, TEFl, and W E K \ co(F U FI). Consider the orbit Wt of W under the action of the one-parameter group t f--+ exp(t(P - Pi)) *, and let w; = Ilwtll-1Wt denote the normalized orbit. Then the proof of Lemma 5.74 (with the concrete compression Up replaced by the abstract compression P) shows that the normalized orbit of any such W is the unique half ellipse that passes through wand has (J and T as endpoints. If the cross-sections described in Definition 9.47 are ellipses, then each such normalized orbit will stay inside K, or equivalently, t f--+ exp(t(P - Pi))' will be a oneparameter group of order automorphisms of V. Thus we have for t E R, PEe, (9.41)
K is elliptic
=}
exp(t(P - Pi))' ~ O.
This is equivalent to requiring exp(t(P - Pi)) ~ 0 for all t E R. This says that P - pi is an order derivation of the order unit space A = Ab(K), cf. (A 66). Note that in the case of a JBW-algebra, ~(I + P - PI)b = (PI) 0 b, so the fact that P - pi is an order derivation follows from the fact that Jordan multiplication by any element of a JB-algebra is an order derivation (Lemma 1.56.) For use below, we recall that if A is any complete order unit space and 15 : A -+ A is a bounded linear map, then by (A 67), 15 will be an order derivation iff for a E A + , (J E (A') + , (9.42)
(a, (J) = 0
=}
(ba, (J) = o.
We will also need the fact that the set of order derivations is closed under commutators (A 68).
9.48. Theorem. A conve1; set K zs affinely isomorphzc to the normal state space of a lEW-algebra iff K zs spectral and ellzptzc.
Proof. We have already shown that the normal state space of a JBWalgebra is spectral and elliptic (Proposition 5.75 and Proposition 8.76). Conversely, assume that K is spectral and elliptic, and let A = Ab(K). We are going to prove that K satisfies the condition (9.29), which by Theorem 9.43 will complete the proof. Let P and Q be arbitrary compressions. We will show that (9.43)
[P - PI,Q - Q/]1 = 0,
CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS
353
which is evidently equivalent to [Te, T f ]1 = 0, and then to (9.29), where e and f are the projective units associated with P and Q. By (9.41) the operators Dl := P - P' and D2 := Q - Q' are order derivations. Consider also the positive operators El := P + P' and E2 := Q + Q'. Note that (9.44) For an arbitrary a E A+ and (J E K we have (Qa, Q'*(J) = 0, so we can use (9.42) with Qa in the place of a and Q' (J in the place of (J to conclude that (D1Qa,Q'*(J) = O. Hence Q'D1Q = O. Similarly QD1Q' = O. An immediate consequence is
(Q - Q')(P - P')(Q - Q') = (Q
+ Q')(P -
P')(Q
+ Q')
(both sides being equal to QD1Q + Q'D1Q'). Clearly the same equality holds with P in the place of Q and D2 in the place of D l . Thus (9.45 ) We next establish (9.46) To verify this, we expand the left side using (9.44):
[Dl' D2]21
= =
D1D2D1D21 D1D2D1D21
+ D2D1D2Dll- D1D~Dll + D2D1D2Dd - D1E2Dd -
D2DiD21 D2E1D21.
Clearly Ell = 1 and E21 = 1, so in the last two terms above we can replace 1 by E21 and Ell (in this order), getting
By (9.45) this gives (9.46). From (9.46) we have (9.47) By the fact that the set of order derivations is closed under commutators, [Dl' D 2 ] is an order derivation. Therefore the left side of (9.47) is positive for all t. From this it follows that [Dl' D2]1 = 0, which proves (9.43) and completes the proof. D
354
9
CHARACTERIZATION OF JORDAN STATE SPACES
Notes The geometric characterization of the normal state spaces of type I JBW-factors (in Theorem 9.33) was first given in [10]' and it was then followed by the characterization of the state spaces of JB-algebras (Theorem 9.38) in the same paper. (Note that the condition in Theorem 9.38 that requires the (j-convex hull of the extreme points to be a split face may be redundant, as there are no known examples to the contrary.) In a paper proposing axioms for quantum mechanics, Gunson l59] showed under various assumptions that the linear span of the atoms formed a Jordan algebra, and the proof of Proposition 9.30 is related to his. The characterization of normal state spaces of general JBW'-algebras (Theorem 9.48) is due to Iochum and Shultz [71]. The Hilbert ball property and the pure state properties were introduced in [10]' and the ellipticity property in [71]. Note that the equivalent condition on the right side of (9.41) formally is very similar to the notion of (facial) homogeneity in self-dual cones due to Connes [32]. (See the discussion following Proposition 5.75.) The characterization of normal state spaces of JB\V-algebras and state spaces of JB-algebras by means of the condition (9.39) was first given (in a slightly different form) in [9], where there is also a brief discussioll of the physical interpretation. For a thorough discussion of an approach to axiomatic quantum mechanics via JBW-algebras, based on Corollary 9.45, see the papers of Kummer [88, 89] and Guz [60]. In the paper [21], the authors Ayupov, Iochum and Yadgorov characterize the state spaces of finite dimensional JB-algebras by three properties. The first one, called "projectivity", is the same as the "standing hypothesis" of Chapter 8 (exposed faces being projective), and the two others are "symmetry" and "modularity". In [22] the same authors show that a sclfdual cone in a real Hilbert space is facially homogeneous iff it is symmetric and modular. TI:ansition probabilities have long played a central role in quantum mechanics. For example, in [135] \Vigner shows that a bijection of the set of pure normal states of B (H) onto itself that preserves transition probabilities is implemented by a unitary or a conjugate linear isometry, and thus induces a *-isomorphism or *-anti-isomorphism of B(H). Araki has given a careful justification of JB-algebras as a model for quantum mechanics in finite dimensions [17]. In this paper he discusses the role of symmetry of transition probabilities (cf. (9.7)), which is one of his axioms, and he also assumes that filters send pure states to multiples of pure states. In the finite dimensional context of Araki's paper, his axioms force the filters to be compressions. The assumption that filters should send pure states to multiples of pure states is also in the paper of Pool [104], and "pure operations" (those that send pure states to pure states) were discussed in Haag and Kastler's influential paper [61]. The lattice-theoretic covering property (Definition 9.6),
NOTES
355
which is equivalent to compressions sending pure states to multiples of pure states (Corollary 9.8), plays a key role in the "quantum logic" approach to axioms for quantum mechanics, e.g., in the work of Piron [102]. Other relevant references for axiomatic foundations of quantum mechanics related to our presentation here can be found in the books of Busch, Grabowski, and Lahti [35], Emch [49]' Landsman [91]' Ludwig [94]' and Upmeier [130]. Finally, we mention briefly JB*-triples. These generalize JB-algebras, and axiomatize the triple product {a,b,c} f--t ~(abc+cba). Recall that JB-algebras are in 1-1 correspondence with symmetric tube domains. Bya theorem of Kaup [83], JB*-triples are in 1-1 correspondence with bounded symmetric domains. Thus they are of considerable geometric interest. There is a Gelfand-Naimark type theorem for JB*-triples, due to Friedman and Russo [54]. Friedman and Russo also showed that most of the geometric properties of JB-algebra state spaces in Theorem 9.38 generalize to preduals of JBW*-triples [53], and they gave a characterization of the preduals of atomic JBW*-triples [55], which generalizes Theorem 9.33. These results are quite remarkable, since they generalize order-theoretic concepts and results to a context where there is no order, requiring totally different proofs.
10
Characterization of Normal State Spaces of von Neumann Algebras
vVe hegin this chapter hy characterizing the normal state space of the von Neumann algebra B(H). Our starting point is Theorem 9.34 which characterizes normal state spaces of JBW-factors of type I by geometric axioms, among those the Hilbert ball property by which the face generated by each pair of extreme points is a Hilbert ball. In the case of B (H) these balls will be 3-dimensional (A 120), and this "3-ball property" is the single additional property we need to characterize the normal state space of B(H) (Theorem 10.2). Since general von Neumann algebras may admit no pure normal states, assumptions involving pure states are not useful to characterize their normal state spaces. Our starting point for general von Neumann algebras is Theorem 9.48 which characterizes normal state spaces of general JBWalgebras, and we show that a "generalized 3-ball property" that involves pairs of norm closed faces rather than pairs of extreme points (Definition 10.20) is the single additional property needed to characterize the normal state spaces of general von Neumann algebras (Theorem 10.25).
The normal state space of B (H) 10.1. Definition. A 3-ball is a convex set affinely isomorphic to the standard 3-ball
Note that a 3-ball is a special case of a Hilhert ball.
10.2. Theorem. Let K be the base of a complete base norm space. Then K 7S affinely 7somorphic to the normal state space of B (H) for a complex Hz/bert space H 7ft all of the followmg hold.
(i) Every norm exposed face of K 7S pro)ectwe. (ii) The cy-convex hull of the extreme pomts of K equals K. (iii) The face generated by every pozr of extreme pomts of K 7S a 3-ball and 7S norm exposed. Proof. Assume first that K is affinely isomorphic to the normal state space of B(H). Since B(H)sa is an atomic lBW-algebra, then properties (i) and (ii) follow from Theorem 9.34, as does the Hilbert hall property. If
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iJ, T are extreme points of K, then the face they generate is norm exposed by the Hilbert ball property. The fact that the face generated by iJ, T is a 3-ball is (A 120). Now assume K satisfies (i), (ii), (iii). By Theorem 9.34, K is affinely isomorphic to the normal state space of an atomic JBW-algebra IvI. Suppose that K contains a proper splIt face F. Since AI is atomic, then F and F' contain minimal projective faces, which are then singleton extreme points (Propositions 8.36 or 5.39). Let iJ E F and TEF' be such extreme points. Then the face of K generated by iJ and T is the line segment joining these two points (Lemma 5.54 or (A 29)), which contradicts (iii). Thus K contains no proper split face, and so by Theorem 9.34, K is affinely isomorphic to the normal state space of a type I JBW-factor M. By (iii), M is isomorphic to 13 (H)sa for a complex Hilbert space H (Proposition 9.36). D
3-frames, Cartesian triples, and blown up 3-balls Let (iJ, T) be a pair of antipodal points in the standard Euclidean 3ball B3. Then there is a unique reflection (i.e., period 2 automorphism) of B3 whose set of fixed points is the axis determined by (iJ, T) (i.e., the line segment with endpoints iJ, T). Two such axes are orthogonal iff the reflection associated with each axis reverses the other axis, i.e., exchanges the corresponding pair of antipodal points of that axis. If we have three orthogonal axes, then the product of the associated reflections will be the identity map. This is characteristic of dimension 3: for higher dimensional Hilbert balls the product of any such triple of reflections will invert any fourth axis orthogonal to the first three, and thus cannot be the identity map. These observations motivate our generalization of 3-balls. We are going to define generalued axes, 3-frames, and blown up 3-balls. (See the definitions (A 185), (A 190), and (A 194) for a discussion of these concepts for normal state spaces of von Neumann algebras.) Our context in the rest of this section will be a JBW-algebra IvI and its normal state space K. Recall that there is a 1-1 correspondence of projections in AI and norm closed faces in K given by p f--t Fl' = {iJ E K I iJ(p) = I} (Corollary 5.33). The faces Fl' and F~ = Fl" (where pi = I-p) are said to be complementary.
10.3. Definition. If K is the normal state space of a JB\V-algebra, an ordered pair (F, F') of complementary norm closed faces is called a generallzed aXlS of K. Observe that if p is a projection and pi = 1 - p, then s = p - pi is a symmetry (and every symmetry arises in this way). If F = Fl" we will call (F, F') the generalzzed axis assocwted with the symmetry s = p - pi .
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Recall that a reflection of a convex set is an affine automorphism of period 2, and that the map U; (dual to Us : x f--+ {sxs} ) acts as a reflection of K with fixed point set co(F U F') (Proposition 5.72). We now show that this reflection is unique, and will then refer to it as the refiection about the generallzed aXlS (F, F'). This uniqueness result in the von Neumann alge bra context is (A 174).
10.4. Proposition. If p lS a proJectwn m a lEW-algebra !vI wlth assocwted pTOJectzve face F, and s = p - pi where pi = 1 - p, then U; lS the umque refiectwn of the nOTmal state space K wlth fixed pomt set co(F U F'). Proof. Let R be any reflection of K with fixed point set co(F U F'). Then ~(I +R) is an affine projection of K onto co(FUF ' ). Since Up, Up' are complementary compressions (Theorem 2.83) with associated projective faces F, F' (cf. (5.4)), then (Up + Up')* is the unique affine projection of K onto co(F U F') (Theorem 7.46). Thus ~(I + R) = (Up + Up')*' so R = (2(Up + Up') - 1)*. Then R = follows from (2.25). 0
u.:
If F is a projective face of K with complementary face F ' , and an affine automorphism ¢ of K exchanges F and F ' , then we will say that ¢ reveTses the generalued aXlS (F, F'). \Ve can use this to define a notion of orthogonality for generalized axes.
10.5. Definition. If !vI is a lBW-algebra with normal state space K, a pair of generalized axes is orthogonal if the reflection about the first generalized axis reverses the second generalized axis. By the following result, the reflection about the second generalized axis will also reverse the first, so orthogonality is a symmetric relation as one would hope.
10.6. Lemma. Let p and q be pTOJectwns of a lEW-algebra M with complementary pToJectwns pi and q'. Let F and G be the norm closed faces associated wlth p and q, and let RF and Rc be the refiectwns of K about the generahzed axes (F, F') and (G, G ' ). The following are equivalent.
(i) The Tefiectwn RF reverses the generallzed aXlS (G, G ' ). (ii) The Tefiection Rc reverses the generallzed aXlS (F, F ' ). (iii) (p - pi) 0 (q - q') = O. Pmof. Let s = p - pi and t = q - q'. Then sand t are symmetries and RF = U; and Rc = Ut (Proposition 10.4). By Corollary 5.17, (i) is equivalent to
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(i ') Us exchanges q and q' while (ii) is equivalent to (ii ') Ut exchanges p and p'. Each of (i') and (ii') is equivalent to (iii) by Lemma 3.6. 0 We say that symmetries s, t are Jordan orthogonal if sot = o. Thus orthogonality of generalized axes is equivalent to Jordan orthogonality of the associated symmetries. The following definition abstracts the key properties of the set of three orthogonal axes of B:3.
10.7. Definition. Let K be the normal state space of a JBvV-algebra M. A 3-fmme is an ordered triple of mutually orthogonal generalized axes of K such that the product of the associated reflections of K is the identity map. Recall that if F is the projective face associated with a projection p, then F can be identified with the normal state space of the JBVIsubalgebra im Up, d. Proposition 2.62. Therefore we can apply the notions of generalized axes and 3-frames to F as well as to K. Recall that a Cartesian triple of symmetries is an ordered triple (r, s, t) of Jordan orthogonal symmetries such that UrUsUt is the identity map (Definition 4.50). We now show that Cartesian triples are the algebraic objects corresponding to 3-frames. 10.B. Definition. If ct = (r, s, t) is a Cartesian triple, then denotes the ordered triple of generalized axes associated with r, s, t.
ct.
10.9. Lemma. Let AJ be a JEW-algebm, The map ct f--+ ct. lS a 1-1 correspondence of Carteszan trzples of symmetrzes and 3-fmmes of K. Proof. Let ct = (r, s, t). By Lemma 10.6, the orthogonality of the generalized axes of ct. is equivalent to Jordan orthogonality of r, s, t. The refiections about the three generalized axes are U;, U; ,Ut' respectively. Then the product of the three refiections is the identity iff U;U;Ut = Id, which in turn is equivalent to UtUsUr = I d. By the remarks after Definition 4.50, if (r, s, t) is a Cartesian triple of symmetries, then the maps Un US) Ut commute, so this is equivalent to (1', S, t) being a Cartesian triple. 0 From thc proof above and Proposition 10.4, it follows that the reflections associated with the generalized axcs in a 3-frame commute (and therefore their product in any order is the identity).
Remark. By Lemma 10.9, the existence of 3-frames in K is equivalent to the existence of Cartesian triples in M. By Definition 10.5 and
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Lemma 10.6, a necessary condition for this is the existence of a projection p E !vI such that p and pi can be exchanged by a symmetry (as in (ii ') of the proof of Lemma 10.6). (In the case of von Neumann algebras, the above condition is sufficient as well as necessary for the existence of 3-frames in K, cf. (A 193) where this condition is referred to as "halvability of the identity element".) Note that the above exchangeability condition is satisfied in the Jordan matrix algebra Mn(C)sa iff n is an even number. Therefore the state space K of Mn(C)sa has no 3-frame when n is odd. But K has closed faces with 3-frames for all n > 1, namely all faces that are isomorphic to state spaces of subalgebras of the form im Up ~ !vh(C)sa where k = rank(p) is even. We are now ready to define our generalization of 3-balls.
10.10. Definition. Let M be a JBW-algebra with normal state space K. A face F of K (possibly F = K) is a blown up 3-ball (of K) if it is a norm closed face admitting a 3-frame. We will see later that this implies that the facial structure and automorphism group of F are quite similar to those of the standard 3-ball
B3. 10.11. Example. In the special case of M = !vh(C)sa, we can identify the state space of !vI with the positive matrices of trace 1, and thus with B3 as follows (see (A 119)):
(10.1) Let 0O. be the ordered triple of pairs of singleton faces of B3 located along each of the three coordinate axes. Each of these pairs is a "generalized axis" in our current terminology. We will refer to 0O. as the standard 3-fmme of B3 . The reflections corresponding to each generalized axis will be exactly the reflections of K ~ B3 in each of the coordinate axes. The associated 3-frame is
(10.2) which are the Pauli spin matrices. If a JBW-algebra M possesses a Cartesian triple of symmetries, then M is isomorphic to the self-adjoint part of a von Neumann algebra M (Theorem 4.56). Furthermore, this von Neumann algebra M will contain
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a set of 2 x 2 matrix units, so that M "" lvh(C) I8i B for a von Neumann algebra B. Using the correspondence of 3-frames and Cartesian triples, we can characterize normal state spaces of such von Neumann algebras.
10.12. Proposition. If the normal state space K of a lEW-algebra M contams a 3-frame, wzth assocwted Carteswn tnple of symmetnes (r, s, t), then M = M + ~M can be eqUIpped w~th a umque von Neumann algebra product compat~ble w~th the gwen lordan product and satzsfymg ~rst = 1. For th~s product, the complex lmear span of 1, T, s, t ~s a *subalgebra *-~somorph~c to M2 (C)! and thus M "" M2 (C) I8i B fOT a von Neumann algebra B. Proof. The proposition follows from the correspondence of 3-frames and Cartesian triples (Lemma 10.9) and Theorem 4.56. 0 The following result generalizes the fact that the state space of lvI2 (C) is a 3-ball, cf. (A 119).
10.13. Corollary. The normal state space K of a lEW-algebra M is a blown up 3-ball ~ff M ~s ~somoTph~c to the self-adJomt pad of a von Neumann algebra M of the fOTm M "" Jvh(C) I8i B for a von Neumann algebra B. Proof. This follows at once from Proposition 10.12 and the definition of a blown up 3-ball. 0 We will now present some elementary concepts and facts that will be used in the proof of our next result (Proposition 10.15). This result is not needed in the sequel, but provides a further illustration of the appropriateness of the term "blown up 3-ball".
10.14. Definition. If K is the normal state space of a JBW-algebra, then Auto(K) is the (norm-continuous) path component of the identity in the group of affine automorphisms of K. (Here we extend affine automorphisms of K to V = lin K and give them the norm topology as bounded linear maps on V.) Note that affine automorphisms of the standard 3-ball B3 extend uniquely to orthogonal transformations of R 3. The latter have determinant ±1. Those with determinant +1 are precisely the rotations, while those with determinant -1 are the reversals (i.e., compositions of a rotation and a reflection in the origin.) Rotations are evidently connected by a path to the identity, while reversals are not (by continuity of the determinant map.) Thus Auto(B3) can be identified with the rotation group SO(3). By (A 131), the rotations of B3 are precisely the duals of the
3-FRAMES, CARTESIAN TRIPLES, AND BLOWN UP 3-BALLS
363
maps x f---> 'uxu* for unit aries u in IvI2 (C), while the reversals arise as the duals of the maps x f---> vx*v- l , where v is conjugate unitary. Thus rotations of B3 are the duals of *-automorphisms of M 2 (C), while reversals correspond to *-anti-automorphisms (A 124). Recall that if K is the normal state space of a JBW-algebra, then the projective faces and norm closed faces of K coincide, and the lattice of norm closed faces of K is an orthomodular lattice (Corollary 5.33). An ortho-lsomorphlsm from one orthomodular lattice into another is an injective map that preserves complements and (finite) least upper bounds and greatest lower bounds. Let NIl and 11:12 be JBW-algebras with normal state spaces K I , K 2 . Recall that the map T f---> T* is a 1-1 correspondence of positive unital (Jweakly continuous maps T: M] --+ IvI2 and affine maps from K2 into KI (Lemma 5.13). If T is also a Jordan homomorphism, and p is a projection in MI with associated projective face Fp , then by Lemma 5.14,
(10.3) and if T is a Jordan isomorphism from NIl onto NI2 , and q is a projection in M 2 , then
(10.4) We will make use of these results several times in the next proof, in a context where T will be a unital *-homomorphism between von Neumann algebras. Recall that for each closed face F of the normal state space of a JBWalgebra, there is a reflection RF with fixed point set co(FUF'), d. Proposition 10.4.
10.15. Proposition. Let K be the normal state space of a lBWalgebra IvI. If K lS a blown up 3-ball, then there ],s an affine map ¢ from K onto B3 such that
(i)
an ortho-lsomorphlsm from the lattlce of faces of B3 mto the lattzce of norm closed faces of K, (ii) there lS an lsomorphzsm (J --+ (j of the rotation group Auto(B3) mto the group of affine automorphlsms of K WhlCh takes RF to Rq,-l(F) for each face F of B 3 , and WhlCh satzsfies (J 0 ¢ = ¢ 0 (j. ¢-l
lS
Proof. Since K is a blown up 3-ball, there is a von Neumann algebra B such that M is isomorphic to the self-adjoint part of M = Ivh (C) ® B (Corollary 10.13). Identify M and M sa, and identify K with the normal state space of M. Define T : M 2 (C) --+ M by Tx = x ® l. We identify the state space of M 2 (C) with B 3 , and let ¢ : K --+ B3 be T* restricted to K.
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To see that ¢ is surjective, it suffices to show ¢(K) contains every pure state on M2(C), Recall that there is a 1-1 correspondence of pure states on lvh(C) and minimal projections in M 2(C) (e.g., Proposition 5.39). Let p be a minimal projection in lvh(C) and w the unique state with w(p) = 1. If T is any normal state on M with T(p ® 1) = 1, then ¢(T)(p) = T(Tp) = T(p ® 1) = 1, so ¢(T) = w. Thus ¢ is surjective. Since ¢ is affine and norm continuous, then ¢-l maps faces of B3 to norm closed faces of K. For faces F and G of B 3 ,
Thus ¢-l preserves greatest lower bounds. Since x f---> x ® 1 is a unital *-homomorphism from lvI2(C) into lvI2(C) ® B, then ¢-l preserves complements (Lemma 5.15). Since (F!\ G)I = FI V G I for projective faces F, G, then ¢-l also preserves least upper bounds, and thus is an orthoisomorphism from the lattice of faces of B3 into the lattice of norm closed faces of K. This proves (i). For each !J E Auto(B3) there is a unique *-automorphism S of .M2(C) such that !J = S*. (See the discussion preceding this proposition.) Let I d be the identity automorphism of B. Then S ® I d is a *-automorphism of M, so (S ® Id)* is in the space Aut(K) of affine automorphisms of K. Observe that !J f---> S is an anti-isomorphism from the group Aut o(B 3) onto the group of all *-automorphisms of M2(C), Define 7r : Auto( B 3 ) -+ Aut(K) by 7r(!J)
= (S®Id)*,
where !J = S*. Then 7r is a group isomorphism into Aut(K). Let F be an arbitrary face of B 3 , with associated projection p. Now we show that 7r(RF) = R¢-l(F)' Then RF is the reflection U;, where s = p - pi (Proposition 10.4). Therefore
Note that s ® 1
= P ® 1 - pi ® 1. Thus 7r(RF) is the reflection Rc where
G is the face of K associated with the projection p ® 1
= Tp. By (10.3), G = FTp = (T*)-l(Fp) = ¢-l(F), so 7r(RF) = R¢-l(F)' Next we show !J 0 ¢ = ¢ 0 7r(!J) for each !J E Auto(B3). Write !J = S*, with S a *-automorphism of lvI2(C), For x E M 2 (C), we have
T(S(x)) = S(x) ® 1 = (S ® Id)(x ® 1) = (S ® Id)(Tx) ,
THE GENERALIZED 3-BALL PROPERTY
so To S
365
= (S 181 Id) 0 T. Dualizing gives U
Setting (j
0 9 = 9 0 (S 181 I d) * = 90 7r( u).
= 7r(u) proves (ii) and completes the proof of the proposition.
0
Remark. By Proposition 10.15 a blown up 3-ball can be mapped onto the standard 3-ball B3 by an affine map which "pulls back" the face lattice of B3 into the face lattice of K and "lifts" the rotation group of B3 into the group of all affine automorphisms of K. This shows that a blown up 3-ball possesses much of the affine structure of the standard 3-ball, and is another reason for our choice of that name. Note that it is not the case that each affine surjection 9 : K ----) B3 which admits a group isomorphism from Auto(B3) into Aut(K) with these properties will admit a group isomorphism from Aut(B 3) with the same properties. \Ve will show this by an example where 9 is as in the proof above and B is chosen to be a von Neumann factor not *-anti-isomorphic to itself, (e.g., [37]). Then it is straightforward to check that the von Neumann algebra M 2 (C) 181 B is also a von Neumann factor. As in the proof above, define T: lvh(C) ----) M by Tx = x 1811, and recall that 9 is the dual map from K onto B 3 , and has the properties described in (ii). Now let 1j; be the transpose map on Jl.h (C). Then 1j;* is an affine automorphism of B 3 , but since 1j; is a *-anti-automorphism of B 3 , then U = 1j;* is in Aut( B 3 ) but is not in Auto(B3). Now suppose that there is a lift (j E Aut(K), i.e., 9 0 (j = U 0 9. Let 1.{t : M ----) M be the Jordan isomorphism such that 1.{t* = (j (Proposition 5.16). By Kadison's theorem, cf. (A 158), since M is a factor, 1.{t is either a *-isomorphism or an *-anti-isomorphism. Since 90 (j = U 0 9, then T* 0 1.{t* = 1j;* 0 T*. Therefore 1.{t 0 T = To 1j;, so for all x E lvh(C) we have 1.{t(x 181 1) = 1j;(x) 181 1. Hence 1.{t restricts to a *-anti-isomorphism on the subalgebra lVh(C) 181 C1. Since 1.{t is either a *-isomorphism or a *-anti-isomorphism on M, the latter must hold. Since 1.{t leaves invariant the *-subalgebra lvI2 (C) 181 C1, it must leave invariant the relative commutant of this subalgebra, namely 1181 B. Then 1.{t induces a *-anti-isomorphism of B, a contradiction. Thus in Proposition 10.15, Auto(B3) cannot be replaced by Aut(B 3 ). It follows that in the definition of global 3-balls in [71], Aut( B 3 ) should be replaced by Auto (B 3 ). With this change in definition, the results and proofs in [71] hold without other change. The generalized 3-ball property
In this section we will define the key property needed to insure that a JBW-algebra normal state space is in fact a von Neumann algebra normal state space. We start by establishing some useful properties concerning equivalence of orthogonal projections. These properties generalize known
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properties of projections in von Neumann algebras, cf. (A 175) for Lemma 10.16, and (A 179) and (A 195) for PropositIOn 10.17.
10.16. Lemma. If p and q aTe oTthogonal, equwalent PToJcctwns m a JB W-algebm M, then p and q aTe exchanged by a symmetTY.
Pmoj. Suppose first that A1. has no 12 or 13 direct summand. By Theorem 4.23, we may assume IvI is represented as a concrete JW-subalgebra of B(H)sa, and is reversible (Corollary 4.301. By assumption there are symmetries 81, ... ,8" with 8182'" SnP8 n '" 81 = q. Let v = 8182 ... 8n p. Then vv* = q and v*v = p, so v is a partial isometry with initial projection p and final projection q. Let 8 = V + v*. Then 8 is a (p + q)symmetry in B(H)sa that exchanges p and q. By reversibility of AI, 8 = 8182 ... 8n P + p8 n ... 8281 E M. By Lemma 3.5, p and q are exchangeable by a symmetry. It remains to establish the lemma's conclusion when p and q are orthogonal, equivalent projections in a JBW-algebra ]1.1 of type 12 or 1 3 . We claim that in this case p and q are abelian projections, and so are exchangeable (Corollary 3.20). Since factor representations separate the points of M (Lemma 4.14), it suffices to show that 7r(p) and 7r(q) are abelian projections for every factor representation 7r. Let 7r be a factor representation of ]1.1., and define N = 7r(lvf) (a-weak closure). Here N will be a type 12 or 1:3 JBW-factor (Lemma 4.8). Then 7r(p) and 7r(q) are exchangeable orthogonal projections in N. If they were not zero and not minimal projections, then each would dominate at least two non-zero orthogonal projections, so N would contain at least four orthogonal non-zero projections, contrary to the properties of factors of type h or 13 (Lemma 3.22). Thus 7r(p) and 7r(q) must be either minimal projections or zero, and so in either case are abelian. This completes the proof that p and q are abelian, and thus are exchangeable by a symmetry. D A key step in the next proof is to show that if p and q are projections with lip - qll < 1, then p and q can be exchanged by a symmetry. (See (A 178) for the corresponding von Neumann algebra result.) We will see that this follows from the fact that for any projections p and q, the projections p - p 1\ q' and q - q 1\ pi can always be exchanged by a symmetry. For von Neumann algebras, a proof of the latter fact can be found in (A 177). For JBW-algebras, note that
= P 1\ (pi V q) = T(Upq), q - (q 1\ pi) = q 1\ (q' V p) = T(Uqp),
p - (p
1\
q/)
where the last equality in each equation follows from Proposition 2.3l. There is a symmetry 8 that exchanges Upq and Uqp (Lemma 3.46). Since Us is a Jordan automorphism, it will also exchange the range projections of Upq and Uqp, and therefore will exchange p - P 1\ q' and q - q 1\ p'.
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10.17. Proposition. If p and q are orthogonal projectIOns m a JBWalgebra AI, then p and q are equwalent Iff there IS a (norm continuous) path of projectIOns from p to q. Proof. Suppose first that p and q are equivalent, orthogonal projections. By Lemma 10.16, p and q are exchanged by a symmetry. If e = p+q, by Lemma 3.5 we can choose an e-symmetry s that exchanges p and q, i.e., Us(p) = q. Then Us(p - q) = q - p, so by the equivalence of (iii) and (iv) in Lemma 3.6 (applied to Me), we have so (p - q) = o. Now let St = (cos 1ft )(p - q) + (sin 1ft)s for 0 S; t S; 1. Then for each t, the element St is an e-symmetry, and So = P - q, while SI = q - p. Thus there is a path of e-symmetries from p - q to q - p. Then Pt = ~ (St + e) will be a path of projections from p to q. Conversely, suppose that p and q are projections such that there exists a path of projections from p to q. Then we can find a finite sequence of projectionll> Po = p, PI, ... ,Pn = q such that Ilpi - Pi+lll < 1 for 0 S; i S; n - 1. We will be done if we show that there is a symmetry Si such that Us,Pi = PHI for 0 S; I S; n - 1. For simplicity of notation we may assume without loss of generality that n = 1, so that lip - qll < 1. We are going to show that p 1\ q' = q 1\ p' = O. Since p 1\ q' S; q', then p 1\ q' is orthogonal to q, so (p 1\ q') 0 q = o. Thus by the lB-algebra norm requirement (1.6),
lip 1\ q'll = II(p 1\ q') 0
(p -
q)11
S;
II(p 1\ q')llllp - qll
S;
lip - qll < 1.
Since the projection p 1\ q' must have norm 1 or 0, then p 1\ q' = o. By a similar argument, q 1\ p' = o. As observed before the statement of this proposition, the projections p-pl\q' and q-ql\p' can always be exchanged, so we conclude that p and q can be exchanged by a symmetry whenever lip - qll < 1. This completes the proof of the proposition. 0
10.18. Corollary. If F and G m·e orthogonal norm closed faces of the normal state space of a JB W-algebra !vI, wIth associated projectIOns p and q, the followmg are eqUIValent.
(i) p and q (ii) p and q (iii) There IS (iv) There IS that RH
are equwalent. are exchangeable by a symmetry. a (norm contmuous) path of prOjectIOns from p to q. a norm closed face H WIth assocwted reflectIOn RH such exchanges F and G.
Proof. The equivalence of (i) and (ii) is Lemma 10.16. The equivalence of (i) and (iii) is Proposition 10.17. A symmetry s exchanges p and q iff U; exchanges F and G (Corollary 5.17). Since the set of maps U; for s a symmetry is the same as the set of reflections RH for H a projective face (Proposition 10.4), this establishes the equivalence of (ii) and (iv). 0
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VON NEUMANN ALGEBRA NORMAL STATE SPACES
In a von Neumann algebra, projections can be exchanged by a finite product of symmetries iff the projections are unitarily equivalent (A 179), so the statements above correspond to unitary equivalence of F and G. For von Neumann algebras, the statement (iii) above is equivalent to there being a (norm continuous) path from F to G in the Hausdorff metric (A 196). 10.19. Definition. Let M be a JBW-algebra with normal state space K. Projective faces F, G of K are equwalent if their associated projections are equivalent. Note that if F and G are orthogonal, then F and G will be equivalent iff any of the conditions (i), (ii), (iii), or (iv) of Corollary 10.18 hold. If K is the normal state space of a JBW-algebra, we topologize the norm closed faces of K by using the 1-1 correspondence of norm closed faces and projections and carrying over the norm topology from the set of projections. Then by Corollary 10.18 orthogonal norm closed faces are equivalent iff there is a norm continuous path from one face to the other. We are now ready for the key definition. By (A 120), the face generated by any two extreme points of the normal state space of B (H) is a 3- ball, and by Theorem 10.2 and Theorem 9.34, this property distinguishes the normal state space of B(H) among the normal state spaces of JBW-factors of type I. More generally, if two pure states on a C* -algebra are unitarily equivalent, then the face they generate in the state space will be a 3ball (A 143). The following definition can be thought of as generalizing this property to the context of general norm closed faces, and is useful in situations where there may be no extreme points. Recall that norm closed faces F, G are orthogonal if they are orthogonal in the orthomodular lattice F, i.e., if F c G' , or equivalently, if the associated projections are orthogonal.
10.20. Definition. Let K be the normal state space of a JBWalgebra M. Then K has the generalzzed 3-ball property if every norm closed face E = F V G generated by a pair F, G of orthogonal, equivalent projective faces is a blown up 3-ball. 10.21. Lemma. Let M be a lEW-algebra wzth normal state space K wzth the generalzzed 3-ball property, let p and q be orthogonal proJectwns exchanged by a symmetry, and let e = p+q, and r = p-q. Then there zs an e-symmetry s that exchanges p and q, and for each such partwl symmetry s there zs a partwl symmetry t such that (r, s, t) zs a Carteswn trzple of e-symmetries. Proof. Let F, G be the projective faces associated with p, q respectively. By Corollary 5.17, since p and q are exchanged by a symmetry w,
THE GENERALIZED 3-BALL PROPERTY
then
F
V
369
U,7J exchanges G and F, so F and G are equivalent. We can identify
G with the normal state space of !'vie (Proposition 2.62). By the gen-
eralized 3-ball property, F V G is a blown up 3-ball, and so by Corollary 10.13, lvIe is Jordan isomorphic to the self-adjoint part of a von Neumann algebra M. We identify Me with Msa. Let l' = P ~ q. Since p and q are exchanged by a symmetry, by Lemma 3.5 there is an e-symmetry s exchanging p and q, so by Lemma 3.6, ros = O. Then 1', s anticommute in M. Define t = zrs E Msa = Me. It is straightforward to verify that (1', s, t) is a Cartesian triple of e-symmetries, cf. (A 192). D
10.22. Lemma. If M zs a lEW-algebra whose normal state space
K has the generalzzed 3-ball property, and !'vil zs any dzrect summand of M, then the normal state space Kl of lvh also has the generalzzed 3-ball property. Proof. Let F and G be orthogonal equivalent norm closed faces of K l . If p and q are the projections in !'vh corresponding to F and G, then p and q are orthogonal and equivalent in M l , and then also in lvI (e.g., by Lemma 3.5). Thus F and G are equivalent in K, so F V G is a blown up 3-ba11. (Note that the latter property is the same whether F V G is viewed as a projective face of K or of K l , cf. Definition 10.10). D
10.23. Proposition. If M zs a von Neumann algebra wdh normal state space K, then K has the generalzzed 3-ball property. Proof. Let F and G be orthogonal equivalent projective faces of K with associated projections p and q. Let e = p + q. There is a symmetry that exchanges p and q (Lemma 10.16), and then by Lemma 3.5 there is an e-symmetry s that exchanges p and q. By Lemma 3.6 the e-symmetries l' = P ~ q and s satisfy l' 0 S = O. By direct calculation or (A 192), if we define t = zrs, then (1', s, t) is a Cartesian triple of e-symmetries. It follows that the face E = F V G associated with e is a blown up 3-ball (Lemma 10.9). Thus K has the generalized 3-ball property. D
10.24. Lemma. If a lEW-algebra .M has the generalzzed 3-ball property, then lvI zs a lW-algebra, and zs reverszble m every representatwn as a concrete lW-algebra. Proof. We first show that lvI is a JW-algebra. Write M = Mo EEl M3 where M3 is the type 13 summand of lvI. Then Mo is a JW-algebra (Theorem 4.23). Let Pl, P2, P3 be equivalent abelian projections in Nh with sum the identity of M 3. By Corollary 3.20, Pl and P2 can be exchanged by a single symmetry. By Lemma 10.21 there exists a Cartesian triple (1', s, t) of (Pl + P2)-symmetries with l' = Pl ~ P2.
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VON NEUl\IANN ALGEBRA NORMAL STATE SPACES
Now let 7r be a factor representation of M 3 , and define N = 7r(M3) (aweak closure). Since P1, P2, P3 are exchangeable and have sum the identity (of M 3 ), then their images are exchangeable with sum 1. In particular, none of the images can be zero, and so (7r (r), 7r ( S ), 7r (t)) is a non-zero Cartesian triple of partial symmetries in N. By Lemma 4.52, H3(O) contains no such triple, so N cannot be the exceptional .1BW-factor. By Corollary 4.20, M3 is a .1W-algebra, and thus so is M. Now assume that M is represented as a concrete .1W-algebra, and let !vh be the 12 direct summand of M. By Corollary 4.30, M is reversible iff M2 is reversible. By definition of type 1 2, the identity of M2 is the sum of equivalent abelian projections P and q, which can be exchanged by a symmetry. By the generalized 3-ball property and Lemma 10.21, AI contains a Cartesian triple of symmetries. Thus by Lemma 4.53, 11/[2 is reversible, and hence AI is reversible. D
10.25. Theorem. A convex set K zs affinely zsomorphzc to the normal state space of a von Neumann algebra zff K IS spectral, ellzptIc, and has the generalned 3-ball property.
Proof. The necessity of the first two of these conditions follows from Theorem 9.48, and the third follows from Proposition 10.23. Conversely, assume that K is spectral, elliptic, and has the generalized 3-ball property. From Theorem 9.48, K is affinely isomorphic to the normal state space of a .1BW-algebra M. Identify K with the normal state space of M. We will show that Ai is isomorphic to the self-adjoint part of a von Neumann algebra. By Lemma 10.24, we may assume that !v[ is represented as a reversible concrete .1W-subalgebra of B (H),a' Let Ro(J'vl) denote the real *-algebra generated by AI; by reversibility, Ro(AI)sa = AI. Recall that by Lemma 10.22 the normal state space of each direct summand of M also has the generalized 3-ball property. If M is of type I 1, then M ~ CR(X) for a compact Hausdorff space X, so Iv[ ~ Msa where M = Cc(X). Since AI is monotone complete and has a separating set of normal states, then M is a von Neumann algebra (A 95). If Mis of type Ik for 2 -s: k -s: 00, or has no type I part, then by Propositions 3.24 and 3.17, the identity is the sum of a finite set Pl, P2, ... ,pn of n exchangeable projections, with n 2 2. For z = 1,2, ... ,n let Si be a symmetry that exchanges P1 and p,. Define
By Lemma 4.28 the family {eij} is a system of matrix units for Ro(AI), and eij + eji = SiP1Sj + SJP1Si E M (by reversibility, or by (1.12)). Now S = e12 + e21 exchanges ell and e22, and is a (ell + e22)-symmetry. By the
THE GENERALIZED 3-I3ALL PROPERTY
371
generalized 3-ball property and Lemma 10.21, there is a partial symmetry t such that (ell ~ e22, e12 + e21, t) is a Cartesian triple of partial symmetries. Thus we can apply Lemma 4.57 to find J central in Ro(M) such that J* = ~J and J2 = ~l. Now by Lemma 4.55, M is isomorphic to the self-adjoint part of a von Neumann algebra. Finally we consider the case where !vI is an arbitrary JBW-algebra. Recall that !vI can be written as the direct sum of JBW-algebras of type I and a JBW-subalgebra with no summand of type I (Lemma 3.16), and the former can in turn be written as the direct sum of JBW-algebras of type Ik for 1 k CXJ (Theorem 3.23). We have shown that each of these summands is isomorphic to the self-adjoint part of a von Neumann algebra, so it follows that the same is true of !vI. This completes the proof of the theorem. 0
s: s:
Remark. If K is affinely isomorphic to the normal state space of a von Neumann algebra M, then K determines the Jordan product on M. In fact, the evaluation map is a bijection from M ,a onto the bounded affine functions on K (A 97), and the functional calculus for such bounded affine functions was defined in Proposition 8.67. Then the Jordan product on Msa can be recovered from ~(ab+ba) = i((a+b)2~(a~b)2). However, the associative product on M is not determined by the affine structure of K. For example, the transpose map on !vh(C) is a *-anti-automorphism, although its dual map is an affine automorphism of the state space. Thus some additional structure on K is needed to determine the associative product on M. One such structure is that of an onentatwn of K (A 201). We recall: 10.26. Theorem. If M lS a von Neumann algebra, then there lS 1-1 correspondence between Jordan compatlble assocwtwe products in M and global orlentatwns of the normal state space K of M. Proof. [AS, Thm. 7.103]. 0 For a discussion of the geometric interpretation of orientations on the normal state space of a von Neumann algebra, see the remarks at the end of Chapter 6. State spaces of C* and von Neumann algebras can also be characterized among state spaces of Jordan algebras (JB and JBW) by the existence of a dynamical correspondence (cf. Definition 6.10), which also determines the associative product. This is explained in the proposition below, which is essentially a reformulation of information from Theorem 6.15 and Corollary 6.16, restated in terms of state spaces instead of algebras. 10.27. Proposition. The state space K of a fE-algebra A lS affinely homeomorphic to the state space of a C*-algebra A iff there is a dynamical
372
10
VON NEUMANN ALGEBRA NORMAL STATE SPACES
correspondence on A. The nor'mal state space J( of a lEW-algebra M zs affinely zsomorphzc to the state space of a von Neumann algebra M ziJ there zs a dynamzcal correspondence on A-1. In these cases, the dynamzcal correspondence determznes a unzque C*-product on A + zA (respectzvely, W*-product on M + zM ). Proof.
J(
is affinely homeomorphic to the state space of a C*-algebra
A iff A is isomorphic (as a JB-algebra) to the self-adjoint part of A (Proposition 5.16). Furthermore, this 0CCurS iff A admits a dynamical correspondence, and dynamical correspondences are in 1-1 correspondence with C*-products on A + zA (Theorem 6.15). The JBW-algebra result is proven in the same way by means of Corollary 6.16. 0 Recall from Theorem 6.18 that there is a 1-1 correspondence between dynamical correspondences and Connes orientations (Definition 6.8) on a JBW-algebra M. Thus we also have the following characterization.
10.28. Proposition. The normal state space
J(
of a lEW-algebra M
zs affinely zsomorphzc to the normal state space of a lEW-algebra M ziJ there zs a Connes orzentatzon on M. In thzs case such a Connes orzentatzon
determines a unzque W* product on M
+ zA1.
Proof. This follows from Proposition 10.27 and Theorem 6.18. 0 Note that a Connes orientation holds more information than the generalized 3-ball property, since it provides a uniquely determined von Neumann algebra whereas the W*-product constructed from the generalized 3-ball property is not unique. (It depends on the choice of symmetries used to construct matrix units in the proof of Theorem 10.25.) Since a Connes orientation determines the associative product when the Jordan product is known, then in this respect it serves the same purpose as the concept of orientation defined for a von Neumann algebra (or its normal state space) in (A 200) and (A 201). Otherwise the two concepts of orientation have little in common. The concept of a Connes orientation is algebraic and global in nature, while the concept of orientation from [AS] is geometric and locally defined in terms of facial blown up 3-balls (A 194). The relationship between these concepts of orientation was discussed in greater detail in the last section of Chapter 6. A dynamical correspondence provides a physically meaningful way to single out C*-algebra state spaces among those of lB-algebras. In the next chapter we will show how one can characterize state spaces of general C*algebras by a set of geometric axioms involving a concept of "orient ability" defined in terms of facial 3-balls as in (A 146), but for convex sets more general than state spaces of C*-algebras.
NOTES
373
Notes The geometric characterization of the normal state space of B (H) (Theorem 10.2) follows easily from the characterization of normal state spaces of type I JBW-factors given in Theorem 9.34, which first appeared in [10]. The characterization of von Neumann algebra state spaces (Theorem 10.25) is due to Iochum and Shultz [62]. It was actually preceded by the characterization of state spaces of C* -algebras (Theorems 11.58 and 11.59 in the next chapter), which appeared in [11]. Theorem 10.26 was announced in [13]' and the proof first appeared in [AS]. In the C* -algebra theorem in Chapter 11, we will see that the Hilbert ball property used in the characterization of JB-algebra state spaces is replaced by the 3-ball property. However, this property is of no use in characterizing normal state spaces of von Neumann algebras, since there may be no extreme points for the normal state space, and thus no facial 3-balls. Therefore, the 3-ball property has been replaced by the concepts of a "blown up 3-ball" and of the "generalized 3-ball property" (Definitions 10.10 and 10.20), which were first given in [71]. There the blown up 3-balls were called "global 3-balls", and they were defined, not by the existence of a 3-frame as in this book, but by the existence of an affine surjection onto a bona fide 3-ball which pulls back the lattice of faces and the group of affine automorphisms in an appropriate way, as in Proposition 10.15.
11
Characterization State Spaces
of
C*-algebra
In this chapter we will reach our final goal: a characterization of the compact convex sets that are state spaces of C*-algebras. (We will assume our C*-algebras have an identity, but our characterization can easily be adapted for non-unital algebras). We will start with our previous characterization of state spaces of JB-algebras (Theorem 9.38). Then we will add two additional properties that characterize C*-state spaces among state spaces of JB-algebras. A key axiom for state spaces of JB-algebras is the Hilbert ball property, by which the face generated by any two pure states is a norm exposed Hilbert ball (Definition 9.9). These balls may be of any finite or infinite dimension (as can be seen from Proposition 5.51). But in the case of C*-algebras, they have dimension 0, 1 or 3, with the last occurring iff the two states are distinct and unitarily equivalent, or which is the same, iff they generate the same split face, cf. (A 142) and (A 143). This is the 3-ball property (Definition 11.17), which will be a key axiom for our characterization of state spaces of C* -algebras. We know (from Corollary 5.42) that each pure state p on a JB-algebra A determines a unital Jordan homomorphism 1fp : A ----+ !v! where M is a JBW-factor of type I and 1fp(A) is o--weakly dense in M. Briefly stated: 1fp is a dense type I factor representatwn. The dense type I factor representations separate the points of A (Lemma 4.14), and each of them is Jordan equivalent (Definition 11.4) to 1fp for a pure state p on A (Lemma 11.16). We show in this chapter that the 3- ball property for the state space K of a JB-algebra A is equivalent to A being of complex type, i.e., that for each dense type I factor representation 1f : A ----+ !v! we have M ~ B (H)sa with H a complex Hilbert space (Theorem 11.19). Thus if the state space K of A has the 3- ball property, then we can associate to each pure state P E K a concrete representation of A on a complex Hilbert space. This concrete representation is irreducible (Lemma 11.3), but it is not generally unique (up to unitary equivalence). By associating an irreducible concrete representation to each pure state as described above, and by forming the direct sum of all these representations, we can obtain a faithful representation 1f : A ----+ B (H)sa on a complex Hilbert space. But this does not solve our characterization problem, since the lB-algebra 1f(A) is not in general equal to the self-adjoint part of the C*-algebra it generates in B (H). However, after imposing a
376
11
C*-ALGEBRA STATE SPACES
second key axiom on K, we can choose concrete representations associated with the pure states in such a way that this is in fact the case. This second axiom involves the concept of a "global orientation" , which is a direct generalization of the same notion for the state space of a C*algebra (A 146). Thus a global orzentatwn of the state space K of a JBalgebra is a continuous choice of orientation of all those faces of K that are 3-balls, and K is said to be oT'zentable if it admits such a global orientation (Definition 11.50). Together, the 3-ball property and orientability will characterize state spaces of C* -algebras arlOng all state spaces of JB-algebras. Our proof of this result involves the non-trivial fact that a JB-algebra of complex type is a universally reversible JC-algebra (Theorem 11.26). A universally reversible JC-algebra is imbedded in its universal C*-algebra in a particularly nice fashion (Corollary 4.42), and we use this fact to show that in our case each pure state is associated with no more than two inequivalent concrete representations. More specifically, if the state space K of a JB-algebra A has the 3-ball property, then each pure state p on A is the restriction to A of just two pure states a and T on the universal C*-algebra U of A (with a = T iff P is "abelian"). Then the GNS-representations of U associated with a and T restrict to a pair of conjugate irreducible representations of A, and those are (up to unitary equivalence) the only irreducible concrete representations associated with the pure state p on A (Propositions 11.32 and 11.36). Now the problem is to select for each pure state p on A either the one or the other of the two pure states a and T on U in such a way that the direct sum of the corresponding irreducible concrete representations maps A onto the self-adjoint part of a C*-algebra. We show that if K is orientable, then this is possible; in fact, each global orientation of K will determine such a selection, so in this case A is isomorphic to the self-adjoint part of a C*-algebra. Here the orient ability condition is necessary since the state space of every C* -algebra is orientable (A 148), and is irredundant since there are examples of JB-algebras of complex type with non-orientable state space (Proposition 11.51). Thus a JB-algebra is isomorphic to the self-adjoint part of a C* -algebra iff its state space has the 3-ball property and is orient able (Theorem 11.58). Combining this result with the characterization of state spaces of JBalgebras (Theorem 9.38), we obtain a complete geometric characterization of the compact convex sets that are state spaces of C*-algebras (Theorem 11.59). By a short argument this also gives the 1-1 correspondence of Jordan compatible C*-products and global orientations (Corollary 11.60), which was proven without use of Jordan algebra theory in (A 156). States and representations of JB-algebras
Recall that a factor representation of a lB-algebra A is a unital homomorphism from A into a lBW-algebra such that the a-weak closure of 1T(A)
STATES AND REPRESENTATIONS OF JB-ALGEBRAS
377
is a JBW-factor (Definition 4.7). A factor representation is fmthful if it is 1-1. Since there are JB-algebras that admit no non-zero homomorphisms into B(H) (e.g., the exceptional algebra H 3 (O)), we need a substitute for the notion of irreducibility of a representation. For a C* -algebra A, a representation 7T : A -+ B (H) is irreducible iff the image is a-weakly dense, cf. (A 80) and (A 88). That motivates singling out the following class of representations.
11.1. Definition. Let A be a JB-algebra and M a JBW-algebra. A factor representation 7T : A -+ !vI is dense if the a-weak closure of 7T(A) equals M. Recall from Proposition 2.68 that the a-weak and a-strong closures of convex sets in a JBW-algebra coincide, so we could use either topology in the definition above. In general, for subspaces X of a JBW-algebra, X will denote the a-weak closure.
11.2. Definition. A concrete representation of a JB-algebra A is a Jordan homomorphism 7T from A into B(H)sa for a complex Hilbert space H. Such a representation is zrreduczble if there are no non-trivial closed subspaces invariant under 7T(A). 11.3. Lemma. Let A be a JB-algebra. Then every dense concrete representatwn 7r: A -+ B (H)sa zs lrreduclble. Proof. If p is a projection in B (H)sa such that the closed subspace pH is invariant under 7r(A), then p7r(a)p = 7r(a)p for all a in A. Then taking adjoints gives 7r(a)p = p7r(a) , and by density of 7r(A) in B(H)sa, p must be central in B(H), and thus equal to 0 or 1. Thus 7r(A) has no non-trivial closed invariant subspaces. 0 Note that the converse to Lemma 11.3 is not true. For example, the JB-algebra of real n x n symmetric matrices acts irreducibly on C n (since the complex algebra generated includes all matrix units and thus all of Mn(C)), and yet is not dense in Mn(C)sa. Recall that representations 7rl, 7r2 of a C* -algebra A are said to be quasz-equivalent if there is a *-isomorphism from 7rl (A) onto 7r2 (A) carrying 7rl to 7r2 (A 136). Now we define the analogous Jordan notion.
11.4. Definition. Let A be a JB-algebra, Ml and lvh JBW-algebras, and let 7Tl : A -+ Ml and 7r2 : A -+ M2 be homomorphisms. Then 7rl and 7T2 are Jordan equivalent if there exists a Jordan isomorphism from 7rl(A) onto 7r2(A) such that (7rl(a)) = 7r2(a) for all a E A.
378
11.
C*-ALGEBRA STATE SPACES
Recall that if A is a JB-algebra and M a JBW-algebra, then each homomorphism 11' : A ---> M admits a unique extension to a normal homomorphism n: A** ---> AI (Theorem 2.65). Then n is a-weakly continuous (Proposition 2.64), so its kernel is a a-weakly closed ideal of A **. Thus by Proposition 2.39, the kernel has the form cA ** for some central projection c in A**.
11.5. Definition. Let A be a JB-algebra, M a JBW-algebra, and 11' : A ---> AI a homomorphism. The central cover of 11' is the central projection c(11') in A** such that kern = (1 - c(11'))A**.
n,
Note that 1 - c( 11') is the largest central projection in A ** killed by and thus c(11') is the smallest central projection c such that n(c) = 1. Most of the remaining results in this section are close analogs of similar results for C*-algebras, d. [AS, Chpt. 5].
11.6. Lemma. Let A be alE-algebra, M a lEW-algebra, 11' : A ---> IvI a homomorphzsm, and n : A ** ---> /vI the umque normal extension. Then n restrzcted to c(11')A** zs a *-zsomorphzsm onto 11'(A) eM.
Proof. By Proposition 2.67, n(A**) = 11'(A). Since (kern) nC(11')A** = (1 - c(11'))A** n c(11')A** = {O}, then n is faithful on c(11')A**. For each element a E A** we have n(a) = n(c(11')a + (1 - c(11'))a) = n(c(11')a), so n maps c(11')A** onto n(A**) = 11'(A). Thus n restricted to c(11')A** is a *-isomorphism onto 11'(A). 0 11.7. Proposition. Let A be a lE-algebra and 11'1 and 11'2 homomorphisms of A mto lEW-algebras. Then 11'1 and 11'2 are lordan equwalent zff c(11'd = C(11'2)'
Proof. Let AI, = 11'i(A) for z of M1 onto IvI2 such that
= 1,2. Let
be a Jordan isomorphism
(11.1) for all a in A. Every Jordan isomorphism between JBW-algebras is normal, and therefore a-weakly continuous (Proposition 2.64). By a-weak continuity of n1 and n2, and a-weak density of A in A **, the equality (11.1) extends to a E A** if we replace 11'; by ni for z = 1,2. It follows that ker n1 = ker n2, and so by definition c( 11'd = c( 11'2)' Conversely, if c(11'd = C(11'2)' then kern1 = kern2, and so we can define : n1(A**) ---> n2(A**) by
(n1(x)) = n2(x).
STATES AND REPRESENTATIONS OF JB-ALGEBRAS
379
The agreement of the kernels is exactly what is needed to make well defined and a *-isomorphism. By Proposition 2.67, 7ri(A**) = Jri(A) = Mi for ~ = 1,2. Thus is a Jordan isomorphism from M1 onto ]vh carrying Jr1 to Jr2. 0 Proposition 11. 7 can be thought of as generalizing the fact that representations of a C*-algebra are quasi-equivalent iff their central covers coincide (A 137).
11.8. Lemma. Let A, B be lB-algebras and Jr : A -> B a lordan from A onto B. If 1 ~s the kernel of Jr and F ~s the anmh~lator of 1 m the state space of A, then Jr* ~s an affine homeomorphism from the state space of B onto the splzt face F. homomorph~sm
Proof. This follows from Corollary 5.38. 0
Recall that if A is a JB-algebra, each state on A has a unique extension to a normal state on A**, so the restriction map is an affine isomorphism from the normal state space of A ** onto the state space of A (Corollary 2.61). As usual we identify the state space of A and the normal state space of A **. Thus if p is a projection in A **, Fp denotes the set of normal states on A ** (or equivalently, states on A) with value 1 on p.
11.9. Lemma. Let A be alE-algebra wdh state space K and let A -> M be a dense factor representatzon mto a lE W-algebra M. Then Jr* ~s an affine ~somorphism from the normal state space of M onto Fe(IT) c K.
Jr :
Proof. Let 7r : A** -> ]vI be the normal extension of Jr. For each normal state a on M, a 0 7r is the unique normal extension of a 0 Jr = Jr* a. Thus to prove the lemma we need to show that a f---> a 0 7r is an affine isomorphism from the normal state space of AI onto Fe(IT). Since the annihilator of ker 7r in K is
((1 - c(Jr))A**r n K
= (im U 1- c (IT)r n K = (ker U~-e(IT)) n K =
Fe(IT)
(cf. equation (5.4)), the lemma follows by an argument similar to that in the proof of Corollary 5.38. 0 11.10. Proposition. Let A be a lE-algebra, and let Jr1 : A -> M1 and Jr2 : A -> M2 be dense factor representatzons. Let K1 and K2 be the normal state spaces of M1 and M2 respectwely. Then Jr1 and Jr2 are Jordan equwalent iff Jri(Kd = Jr2(K2). Proof. We have Jri(KI) = Jr 2(K 2 ) iff Fe(IT,) i.e., iff C(Jrl) = C(Jr2). This in turn holds iff Jr1 and (Proposition 11. 7). 0
= Jr2
F e(1I"2) (Lemma 11.9), are Jordan equivalent
380
11.
C*-ALGEBRA STATE SPACES
For our purposes, we will be most interested in concrete dense representations. In this case, Jordan equivalence is closely related to unitary equivalence as we shall now see.
11.11. Definition. Let A be a JB-algebra. Then concrete representations 7ri : A ----) B(Hd and 7r2 : A ----) B(H2 ) are umtarzly equwalent (respectively, conJugate) if there is a complex linear (respectively, conjugate linear) isometry u: Hi ----) H2 such that
for all a E A.
11.12. Proposition. Let 7ri : A ----) B(Hdsa and 7r2 : A ----) B(H2)sa be concrete representatwns of a JB-algebm A. If 7ri and 7r2 are umtan.ly equwalent or conJugate, then they are Jordan equwalent. If 7ri and 7r2 are Jordan equwalent and dense, then they are ezther umtarzly equwalent or conJugate, with both posszbziztzes occurrmg zff the representatwns are one dzmenswnal. Proof. Suppose that 7ri and 7r2 are either unitarily equivalent or conjugate. Let u be as in Definition 11.11. Define : B(Hd ----) B(H2) by (x) = ux'u- i if u is complex linear, and by (x) = ux*u- i if u is conjugate linear. In the former case, is a *-isomorphism, and in the latter case it is a *-anti-isomorphism (A 124). Thus in either case is a Jordan isomorphism of B(Hdsa onto B(H2 )sa. A Jordan isomorphism is normal, and thus and -i are cr-weakly continuous (Proposition 264). Hence maps the cr-weak closure of 7ri(A) onto the cr-weak closure of 7r2(A). It follows that concrete representations that are either unitarily equivalent or conjugate are also Jordan equivalent. Let 7ri : A ----) B(Hdsa and 7r2 : A ----) B(H2)sa be dense and Jordan equivalent. Let be a Jordan isomorphism from B(Hdsa onto B(H2 )sa carrying 7ri to 7r2. Extend to a complex linear Jordan isomorphism from B(Hi ) onto B(H2), By Kadison's decomposition theorem for Jordan isomorphisms of von Neumann algebras (A 158), is either a *-isomorphism or a *-anti-isomorphism. Thus by (A 126) there is an isometry v : Hi ----) H2 which is either complex linear or conjugate linear such that
Therefore
and so
7rl
and
7r2
are either unitarily equivalent or conjugate.
STATES AND REPRESENTATIONS OF JB-ALGEBRAS
381
Finally, if 1f1 and 1f2 are unitarily equivalent and conjugate, then there is a *-isomorphism and a *-anti-isomorphism IlJ from B(H 1) onto B(H2) such that
(1f1(a)) = 1lJ(1f1(a)) = 1f2(a)
for all a E A
(A 124). Then 1lJ- 1 0 fixes 1f1(A). By density of 1f1(A) in B(Hdsa and complex linearity, 1lJ-1 0 is the identity on B(H1)' Since 1lJ- 1 0 is a *-anti-isomorphism, this implies that B(Hd is abelian. Thus H1 and H2 must be one dimensional. Conversely, suppose 1f1 and 71'2 are Jordan equivalent, and that H1 and H2 are one dimensional. Let : B(Hdsa ---> B(H2)sa be a Jordan isomorphism carrying 1f1 to 71'2' Then it is straightforward to check that any complex linear or conjugate linear isometry from H1 onto H2 implements , so 71'1 and 71'2 are both unitarily equivalent and conjugate. D Let CJ be a state on a JB-algebra A. Recall that the central carrier c(CJ) is the least central projection with value 1 on CJ (Definition 4.10), and that 71'" : A ---> c(CJ)A** is defined by 71',,(a) = c(CJ)a (Definition 4.11).
11.13. Lemma. Let CJ be a state on a fB-algebra A. Then c(71',,) equals c( CJ).
Proof. The unique normal extension of 71'" to A** (cf. Theorem 2.65) is given by 1i'" (a) = c( CJ)a for a E A **. The kernel of this extension is (1 - c(CJ))A**, which must by definition equal (1 - c(71',,))A**, and so c(71',,) = c(CJ) follows. D 11.14. Corollary. If A zs a fE-algebra wzth state space K, and CJ and T are states on A, then the followmg are equwalent.
(i)
71'"
and 71'T are fordan equwalent.
(ii) c(CJ) = C(T). (iii) The spilt faces of K generated by CJ and T coinczde.
Proof. By Proposition 11.7, 71'" and 71'T are Jordan equivalent iff c(1f,,) = C(1fT)' By Lemma 11.13, this is equivalent to (ii). By Proposition 5.44, the minimal split faces generated by CJ and Tare F e (,,) and Fe(T) respectively. By the correspondence of projections and norm closed faces (Corollary 5.33), Fe(,,) = Fe(T) iff c(CJ) = C(T), which implies the equivalence of (ii) and (iii). D
Remark. If CJ is a state on a C*-algebra A, temporarily denote the associated GNS representation (A 63) by 1/;" : A ---> B(H). Identify (A**)sa with (Asa)** (cf. Lemma 2.76). Then the central cover of the Jordan representation 1/;O'IAsa is c(CJ) (cf. (A 135)). Define 71'0' : Asa ---> (Asa)** by 1fO'(a) = c(CJ)a. The central cover of 1f0' is c(CJ) (Lemma 11.13), so by Proposition 11.7, 1/;0'1 Asa is Jordan equivalent to 71'0"
11
382
C*-ALGEBRA STATE SPACES
11.15. Definition. Let A be a JB-algebra, and M a JBW-algebra. A homomorphism 7r : A --> M is a type I factor representation if the 0'- weak closure of 7r(A) is a type I JBW-factor. Recall that if~ a pure state, then 7r a is a dense type I factor representation, i.e., 7r a (A) is a type I JBW-factor (Corollary 5.42). In fact, up to Jordan equivalence all dense type I factor representations arise in this way, as we now show. Recall that Fa denotes the smallest split face containing 0'. Lemma. Let A be a lE-algebra with state space K, and M a dense type I factor representatwn mto a lEW-algebra /vI. Then Fe(K) contains at least one pure state, and for every such pure state 0' m Fe(K)' 7r zs lordan equwalent to 7ra, and Fe(K) = Fa.
11.16.
7r :
A
-->
Proof. By Lemma 11.9, 7r* is an affine isomorphism from the normal state space of M onto Fe(K)' Since M is of type I, by Corollary 5.41 the normal state space of M contains an extreme point, and therefore so does Fe(K) . Let 0' be an extreme point of Fe(K)' Since M is a factor, it contains no non-trivial central projections, and thus its normal state space contains no proper split face (Corollary 5.35). Hence the split face Fa = Fe(a) generated by 0' must equal Fe(K) , so c(O') = c(7r). Since c(O') = c(7r a ) (Lemma 11.13), then c(7r) = c(7r a ). Therefore 7r is Jordan equivalent to 7r a (Proposition 11. 7). 0
Reversibility of JB-algebras of complex type We now define one of the key properties that distinguishes C*-algebra state spaces. Recall that a 3-ball is a convex set affinely isomorphic to the closed unit ball B3 of R3 (Definition 10.1). In the case of C*-algebras, the faces generated by distinct pure states are either line segments (iff the states are unitarily inequivalent), or 3-balls (A 143). We now abstract this property. Recall that for the state space of a JB-algebra, the face generated by any pair of pure states is a Hilbert ball (Corollary 5.56), but that this Hilbert ball might be of any dimension (cf. Proposition 5.51). If 0' and 7' are distinct pure states, the split faces Fa and FT generated by 0' and 7' will be minimal split faces (Lemma 5.46). These must either coincide or be disjoint, since the intersection of split faces is also a split face (A 7). The face generated by 0' and 7' will degenerate to a line segment iff Fa and FT are disjoint (Lemma 5.54), or equivalently, iff 7r0' and 7rT are not Jordan equivalent (Corollary 11.14).
REVERSIBILITY OF JB-ALGEBRAS OF COMPLEX TYPE
383
11.17. Definition. If K is a convex set, then K has the 3-ball property if the face generated by every pair of extreme points of K is norm exposed, and is either a point, a line segment or a 3-ball. Note that in Definition 11.17 we allow the two extreme points to coincide, in which case the generated face will be a single norm exposed extreme point. Note also that the convex sets with the 3-ball property are a subclass of the convex sets with the Hilbert ball property (for which every extreme point is norm exposed, cf. Definition 9.9). Of course, the 3-ball property will be useful only when there are many extreme points, e.g., when K is compact, or K is the normal state space of an atomic JBW-algebra (cf. Lemma 5.58).
11.18. Definition. A JB-algebra is of complex type if for every dense type I factor representation 7f : A ---+ A1 we have M ~ B (H)sa for a complex Hilbert space H. 11.19. Theorem. A iE-algebra A lS of complex type lff ltS state space has the 3-ball property. Proof. Suppose first that A is of complex type, and let 0' and T be pure states. If 0', T are separated by a split face, then face ( 0', T) is the line segment [0', T], cf. Lemma 5.54 or (A 29). Now suppose 0' and T are not separated by any split face, and let F = FO' = FT' Let 7f (J : A ---+ c( O')A ** be multiplication by c( 0'). Then 7f (J is a type I factor representation (Corollary 5.42). Since A is of complex type, then c(O')A** ~ B(H)sa. The normal state space of c(O')A** is F = FC((J) = F(J (Proposition 2.62). Hence we conclude that F is affinely isomorphic to the normal state space of B (H)sa. The latter has the 3- ball property (A 120), and thus so does F. The face of K generated by T and 0' is also a face of F, and so we conclude that it is a 3-ball. Thus we have shown that K has the 3-ball property. Conversely, assume K has the 3-ball property, and let M be a JBWalgebra, and 7f : A ---+ M a dense type I factor representation. Then M = 7f(A) is a type I JBW-factor. By Lemma 11.9, 7f* is an affine isomorphism from the normal state space of M onto FC(7r)' The normal state space of M is a split face of K, and thus inherits the 3-ball property from K. Let 0' and T be distinct extreme points of the normal state space of M. Since M is a factor, its normal state space has no proper split faces (Corollary 5.35). Then 0' and T cannot be separated by a split face, so face(O', T) is not a line segment (Lemma 5.54). By the 3-ball property, face(O', T) is a 3-ball. Thus M is a type I JBW-factor for which the face of the normal state space generated by each pair of distinct pure normal states is a 3-ball. Therefore M is isomorphic to B (H)sa for some complex Hilbert space H (Proposition 9.36). Hence A is a JB-algebra of complex type. 0
11.
384
11.20.
C*-ALGEBRA STATE SPACES
Corollary.
The self-adJomt part of a C*-algebra IS a JE-
algebra of complex type.
Proof. This follows from Theorem 11.19 and the fact that the state space of a C*-algebra has the 3-ball property, cf. (A 143) and (A 108). 0 The following simple observation will be used frequently.
11.21. Lemma. Every JEW-algebra M of fype Jordan orthogonal symmetrIes.
h
contams a pazr of
Proof. By the definition of type I2 (Definition 3.21), there are exchangeable projections P1 and P2 in M with sum 1 (Corollary 3.20). Let 8 be a symmetry that exchanges P1 and P2. Then by Lemma 3.6, sand the symmetry r = P1 - P2 are Jordan orthogonal. 0 Our next lemma involves Jordan polynomials in several variables. Such a polynomial is a sum of products, just like a standard polynomial. But by non-associativity, parentheses are needed to indicate how these products are to be multiplied out. Therefore each standard polynomial corresponds to a whole class of Jordan polynomials (obtained by inserting parentheses). Clearly all those take the same values as the given standard polynomial on every associative subalgebra, and then in particular on the center.
11.22. Lemma. Let S be any spm factor and let Sl and 82 be any two Jordan orthogonal symmetrIes m S. For each natural number k we can find a Jordan polynomzal P k m k + 2 varzables, with P k mdependent of the chozces 81, 82, such that k elements a1,"" ak of S are 1m early dependent Iff (11.2)
Proof. Recall that S = R1 EB N (vector space direct sum), where N is a real Hilbert space and the Jordan product in S is defined by (11.3)
(AI
+ x)
0
(p,1
+ y) =
(Ap,
+
(xly))l
+
(AY
+ p,x)
for x, YEN, and that N consists of all scalar multiples of symmetries in S other than ±1 (Lemma 3.34). We extend the inner product from N to all of S by setting (11.4)
(AI
+ x I p,1 + Y)
= Ap,
+
(xly)
for x, YEN, so that S becomes the Hilbert space direct sum of R1 and N. By the Gram criterion for inner product spaces, k elements a1, .. . ,ak
REVERSIBILITY OF JB-ALCEBRAS OF COMPLEX TYPE
385
of 5 are linearly dependent iff' det( (ai la J )) = O. We will now use this criterion to construct Pk . Observe first that if ai = Ail + x, with Xi E N and Ai E R for I = 1, ... , k, then by (11.4) (11.5) Furthermore, for any a = Al by means of (11.3) gives
+x
with x E N and A E R, multiplying out
(11.6) and then also (11. 7) Now it follows from (11.5), (11.6) and (11.7) that there is a Jordan polynomial Q such that (11.8) Choose a Jordan polynomial f in k 2 variables which takes the same values as the determinant of order k on the center Rl of 5, i.e.,
for all scalars a'j with 1 ~ I, J ~ k. (See the remark in the paragraph preceding this lemma). Substitute Q(sl,s2,ai,a J ) for the (IJ)th variable of f for 1 ~ I, J ~ k, and then denote the resulting Jordan polynomial by Pk . By (11.8),
Recall that the JB norm and the Hilbert space norm on a spin factor are equivalent (cf. equation (3.10)), so closed subspaces for the two norms coincide. Observe for use in the next proof that if 5 is a spin factor and B is a norm closed subspace of dimension 3 or more, containing the identity 1, then B is also a spin factor for the inherited product and norm. In fact, if 5 = Rl E8 N where N is a real Hilbert space, then B is seen to be the spin factor R1 E8 (B n N).
386
11
C*-ALGEBRA STATE SPACES
11.23. Lemma. Let A be a lB-algebra of complex type, and c the central proJectzon such that cA ** IS the type 12 summand of A **. Then every factor representatzon of cA ** IS onto a spzn factor of dzmenszon at most 4.
Proof. Let 1f be a factor representation of cA **, and let S = 1f( cA **) Note that S is of type I2 (Lemma 4.8), and thus is a spin factor (Proposition 3.37). We must show S has dimension 4 or less. Vie will first show 1f(cA) has dimensi0n 4 or less. Suppose 1f(cA) has dimension 4 or more. The homomorphic image 1f(cA) ofthe JB-algebra cA is a norm closed unital sub algebra of the spin factor S (Proposition l.35), so it is itself a spin factor. (See the observation preceding this lemma.) Therefore a f----> 1f( ca) is a dense type I factor representation mapping A onto the spin factor 1f(cA). Since A is of complex type, 1f(cA) ~ 6(H)Sd for a complex Hilbert space H. But 6(H)sa is a spin factor iff H is two dimensional (Theorem 3.39). Hence 1f(cA) is four dimensional. We have now shown that 1f( cA) is of dimension 4 or less. It remains to prove the same result for 1f(cA**) in place of 1f(cA). Since cA** is of type I 2 , it contains a pair r, s of Jordan orthogonal c-symmetries (Lemma 1l.21). Then since 1f is unital, 1f(r) and 1f(s) are seen to be Jordan orthogonal symmetries in S. Let al,"" a5 E A. Since the dimension of 1f(cA) is 4 or less, then 1f(ad, ... , 1f(a5) will be linearly dependent in S, so by Lemma 1l.22, with P5 as described in Lemma 11.22, (1l.9) Since 1f is an arbitrary factor representation of cA ** and the factor representations of a JB-algebra separate points (Lemma 4.14), then (1l.1O) By the Kaplansky density theorem for JB-algebras (Proposition 2.69), the unit ball of A is a-strongly dense in the unit ball of A **, so by a-strong continuity of Jordan multiplication on bounded sets (Proposition 2.4), the equality (11.10) will hold not only for arbitrary al,'" ,a5 in cA, but also for all al,"" a5 in cA **. Therefore (1l.9) will hold for every factor representation 1f of cA ** and all aI, ... ,a5 in cA **. By Lemma 11.22, 1f(ad, ... ,1f(a5) are linearly dependent, so the dimension of 1f(cA**) is at most 4. We are done. 0 Recall that an element I of a JBW-algebra M is central iff UsI = I for all symmetries s in M (Lemma 2.35). We need the following slightly stronger result for JBW-algebras of type h.
REVERSIBILITY OF JB-ALCEBRAS OF COMPLEX TYPE
387
11.24. Lemma. Let M be a JEW-algebra of type h, and let r, S be a pazr of Jordan orthogonal symmetrzes zn M. If x E M satisfies Urx = Usx = x, then x zs central zn lVI.
Proof. Let 7T be a factor representation of M, and S = 7T(M). We will show 7T(X) is central in S. By Lemma 4.8, S is of type 1 2 , and thus is a spin factor. Recall that S = Rl EB N where N consists of all scalar multiples of symmetries in S other than ±1 (Lemma 3.34). Since 7T is unital, 7T(r) and 7T(S) are Jordan orthogonal symmetries. Now 7T(r) 07T(S) = 0 implies that neither 7T(r) nor 7T(S) can equal ±1, and so in particular 7T(r) E N. Thus we can write 7T(X) = Al
(11.11)
+ Q7T(r) + w,
where Q, A E R, and wEN is orthogonal (and then Jordan orthogonal) to 7T(r). By Lemma 3.6, Un(r)W = -w. By hypothesis, x = Urx = {rxr}, and since 7T preserves Jordan products, then 7T(X) = Un(r)7T(X). Substituting (11.11) into this last equality gives Al
+ Q7T(r) + W =
Al
+ Q7T(r)
- w.
Thus w = 0, so 7T(X) E R1 + R7T(r). Similarly 7T(X) E Rl + R7T(S), so 7T(X) E R1. Thus 7T(X) is central for all factor representations of lVI, and therefore x is central in M. 0 We will now prove a technical result that will playa key role in showing that JB-algebras of complex type are universally reversible, cf. Definition 4.32. We begin by listing a few elementary general facts. By common usage, 0 is often omitted when multiplying by a central element in a Jordan algebra. (See the remarks following Proposition 1.52). Let q be a central projection in a JBW-algebra lVI. If x is a general element of lV!, then (11.12)
qx = qox = {qxq} = Uqx.
(cf. (1.65)). Furthermore, multiplication by q is a unital Jordan homomorphism from M onto the JBW-algebra Mq (cf. (1.66)). Note also that if rand S are orthogonal symmetries in a JB-algebra, then Ur and Us will commute. In fact, Urs = -s (Lemma 3.6), so by the identity (1.16) (11.13) and then multiplying both sides by Ur gives UrUs = UsUr . Recall that every JBW-algebra of type 12 contains a pair of Jordan orthogonal symmetries (Lemma 11. 21).
11
388
C*-ALGEBRA STATE SPACES
11.25. Lemma. Let M be a lEW-algebra of type 12 , and let r, s be a pa2r of lordan orthogonal symmetnes m M. If M admlts a homomorph2sm onto the four dlmenswnal spm factor, then M con tams a central projectwn q and an element t such that qr, qs, tare lordan orthogonal q -symmetnes. Proof. Let 7r : M --) S be a representation onto a four dimensional spin factor S. Recall that S = Rl EB N, where N is the span of the symmetries other than ±1, and N is a three dimensional Hilbert space such that a 0 b = (alb)1 for a, bEN. We begin by constructing an element y E M, Jordan orthogonal to rand s. Note that 7r(r) and 7r(s) are not equal to ±1 (since 7r(r) and 7r(s) are Jordan orthogonal symmetries), and so 7r(r) and 7r(s) are in N. Let v E N be a non-zero vector orthogonal to 7r( r) and 7r( s), and choose x E M such that 7r(x) = v. By the definition of the Jordan product on S, 7r (x) = v is Jordan orthogonal to 7r (r) and 7r (s ) . Note then by Lemma 3.6, U 1T (r)7r(x) = ~7r(x) and U7C(S)7r(X) = ~7r(x). Thus
(11.14)
(1
~ U7C (r))(1 ~ U7C(S))7r(x)
=
47r(x)
i- O.
Define (11.15) By (11.14), Y is not zero. Since U; = Id, then Ury = ~y. Now (11.15), together with the fact that Ur and Us commute (see the remarks preceding this lemma), implies that UsY = ~y. Then Ury = ~Y and Usy = ~Y imply that Y is Jordan orthogonal to rand s (Lemma 3.6). Thus we have succeeded in finding an element of M Jordan orthogonal to rand s. Next we construct a non-zero central projection q, and a q-symmetry t E M Jordan orthogonal to rand s. Define
where r(a) denotes the range projection of a positive element a (cf. Definition 2.14), and Y = y+ ~ y- is the unique orthogonal decomposition of y (Proposition 1.28). By Proposition 2.16, r(y+) and r(y-) are orthogonal, and since y i- 0, they are not both zero. Hence q is a non-zero projection, and t is a q-symmetry. Since U r and Us are Jordan automorphisms of M (Proposition 2.34), they preserve orthogonal decompositions. The unique orthogonal decomposition of ~y is y- ~ y+, and as shown above Ur exchanges y with ~y, so it follows that Ur exchanges y+ and y-, and then also exchanges their range projections. Hence Urt = ~t, and similarly Ust = ~t. Thus t is a q-symmetry, Jordan orthogonal to rand s.
REVERSIBILITY OF JB-ALGEBRAS OF COMPLEX TYPE
389
We finally show that q is central. Since Ur and Us exchange r(y+) and r(y-), then by the definition of q, (11.16) Thus q is central (Lemma 11.24), and so a f-* qa is a unital Jordan homomorphism from M onto the JBW-algebra M q . A unital Jordan homomorphism preserves Jordan orthogonality, so qr, qs, qt = t are Jordan orthogonal. A unital homomorphism also takes symmetries to symmetries, so qr, qs, qt = t are the desired Jordan orthogonal q-symmetries. 0 The following theorem is a key property of JB-algebras of complex type that will playa crucial role in what follows. For use in the proof below, we review the notion of a Cartesian triple (Definition 4.50). If e is a projection in a JBW-algebra lvI, a Cartesian tnple of e-symmetnes is a triple (r, s, t) such that (i) ros = sot = tor = 0, (ii) r2 = s2 = t 2 = e, and (iii) UrUsUt is the identity on Me. Any JBW-algebra admitting a Cartesian triple of symmetries is a JW-algebra, and is reversible in any faithful representation as a concrete JW-algebra (Lemma 4.53). The condition UrUsUt = I in the definition of a Cartesian triple of symmetries is not redundant. (A counterexample is given in the remark after [AS, Thm. 7.29]). But note for use in the next proof that it is redundant in case M is a four dimensional spin factor. Indeed, if M = Rl EB N where N is a three dimensional Hilbert space, then by using the general fact that Urs = -s for any pair r, s of Jordan orthogonal symmetries (Lemma 3.6), and the fact that a given Jordan orthogonal triple r, s, t of symmetries is a basis for N, one readily computes that UrUsUtx = x for x E N, so that UrUsUta = a for every a = Al + x E N.
11.26. Theorem. Every iE-algebra A of complex type sally reversible iC-algebra.
lS
a umver-
Proof. We first show that A is a JC-algebra. For each pure state a of the state space of A, the map 7r a : A ---> c( a) A * * is a dense type I factor representation (Corollary 5.42), and such representations separate elements of A by Lemma 4.14. Since A is of complex type, then by definition c(a)A** must be isomorphic to B(H)sa for some complex Hilbert space H. The direct sum of such representations then provides a faithful representation of A as a JC-algebra. The bidual of a lC-algebra is a JW-algebra, so A** is a JW-algebra (Proposition 2.77). We will show that A** is reversible in any faithful representation as a concrete lW-algebra on a Hilbert space H. By Lemma 4.33, this implies that A is universally reversible, which will complete the proof. Assume A** C B(H)sa. Let c be the central projection such that cA ** is the b summand of A **. By Corollary 4.30 it suffices to prove that
390
11
C*-ALGEBRA STATE SPACES
cA ** is reversible. We are going to show cA ** contains a Cartesian triple of symmetries, and thus is reversible (Lemma 4.53), which will complete the proof. Let {(rc;, Sc;, tc;)} be a maximal collection of Jordan orthogonal triples of partial symmetries in cA** such that the projections Cn = T~ = s; = are orthogonal and central. Let TO = Lc; r c;, So = La Sa, to = Lc; t a , Co = La Ce;. (For each of these sums, the terms live on orthogonal subspaces of H, so the sums converge strongly. Therefore the sums also converge a-weakly in B(H)sa, and then in M as well.) Then ro, So, to are Jordan orthogonal co-symmetries. We will show that Co = c, and then show that ro, So, to form a Cartesian triple. Let w = C - Co and suppose that w is not zero. Since wA ** is a direct summand of cA **, it is also of type 1 2 . We now show that wA ** admits a homomorphism onto the four dimensional spin factor. Suppose not, and let 7r be any factor representation of wA **. Since wA ** is of type 12, then 5 = 7r(wA**) (a-weak closure) is an 12 factor (Lemma 4.8), i.e., 7r( wa) is a spin factor representation of cA", a spin factor. Then a so by Lemma 11.23 the dimension of 5 is 4 or less. We are assuming the dimension of 5 is not 4, so the dimension is 3 or less. Since A is of complex type, it has no spin factor representations onto the three dimensional spin factor. Hence 7r(wA) is of dimension 2 or less. Then 7r(wA) is spanned by the identity and at most one other element, so must be associative. Thus 7r( wA) is associative for all factor representations 7r of wA **. Since factor representations of wA ** separate points, we conclude that wA is associative. Then wA ** must also be associative (by a-weak density of wA in wA** and separate a-weak continuity of Jordan multiplication). This contradicts wA** being of type 1 2 . Thus wA** admits at least one representation 7r onto a four dimensional spin factor. Let r, S be a pair of Jordan orthogonal w-symmetries in wA** (cf. Lemma 11.21). Since wA** admits a homomorphism onto the four dimensional spin factor, wA** contains a non-zero central projection q and an element t such that qr·, qs, t are Jordan orthogonal q-symmetries (Lemma 11.25). This contradicts the maximality of the collection {(ra, Sa, ta)}, and thus proves that w = o. Hence Co = c. We finally show that TO, So, to form a Cartesian triple. All that remains is to show that Uro Us[) Uto is the identity on cA **. We prove this by considering factor representations of cA **. By Lemma 11.23, an arbitrary factor representation 7r maps cA ** onto a spin factor 5 of dimension 4 or less. Thus 5 = R1 EEl N where N is a real Hilbert space of dimension 3 or less. In fact, N must be of dimension 3, since 7r(ro), 7r(so), 7r(to) are Jordan orthogonal symmetries in 5, hence orthogonal vectors in N. Therefore 5 is four dimensional. Now it follows from the observation in the paragraph before the theorem that (7r(ro), 7r(so), 7r(to)) will be a Cartesian
t;
f--)
REVERSIBILITY OF JB-ALGEBRAS OF COlVIPLEX TYPE
391
triple in S. Therefore for all x E cA **,
Hence UTa Usa Uto = I d on cA ** . This completes the proof that cA** admits a Cartesian triple. We are done. 0 We have worked hard to prove reversibility of JB-algebras of complex type. The following corollary is the key consequence of reversibility that we will use in this chapter. Recall (Definition 4.37) that with each JBalgebra there is associated its universal C*-algebra C~(A), and a canonical homomorphism from A into C~ (A) such that the image of A generates C~ (A). If A is a JC-algebra, this homomorphism is 1-1, and we identify A with its image under this imbedding. There is also a canonical *-antiautomorphism
Asa be a iordan zsomorphzsm from a iE-algebra A onto the self-adJomt part of a C*-algebra A. Then there zs a umque C*-product on A + zA makmg 1f : A + zA -> A into a *-zsomorphzsm. Two such *-zsomorphzsms 1f1 : A + zA -> A I and 1f2 : A + zA -> A2 mduce the same C*-product on A + zA zff 1f2 01f1 1 zs a *-zsomorphzsm from Al onto A 2 . Proof. The product (x,y) f--+ 1f- I (1f(x)1f(Y)) on A + zA is easily seen to be a C* -product making 1f : A + zA -> A into a *-isomorphism. In fact, it is the only such product, for if * is any product on A + zA such that 1f(x * y)
= 1f(x)1f(Y)
for X, yEA + zA, then applying 1f- 1 to both sides gives x * y = 1f- I (1f(x),1f(Y))· For the second statement of the proposition, define = 1f2 01f 11 and let \jJ be the identity map from A + zA equipped with the C*-product determined by 1f1 onto A + zA equipped with the C*-product determined
THE STATE SPACE OF THE UNIVERSAL C*-ALGEBRA
401
If these two products coincide, then it follows from the equality W 07r 1 1 that is a *-isomorphism. Conversely, if is a *isomorphism, then it follows from the equality W = 7r21 0 0 7rl that the identity map W is a *-isomorphism, so that the two products coincide. 0 by
7r2.
= 7r2 0
Thus for a JB-algebra A, specifying a C*-product on A + zA is equivalent to giving a Jordan isomorphism of A onto the self-adjoint part of a C* -algebra. We next give a necessary and sufficient condition for a JBalgebra of complex type to admit such a C* -product, in terms of the state space K of the universal C*-algebra.
11.42. Proposition. Let A be a iE-algebra of complex type, and let U, K, K and r be as defined at the begmmng of thzs sectwn. Then A zs zsomorphzc to the self-adJomt part of a C*-algebra zff there exists a w*-closed splzt face F of K such that r maps F bZJectwely onto K. Furthermore, there zs a 1-1 correspondence of such splzt faces F and C*products on A + tA.
Proof. Let F 0 be the set of w* -closed split faces F of K that are mapped bijectively by r onto K. We will establish the 1-1 correspondence of C*-products and split faces F in F o. Fix a C*-product on A + tA. This product organizes A + tA to a C*-algebra, which we will denote by A for short. Let L be the imbedding of the JB-algebra A into its universal C*-algebra U, and let 1/J be the unique lift of the inclusion 7r : A -> A + tA to a *-homomorphism from U onto A. Thus 1/J 0 L = 7r. Let J be the kernel of 1/J. Then 1/J* is an affine isomorphism from the state space S (A) of A onto the annihilator F of J in K, which is a w*-closed split face of K (A 133). On the other hand, 7r* is an affine isomorphism from S(A) onto K (Proposition 5.16). Dualizing 1/J 0 L = 7r gives i* o1/J* = 7r*. But i* = r, so r 0 1/J* = 7r*. Since 1/J* is an affine isomorphism from S( A) onto F and 7r* is an affine isomorphism from S (A) onto K, we conclude from this that r is an affine isomorphism from F onto K. Thus to each C*-product we have associated a split face in Fo. Suppose that two C*-products lead to the same split face FE F 0, and let 7rJ : A -> A + zA for J = 1,2 be the inclusions of A into the C* -algebra A + tA equipped with each product. Then the kernels of the associated *-homomorphisms 1/Jl : U -> A + tA and 1/J2 : U -> A + tA must coincide, since both are the annihilators of F in U (cf. (A 114)). Hence there is a *-isomorphism from A + zA with the first product, to A + zA with the second product, such that 01/Jl = 1/J2. By definition of 1/Jl and 1/J2, we have 7rl = 1/Jl 0 Land 7r2 = 1/J2 0 i. Thus 0 7rl = 7r2, so = 7r2 07rll. By Proposition 11.41, the two C*-products must coincide. Thus we have shown that the map from C*-products into F 0 is injective. To prove surjectivity, let F E F 0 and let J be the annihilator of F in U. Recall that J is a closed ideal in U and that F is the annihilator
11
402
C*-ALGEBRA STATE SPACES
of J m K (A 114). Let 'ljJ : U --> U / J be the quotient map, and define = 1j; 0 i : A --> U / J. We will show that 1i' maps A bijectively onto (U / J)sa. Recall that ;j;* is an affine isomorphism from the state space of U / J onto F (A 133). Since!' E F 0, then T is an affine isomorphism from F onto K, so 1i'* = TO 'ljJ* is an affine isomorphism from the state space of U / J onto K. It follows that 1i' is a Jordan isomorphism from A onto (U / J)sa (Proposition 5.16). Give A + zA the C*-product indpced by 1i' (cf. Proposition 11.41). Then 1i' : A --> U / J extends to a *-isomorphism 1i' : A + zA --> U / J Define ¢ = (1i') -1 : U / J --> A + zA, and 'ljJ = ¢ 0 ;j; : U --> A + zA. Let 7f : A --> A + zA be the inclusion map.
1i'
Then 'ljJ is a *-homomorphism, and for all a E A, ('ljJ
0
t)(a) = ¢(;j;(i(a))) = ¢(1i'(a)) = a = 7f(a),
~ 'ljJ 0 i = 7f. Since ¢ is a *-isomorphism, the kernels of 'ljJ = ¢ o;j; and 'ljJ coincide, and thus both equal F. Hence the split face associated wIth the given product on A + zA is F, which completes the proof of a 1-1 correspondence of C*-products and split faces in F o. 0 Lemma 1l.39 and Proposition 1l.42 will playa key role in the proof of our characterization of C* -algebra state spaces.
Orient ability The 3-ball property is not sufficient to guarantee that a JB-algebra state space is a C*-algebra state space, since there are JB-algebras of complex type that are not the self-adjoint part of C*-algebras, as we will see in Proposition 11.5l. An extra condition is needed ("orientability"), which we will now discuss. The notion of orientation of 3-balls and of C*-algebra state spaces was introduced in [AS, Chpt. 5], where it was shown that orientations of the state space of a C*-algebra are in 1-1 correspondence with C*-products
ORIENTABILITY
403
(A 156). C*-algebra state spaces are always orientable (A 148), and we will see that this property (extended to JB-algebra state spaces) is exactly what singles out C*-algebra state spaces among all JB-algebra state spaces satisfying the 3-ball property. We briefly review the relevant concepts, and extend them to our current context. Recall that B3 denotes the closed unit ball of R 3, and that a 3-ball is a convex set affinely isomorphic to B3.
11.43. Definition. Let K be a convex set. A facial 3-ball is a 3-ball that is a face of K. Our primary interest will be in the facial 3-balls of the state space of a JB-algebra of complex type. However, we will first review some elementary notions concerning general 3-balls, d. (A 128).
11.44. Definition. If B is a 3-ball, a parametenzatwn of B is an affine isomorphism from B3 onto B. Each parameterization eP of a 3-ball B is determined by an orthogonal frame (i.e, an ordered triple of orthogonal axes), given by the image under eP of the usual x, y, z coordinate axes of B3. If ePl and eP2 are parameterizations of a 3-ball B, then eP;;l 0 ePl is an affine isomorphism of B 3 , which extends uniquely to an orthogonal transformation of R3.
11.45. Definition. If B is a 3-ball, an oT'ientatwn of B is an equivalence class of parameterizations of B, where two such parameterizations ePl and eP2 are considered equivalent if det (eP;;l 0 ePd = l. If eP : B3 ---+ B is a parameterization, we denote by [eP] the associated orientation of B. An orientation of B corresponds to an equivalence class of orthogonal frames, with two frames being equivalent if there is a rotation that moves one into the other. Note that each 3-ball admits exactly two orientations, which we will refer to as "opposite" to each other.
11.46. Definition. If Bl and B2 are 3-balls equipped with orientations [ePl] and [eP2] respectively, and 1jJ : Bl ---+ B2 is an affine isomorphism, we say 1j; preserves onentatwn if ['l,6 0 ePl] = [eP2], and reverses onentation if the orientation [1jJ 0 ePl] is the opposite of [eP2]' We now outline how we will proceed. In the special case of the JBalgebra A = M 2 (C)sa, the state space is a 3-ball. The affine structure of the state space determines the Jordan product on A uniquely. There are two possible C*-products on A + ~A compatible with this Jordan product, namely, the usual multiplication on M 2 (C) and the opposite one. This choice of products corresponds to a choice of one of the two possible orientations on the state space (A 132).
404
11.
C*-ALGEBRA STATE SPACES
More generally, if K is the state space of a JB-algebra A of complex type, then a choice of orientation for each facial 3-ball may be thought of as determining a possible associative multiplication locally (i.e., on the affine functions on that 3-ball). It isn't necessarily the case that these choices of local multiplications can be extended to give an associative multiplication for A. We will see that this is possible iff the orientations of the facial 3-balls can be chosen in a continuous fashion. To make sense of a continuous choice of orientation, we need to define topologies on the set of facial 3-balls and on the set of oriented facial 3balls. We will then have a Z2 bundle with the two possible orientations (or oriented balls) sitting over each facial 3-ball. A continuous choice of orientation then will be a continuous cross-section of this bundle. The precise definition of this concept will be given for convex sets with the 3-ball property. Since each face of a convex set with the 3-ball property also has the 3-ball property, this definition will apply not only to state spaces of JB-algebras of complex type, but also to all faces of such state spaces.
11.47. Definition. Let K be a compact convex set with the 3-ball property in a locally convex linear space. Param(K) denotes the set of all affine isomorphisms from the standard 3-ball B3 onto facial 3-balls of K. We equip Param(K) with the topology of pointwise convergence of maps from B3 into K. Note that 0(3) and SO(3) acting on B3 induce continuous actions on Param(K) by composition. Two maps ePl, eP2 in Param(K) differ by an element of 0(3) iff their range is the same facial 3-ball B, and differ by an element of SO(3) iff they induce the same orientation on B. Thus we are led to the following definitions.
11.48. Definition. Let K be a compact convex set with the 3-ball property in a locally convex linear space. We call 03 I( = Param(K) / SO(3) the space of on.ented facwl 3-balls of K. We equip it with the quotient topology. If eP E Param( K), then we denote its equivalence class by [eP]. (Note that [eP] is an orientation of the 3-ball eP(B 3 ).) We call 3 I( = Param(K) / SO(3) the space of facwl 3-balls, equipped with the quotient topology from the map eP f---+ eP( B 3 ). When there is no danger of confusion we will write 03 for 03I( and 3 for 3 I(. Note that the definition above generalizes the corresponding one for C*-algebra state spaces (A 144). But the proposition below, which involves the canonical map from 03 to 3 that takes an oriented facial ball to its underlying (non-oriented) facial ball, is only a partial generalization of the corresponding result for C*-algebra state spaces (A 145) and (A 148), since in the C*-algebra case, this canonical map defines a trivial bundle, and not merely a locally trivial one.
o RIENTABILITY
405
11.49. Proposition. Let K be a compact convex set with the 3-ball property m a locally convex lmear space V. The spaces 0 B K and B K are Hausdorff, the canonical map from OB K onto B K ~s contmuous and open, and OB K -> B K ~s a locally trivial prmc~pal Z2 bundle.
Proof. By definition of the quotient topologies, the quotient maps from Param(K) onto Param(K)/50(3) and onto Param(K)/0(3) are continuous. Since these are quotients of a Hausdorff space by the action of compact groups, both are Hausdorff [32, Props. III.4.1.2 and III.4.2.3]). Since the relevant equivalence relations on Param(K) are given by continuous group actions, the quotient maps are open. Since the quotient maps Param(K) -> OB and Param(K) -> B are continuous and open, it is straightforward to check that the canonical map OB -> B is continuous and open. Now let cPo be any element of 0(3) that has determinant -1 and whose square is the identity. Then the continuous action of the group {cPo, I} on Param(K) induces a continuous Z2 action on OB = Param(K)/50(3) that exchanges the two elements in each fiber. To finish the proof we need to show that for each 3-ball B in B there is an open neighborhood W containing B such that the bundle OB -+ B restricted to W is isomorphic to the trivial bundle Z2 x W -+ ~V. This is equivalent to showing that for each cP E Param(K) there is an open neighborhood 5 of [cP] that meets each fiber of the bundle OB -+ B just once. Let A(B 3 , K) denote the set of affine maps from B3 into K, with the topology of pointwise convergence. We may assume without loss of generality that is not in the affine span of K. (Otherwise translate K so that this is true). We claim that the set of injective maps is open. To see this, let cPo E A(B 3 ,K) be injective. Let {ai, ~ = 1,2,3,4} be affinely independent elements of B3. Then {cPO(ai), ~ = 1,2,3,4} are affinely independent, and since 0 is not in the affine span of K, then these vectors in V are also linearly independent. Thus we can pick continuous linear functionals al,a2,a3,a4 such that ai(cPo(aj)) = 6iJ (the Kronecker delta). Then the determinant of the matrix (ai (cPo (a j ) )) will be 1. If cP is near enough to cPo then the determinant of the matrix (ai(cP(aj))) will be nonzero, and so {cP(ai), ~ = 1,2,3,4} must be linearly independent. It follows that the affine span of {cP(ai), ~ = 1,2,3,4} must be three dimensional, and so cP must be injective. Note further that for each E > 0 the set of cP such that 16ij - (ai (cP( a J ))) 1< E is a convex open neighborhood of cPo consisting of injective maps for E sufficiently small. Now choose a convex open neighborhood No of cPo in A( B3 , K) consisting of injective maps, and let N be the intersection of this neighborhood with Param(K). Let S = {[cP] cP EN}. Then 5 is an open subset of OB K , which we claim meets each fiber of our bundle at most once. Suppose that cPl and cP2 are in N and [cPt] and [cP2] are in the same fiber. Then cP1 and cP2 map B3 onto the same facial 3-ball B. For each t E [0,1] define
°
1
406 'Yt : B3
11 --->
C*-ALGEBRA STATE SPACES
K by 'Yt
= t¢l + (1 - t)¢2.
Then each 'Yt is in the convex neighborhood No, and so must be an injective map of B3 into B. For each t E [0, 1], ¢:;1 0 'Yt is an injective affine map from B3 into B 3 , taking 0 to 0, which then admits a unique extension to an invertible linear map from R3 into R3. Thus t f-> det(¢:;l 0 'Yt) is a continuous map from [0,1] into R \ {O}. Since its value at t = 0 is 1, its value at t = 1 cannot be -1, and so ¢1 and ¢2 must determine the same orientation. We have shown the bundle VB ---> B over W = {¢( B 3 ) I ¢ E N} is isomorphic to the trivial Z2 bundle Z2 x W ---> W, and thus is a locally trivial Z2 principal bundle. 0 Let 1r : VB ---> B be the bundle map that takes each oriented 3-ball to the underlying facial 3-ball. To specify an orientation for each facial 3-ball, we can give a map 8 : B ---> VB such that 1r(8(F)) = F for all FEB. Such a map is determined by its range, which will be a subset of VB that meets each fiber exactly once, i.e., includes exactly one of the two oriented 3-balls sitting over each facial 3-ball. We refer to both the map 8 and its range as cross-sectwns. If 8 is a continuous crosssection, then its range is a closed cross-section, since VB is Hausdorff and 8(B) = {[¢] I 8(1r([¢])) = [¢]}. Conversely, suppose X C VB is a closed cross-section. Then the complementary set Y of oriented 3-balls is also closed (since it is the image of X under the Z2 action of the bundle). We refer to Y as the opposzte cross-sectwn to X. Then X and its opposite cross-section Yare both open and closed. If 8 : B ---> VB is the crosssection such that 8(B) = X, then 8 will be continuous, and there is a 1-1 correspondence between closed cross-sections and continuous crosssections. Note that a closed cross-section X provides a trivialization of the bundle: the bundle is isomorphic to the trivial bundle X x Z2 ---> X. Clearly each trivialization of the bundle gives a closed cross-section.
11.50. Definition. Let K be a convex set with the 3-ball property in a locally convex space. Then K is orzentable if the bundle VB K ---> B K is trivial. A continuous cross-section of this bundle is called a global orzentation, or simply an orzentatwn, of K. Note that the above definition is a direct generalization of the corresponding definition for C*-algebra state spaces (A 146). Characterization of C*-algebra state spaces among JB-algebra state spaces We now pause to give an example showing that orient ability is not automatic for state spaces of JB-algebras of complex type.
C*-ALGEBRA STATE SPACES AMONG JB-ALGEBRA STATE SPACES 407
11.51. Proposition. Let T be the umt clrcle and let A consist of all contmuous functions f from T mto M 2 (C)sa such that f( ->.) = f(>.)t for all >. E T, where f(>.)t denotes the transpose of f(>.). Then A lS a iE-algebra of complex type whose state space is not onentable.
Proof. A is a norm closed Jordan sub algebra of the self-adjoint part of the C*-algebra C of all continuous functions from T into M 2 (C), and thus is a JB-algebra. Let a be a pure state of A. \Ve will show that the split face of the state space K of A generated by a is a 3-ball. The set of states of C that restrict to a form a w* -closed face of the state space of C, and so by the Krein-Milman theorem there is a pure state 0- of C that restricts to a. Let 7r be the GNS representation and ~ the cyclic vector associated with 0-. Since 0- is pure, then 7r is an irreducible representation of C (A 81). We will now verify that 7r is unitarily equivalent to evaluation at some point in T. Let Co be the subalgebra of C consisting of all f E C whose values are scalars (i.e., are in C1 where 1 is the identity matrix) at each point in T. Then 7r(Co) is central in 7r(C). Since 7r is irreducible, 7r(C) is a-weakly dense in B(H) (A 141), so 7r(Co) is also central in B(H). Thus 7r(Co) must consist of scalar multiples of the identity. Therefore 7r is a multiplicative linear functional on Co. By a well known theorem, every multiplicative linear functional on the commutative C*-algebra of all continuous complex valued functions on a compact set is evaluation at a point. Therefore there exists >'0 E T such that 7r(J) = f(>.o) for all f E Co. Write M2 for M2(C)' For 1 :S l,] :S 2 let mil E C be the constant function whose value at each point of T is the standard matrix unit ei) of lVh. Then every element of C admits a unique representation in the form Li) fi]mi] where each fij E Co. Thus
7r(
2.:= ];jmij ) ']
=
L
fij(>'O)7r(mi])'
']
Then 7r( mij) are 2 x 2 matrix units and generate the image 7r( C) of C, so 7r(C) must be isomorphic to M 2 . Identify 7r(C) with lvh. Choose a unitary v E 11/[2 such that V7r( mij )v* = e,] for all l,]. Then v7rv* is exactly evaluation at >'0. Thus 7r is unitarily equivalent to evaluation at some point in T, and we assume hereafter that 7r is given by evaluation at such a point >'0. We claim that 7r(A) = M 2 (C)sa' Choose a continuous function f : T ~ R so that f(>.o) = 1 and f( ->'0) = O. Given m E (M 2 )sa, the map a : >. f-> f(>.)m + f( _>.)m t is in A, and satisfies 7r(a) = m. Thus 7r(A) = M 2 (C)sa' Therefore (7rIA)* is an affine isomorphism of the state space of M2 (i.e., the 3-ball B 3 ) onto a split face F of K (Lemma 1l.8). Since a(a) = 0-( a) = (7r( a)e I e) for a E A, this split face contains (J. Thus the split face
408
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generated by each pure state a is a 3-ba11. If T is another pure state of A, then either T E F, in which case face(T,a) = F is a 3-ball, or else T and a are separated by the split face F, in which case face(T, a) is the line segment [a, T], cf. Lemma 5.54 or (A 29). It follows that the state space K of A has the 3-ball property, and so A is of complex type (Theorem 11.19). Next we are going to construct a path between two distinct oriented facial 3-balls of A in the same fiber of OB --> B. We will see that this implies that K is not orientable. We identify the state space S(M2 ) with B 3 , and thus view an affine isomorphism from the state space of M2 onto a 3-ball of K as an element of Param(K). For each s E [0,1]' define a homomorphism 'irs : A --> M2 by 'irs(f) = f(e i7rs ). Then 'ir; will be an affine isomorphism from S(M2) = B3 onto a split face of K, which will also be a facial 3-ball of K. Thus 'ir; is a parameterization of a facial 3-ball of K; as usual we denote the associated oriented 3-ball by ['ir;]. Thus S f--+ 'ir; is a continuous map from [0,1] into Param(K), and so S f--+ ['ir;] is a continuous path in OB. If 1/; : M2 --> M2 denotes the transpose map, then by the definition of A, 'irl (f) = 1/;( 'ira (f)) for all f E A. Thus 'ira 0 1/;* = 'iri. It follows that the ranges of the orientations 'ira and 'iri are the same facial 3-ba11. Furthermore, since the dual of the transpose map, extended from S(M2 ) = B3 to R 3 , has determinant -1 (A 130), then the orientations given by 'ira and 'iri are different, so ['ira] and ['iril are the two distinct elements of a single fiber of OB --> B. However, if K were orient able , then the bundle of oriented 3-balls would be trivial, and so ['ira] and ['iril would lie in different connected components of OB. This contradicts the fact that we have just found a path in OB from ['ira] to ['iri], so K is not orientable. 0
Remark. In the example above, K is e,ctually affinely isomorphic to a C*-algebra state space, but by Proposition 11.51 the affine isomorphism cannot be chosen to be a homeomorphism, since the state space of a C*algebra is orientable (A 148). We sketch how to construct such an affine isomorphism. Let D be the C*-algebra of continuous functions f from T into M 2 (C) such that f( -A) = f(A) for all A E T. Let J be an ideal of D consisting of functions vanishing at 1, and similarly J the Jordan ideal of A consisting of functions vanishing at 1. For each f E Jsa or f E J, we associate the map S f--+ f(e i7rs ) from [0,1l into M 2 (C)sa. In this fashion, Jsa and J both can be identified with the Jordan algebra of all continuous functions from [0,1] into M 2 (C)sa vanishing at 0 and 1. Let Fv and FA be the annihilators of Jsa and J in the respective state spaces of D and A. Since J and J are the kernels of homomorphisms onto M2 (C) and M 2(C)sa respectively, it follows that Fv and FA are affinely isomorphic to 3-balls. Their complementary split faces F-fy and F;" are then affinely isomorphic to the state space of .:J and J respectively, i.e., the set of positive linear functionals of norm 1 on .:J or J respectively.
C*-ALGEBRA STATE SPACES A:tvIONG JB-ALGEBRA STATE SPACES 409
In particular, since Jsa ~ 1, then F-b and F~ are affinely isomorphic. This isomorphism, combined with the affine isomorphism of Fv and FA, extends uniquely to an affine isomorphism of the state spaces of D and A. We do not know if the state space of an arbitrary JB-algebra A of complex type admits a (possibly discontinuous) affine isomorphism onto the state space of a C* -algebra; nor do we know if A ** is isomorphic to the self-adjoint part of a von Neumann algebra for each JB-algebra A of complex type.
If K is the state space of a JB-algebra of complex type and if F is a split face of K, then the subspace Param(F) denotes the subset of Param(K) consisting of maps with range in F, with the relative topology from Param( K). B F denotes the set of facial 3-balls contained in F with the quotient topology from Param(F). We define VB F in an analogous way. Since Param(F) is saturated with respect to the action of 0(3) and SO(3), and the actions of 0(3) and SO(3) induce open quotient maps, B F has the topology inherited from B K, and VB F has the topology inherited from VBK ([32, I.5.2, Prop. 4]). If N is the normal state space of B (H), we can view N as a split face of the state space of B (H) (A 113). Then B N is path connected (A 152), and more generally we have the following result for state spaces of JB-algebras of complex type. 11.52. Lemma. If A is a lB-algebra of complex type, and F the spilt face generated by a pure state, then the space B F of faczal 3-balls m F zs path connected. Proof. Let a be a pure state on A, and let 71" be an irreducible concrete representation of A on a Hilbert space H associated with a (cf. Corollary 11.33). Then 71"* is an affine isomorphism from the normal state space N of B(H) onto the split face F generated by a (Lemma 11.30). Furthermore, 71"* is continuous for the weak topology on N given by the duality of B (H) and B (H) * and the w* -topology on F. It follows that 71"* induces a continuous surjective map from Param(N) to Param(F), and then also between the quotient spaces B Nand B F. Since B N is path connected (A 152), then B F is the continuous image of a path connected set, and therefore is itself path connected. 0 11.53. Definition. Let Kl and K2 be arbitrary convex sets, with each facial 3-ball in Kl and K2 given an orientation. Let ¢ : Kl -> K2 be an affine isomorphism of Kl onto a face of K 2. Then ¢ preserves onentation if ¢ carries the given orientation of each facial 3-ball F of Kl onto the given orientation of ¢(F), and reverses onentatwn if ¢ carries the given orientation of each facial 3-ball F of Kl onto the opposite of the given orientation of ¢(F).
410
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Let A be a JB-algebra of complex type. We continue the notation of the previous section: U is the universal C*-algebra for A, is the canonical *-anti-automorphism of U, K is the state space of U, K is the state space of A, and r : K -> K is the restriction map.
11.54. Lemma. Let A be a iE-algebra of complex type, and let U, , K and r be as defined above. Then the map rP f--> r 0 ¢ zs a contmuous map from Param(K) onto Param(K) , and mduces contmuous maps [rPl f--> [r 0 rPl from OS K to K, and B f--> r(B) from S K onto SK
os
Proof. Recall that r maps oeK onto oeK (Corollary 11.37), and for each pure state a on U, r maps the minimal split face Fa bijectivelyonto the minimal split face Fr(a) (Proposition 11.31). Therefore r maps facial 3-balls of K bijectively onto facial 3- balls of K, and every facial 3- ball of K is the image of a facial 3-ball of K. Thus the map rP f--> r 0 ¢ is a continuous map from Param( K) onto Param( K). This map commutes with the action of 0(3) and 50(3), and hence induces a continuous map [¢l f--> [r 0 rPl from OS K to OSK, and a continuous map B f--> r(B) from SK to SK. D We now make use of the fact that the state space of any C*-algebra (in particular, that of U) has a global orientation induced by the algebra (A 148), i.e., an orientation assigned to each facial 3-ball in such a way as to give a continuous cross-section to the bundle OSK -> SK. This orientation is defined in (A 147), and a key property of this orientation, which will be used below, is that the dual * of a *-anti-isomorphism of U reverses the orientation of K (A 157).
11.55. Lemma. Let A be a iE-algebra of complex type whose state space K zs onentable, let U, , K and r be as defined before Lemma 11.54, and let a global onentatwn of K be gwen. Then for each pure state a on A, there IS a umque pure state (j of U such that the restnctwn map r zs an orzentatwn preservmg affine zsomorphzsm from the spld face F(j onto the splzt face Fa, and orzentatwn reversmg from Fip'(j onto Fa.
Proof. Let w be any extension of a to a pure state on U. Then r is an affine isomorphism from Fw onto Fa (Proposition 11.31). Equip each facial 3-ball in K with the orientation induced by U (A 148). Let [rPol and [¢ll be two of the resulting oriented 3-balls in Fw. By Lemma 11.52 there is a path of facial 3-balls from rPo (B 3 ) to rPl (B 3 ). Composing with the orientation induced by U gives a path of oriented 3-balls [rPtl from [rPol to [rPd· By continuity of the map [rPl f--> [r 0 rPl (Lemma 11.54), [r 0 rPtl is a continuous path in OB K. Then the path must lie either in the closed
C*-ALGEBRA STATE SPACES AMONG JB-ALGEBRA STATE SPACES 411
cross-section given by the specified orientation of K, or in its opposite, since these cross-sections are both open and closed. Thus [r 0 Fe is 1-1 from the set of global orientations of K into the set of w* -closed split faces F of K for which r maps F bijectively onto K. To prove surjectivity, we suppose that we are given a w*-closed split face F of K which is mapped bijectively by r onto K. The global orientation on K restricts to give a global orientation of F. Since r : F f-> K is an affine homeomorphism, there is a unique global orientation e of K such that r : F --> K is orientation preserving. For each w E oeF, the restriction map r is an orientation preserving affine isomorphism on the generated split face Fw c F. Thus w has the defining property of the pure state ¢(r(w)), so w = ¢(r(w)). Hence oeF c {¢(O") I 0" E oeK}. On the other hand, if 0" E oeK, and w E oeF is the unique pure state such that r(w) = 0", then w = ¢(r(w) = ¢(O"), so {¢(O") I 0" E oeK} c oeF, and thus equality holds. Since Fe is by definition the w*-closed convex hull of {¢(O") 10" E oeK}, we conclude by the Krein-Milman theorem that F = Fe, as desired. Thus the map e f-> Fe is a 1-1 correspondence from
416
11.
C*-ALGEBRA STATE SPACES
global orientations of K to w*-closed split faces F mapped bijectively onto K by r. This completes the proof of the 1-1 correspondence of C*-products on A and global orientations of K. D The above characterization of C*-algebra state spaces was obtained by adding new assumptions to those characterizing JB-algebra state spaces. We did the same thing in Proposition 10.27, where we added the assumption about the existence of a dynamical correspondence. However, the two sets of extra assumptions are quite different in character. The one in Theorem 11.59 is "geometric and local", whereas the one in Proposition 10.27 is "algebraic and global". The former proceeds from local data of geometric nature (a continuous choice of orientations of facial 3-balls) from which a C*-product is constructed, while the latter postulates the existence of a globally defined map a f-> 1/Ja from elements to skew order derivations, from which the Lie part of the C*-product is obtained by an explicit formula (equation (6.14)). Thus, the starting point is closer to the final goal in the latter than in the former of the two characterizations. But the characterization in terms of dynamical correspondences is nevertheless of considerable interest, both because it explains the relationship with Connes' approach to the characterization of O"-finite von Neumann algebras, and because of its physical interpretation, which gives the elements of A the "double identity" of observables and generators of one-parameter groups (explained in Chapter 6). Note, however, that the conditions of Theorem 11.59 also relate to physics. We know that a Euclidean 3-ball is (affinely isomorphic to) the state space of ]l;h(C), which models a two-level quantum system (A 119). Thus the key role that is played by 3-balls in Theorem 11.59 reflects the fact that properties that distinguish quantum theory from classical physics are present already in the simplest non-classical quantum systems, the two-level systems, and that multilevel systems do not introduce any new features in this respect. The state space of a C*-algebra has at least two different global orientations. One is the orientation induced by the given C*-product (A 147), and the other is the opposite orientation, obtained by reversing the orientation of each facial 3-ba11. But there may be many other global orientations. For example, if we have a general finite dimensional C*-algebra A = L.~ Mj(C), then each summand M](C) has just two possible associative Jordan compatible multiplications, each associated with one of the two orientations of the finite dimensional split face corresponding to that summand. In each such split face, there are just two ways to choose a continuous orientation, and so the orientation for each such split face is determined by the orientation of a single facial 3-ball. There are n connected components of the space of facial 3-balls, each consisting of those 3-balls contained in a single minimal split face, and there are 2 n possible global orientations.
NOTES
417
Finally we observe that if the global orientation of a C*-algebra state space is changed to the opposite orientation, then the associated dynamical correspondence a f---> Wa (determined by the associated Lie product) will be changed to the opposite dynamical correspondence a f---> -Wa, and then the generated one-parameter group exp (tWa) will be changed to the oneparameter group exp (-tWa), which has opposite sign for the time parameter t. In this sense, changing to the opposite orientation can be thought of as time reversal.
Notes The results in this chapter first appeared in [11]. These results can be thought of as saying that the state space together with an orientation is a dual object for C*-algebras. Instead of the state space, some papers focus on the set of pure states as a dual object, e.g., [118]' and the papers of Brown [34] and Landsman [90].
Appendix
Below are the results from State spaces of operator algebras: baszc theory, orzentatwns and C*-products, referenced as [AS] below, that have been referred to in the current book. Many of these are well known results; see [AS] for attribution. A 1. A face F of a convex set K c X is said to be semz-exposed if there exists a collection 1{ of closed supporting hyperplanes of K such that F = K n nHEH H, and it is said to be exposed if 1{ can be chosen to consist of a single hyperplane. Thus, F is exposed iff there exists a y E Y and an a E R such that (x,y) = a for all x E F and (x,y) > a for all x E K \ F. A point x E K is a semz-exposed poznt if {x} is a semi-exposed face of K, and it is an exposed pomt if {x} is an exposed face of K. The intersection of all closed supporting hyperplanes containing a given set B c K will meet K in the smallest semi-exposed face containing B, called the semz-exposed face generated by B. [AS, Def. 1.1] A 2. A non-empty subset of a linear space X is a cone if C + C c C and )"'C c C for all scalars)... 2:: o. A cone C is proper if C n (-C) = {O}. Let C be a proper cone C of a linear space eX, and order X by x -::; Y if y - x E C. If F is a non-empty subset of C other than {O}, then F is a face of C iff F is a subcone of C (i.e., F + F c F and )"'F c F for all )... 2:: 0) for which the following implication holds:
It follows that if x E C, then facec(x) = {y E C I y -::;)...x
for some
)... E R+}.
[AS, p. 3] A 3. Two ordered vector spaces X and Yare in separatmg ordered dualzty if they are in separating duality and the following statements hold for x E X and y E Y:
x 2:: 0
¢=}
(x, y) 2:: 0 all y 2:: 0,
2:: 0
¢=}
(x, y)
y
[AS, Def. 1.2]
2:: 0 all x 2:: o.
420
APPENDIX
A 4. We will say that two positive projections P, Q on an ordered vector space X are complementary (and also that Q is a complement of P and vice versa) if ker+Q
= im+ P,
ker+ P
= im+Q.
We will say that P, Q are complementary m the strong sense if ker Q = im P,
ker P
= im Q.
[AS, Def. 1.3]
A 5. We say that a convex set K c X is the free convex sum of two convex subsets F and G, and we write K = FfficG, if K = co(FuG) and F, G are affinely independent. Observe that if K = F ffic G, then the two sets F and G must be faces of K. We say that a face F of K is a spilt face if there exists another face G such that K = F ffic G. In this case G is unique; we call it the complementary spilt face of F, and we will use the notation F' = G. More specifically, F' consists of all points x E K whose generated face in K is disjoint from F, in symbols F' = {x E K I face K (x) n F = 0}. [AS, Def. 1.4] A 6. If F and G are split faces of a convex set K, then every x E K can be decomposed as x = ~~,j=l a'jXi] where aij;::: 0 for 2,J = 1,2 and Xll E FnG, x12 E FnG', X21 E F'nG, X22 E F'nG'. This decomposition is unique in that every aij is uniquely determined and every Xij with nonvanishing coefficient aij is uniquely determined. [AS, Lemma 1.5] A 7. If F and G are split faces of a convex set K, then F n G and co (F U G) are also split faces and
(F n G)'
=
co(F'
U
G').
[AS, Prop. 1.6]
A 8. Let K be a compact convex set regularly embedded in a locally convex vector space X. K is said to be a Choquet simplex if X is a lattice with the ordering defined by the cone X+ generated by K. [AS, Def. 1.8] A 9. An ordered normed vector space V with a generating cone V+ is said to be a base norm space if V+ has a base K located on a hyperplane
421
APPENDIX
H (0 ~ H) such that the closed unit ball of V is co(Ku -K). The convex set K is called the d2stmguzshed base of V. [AS, Def. 1.10] A 10. If V is an ordered vector space with a generating cone V+, and if V+ has a base K located on a hyperplane H (0 ~ H) such that B = co(K U -K) is radwlly compact, then the Mmkowski Iunctwnal
Ilpll = inf{a > 0 I p E aB} is a norm on V making V a base norm space with distinguished base K. [AS, p. 9] A 11. If V is a base norm space with distinguished base K, then the restriction map I f---> 11K is an order and norm preserving isomorphism of V* onto the space Ab(K) of all real valued bounded affine functions on K equipped with pointwise ordering and supremum norm. [AS, Prop. 1.11]
A 12. A positive element e of an ordered vector space A is said to be an order umt if for all a E A there exists A ::::: 0 such that -Ae
Iiall,
A < /1,
1\,",>.\ e,l = e.\ for all A E
R.
The family {e.\} is given by e.\ = 1 - r((a - A1)+). For each increasing finite sequence 1= {Ao, AI,"" An} with AO < -ilall and An > Iiall, define Ihll = max(Ai - Ai-I! and s, = L~=l Ai(e.\, - e'\,_l)' Then lim
1h11~0
lis, - all = O.
[AS, Thm. 1.58] A lattice L with least element 0 and greatest element 1 is
A 40.
orthomodular if there is a map p
f----+
pi, called the orthocomplementatwn,
that satisfies
(i) pI!
= p,
(ii) p::::; q implies p' ?: q', (iii) p V p' = 1 and p A p' = 0, (iv) If p ::; q, then q = p V (q A p').
If p ::::; q' in an orthomodular lattice L, then we say p is orthogonal to q and write p ~ q. If p ~ q, then we sometimes write p + q in place of p V q. [AS, Def. 1.60] A 41. Let L be a complete lattice. A non-zero element p of L is an atom if each element q ::; p either equals p or is the zero element. L is
426
APPENDIX
atomzc if every non-zero element is the least upper bound of atoms, and an element P of L is finzte if p is the least upper bound of a finite set of atoms. [AS, Def. 1.61] A 42. Let p be a finite element in a complete atomic lattice. The minimum number of atoms whose least upper bound is p is called the dzmension of p and is denoted dim(p). [AS, Def. 1.62]
A 43. Let p and q be elements in a complete lattice L. We say q covers p if p < q and there is no element strictly between p and q. We say L has the covermg property if for all pEL and all atoms U E L,
(3)
pV
U
= P or
PV
U
covers p.
We say L has the finzte covermg property if (3) holds for all finite p in L and all atoms U E L. [AS, Def. 1.63] A 44. Let p be a finite element in a complete atomic orthomodular lattice L with the finite covering property. Then p can be expressed as a sum of atoms. In fact, p = PI + ... + Pk for each maximal set of orthogonal atoms PI, ... ,Pk under p, and the cardinality of any set of atoms with sum p is dim(p). Furthermore, every element q ::; p is finite with dim(q) < dim(p). [AS, Prop. 1.66]
A 45. An algebra is said to be power assoczatzve if parentheses can be inserted freely in products with identical factors. Thus the n-th power an of an element a is well defined in this case. We will write An for the set of all n-th powers of elements in a power associative algebra A. [AS, Def. 1.70] A 46. An order unit space A which is also a power associative complete normed algebra (for the order unit norm) and satisfies the following requirements: (i) the distinguished order unit 1 is a multiplicative identity, (ii) a 2 E A+ for each a E A, will be called an order unzt algebra. [AS, Def. 1.72]
A 47. If A is an order unit algebra, then A+
=
A2. [AS, Lemma 1.73]
A 48. If A is an order unit algebra, then the following are equivalent: (i) A is associative and commutative, (ii) abEA+ for each pair a,bEA+,
427
APPENDIX
(iii) A ~ CR(X) for a compact Hausdorff space X. [AS, Thm. 1.74]
A 49. The Jordan product in an associative (real or complex) algebra A is given by
a 0 b = ~ (ab
+ ba)
for a, bE A .
[AS, Def. 1.76]
A 50. Let Ao be a linear subspace of an order unit space A, and suppose that a, b f-7 ab is a bilinear map from Ao x AD into A such that for a, bEAD (and with the standard notation a 2 = aa),
(i) ab
=
ba,
(ii) -1::; a ::; 1
=}
0::; a 2
::;
1.
Then for all a, bEAD,
Ilabll ::; Iiall Ilbll· [AS, Lemma 1.79]
A 51. Suppose A is a complete order unit space which is a power associative commutative algebra where the distinguished order unit 1 acts as an identity. Then A is an order unit algebra iff the following implication holds for a E A:
[AS, Lemma 1.80]
A 52. Suppose A is a real Banach space which is equipped with a power associative and commutative bilinear product with identity element 1. Then A is an order unit algebra with positive cone consisting of all squares, distinguished order unit 1 and the given norm, iff for a, b E A,
(i) (ii) (iii)
Ilabll::; Ilallllbll, IIa 2 1 = Ila11 2 , Ila 2 11::; IIa 2 + b2 11.
Moreover, the ordering of A is uniquely determined by the norm and the identity element 1. [AS, Thm. 1.81] A 53. Let A be a commutdtive order unit algebra that is the dual of a base norm space V such that multiplication in A is separately w*continuous. Then for each a E A and each E: > 0 there are orthogonal
APPENDIX
428
projections P1, ... ,Pn in the w*-closed sub algebra W(a, 1) generated by a and 1 and scalars A1,"" An such that n
,=1 [AS, Thm. 1.84]
A 54. For each pair of elements x, y in a unital *-algebra, sp(yx) \
{O} = sp(xy) \ {O} .
[AS, Lemma 1.90]
A 55. If a is a continuous complex valued function on a compact Hausdorff space X, then the spectrum of a relative to the Banach algebra Cc(X) is equal to the range of a, i.e., sp(a) = {a(s) I SEX}. Moreover, the spectrum of a relative to Cc(X) is the same as the spectrum of a relative to the norm closed *-subalgebra C(a,l) generated by a and the unit function 1. The same statements hold with CR(X) in the place of Cc(X). [AS, Lemma 1.91] A 56. If A is a unital complex *-algebra and the real vector space Asa is an order unit algebra with distinguished order unit equal to the multiplicative identity and with the Jordan product induced from A, then A+ = A2 = {x*x I x E A}. [AS, Lemma 1.92]
A
=
A 57.
A C*-algebra is a complex Banach *-algebra A such that for all x E A. Unless otherwise stated, we will assume that C*-algebras mentioned have an identity. [AS, Def. 1.93]
Ilx*xll = I xl1 2
A 58. If A is a unital C*-algebra, then its self-adjoint part A = Asa is an order unit algebra for the Jordan product and the norm induced from A, and with distinguished order unit equal to the multiplicative identity and positive cone A+ = A2 = {x*x I x E A}. [AS, Thm. 1.95]
A 59. If A is a unital complex *-algebra and Aba is a complete order unit algebra for the Jordan product induced from A with distinguished order unit equal to the multiplicative identity, then the positive cone consists of all elements x* x where x E A, and A is a C* -algebra for a unique norm which restricts to the order unit norm on Asa. [AS, Thm. 1.96], where "order unit space" should read "order unit algebra". A 60. An element of a C*-algebra A is said to be positive if it is of the form x* x for some x E A, and we will denote the set of all positive
APPENDIX
429
elements in A by A +. A linear functional p on a C*-algebra A with identity element 1 is called a state if it is positive on positive elements and p(l) = l. The set of all states on A is called the state space of A, and it will be denoted by S( A), or just by K when there is no need to specify A, and the extreme points of K are called pure states. [AS, Def. l.97] A 61. A representatwn of a *-algebra A is a *-homomorphism 7r : l3 (H) where H is a real or complex Hilbert space. If we want to specify H, then we say that 7r is a representatwn of A on H. If 7r is injective, then we say that 7r is fa~thful. If ~ E H and the linear subspace 7r(A)~ = {7r(x)~ I x E A} is dense in H, then we say that ~ is a cychc vector (or a generatmg vector) for 7r. A representation with a cyclic vector is called a cycl~c representatwn. [AS, Def. l.98]
A
->
A 62. Suppose A is a (real or complex) unital *-algebra whose selfadjoint part A = Aba is a complete order unit space with distinguished order unit equal to the multiplicative identity and with the positive cone A+ = {x*x I x E A}. For each state p on A there is a Hilbert space H p, a cyclic unit vector ~p E Hp and a representation 7r p of A on Hp such that p(x) = (7rp(x)~p I ~p) for all x E A. Moreover, II7rp(x)11 ::; Ilx*xII1/2 for all x E A. [AS, Prop. l.100] A 63. If A is a C*-algebra and p is a state on A, then the representation described above is the GNS-representatwn associated with p. We will denote it by (7r p, H p, ~p), or just by 7rp when there is no need to specify Hp or ~p. Thus
[AS, Def. 1.101] A 64. (Gelfand-Naimark Theorem) Each unital C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of l3(H) for a complex Hilbert space H. One such isomorphism is the direct sum of all GNSrepresentations associated with states on A. [AS, Thm. l.102] A 65. Suppose A is a real *-algebra with identity 1 whose self-adjoint part Asa is a complete order unit space with the distinguished order unit 1 and the positive cone A + = {x* x I x E A}. Then there is a representation 7r of A on a (complex) Hilbert space H which is an isometry on Asa. [AS, Prop. l.103] A 66. Let A be an ordered Banach space (i.e., a real Banach space ordered by a positive cone A + ). A bounded linear operator 8 on A is an order derivation if et8 is a positive operator on A, i.e., if et8 (A+) C A+
APPENDIX
430
for all t E R, or equivalently, if {e tO } is a one-parameter group of order automorphisms. We will write D(A) for the set of all order derivations on
A. [AS, Def. 1.104] A 67. Let A be a complete order unit space, and 0 a bounded linear map from A into A. The following are equivalent: (i) 0 is an order derivation. (ii) If x E A+, 0::::; u E A* and u(x)
= 0, then u(ox) = O.
[AS, Prop. 1.108] A 68. The set D(A) of order derivations of a complete order unit space A is a real linear space closed under Lie brackets [01,02] = 0102 - 0201. [AS, Prop. 1.114] A 69. Let A be a (unital) C*-algebra with state space K. Its selfadjoint part Asa is a complete order unit space under the norm induced from A, the ordering determined by the positive cone A + = A 2 = {x* x I x E A}, and with the distinguished order unit equal to the multiplicative identity 1. [AS, Prop. 2.2] A 70. Let A be a C*-algebra with state space K, and define for each on K by fL(p) = p(a) for p E K. Then the map a f--? is an order and norm preserving linear isomorphism of Asa onto the space A(K) of all continuous affine functions on K. [AS, Prop. 2.3]
a E Asa the function
a
a
A 71. A linear functional p on a C*-algebra A is positive iff it is bounded with Ilpll = p(I). [AS, Prop. 2.11]
A 72. Let A and E be two possibly non-unital C*-algebras. If is a *-homomorphism from A into E, then (A) is a C*-subalgebra of E. [AS, Prop. 2.17] A 73. Let p be a projection in a C*-algebra A and set q = p'. If a E A is self-adjoint, then the following two statements are equivalent:
(i) aq = 0 (by taking adjoints also qa = 0), (ii) ap = a (by taking adjoints also pa = a), and if a ~ 0, then they are also equivalent to each of the following four statements: (iii) (iv) (v) (vi)
qaq = 0, pap = a,
a::::; Iiallp, a E face(p).
[AS, Lemma 2.20]
APPENDIX
wE
431
A 74. Let p be a projection in a C* -algebra A and set q = p'. If A* is self-adjoint, then the following four statements are equivalent:
(i) q. w (ii) p. w
= 0 (by taking adjoints also w· q = 0), = w (by taking adjoints also w . p = w),
and if w ;::: 0, then they are also equivalent to each of the following four statements, (iii) w(q) (iv) w(p)
= 0, = Ilwll,
(v) q·w·q=O, (vi) p. w . p = w. [AS, Lemma 2.22] A 75. Let A be a C * -algebra. Then the extreme points of precisely the projections in .A. [AS, Prop. 2.23]
Ai
are
A 76. If {a,} is an increasing net bounded above in B(H)sa, then {a,} has a least upper bound a E B (H)sa, and a is also a strong (and weak) limit of {a,}. Similarly for a decreasing net and its greatest lower bound. [AS, Prop. 2.29] A 77. Let 7rl be a representation of a C*-algebra A on a Hilbert space Hl with cyclic vector~. Another representation 7r2 of A on a Hilbert space H2 is unitarily equivalent to 7rl iff it has a cyclic vector 6 such that
and then the equivalence can be achieved by a unitary that u6 = 6. [AS, Prop. 2.31]
U E
B(Hl' H 2 ) such
A 78. A representation 7r of a C*-algebra A on a Hilbert space H is said to be zrreducible if {O} and H are the only closed subspaces invariant under 7r(A). [AS, Def. 2.34] A 79. For each subset 5 of B(H) for a Hilbert space H, the set of operators in B (H) that commute with all operators in 5 is called the commutant of 5 and is denoted by 5'. [AS, Def. 2.35] A 80. A representation 7r of a C*-algebra A on a Hilbert space H is irreducible iff it has trivial commutant, i.e., iff 7r(A)' = Cl. [AS, Lemma 2.39]
A 81. The GNS-representation associated with a state p on a C*algebra is irreducible iff p is a pure state. [AS, Cor. 2.41]
432
APPEND~X
A 82. The irreducible representations of a C*-algebra A separate the points of A. [AS, Cor. 2.43] A 83. If A is a C* -algebra with state space K, then EB pEK 7r P (the direct sum of all GNS-representations) is called the umversal representatwn of A. [AS, Def. 2.45] A 84. If A is a *-subalgebra of B(H), then the weak, respectively stT'Ong, (operator) topology of A is the locally convex topology determined by the semi-norms a f---> l(a~I7])I, respectively a f---> Ila~ll, where ~,7] E H; the a-weak topology is determined by the semi-norms
where {~;}and {7];} are two sequences in H such that I::l II~i112 < 00 and I::l II7]i 112 < 00, and the a-stT'Ong topology is determined by the semi-norms
where {~;} is a sequence in H such that I:: l II~d2 < 00. Note that aa ---> a a-strongly iff (aa -a)*(aa -a) ---> 0 a-weakly. [AS, Def. 2.27 and Def. 2.64]
A 85. For each positive operator a E B(H) we define the trace of a as the sum tr(a) = I:,(a~'YI~'Y) (with values in [0,00]), where {~'Y} is any orthonormal basis in H. For arbitrary a E B(H) we define Ilalll = tr(lal)· [AS, Def. 2.51] A 86. An operator a E B (H) is of trace class if Ilalll < 00. The set of trace class operators is denoted by T(H) (or by L 1 (H)). By Lemma 2.53, T(H) is a two-sided ideal in B(H) and Ilalll is a norm on this ideal; we call it the trace norm. [AS, Def. 2.54] A 87. If r is a trace class operator, we define a linear functional on B (H) by Wr (a) = tr( aT'). The map r f---> Wr is an isometric order isomorphism of the ordered Banach space T(H) of trace class operators onto the subspace of B (H)* which consists of all a-weakly continuous linear functionals. [AS, Thm. 2.68]
Wr
A 88. If A is a unital *-subalgebra of B(H), then the bicommutant A" is the closure of A in any of the weak, strong, a-weak, or a-strong
topologies. [AS, Cor. 2.78]
APPENDIX
433
A 89. A unital *-subalgebra M of B(H) is called a concrete von Neumann algebra if it is weakly closed. A C*-algebra which admits a faithful representation as a concrete von Neumann algebra is called an abstract von Neumann algebra. [AS, Def. 2.80] A 90. A positive linear said to be normal if ¢( a) = least upper bound a in A. said to be normal if it is a functionals on A. [AS, Def.
functional (or state) ¢ on a C* -algebra A is lim-y ¢( a-y) for each increasing net {a-y} with More generally, a linear functional on A is linear combination of normal positive linear 2.82]
A 91. Let M be a von Neumann algebra acting on a Hilbert space H. The O"-weakly continuous linear functionals on M form a norm closed subspace of M'. [AS, Lemma 2.85] A 92. If M is a von Neumann algebra acting on a Hilbert space, then the O"-weakly (and O"-strongly) continuous linear functionals on M are precisely the normal linear functionals on M. [AS, Cor. 2.87] A 93. The O"-weak topology on a von Neumann algebra M is determined by the semi norms x f--+ Iw(x)1 where w is a normal state on M, and the 0" -strong topology on M is determined by the seminorms x f--+ w(x'x)1/2 where w is a normal state on M. [AS, Cor. 2.90] A 94. If M is a von Neumann algebra, then the map W : M f--+ (M.)* defined by (Wa)(¢) = ¢(a) for a E M and ¢ E M. is a surjective isometric isomorphism and a homeomorphism from the O"-weak topology on M to the w*-topology on (M.)*. Moreover, Wa ::,. 0 iff a ::,. o. [AS, Thm. 2.92] A 95. A C*-algebra A is a von Neumann algebra iff it satisfies either one of the following two conditions: (i) Asa is monotone complete and the normal states separate the points of A. (ii) There is a Banach space B such that A is (isometrically isomorphic to) the dual of B. Moreover, if condition (ii) is satisfied, then B is unique, in fact it is (up to an isometry) the Banach space of normal linear functionals on A. [AS, Thm. 2.93] A 96. The self-adjoint part (M.)sa of the predual of a von Neumann algebra M is a base norm space whose distinguished base is the normal state space K of M. [AS, Cor. 2.96] A 97. Let M be a von Neumann algebra with normal state space K, and define for each a E Msa the function a on K by ii(p) = p( a)
434
APPENDIX
a
for p E K. Then the map a f--+ is an order and norm preserving linear isomorphism of M,a onto the space Ab(K) of all bounded affine functions on K. [AS, Cor. 2.97]
A 98. The set of projections in a von Neumann algebra is an orthomodular lattice (cf. (A 40)) with p 1\ q = pq and p V q = p + q - pq for a pair of commuting projections p, q. [AS, Thm. 2.104] A 99. To each self-adjoint element a in a von Neumann algebra M there exists a unique resolution of the identity {e)J .\ER such that (i) e.\a::::: >.e.\ and (1 - e.\)a ;::: >.(1 - e.\) for all >. E R, (ii) e.\ = 0 for>. < -ilall and e.\ = 1 for>. > Iiall, (iii) e.\ commutes with a for all >. E R. The projections (iv) a =
e.\
are given by e.\
= 1 - T((a - >'1)+), and
J >.de.\.
Here for x E M +, T(X) denotes the least projection p such that pxp = x, and is called the mnge pTOJectwn of x. [AS, Def. 2.107 and Thm. 2.110] A 100. Let Al and A2 be C*-algebras. Then T f--+ T* is a 1-1 correspondence of unital order isomorphisms T from Al onto A 2 , and affine homeomorphisms from the state space of A2 onto the state space of A 1 . Similarly, if M 1 and M 2 are von Neumann algebras, then there is a 1-1 correspondence of unital order isomorphisms from M 1 onto lvh and affine isomorphisms from the normal state space of M 2 onto the normal state space of M 1 . [AS, Cor. 2.122]
A 101. If A is a C*-algebra with the universal representation 7r (cf. (A 83)), then the von Neumann algebra A = 7r(A) (weak closure) is called the envelopmg von Neumann algebm of A. A is isomorphic (as a Banach space) to the bid ual A ** of A, and the normal state space of A is affinely isomorphic to the state space of A. By common usage, we will denote the enveloping von Neumann algebra by A **. Thus we identify A with A ** equipped with the induced involution and product. [AS, Def. 2.123, Cor. 2.126 and Cor. 2.127] A 102. Let A be a C*-algebra with state space K and enveloping von Neumann algebra A** (cf. (A 101)). Then (A**)sa is isomorphic (as an order unit space) to the space Ab(K) of all bounded affine functions on K under the map a f--+ (with a(p) = p(a) for p E K) which is the unique O'-weakly continuous extension of the corresponding isomorphism of Asa onto A(K) (cf. (A 20)). [AS, Prop. 2.128]
a
A 103. A unital *-homomorphism ¢ : A
-+
M from a C*-algebra
A into a von Neumann algebra M has a unique extension to a normal *-homomorphism ¢: A ** -> M. [AS, Thm. 2.129]
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435
A 104. If u is a normal state on a von Neumann algebra M, then there is a smallest central projection e E M such that u( e) = 1. This projection e is called the central car'r1,er of u, and is denoted e(u). [AS, Prop. 2.130 and Def. 2.131]
A 105. Let J be a u-weakly closed two-sided ideal in a von Neumann algebra M. Then there is a unique central projection e such that J = eM; in fact, e is the unique two-sided identity of J. [AS, Cor. 3.17]
t,
A 106. If M is a von Neumann algebra and F is a subset of M then the least projection p E M such that w(p) = Ilwll (or equivalently w(p') = 0) for all w E F is called the car'r1,er pmJectwn of F and is denoted by carrier (F). [AS, Def. 3.20] A 107. Let F be a face of the normal state space K of a von Neumann algebra M and set p = carrier(F). Then the norm closure F of F consists of all u E K such that u(p) = 1. [AS, Prop. 3.30] A 108. A norm closed face F of the normal state space of a von Neumann algebra is norm exposed. [AS, Prop. 3.34] A 109. Let M be a von Neumann algebra with normal state space K, and denote by :F the set of all norm closed faces of K, by P the set of all projections in M, and by 3 the set of all u-weakly closed left ideals in M, each equipped with the natural ordering. Then there are an order preserving bijection : p f---> F from P to :F and an order reversing bijection ~ : p f---> J from P to 3, and hence also an order reversing bijection e = ~ 0 -l from :F to 3. The maps , ~,e and their inverses are explicitly given by the equations (i) F
= {u E K I u(p) = I}, p = carrier(F),
(ii) J={aEMlap=O},p=r(J)', (iii) J F
= {a E M I u(a*a) = 0 all u E F}, = {u E K I u(a*a) = 0 all a E J}.
Here in (ii) r(J) denotes the right identity of J. [AS, Thm. 3.35] A 110. If p is a projection in a von Neumann algebra M with normal state space K, the associated face Fp = {u E K I u(p) = I} satisfies
Fp = {u
E
Kip· u . p = u}.
[AS, equation (3.14)] A 111. If p is a projection in a von Neumann algebra M, then the following are equivalent:
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APPENDIX
(i) p is central, (ii) a = pap + pi apl for all p EM, (iii) W = p. W . P + pl. W . pi for all wE M *.
[AS, Lemma 3.39] A 112. Let p be a projection in a von Neumann algebra M, let J be the associated a-weakly closed left ideal in M, and let F be the associated norm closed face of the normal state space K of M. Then the following are equivalent: (i) p is a central projection, (ii) J is a two-sided ideal, (iii) F is a split face.
If these conditions are satisfied, then the complementary split face of F is the norm closed face FI associated with pl. [AS, Prop. 3.40] A 113. The normal state space of a von Neumann algebra M is a split face of the state space K of M. [AS, Cor. 3.42] A 114. Let A be a C*-algebra with state space K. Then the canonical 1-1 correspondence between closed two-sided ideals J in A and w*-closed split faces F in K maps the ideal J to its annihilator ;0 n K in K and the face F to its annihilator FO in A. [AS, Cor. 3.63] A 115. If {Fo:} is a collection of split faces in the state space K of a C*-algebra A, then the w*-closed convex hull co(Uo: Fo:) is a split face. [AS, Prop. 3.77] A 116. Let M be a von Neumann algebra and let P be a positive, a -weakly continuous, normalized projection on M sa' There exists a projection p E M such that P = Up iff P is bicomplemented; in this case p is unique: p = PI, and the complement Q of P is also unique: Q = Uq (where q = 1 - p). [AS, Thm. 3.81] A 117. The conditional expectation E = Up + Uq associated with a projection p with complement q = pi in a von Neumann algebra M is the unique normal positive projection of M onto the relative commutant {py = pMp + qMq. [AS, Thm. 3.85] A 118. The extreme points of the normal state space K of B (H) are the vector states wT) (with TJ a unit vector in H), and each a E K is an infinite convex combination of vector states, i.e., a = 2::::: 1AiWT)i (norm convergent sum in B(H)*) where 2:::::1 Ai = 1 and where Ai ? 0 and IITJil1 = 1 for ~ = 1,2, .... [AS, Prop. 4.1] A 119. For each Hilbert space H the normal state space K of B(H) can be inscribed in a Hilbert ball; in fact, K can be inscribed in a ball
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437
defined by the norm of the space HS(H) of Hilbert-Schmidt operators by the map Wr f-> r. If dimH = n < 00, then K is a convex set of dimension n 2 - 1 which can be inscribed in a Euclidean ball of the same dimension. If n = 2, then K is affinely isomorphic to the full Euclidean ball B3; in fact, the positive trace-one matrix representing a state W E K relative to an orthonormal basis {6,6} can be written in the form
where
W
f->
((31, (32,(h) is an affine isomorphism of K onto B3. [AS, Thm.
4.4]
A 120. The face generated by two distinct extreme points of the normal state space of B(H) is a Euclidean 3-ball. [AS, Cor. 4.8]
A 121. A map v from a Hilbert space Hi into a Hilbert space H2 is said to be conjugate linear if it is additive and v( ,\~) = '\v~ for all ,\ E C and ~ E Hi. A conjugate linear isometry from Hi into H2 is called a conjugate umtary. If j is a conjugate unitary map from the Hilbert space H to itself and J2 = 1, then J is said to be a conJugatwn of H. [AS, Def. 4.21] A 122. If v is a conjugate linear isometry from a Hilbert space Hi into a Hilbert space H 2, then (v~IV7)) = (ei7)) for all pairs ~,7) E Hi. [AS, Lemma 4.22]
A 123. Every orthonormal basis a conjugation, namely
{~;}
in a Hilbert space H determines
We will refer to this as the conjugatwn assoczated wzth the orthonormal basls {~;}. [AS, equation (4.13)]
A 124. Let Hi and H2 be Hilbert spaces and let Kl and K2 be the normal state spaces of B (Hd and B (H 2) respectively. If u is a unitary from Hi to H 2, then the map 1> : a f-> Adu(a) is a *-isomorphism from B(Hd onto B(H2) and 1>* is an affine isomorphism from K2 onto K 1 . If v is a conjugate unitary from Hi onto H 2, then the map W : a f-> Adv(a*) is a *-anti-isomorphism from B(Hd onto B(H2) and w* is also an affine isomorphism from K2 onto K 1 . [AS, Prop. 4.24] A 125. Let H be a complex Hilbert space, and {l;aJ an orthonormal basis. Define the map a f-> at on B (H) by at = ja* j where j is the
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conjugation associated with the orthonormal basis. The map a f-7 at is called the transpose map with respect to the orthonormal basis {~,,}. [AS, Def. 4.25] A 126. Let HI and H2 be Hilbert spaces. A map from B(Hd onto B(H2 ) is a *-isomorphism iff it is implemented by a unitary and is a *anti-isomorphism iff it is implemented by a conjugate unitary, cf. (A 124). [AS, Thm. 4.27] A 127. The convex set B3 is the closed unit ball of R3. A convex set (in a linear space) is called a 3-ball if there is an affine isomorphism of B3 onto the set. A parametenzatzon of a 3-ball F is an affine isomorphism ¢ from B3 onto F. [AS, p. 193] A 128. The relation ¢1 rv ¢2 mod 50(3) divides the set of all parametrizations of a 3-ball B into two equivalence classes, each of which is called an onentatzon of B. We refer to each of the two orientations of a 3-ball as the opposite of the other. We denote the orientation associated with a parametrization ¢ by [¢]. An affine isomorphism 'ljJ : BI --> B2 between two 3-balls oriented by parametrizations ¢I and ¢2, is said to be onentatzon preservmg if [¢2] = ['ljJ 0 ¢d, and it is said to be orzentatzon reversmg if ['ljJ 0 ¢d is the opposite of [¢2]. [AS, Def. 4.28] A 129. Let a be a non-scalar self-adjoint operator on a two dimensional Hilbert space H, consider the w*-continuous affine function on the state space K of B(H) (defined by a(w) = w(a)), and assume that attains its maximum J.l at the point (J E K and its minimum l/ at the point T E K. Then the unitary u = e ia determines a rotation Ad: with the angle J.l - l/ about the axis c;:;. [AS, Thm. 4.30]
a
a
A 130. If {6, 6} is an orthonormal basis in a two dimensional Hilbert space H, then the dual of the transpose map with respect to this basis is a reflection about the plane orthogonal to the axis W7)~7)2 where
Thus the dual of the transpose map reverses orientation. [AS, Lemma 4.33] A 131. Let K be the state space of B (H) for a two dimensional Hilbert space H, so that K is a 3-ba11. Then the rotations of K are precisely the maps * where is implemented by a unitary operator u, i.e., : a f-7 uau* = Adu(a), and the reversals of K (i.e., the orientation reversing maps, which are the compositions of a reflection with respect to any diametral plane, and a rotation), are precisely the maps * where
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APPENDIX
is implemented by a conjugate unitary operator v, i.e.,
va*v- I
=
A 132. Let H be a two dimensional Hilbert space and let K be the state space of B(H). If [till is a *-isomorphism of M onto Mn(T). Moreover, eiiMeii is *-isomorphic to T for each ~, so M is also *-isomorphic to Mn(eiiMeii) for each L [AS, Lemma 6.26]
A 170. If M is a von Neumann algebra of type In (n < 00) with center Z and e is an abelian projection with central cover 1, then M is *-isomorphic to both Mn(Z) and Mn(eM e). [AS, Thlll. 6.27] A 171. We say two projections p and q in a C*-algebra are umtar~ly equwalent, and we write p "'u q, if there is a unitary v in the algebra such that vpv* = q. [AS, Def. 6.28] A 172. An element s in a von Neumann algebra M is a symmetry if 8* = 8 and 8 2 = 1, and is a partwl symmetry (or e -symmetry) if s* = s and s2 = e for a projection e EM. We say two projections p and q in M are exchanged by a symmetry, and we write p "'s q, if there is a symmetry 8 E M such that 8p8 = q. [AS, pp. 251-2] A 173. Each e-symmetry 8 in a von Neumann algebra M can be uniquely decomposed as a difference s = p - q of two orthogonal projections, namely p = ~ (e + 8) and q = ~ (e - 8). This is called the canomcal decomposdwn of 8. Conversely, each pair p, q of two orthogonal projections determines an e-symmetry 8 = P - q with e = p + q. Moreover, if 8 is an e-symmetry, then 8 = e8e = e8 = se, and s is a symmetry in the von Neumann subalgebra eMe. [AS, Lemma 6.33] A 174. Let 8 be a symmetry with canonical decomposition 8 = P - q (where q = pi) in a von Neumann algebra M with normal state space K, and let F and G be the norm closed faces in K associated with p and q respectively. Then the set of fixed points of Us is equal to the relative commutant {s y of 8 in M and also to the range space im E = pMp+qMq of the conditional expectation E = Up + Uq, and Us is the unique normal order preserving linear map of period 2 whose set of fixed points is equal to im E. Furthermore, F and G are antipodal and affinely independent faces of K, the restriction of the map E* to K is the unique affine projection (idempotent map) of K onto co(F U G) and the reflection U; determined by s is the unique affine automorphism of K of period 2 whose set of fixed points is equal to co(F U G). [AS, Thm. 6.36]
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445
A 175. Let e and f be two orthogonal projections in a von Neumann algebra M. Then e and f are unitarily equivalent iff e and f are exchanged by a symmetry iff e and f are Murray-von Neumann equivalent. [AS, Prop. 6.38] A 176. To each pair of projections p, q in a von Neumann algebra we assign the element c(p, q) = pqp + p' q' p' (which is a "generalized squared cosine", or "closeness operator" in the terminology of Chandler Davis). [AS, Def. 6.40 and Lemma 6.41] A 177. If p, q is a pair of projections in a von Neumann algebra M, then p - p 1\ q' and q - q 1\ p' can be exchanged by a partial symmetry. [AS, Prop. 6.49] A 178. Let p and q be distinct projections in a von Neumann algebra lip - qll < 1, then p and q can be exchanged by a unitary u E C*(p,q,l) which is the product of two symmetries in M and satisfies the inequality
M. If
111 - ull < hllp - qll· [AS, Thm. 6.54] A 179. Two unitarily equivalent projections p, q in a von Neumann algebra M can be exchanged by a finite product of symmetries. [AS, Prop. 6.56] A 180. If p and q are projections in a von Neumann algebra M, then there is a central symmetry c E M such that cP.:Sscq
and
c'q.:Ssc'p,
i.e., cp can be exchanged with a subprojection of cq by a symmetry, and likewise for c'q and c'p. [AS, Thm. 6.60] A 181. A bounded linear operator
