STATE SPACES OF JORDAN ALGEBRAS
Erik M. Alfsen
and
Frederic W.. Shultz
Introduction
The purpose of this paper is to give a geometric characterization of the state spaces of the class of normed Jordan algebras
named
JB-algebras in [3].
Recall in this connection that by the
generalized Gelfand-Neumark theorem of [3] the study of
JB-algebras
can be reduced to the study of Jordan algebras of self-adjoint operators on a Hilbert space and the exceptional algebra ~ •
the most important examples of a
One of
JB-algebra is the self-adjoint
part of a
C*-algebra; thus the properties we establish for state
spaces of
JB-algebras also give information about state spaces of
C*-algebras.
It is known that state spaces of
JB-aigebras are strongly
spectral compact convex sets in the terminology of [2], but this
property alone does not characterize state spaces of
JB-algebras.
(For example, any two-dimensional strictly convex and smooth compact
convex set is spectral, but it is a state space only if its boundary
is an ellipse.)
In the present paper a characterization is obtained
by adding two geometric conditions: symmetry and the Hilbert ball
property.
First, a spectral convex set
K is said to be
it is symmetric with respect to each set
co(F U F#)
11
symmetricn if
where
F
is a
iv.
The final result is then established in §7, where the results
from § 6 are globalized by the existence of a separating family of
"type- I
representations".
Acknowledgements.
Part of the present paper was completed
while the authors were visiting the University of California,Berkeley.
Both authors would like to ezpress their gratitude to the Mathematics
Department in Berkeley and especially to professors William Bade and
Marc Rieffel for their hospitality during the time each author spent
in Berkeley.
The first author would like to express his gratitude to the
Norwegian Council for Science and the Humanities (NAVF) for a travel
grant which made the visit to Berkeley possible.
The second author
gratefully acknowledges support from a researCh grant from Wellesley
College during a year spent on leave.
.§_j_.
Preliminaries on non-commutative spectral theory.
In this section we will summarize the main results from
(2] which are needed in the sequel, and we will also establish
some useful new results on spectral theory.
Our setting in this
section will be that of (2 ], i.e. we shall consider an order-unit
space
(A,e)
and a base-norm space
(V,K)
in separating order
and norm duality.
Following [2] we say that two positive projections
on
A
(or on
P,Q
are quasicomplementary (q.c.) if
V)
( 1 .1 )
A weakly
with
(i.e. a(A,V)-)
!IPf! < 1
continuous positive projeetion
is said to be a
P: A- A
P-projection if there exists a
(necessarily unique) weakly continuous positive projeetion
p': A -+A
with
ProJ·ectJ.·ons
fiP' !J < 1
p*,p'*
and 2.5 of [ 2]
such that
(on V)
P,P'
are q.c. and the dual
ar e als o q.c.
(See Theorems 1.8
for alternative characterizations of
P-projec-
tions).
To every
up = Pe
P-projection
P
is associated a projective unit
and a projective face
Fp = K n imP*= (p E K I (Pe, p) = 1 } •
(See Theorem 3.5 of [ 2 ]. for a geometric characterization of projective faces).
units in
The sets of
A , and projective faces of
and~ , respectively.
when
P-projections on
A , projective
K are denoted by [JJ , Qt ,
In ~ an ordering is defined by
P ~ Q
PQ = QP = P ; this is equivalent to the natural ordering of
the corresponding elements of
U
Fp ~ FQ
Thus the sets
(cf. [2 ;Lem .. 2.16]).
and
g: ,
i.e. to
?J,
Pe
.s
,fJJ, f/(
Qe
and
are order
- 2 -
isomorphic under the maps
Pe
~
P
~
Fp
(2 ;Th.2.17].
Using
these maps one can transfer the notion of a quasicomplement from
fP
u
to
= Pe
7£
and
and
F
cg; .
= Fp
P E gJ
with
= K n im P' *
F # -- FP'
Specifically, let
•
;
F#
A P-projection P
=
a
if
is.
Q
(P+P' )a •
Note that
P
is compatible with
commute;
u
=
Qe
in this case we say that
Note that if
P,Q
gD
ordering of
P
P~
Q, then
P
and
a
iff
P
fP
P
iff P
Q are comPQ
and
is also
Q in the
It is also easily verified that
Q are compatible.
is said to be bicompatible with an element
patible with
QE
and
are compatible, then
[2 ;Prop.5.2].
a
P-projection
where
P-projection, in fact it is the g.1.b. of
if
and
of convexity theory.
It can be proved (2 ;Prop.5.2] that a
patible.
a
e_- u
say
is said to be compatible with an element
is compatible with a projective unit
and
cg; ,
F# ~ F', cf. [2;p.15]).
Generally
p'
= P' e =
F E
is chosen to avoid con-
F'
fusion with the ordinary complement
a EA
u'
then
(The symbol
u E flJ, and
and with all Q E
6D
P-projection
A
a EA
if
compatible with
P
a •
P
is comA P-
projection is said to be central if it is compatible with all
a EA •
These notions are all transferred from
{jjJ
to
71, and
rg:;
by the natural isomorphisms.
To achieve a spectral theory one must impose an axiom to
ensure that there are "sufficiently many" projective faces of
(V,K)
Specifically,
(A,e)
duality if
is pointwise monotone a-complete (viewed as a func-
A
tion space on
>.EJR
are said to be in spectral
K) and if there exists for every
a projective face
F
such that
on
F
a >A.
( 1 .2)
( 1. 3)
and
K •
F
is bicompatible with
on
a •
a EA
and every
~
3 -
It can be proved that the requirements (1.2) and (1.3) determine
FE
Y.F'
uniquely [ 2 ;Lem. 7. 1].
The corresponding projective unit
is termed the spectral unit of
e~
denoted by
the element
If
or simply by
eA
for the value
A , and it is
when there is no need to specify
a EA •
(A,e)
(hence also
a
U
and
(V,K)
and fJ( )
00
are in spectral duality, then
is a
cr-complete orthomodular lattice.
(See Theorem 4. 5 of [ 2 ] ) •
If
(A,e)
and
(V,K)
are in spectral duality, then every
a E A can be written as an abstract Stiel tj es integral
a =
JAdeA
defined by the corresponding (numerical) integrals
(1.4)
(a, p)
(Cf .• [ 2 ;Th.6.8]).
= J). d ( eA , p)
all
Accordingly we will term the family
the spectral resolution of the given element
for every
functions on R •
~
(eA.}
a EA •
Now one can define a functional calculus in
~(a) = s~(A.)deA.
pEK •
in the class OcJ
A by writing
of bounded Borel
This functional calculus will satisfy all
customary requirements ((8.22)-(8.27) in [ 2 ]), and it will be
the only such calculus which is "extreme point preserving" in
that
XE(a)
is an extreme point of
every Borel set
E
A~ = {a E A I 0.::; a.::; e
(i.e. for every extreme point
XE
for
of 6.3~)
[2 ;Th.8.9]. Note in this connection that the extreme points
of A+1 are exactly the projective units, and they are in turn
just those elements
i.e.
a 2 =a
[2 ;Prop.8.7].
where
a E A which are "idempotent under squaring",
a2
= cp(a)
with
cp(A)
= A2
for
A E.:R
(Observe that spectral duality is essential for
the identification of projective units with extreme points.
In
more general cases the projective units can form a proper subset
of the extreme points of
A~ , cf. [ 2 ; p.14]).
- 4 -
Assuming that
we can ext.end
elements of
(A,e)
and
(V,K)
are in spectral duality,
the notion of compatibility to arbitrary pairs of
A •
For given
compatible if all pairs e a
-
A.
are compatible.
a, bE A
'
eb
~
we say that
a
and
are
from their spectral resolutions
Then we define an abelian subspace of
a norm closed subspace containing
b
to be
A
e , closed under the map
a~ a 2 , and all of whose members are mutually compatible.
abelian subspace of
X
A
is isometrically isomorphic to
Every
C(X)
for
compact Hausdorff when equipped with the product
( 1 • 5)
(Cf. [ 2
;Prop.9~8]).
For every subset of
A
consisting of mutu-
ally compatible elements (in particular for every single element
of
A) there is a smallest abelian subspace of
A
containing it
(and a smallest weakly closed abelian subspace containing it).
The set
of
A
of
A .
~ith
a
Z(A)
of elements in
A
compatible with every element
form a weakly closed abelian subspace called the center
(Note that this definition of center is consistent
that of
[23] , cf. [2; Th. 9.19], and note also: that
P-projection is central in the previous sense of the word iff
Pe E Z (A) , cf. [ 2 ; p. 7 9]).
In most of the important applications the spaces
will satisfy the requirement
hence
£P
when
(A,e)
then every
and
fF)
and
A
~
Then the lattice
is complete [ 2 ;Cor.12.5].
(V,K)
projective [2 ;Cor.12.4].
projective faces of
K
K
is
and
q£
V
(and
Recall also that
are in spectral duality with
A-semi-exposed face of
exposed faces of
V* •
A
A~ v*
A-exposed, therefore
Hence in this case the collection of
coincides with the collection of
A-semi-
K , and so it is closed under arbitrary inter-
~
sections.
5 -
It follows that every subset
smallest projective face
.
we write simply
F(p)
by writing
= F(m)
F(p)
=
F
F(E) •
E c K is contained in a
If
E
p E v+,[o}
For
, and we set
=
for some
(p}
p E K,
we extend this notation
=¢ .
F(O)
In the applications to convexity theory one starts out
with a convex set
K which can be embedded in a linear space
(uniquely up to a linear isomorphism) in such a way that
becomes a base-norm space.
(Recall from [1 ;p.77] that
is a base-norm space if the affine span of
not passing through the origin and
K
co(KU -K)
V
(V,K)
(V,K)
is a hyperplane
is radially compact,
this set being the unit ball of the base ...norm). As usual we will
K by
denote the space of bounded affine functions on
We will say that a convex set
embedded in a linear space
V
K is spectral if it can be
in such a way that
a base-norm space in spectral duality with
A = Ab(K) ~= v* •
Wh en wor k"1ng
Wl"th
Ab(K).
(A,e)
(V,K)
becomes
where
spec t ra 1 convex se t s, we
Wl"11
always assume that this embedding is performed, and we will make
free use of notions from general spectral theory (e.g. that of a
"projective face") which will then refer to the duality of
and
(A,e)
(V ,K).
The above definition is more general than that of
which is confined to compact convex sets.
( 2 ]
Note, however, that the
new definition will agree with the old one when
K is compact,
since every compact convex set can be embedded in a locally convex
space
V in such a way that
(V,K)
becomes a base;...norm space
(the "regular embedding" [ 1 ;Ch.II,§ 2]).
Following [ 2] we say that a compact convex set
K
is
strongly spectral if it is spectral and if in addition
is
u.s.c. in the given topology of
a
the space
A(K)
K
for all
A. EJR
of continuous affine functions on
and all
K.
in
A spectral
- 6 compact convex set
K is strongly spectral iff
A(K)
is closed
under functional calculus by continuous functions [2 ;Th.10.6].
In particular, if
K is strongly spectral, then
under the squaring map
a 1-> a
2
A(K)
is closed
Examples of spectral compact
•
convex sets are the closed unit balls of
LP
for
1 < p < oo
the weak topology and also all Choquet simplexes.
in
The former are
all strongly spectral, the latter are strongly spectral iff they
have closed extreme boundaries (i.e. if they are Bauer simplexes).
[ 2 ;Ths.10.4-10. 5 and Prop.10.9].
Now let
that
A
~v
*.
to
K •
of
K
For a given
FCK
jective face
P-projection
p
with associated pro-
we consider the restriction
(P+P') *
of
1.\rF
This map can be shown to be the unique affine retraction
onto
co(FUFif) [ 2; Th.3.8].
(this occurs exactly when
Prop.10.2]), then
with
be in spectral duality and suppose
(V,K)
(A, e) and
pE
F
and
K
P
If
is a central
F
is a split face of
P-projection [2;
[a,T]
is the union of all line segments
a E Fif.
For a general projective face
K is the union of the fibers
w]: 1 ([p,a])
K
with
pE
F
F , then
and
a E F4f.
Note that two different fibers will either be disjoint or they
will meet at an "end point"
pEF
or
a E F# •
Later on we shall
see that the geometry of these fibers holds important information
about
K ..
An important special case is obtained by taking
A
to be
the self-adjoint part of a von Neumann algebra 0~ with identity
element
space
(V,K)
e , and
(~*
with
to be the self-adjoint part of the predual
V
K
the normal state space.
are in spectral duality with
tions are precisely the maps
projection in
a~>
OL [ 2 ; Prop.11.4].
Then
A= V* , and the
pap with
p
(A,e)
and
P-projec-
a self-adjoint
In this case the notions of
- 7 -
compatibility
and biyompatibility will coincide with commutation
and bicommutation ( 2; Cor.11.3].
Applying these results to the
enveloping von Neumann algebra, one easily sees that the state
space of a
c*-algebra is a strongly spectral compact convex set
( 2 ; Th. 11 • 6 ] •
After this summary of results from [ 2 ] we shall present
some new results on non-commutative spectral theory which will be
needed in the sequel.
Our first proposition will be a new charac-
terization of spectral duality.
if
(A,e)
and
such that
A
(V,K)
In this connection we recall that
are in separating order and norm duality
is pointwise monotone
posed face of
K
a-complete and every
is projective (this is implied by spectral
duality [ 2 ; Prop. 6 .2]), -l:;hen every
element
r(a) E
U
such that
denotes the face of
In fact,
( 1. 6)
r( a)
A+
a E A+
generated by
=0
<==>
Recall also that two elements
a~
b , if
admits a smallest
aE face(r(a)) •
(a,p)
a, bE A+
~
r(a)+ r(b)
P-projections corresponding to
r(a)
(Here face(r(a))
r(a) , of. (2; Prop.4.7]).
?.£
is the unique element of
(r(a),p)
in symbols
A-ex-
=0
such that for
p
EK :
•
are said to be orthogonal,
e , or equivalently if the
and
r(b)
annihilate each
other [2; Prop.4.4].
Proposition 1.1.
Let
(A,e)
order and norm duality such that
plete and every
and
A
A-exposed face of
(V,K)
be in separating
is pointwise monotone a-comK is projective.
two spaces will be in spectral duality iff every
unique decomposition
a
= a+- a-
where
a+,a- E A+
Then the
a E A admits a
and
a+ .J...a- •
.,.. 8 -
Proof.
1.)
To prove that the condition is sufficient, we
assume that every
a EA
has a decomposition of the type described.
By proposition 6. 3 of [ 2 J
(A, e)
and
are in "weak"
(V ,K)
spectral duality, in that they satisfy all requirements for spectral duality except the bicompatibility in (1.3) which is replaced
by compatibility only.
But by Theorem 7.5 of [2] the desired hi-
compatibility will follow if we can prove that for any given
a E A and
which satisfies ( 1. 2).
may as well assume
only one
FE
fF
on
Now let
FE
fJ;'
Then
F ,
and
a
by
a
a- Ae , we
Thus, we shall prove that there exists
compatible with
a < 0
compatible with
Since we can replace
A= 0 •
-
(1.7)
FE (/fi
there is a unique
A. EJR
a
such that
on
a > 0
P E fP
F:/1= •
correspond to the projective
a+= P'a
and
a
= -Pa , and by (1.6)
we have
F = {pE K I (r(a+),p)= 0
( 1 .8)
Hence
and
Thus since· a= P'a =a+
on:,
a = Pa =-a
-
= {pE Kl (a+,p)= 0}.
on
F , then the projective face
on
must
F
satisfy the relations of (1.7).
Now suppose that
ible with
(Such a
a
G is any projective face of
and satisfying (1.7) with
G.
Then
a = Q' a- (-Qa)
on
G, and so by (1.8)
Let
QE
F •
q:>
corre-
will be a decomposition of
the type mentioned in the theorem, so
a+= 0
G in place of
G exists by weak spectral duality)..
spond to
K compat-
G c F.
Q' a •
Now
F
It follows that
and
G are
compatible, and it follows by the argument of (2; Lem.7.1] that
F
is the only member of ~ which is compatible with
satisfies (1.7).
a
and
- 9 2.)
To prove necessity we assume
spectral duality.
a decomposition
Taking
F
and
F
holds.
P
to be the projective face and
e-r(a+)
(V,K)
in
P-projection
and arguing as before, we conclude
is a projective face compatible with
a
for which (1~7)
Spectral duality implies that such projective faces are
uniquely determined [ 2 ; Th. 7. 2], and so
where
Now
and
By Proposition 6. 3 of [ 2 ] every a E A admits
a n d a + l a- •
a= a + -a
corresponding to
that
(A,e)
F0 E
e.g;
and
P = P0
P 0 E qJ correspond to the spectral unit
and
a+= p'a = p'a •
Hence
0
a+
is uniquely determined, and
0
we are done.
Remark.
F = F0
The above Proposition 1e1 may be considered a supplement
to Proposition 6.3 of [ 2 ], stating that uniqueness of the decemposition
a= a + -a
is exactly what is needed to pass from weak
spectral duality to spectral duality.
Our next result is a technical lemma.
For the proof of
this lemma we recall two useful formulas from [ 2 ].
u 1 ,u 2 , ••• ,un
If
are mutually orthogonal projective units, then by
[2; (4.13)]:
n
( 1 • 9)
v u ~.
. 1
= u1
+ ••• + u
n
~=
•
Secondlyt by the orthomodular identity [ 2; Th.4.5 (iv)] and (1.9),
the following holds for projective units
u
(1.10)
Lemma 1.2.
then for given
(1.11)
~
v
implies
If
(A,e)
and
a EA +
and
P E f!P
r(Pa) = (r(a)
V
u,v :
v-u=v"u.
'
(V,K)
are in spectral duality,
p' e)/\ Pe .
- 10 Proof.
We claim that it suffices to prove
( 1 • 12)
= r ( Pa) +
r (a) V P' e
P1e •
Then also
v
= r(Pa) + P ' e
, and since
= VAU '
(r(a)VP 1 e)APe
=
p'
since
also be.
and
r(a)
and
.:f
v , we obtain from (1.10)
P
I
and
P v Q
I
=
Q
and
R
= Pa
im+P0
equals the face of
A+
=
r(Pa).
the
P-projections
P
Note that
and
I
P v Q will
(Recall that for
generated by
P0 e
P0 E
YJ
[ 2; Cor.2.12]).
P a E im ( P 1 v Q ) , and so
r ( Pa) ~ ( P' v Q) e
= (P' e) v
r (a) •
r(Pa)~Pe); therefore (by (1.9)):
Clearly
r(Pa).LP 1 e (since
(1.13)
r(Pa)+P e = r(Fa)vP e < (P e)vr(a).
I
I
I
To prove the other half of (1.12) we note that
implies
R~F •
Hence
I
(PAR )a
1
(P v R)a
PAR
=R
=
I
I
I
= R P
=R
Pa
I
e•
,
im+Q S im(P 1 v Q) •
a E face(r(a))
= r (a) v P
commute, and so
I
since
v
V-U
P 1 v Q are compatible, then
Hence
and
e
r(Pa) , respectively.
(P v Q)Pa = P(P v Q)a
Now
=
I
=P
(r(Pa)+P 1 e)-P 1 e
To prove (1.12) we denote by
corresponding to
u
I
u
In fact, given this inequality we write
I
(FAR ) a
I
and
RF
=
0 •
(RP.a)
=a
•
=
F •
r(Pa) ~Fe
Thus we have
- 11 -
a E im(P' v R) = .fa9e( (p' v R)e) , and so
Hence
r ( a ) ~ ( P ' v R) e = ( P 1 e ) v r ( Pa ) •
Hence we have
(cf~
(1.13)):
I
(1.14)
~
(P e) v r( a)
I
=P
(P e) v r(Pa)
I
e + r(Pa) •
0
By (1.13) and (1.14) the proof is complete.
One can dualize the proof of Lemma 1.2 by replacing
p*
a
by
(Recall that
~) •
crE v+,{o}) , r(a)
(where
a
F(cr)
by
P
by
F(cr) , and so on.
is the projective face of
K generated by
This gives:
Corollar;y 1.3.
dualit;y with
A
Erojective face
Let
and
(Az e}
= v * . For given
F = K n imP* and
(V 2K)
fP
PE
be in s:12ectral
with corresEonding
for each
cr E v+
one has
We now turn to the problem of relativizing spectral duality.
Proposition 1.4.
A= v*
duality with
V0 = imP *0
,
and
Let
(A,e)
Let
P 0 E (!P
K 0 = V0 nK.
and
Then
(V,K)
A0
=
imP 0
(A~ 0 )
in spectral duality for the induced pairing and
P-projections on
PEf}?
, p==:;: P 0
A0
I
those
=
P
I
A
(such maps leave
P 0 1Ao
)
•
FE q:; such that
are just those
uE
U
,
A0
such that
= P~
,
(Vo.z.,!f 0 ) are
A0 = V* • The
PIAo
with
invariant); for such a
P'
jA 0
(which is the same as
The projective faces of
F s;;; K0
e0
and
for this duality are the maps
P-projection the Quasicomplement is
E_E 0 1Ao
be in spectral
K0
are just
and the proj ecti veuni ts in
u < eo-.
A0
- 12 Proof.
By [ 2; Prop.2.11]
(A 0 ,e 0
(V0' K0 )
and by [ 2 ; Prop. 2. 14]
is an order-unit space,
)
J.·s a base-norm space.
We omit
the easy verification that they are in separating order and norm
duality with
A0
V*0 •
=
Note, the latter fact implies that
is pointwise monotone complete.
p E q{J and
Recall that if
PI
compatible, and so are
I
P 1 AP 0 = PIP 0 = pop •
It follows that
invariant, and clearly
and
Also
and
,
P ~Po
and
then
Moreover,
Po •
p
IAo
PI
and
P* jv 0
P I* lvo •
and
p
= p0p
p'
and
leave
A0
~p 0
the restriction
,
with quasicomplement
P jA 0
on
A0
and
V0 , and with
Hence for each
PjA 0
1
A0
The latter two projections
are seen to be quasicomplementary.
such that
p
are
Po
are seen to be positive, of norm at most 1,
dual projections
V0
and
are quasicomplemen t ary.
weakly continuous in the induced duality of
on
p
will be a
P E gD
P-projection
•
In order to apply Proposition 1.1 we will now verify that
each
A0 -exposed face of
and
(Ao,eo)
there exists
We define
(b, p)
to
=0
(a,p)
aE A+
= a+ P 0' e = a+ e- a 0 E A+
=0
iff (a, p) = (P'e,p)
0
= 0 and P E K0 i.e. to
'
F = Kn im * P •
that
A0 -exposed face
F
on
a > 0
'
pEK
For
F
on
of
Let
PE
Clearly
P
A0
fP
P0
since
.
we then have
which in turn is equivalent
PEF
correspond to
~
Ko
K0 ' F
..
Thus, F
will be an
K and therefore also a projective face of
P-projection on
F
a = 0
b
[2; Prop.6.2].
F
For a given
such that
0
A-exposed face of
is a
.
(V o'Ko))
is projective (in the duality of
Ko
F
~
K
F, i~e. let
K0
•
Therefore
PjA 0
and it follows from the equalities
= K n im p * = K n ( im p *0) n ( im p * ) = K n im ( p * Iv 0 )
is a projective face of
K0
corresponding to the
P-pro-
- 13 jection
(with dual
PjA 0
From the above argument it also follows that the projective
faces of
Ko
K which
are precisely those projective faces of
are contained in
Ko
,
cisely the maps
PtAo
projective units in Ao
that the
with
P-projections on
PEf/J
and
P~ Po
,
are pre-
Ao
and that the
are precisely those projective units in A
which are majorized by
e0
Hence it only remains to verify the
•
criterion for spectral duality given in Proposition 1.1.
To this end we consider an arbitrary a E A 0
unique decomposition a= a + -a
with a + ,a - ~ 0
in the duality of
a
(A,e)
are actually in
A0
and
(V,K) •
(Ao,eo)
and
norm, then
(since
a+
with
(~\,eo)
a_s !lajjeo
a E A0 = im P0
is the l.u.b. of
= A0
r(a+)
.
and
a
a
Clearly
P0
and
and
0
r(a-) , and hence
where
(A, e)
and
(V,K) •
are compatible
eo
0 < a+ _s fla'!e 0
•
Hence also
By [ 2 ; Prop.9.3]
In particular
a
=a+- a EA 0
0
With the assumptions and notation of Propo-
p~
p(a)
the functional calculus on
for all
A
a E A0
(A~ 0 )
is given by
is the functional calculus on
For those continuous functions
P 0 ( p (a) ) = p ( a)
and
among those elements compatible
sition 1.4, the functional calculus on
(A,e) •
and
are compatible).
(cf. [ 2; Cor.2.12]).
Corollary 1.5.
P 0 (~(a))
a+
is an order-unit space for the relativized
and the proof is complete.
p ->
We will show that
as in the duality of
a , so we conclude that
a+E imP0
and
a+ ~a- , will be the same in the duality of
(V o'Ko)
Since
having the
This will complete the proof, for by
•
the above results the definition of
of the relation
,
p
such that
agrees with that of
•
A
~(0)
= 0
-· 14- -
Proof.
Fix
a E A0
Borel functions on E
The map
•
into
A0
cp
t->
P 0 (cp (a))
from bounded
will satisfy all the requirements
for an extreme point preserving functional calculus (as specified
in ( 2; Th.8,9]), and by the uniqueness of such a functional calculus,
cp~P 0 (cp(a))
defined on
must coincide with the functional calculus
(A 0 ,e 0 )
Now assume
•
cp
exists a sequence
is continuous, with
Then there
{cpn} of continuous functions, each vanishing
in a neighbourhood of zero, such that
(-Hall,!la!IJ •
cp(O) = 0 •
For each index
n
such that
cp n -> cp
there exists a positive constant
for all
the functional calculus on
A
uniformly on
AE
[-II all, Hall]
•
Since
preserves order, we have the rela-
tions
By Proposition 1.4,
a+
cpn(a) EA 0
for
calculus,
cpn(a) -> cp(a)
as claimed.
and
n = 1 '2' •••
,
a
tl
so
are in
A0
and therefore
,
By norm continuity of the functional
cp(a) E A0
Therefore
•
P 0 (cp(a))= cp(e.)
0
From Corollary 1.5 we immediately obtain the following:
Corollary 1.6.
With the assumptions and notation of Propo-
sition 1 .4, the squaring operation
a J-> a 2
is the same one whether calculated in
A
for elements
or in
A0
a EA0
•
Our next corollary concerns the relativization of central
P-projections.
Corollary 1.7.
sition 1 .4, let
is central for
With the assumptions and notation of Propo-
P 0 E <[p be central.
A
Then a
iff it is central for
A •
P-proj ection
P ~ P0
- 15 Proof.
iff
Pa _::; a
If
a
0
E A+ •
0
tral.
for all
Pa .::; a
a
for all
By [ 2; Prop.5.1]
0
a E A+
for all
E A+ c A+
0-
a E A+ , then clearly also
0
duality with
= P(P 0 a)
A
= V*
jection other than
< P a < a
-
PE
f/J
-
(A,e)
for all
since
-
P0
is cen-
V)
'
and
Then we say
•
0
0
(V,K)
are in spectral
is a factor (with respect
A
if it does not admit any central
and
I •
is a factor if
Also we say that a subset
A0
A0 = im P 0
is central, will be a factor iff the only central
0
and
P0
A
= V* •
Let
such that
=
=1
such that
P
where
P0
P-projections
be in spectral
there exists a smallest
(c(p),p)
is the smallest
=
1 ;
the
P-projection
p ; and the corresponding projective face is the
K
which contains
Recall first that for
iff
(V,K)
c(p)
smallest split face of
Proof.
and
PE K
P-projection
P* p
(A,e)
For each
central projective unit
corresponding
= imP
•
Proposition 1.8.
duality with
A0
is a factor in the above sense
Note that by Corollary 1.7 the subset
are
P-pro-
V 0 = imP* •
with respect to the duality with
(Pe,p)
Pa 0 .$ a 0
0
Suppose
to the duality with
P0
A0 )
Hence
Definition.
~
(or
Pa 0 .$ a 0
P a < a
we have
which completes the proof.
P
A
A E A~) •
(respectively
a E A+
Pa
where
is central for
Conversely, assume
•
For given
P
P* p
=
p
pEK
[ 2 ; Lem .. 2. 3 J.
p •
and
P E
£P
one has
- 16 -
Now, if
to central
c 1 ,c 2
are central projective units corresponding
P-projections
P 1 ,P 2 , and if
= (c 2 ,p) = 1
(c 1 ,p)
,
then
and so
units
=
(c 1 A c 2 ,p)
c
such that
1 •
Therefore the set of central projective
=1
(c,p)
, is directed downward.
Lem.12.1] the pointwise limit
c(p)
of this directed set exists
and is a projective unit; by continuity
all
P E
fP ,
c(p)
so
is central.
central projective unit such that
(P+P 1 )c(p)
Thus,
c(p)
(c(p),p)
=
= c(p)
corresponding to
P* p
such that
PE
p •
1.
[j)
is the smallest central
A= V * •
Let
(A,e)
For given
and
pEK
P-projection corresponding to
c(p)
= Pe
(By l:2; Cor.2.12]
A
is
[]
(V,K)
be in spectral duality
c(p) ; thus
p
F
A
the range of
p
A
p
= imP
where
is also the order ideal of
A
c(p)) •
Proposition 1.9.
A
-p
is a split face
p •
we denote by
the
duality with
P-projection
is central [ 2 ; Prop. 1 o·. 2], we conclude that
Definition.
generated by
KnimP *
Recalling that
the smallest split face containing
with
P-projection
By definition the corresponding projective
*
F = Kn imP.
face is
iff
=
c(p)
for
is the smallest
By the introductory remark of this proof, the
P
By [2;
A
= V*
Let
If
(A,e)
p
and
(V,K)
be in spectral
is an extreme point of
K
then
is a factor.
Proof.
P-projection
is central,
Suppose for contradiction that there exists a central
QE
fP
Q+ Q'
=
such that
I •
Hence
Q~ P , Q
f.
0 , Q
' *
(Q+Q)p=p
f.
P •
Since
Since
p
is
Q
- 17 Q* p
extreme,
Q'* p
and
Q* p = 0
necessarily
or
c1 = Qe
and
0 < c1 < c(p)
c1 + c2 = c{p)
p •
Q'* p = 0 ; otherwise
would be a
Q'* p
non-zero multiple of
Now let
must both be multiples of
which is impossible.
and
Q*p = 0
then
(c1 ,p) = (Qe,p) = 0
Q'* p = 0
If
( c2 ,p) = (c(p)- c 1 ,p) = 1 •
(c 1 ,p) = 1 •
By hypothesis
c2 = c(p)-Qe •
are central projective units such that
c1,c2
If
Q* p
then we similarly get
'
c(p) •
[]
1t .
We now turn to the study of minimal elements of
Proposition 1.10.
Let
A= V* •
duality with
7£ ,
If
(A,e)
imP
where
~R
I (a, u)
= u,
Pe
of the minimal elements of
Proof.
ing
Let
uE
7£
P-projection, i.e.
order ideal of
A
qe
A E lR+
0
a
0
onto the
where
u~
p EK
p
u •
0 < a
~
1-1 map
A-exposed points of
P
Therefore
u
and
is a
implies
K.
be the correspond-
By ( 2 ; Cor. 2. 12] ,
u = Pe
generated by
.
Since
eA
=0
e - eA
~
e- e 0 = u
for
A. 0 ElR+
,
a < u
~
have
exists
f p}
im P is the
imP~
a
1R
will
= A0 u
for
.
Now let
tion of
=
be minimal and let
follow if we can prove that
some
F
1-dimensional
and the corresponding pro-
Moreover, the map
= 1
be in spectral
P-projection has a
jective face is a singleton, i.e.
F = fa E K
(V,K)
and
is a minimal (non-zero) element
u
then the corresponding
range, i.e.
so
'
In either case we have a contradiction with the
minimality requirement defining
of
Then
a
{e~)
is positive and
A. < 0
for
and let
and
e0
A> 0 •
such that
eA. = 0
=
be the spectral resoluuE
e- r(a)
U
=
is minimal, we
e - u •
Hence also
From this it follows that there
for
A < A0
and
eA.= u
for
- 18 -
From the above it also follows that
sist of just one point
Since
face.
F
Hence
is a projective face, it is also an
p
is an
points in this way.
[p}
= K n imP*
is an
must con-
p EK •
A-exposed point of
different minimal elements of
then
F
U
Finally, if
K •
A-exposed
Clearly two
determine different
p
is any
A-exposed face; hence
A-exposed
A-exposed point,
[p}
is a projective face.
Clearly it is minimal among the (non-empty) projective faces;
hence it corresponds to a minimal projective unit.
Remark.
(]
It follows from the above proposition that the
minimal projective faces are (singletons consisting of) extreme
points.
It is not clear if the converse holds in this generality.
But in § 3 we shall prove that it does hold in the
JB-algebra
setting.
Definitions. Let (A,e) and (V,K) be in spectral duality
A = V*
Then a minimal non-zero projective unit in A
with
will be called an atom.
unique
If
that
u
we denote by
u"
the
(A-exposed and extreme) point in the corresponding pro-
jective face.
(1.16)
For each atom
Thus
....
u
is the unique point of
(u,U.>
K
such that
= 1 •
A is a factor containing at least one atom, then we say
A is a factor of type I.
- 19 -
i_g.
JB-algebras and spaces in spectral duality.
Following [ 3 ]
algebra
B
we define a
JB-algebra to be a Jordan
over the reals with identity element
with a complete norm such that for
l!aob!l
<
/la!!·llbl!
(2.2)
lla2 !1
~
llall 2
(2.3)
Jla211 ~ lla2+b2fl
squares in
B
B
is a
equipped
a, bE B :
( 2. 1)
Recall that if
e
B2
JB-algebra, then the set
is a proper convex cone organizing
B
of all
to a (norm)
complete order-unit space whose distinguished order-unit is the
multiplicative identity and whose norm is the given one, and such
that for
a EB
-e~ a ~ e
(2.4)
Conversely, if
B
0 ~ a2 < e .
implies
is a complete order-unit space equipped with
a Jordan product for which the distinguished order-unit acts as
identity element and such that (2.4) is satisfied, then
a
is
JB-algebra in the order-unit norm [ 3 ; Th.2 .1] •
In § 3 of ( 3 ]
JB-algebra
B
it is shown that one can associate to every
a monotone complete enveloping
and in [20] it is proved that
bidual
B**
dual spaces.
element
e
JB-algebra
~
B can be identified with the
equipped with the Arens product and the usual norm.
In (20] there is also an investigation of
K
B
Specifically, let
such that
A
= V*
A be a
normal if whenever
JB-algebra with identity
for some Banach space
be the set of normal states for
linear functional such that
JB-algebras which are
(e,p)
A •
=1
(A state p
V , and let
is a positive
, and it is said to be
is an increasing net in
A with least
- 20 upper bound
a, then
(a,p) = lim(aa.,p)). It is shown [20;Th.2.3]
a.
is unique, in fact it can be identified with
that the predual
V
the space
of all normal linear functionals in
linK
Definition.
In the sequel we will refer to
which are dual spaces, as
A*
JB-algebras
the monotone complete enveloping
Also we will refer to
**
JB-algebra B ~ B
of a
JB-algebra
JBW-algebra of
B
JBW-algebras.
as the enveloping
I'J
It is natural to expect that a
are in spectral duality.
B •
JBW-algebra and its predual
We are now going to prove this result
which will generalize Theorem 1 2.13 of [ 2 ] •
Proposition 2.1.
predual
Let
A be a
V , and normal state space
K •
are in spectral duality, and the map
algebra
A
JBW-algebra with identity e,
Then
a~> a 2
(A,e)
and
(V,K)
in the Jordan
coincides with the squaring map defined by the func-
tional calculus.
The projective units in
idempotent elements, and the
A are precisely the
P-projections are the maps
Up
defined by Jordan triple products
(2.5)
=
up a
with
p
tion for
an idempotent in
uP
is
all
(pap.}
A
the quasicomplementary
aEA
'
P-projec-
ue-p •
Proof.
In [20; Th.2.3] it is shown that under the natural
embedding of V in V** = A* the image of (V,K) is of the form
""' n im U*c) where c is a central idempotent in A = A **
( im U*c , K
and
I'J
K
is the state space of
Prop.2.14] that
(V,K)
shown that each map
Ua
A
Now it follows from [2;
is a base-norm space.
with
a EA
In [20]
'
it is also
takes normal functionals to
- 21 (i.e. a(A,V)-) continuous.
normal functionals, and so is weakly
Given these results we complete the verification as in the proofs
D
of Theorems 12.12 and 12.13 of [2 ].
Corollary 2.2.
The state space of a
JB-algebra is a
strongly spectral compact convex set.
Proof.
Let
B
the order-unit space
be an arbitrary
(A,e) where
algebra) and the base-norm space
the state space of
in spectral duality.
Hence
K
B •
A
JB-algebra and consider
,..,
=B
(the enveloping
(V,K) where
V
= B*
and
JEWK
is
By Proposition 2.1 these two spaces are
(This also follows from [ 2; Th-.12.13]).
is a spectral compact convex set.
It follows from the uniqueness of the spectral functional
calculus [ 2; Th.8.9] that the functional calculus defined by the
spectral duality of
[ 3; § 4].
(A,e)
In particular
and
B ;' A(K)
calculus by continuous functions.
Having explained how
(V,K) , will agree with that of
is closed under functional
Hence
K
is strongly spectral.O
JB-algebras give rise to spaces in
spectral duality, we will now consider the converse problem of
deriving Jordan structure from spectral duality under appropriate
hypotheses.
Henceforth we assume that
(A,e)
and
and base-norm spaces in spectral duality.
candidate for a Jordan product in
(V,K)
are order-unit
Then there is a natural
A , namely
(2.6)
where the squares are defined by the functional calculus.
operation will coincide with the customary Jordan product
This
- 22 aob
= i(ab+ba)
when we specialize to operator algebras (cf.
[ 2; Prop.11.4, Th.11.6].
But in the general case the product
(2.6) can fail to be bilinear.
Hm-vever, it is proved in [ 2;
Th.12.12] that if it is bilinear, then it organizes
JB-algebra.
A
to a··
We will now proceed to give a necessary and sufficient
condition for bilinearity of the operation (2.6).
First we recall that if
(2.6) and if
jection
P
u
on
A
is a
JB-algebra for the product
is a projective unit corresponding to a
A , then by Theorem 12.12 of (2]
ing multiplication operator
(2.7)
u oa
a~>
= t (a+ Pa -
uoa
P-pro-
the correspond-
is given by
P 'a) •
This formula motivates the following general definition of
uE
U
with
A-> A associated with a given projective unit
Tu :
the operator
u
= Pe
for
PE
t[P :
(2.8)
For later references we state the following:
Lemma 2.3.
Let
(A,e)
and
(V,K)
be an order-unit space
and a base-norm space in spectral duality.
u
= Pe
(2.9)
, v
=
Qe
[Tu,Tv]e
Proof.
for
P ,Q E fP
= 41 [P-P
I
I
,Q-Q ]e
,
If
u,vE
'lt
with
then
1
= ~([P,Q]+
t
[P ,Q
']
)e .
By linearity of commutators in each variable,
[Tu,Tv J e
1
= 4 [I+P-P
f
I
,
= 4 (P-P
,I+Q-Q ]e
I
I
,Q-Q ]e •
Again by linearity,
1
I
4 [P-P',Q-Q ]e
=
1
I
I
I
I
4 ([P,Q]e+[P ,Q ]e-[P,Q ]e-(P ,Q)e).
- 23 f
I
Substituting
Qe = e -:- Q e ,
Pe = e - P e
in the last two terms,
we get
1
I
1
I
I
I
4 [P-P ,Q-Q ] = 2([P,Q]+ [P ,Q J)e •
Lemma 2.4.
and assume that
for each pair
Let
A
be in spectral duality
Then
.CfJ :
[P,Q]e
Proof.
(V,K)
and
JB-algebra for the product (2.6)
is a
P, Q E
(2.10)
(A,e)
0
I
= [Q ,P
I
]e
By commutativity of the Jordan product we obtain
[J
By Lemma 2.3, this completes the proof.
We will now prove that the condition (2.10) is sufficient
as well as necessary in order that
Lemma 2.5.
Let
(A,e)
and
A be a
(V,K)
JB-algebra.
be in spectral duality
and assume that the condition (2.10) is satisfied for all pairs
P, Q E fl>
•
Then the product ( 2. 6) is bilinear on the space
of all finite linear combinations of elements of
for
u E 2£
and
a, bE A0
7£ ;
A0
moreover,
:
(2.11)
(2.12)
Proof.
with
We first observe that if
u 1 , ••• , ~ E
n
?.£
and if
n
vE
( E A·T
)v = . E 1 >.,.T
T e
~ u. v
1. = 1 1 u.~
~=
~
U ,
a E A0
,
say
n
E
. 1
~=
then by Lemma 2.3
n
= . E 1 >...T
T e =Tva •
~ v u.
~=
a=
~
>...u.
~
~
- 24 n
(_~
Hence the value of
representation
n
AiTu_)v
l=1
is independent of the particular
l
~ A·U· •
By linearity and continuity this
. 1 l l
l=
result subsists with an arbitrary element bE A in place of v •
~
=
Thus, for every
Ta: A -> A
a EA0
there is a well defined opera tor
such that
n
(2.13)
~
. 1
l=
for any representation
Note that if
a=
a=
n
~
. 1
l=
A..T
l
n
~
i=1
A.u.
l
with
l
and
x.u .
l
u.l
l
b
=
u 1 , ••• , ~ E
n
~
'J·V·
j=1 J J
U •
with
u 1 , ••• , ~, v 1 , ••• , vm E (t , then
n
m
m
n
A·T
(
~ 'J·T
e)=
E
'J.T
(
E A..T e)= Tba •
. 1 l u . . 1 J v.
.
1 J v . . 1 l u.
l=
l J=
J
J=
J l=
l
Tab=~
Hence for all
a, b E A 0
The next, and crucial, step is to prove that for all
a E Ao·•
(2.15)
Observe first that if
n
a =!: x.u.
with ui = P.e
l
.l= 1 l l
i t- j
then (2.15) holds.
'
n
Taa:
=
n
a
for
has finite spectrum, i.e. if
PiE
fP
and with
u l. .J..u.
.. J
for
In fact,
n
~
A_.T (~X .u.) = !: X-X .T u .
.l= 1 l u l. .J= 1 J J
. .J= 1 l J u.l J
l'
n
n 2
2
X.A.P.u. = ~ A..u. =a •
i, j = 1 l J l J
i= 1 l l
~
Now observe that by definition the map
al-> Ta
is linear
~
on
A0
and so the map
,
into
A •
Thus for all
25 -
(a,b) a-> Tab
is bilinear from
A0 XA 0
a, bE A0
(2.16)
Now if
a+b
a
and
b
are compatible and have finite spectrum, then
has finite spectrum (consider the abelian subspace
generated by
a
and
b), and so by (2.16) and the remarks above:
a E·A 0 , let
For arbitrary
closed abelian subspace
in norm and each
an
M(a)
(an} be a sequence in the weakly
generated by
has finite spectrum.
exists by spectral theory).
of the maps
- C(X)
a , such that
a n ->a
(Such a sequence
Then by (2.14) and norm continuity
Tb
with
b E A0
Taa
=
limn Ta an = limn limm
=
limn lim_m a n oam
T~am
= a2
(where the last equality follows from continuity of the product
on
M(a)
~
C(X)) •
This establishes (2.15).
Combining (2.15) and (2 .. 16) we now obtain for
Tab =
t ( (a+ b ) 2 - a 2 - b 2 )
a,b E A0
= a ob •
This proves (2.11) as well as bilinearity of the product
on
A0
aob
•
It remains to prove (2.12).
and
:
Hall, lib!!
~
1 •
To this end we assume
a,b EA 0
Recall also the general inequality
valid for any two positive elements
c,d
of an order-unit space,
..., 26 and the equality
following from spectral theory [ 2; Prop.8.6, formula(8.25)].
Now
11 a ob 11 = !11 ca+ b ) 2 - <a-b ) 2 !1
~
1
4 max ( II a+ b II
from which (2.12) follows.
Theorem 2.6.
Let
2
, II a-b II 2 )
_s 1 ,
(]
be an order-unit space in spectral
(A,e)
duality with a base-norm space
(V,K).
Then
A
is a
JB-algebra
for the product
(2.17)
iff
I
(2.18)
I
[P,Q]e = [Q ,P ]e
for all pairs
Proof.
P,Q
of
P-projections on
A •
The condition (2.18) is necessary by Lemma 2.4.
To prove that (2.18) is sufficient, it is enough to prove
that the product
aob
is bilinear [ 2; Th.12.12].
We have alieady
shown that this product is bilinear on the dense subspace
of
A
(Lemma 2.5).
Hence we shall be through if we can prove
that for two given sequences
to
a E A and
{an}
and
{bn}
in
A0
converging
b EA respectively, the product sequence
will converge to
A0
{an obn}
aob •
Thus, we assume
an, bn E A0
and
I! a-ani!
->. 0 , llb-bnll -> 0
as
n -> oo •
By spectral theory there exists a sequence
in
M(a)nA 0
such that
' -> 0
lJa-anll
as
n -> oo.
{a~}
By continuity
- 27 -
of squaring in
(an')2 -> a2 •
M( a ) , we also have
Since
I
llan-anl! -> 0 , we obtain from Lemma 2.5
The last expression tends to zero as
b~
Similarly we prove that
-> b 2 •
Corollary 2.7.
Hence
oo •
Finally
= t ( (a+ b) 2 and the theorem is proved.
n ->
2
2
a - b )
=a
o
b
t
[]
For a given convex set
K
the following
are equivalent:
(i)
K
is affinely isomorphic to the normal state space
of a
(ii)
JBW-algebra.
K is spectral and
of
P-projecti·ons on
Proof.
[P,Q]e
A
=
I
I
[Q ,P ]e
= Ab(K)
for every pair
P,Q
•
If (i) holds, then (ii) will hold by Proposition 2.1
and Lemma 2.4.
Conversely, if (ii) holds, then
linear space
V
in such a way that
space in spectral duality with
By Theorem 2.6
JB-algebra.
A
Since
(A,e)
K
can be embedded in a
(V,K)
where
becomes a base-norm
A
= Ab(K) = v* •
can be equipped with a product making it a
A
is a dual space, it is in fact a
JBW-
- 28 algebra, and py [20; Th.2.3] the normal state space of
affinely isomorphic to
Corollary 2.8.
A
is
[J
K •
For a given compact convex set
K the
following are equivalent:
(i)
K is affinely isomorphic and homeomorphic to the state
space of a
(ii)
K
JB-algebra equipped with the usual
is strongly spectral and
pair
P,Q
Proof.
of
w*-topology.
f
I
[P,Q]e = [Q ,P ]e
P-projections on
A
= Ab(K)
for every
•
If (i) holds, then (ii) will hold by Corollary 2.2
and Lemma 2.4.
Conversely, if (ii) holds, then we consider the regular
embedding of
K
[1; Ch II, §2].
duality with
2.6,
A
into the base-norm space
Since
(A,e)
K is spectral,
where
A
(V,K)
(V,K)
= Ab(K) ~ v* = A(K)**
JEW-algebra).
•
By Theorem
JB-algebra
The Jordan product of
given by (2.17); hence the squaring map
= A(K)*
is in spectral
can be equipped with a product making it a
(and in fact even a
V
where
a~> a 2
A
is
determined by
this Jordan product, is the same as that determined by functional
calculus.
Since
K
is supposed to be strongly spectral,
is closed under the squaring map.
Jordan subalgebra of
JB-algebra.
A= Ab(K)
Hence
= A(K)
must be a
containing e, and thus is a
By elementary properties of compact convex sets
[ 1 ; Ch II,§ 2], the convex set
K is linearly isomorphic and
= A(K)
proof.
[J
homeomorphic to the state space of
w*-topology.
B
A(K)
This completes the
B
Remarks on the physical interpretation.
equipped with the
We do not know of
any natural geometric interpretation of the condition
- 29 -
= [Q
[P,Q]e
t
I
,P ]e ; our main concern in the sequel will be to
replace it by requirements which are more geometric.
However,
the above condition does admit an interesting physical interpretation, which we will now briefly discuss.
Following [14] and [ 7 ] we may view each
senting a beam of particles with intensity
the elements of
Q,Q
I
P E v+
as repre-
IIPII = (e,p)
K represent beams of unit intensity.
of quasicomplementary
, so that
Each pair
P-projections represent complementary
physical filters which serve to measure whether a particle does
or does not possess a certain property
coming beam
p entering the
Q* p
outgoing beam
"Q" •
Thus, for an in-
Q-filter, the result will be an
consisting of those particles which definitely
do possess property
Q •
(In operative terms, this means that
with probability 1 they will pass through any
after the original one).
Q-filter placed
Similarly, for an incoming beam
p
I
entering the
Q -filter, the result will be an outgoing beam
I*
Q p
consisting of those particles which definitely do not possess
property
by any
IJQ * e!l
Q •
(With probability 1 these particles will be stopped
Q-filter·placed after the original
= ( e, Q* p)
JIQ*pll/!IP!I
is the probability that a
particle in the beam will pass through the
Q •
Note that
is the intensity of the beam emerging from the
Q-filter; hence the ratio
property
I
Q -filter).
In particular, if
Q-filter, i.e. have
P E K , then
l!Q * p 11
probability that a particle in the beam has property
is the
Q •
Now consider a sequence of two measurements, first by either
one of the filters
Q
or
·.
Q
I
•
or
pEK
and then also through the
P
I
, then by either one of the filters
Note that for example Q*p * p represents that portion
of an incoming beam
P
which will pass through the P-filter
Q-filter. Thus, lJQ *p * Pll represents
- 30 -
the probability that a particle in the beam
both properties
P
and
out in this order.
Q
1
Similarly,
Q , and so on.
E K will possess
when the measurements are carried
IIQ '* P* Pll
lity that a particle in the beam
but not
p
represents the probabi-
pE K will possess property
Starting from this, we can express the
probability of any logical combination of the properties
Q
P
(keeping the assumption that the measurements of
formed before the measurements of
Q) .
P
P
and
are per-
In particular, we are
interested in the "exclusive disjunction" (either one or the other,
but not both).
(2.19)
Here we obtain:
Prob(P and(not Q), or, (not P) and Q)
= 1l Q '* P * P II
=
+ II Q*P '* P11
(e,Q '* P* p) + (e,Q*·p '* p)
= ((PQ
1
+ P ' Q)e,p) •
Suppose we now reverse the order of the measurements.
Then we get:
(2.20)
Prob(Q and(not P), or, (not Q) and P)
= liP '* Q* PIJ + !IP*Q'* PI!
= (e,P '* Q* p) + (e,P* QI* p)
= ((QP ' + Q' P)e,p)
•
Finally observe that by replacing
dition
(P,Q]e
= [Q
I
Q by
I
,P ]e , we can convert it into
(Q,P ' ]e , which is equivalent to
I
( 2. 21 )
Q'
I
( PQ + P Q) e
= (QP
I
I
+ Q P) e •
in the conI
(P,Q ]e
=
- 31
~
Comparison with (2.19) and (2.20) shows that the condition (2.21)
is equivalent to the requirement that the probability of the
exclusive disjunction of
of the measurements of
P
P and
and
Q
Q •
is independent of the order
Thus the key property (2.18)
needed for Jordan structure is equivalent to this physical property.
Note in particular that this requirement is satisfied in the usual
model of quantum mechanics, in which
K is the normal state space
of the algebra of all bounded operators on a Hilbert space.
(This follows from Lemma 2.4, but it can also be verified by
elementary operations with Hilbert space operators).
- 32 -
1_2.
The ball and symmetry properties.
Our purpose in this section is to show that normal state
spaces of
JBW-algebras (and thus also state spaces of
JB-alge-
bras) possess two geometric properties which we call the Hilbertball property and the symmetry property, respectively.
These
properties are not generally enjoyed by spectral convex sets,
and in a later section we will show that they do in fact characterize the state spaces of
JB-algebras among the strongly spec-
tral compact convex sets.
We begin by recalling some general notions relating to an
arbitrary convex set
we denote by
on
K.
K
(in some linear space), and for brevity
A the space
Ab(K)
of all bounded affine functions
(This conforms with the notations used in§§ 1,2).
any subset
of
E
which contains
K
there exists a smallest face
(possibly
E
F
= K).
A face
to be
A-exposed if there exists a function
a > 0
on
be
K'F
and
A-exposed if
a
{p}
=0
on
is an
A-exposed point is extreme.
A-exposed face.
A face
F
of
of
a EA
A point
F
F
F
K
= face(E)
K
is said to
Clearly, every
is said to be a split
F'
such that
every element of
K can be written uniquely in the form
A. p + ( 1-A. )a
O~A. ,.=:;
I
1, PEF, aEF
a split face with complementary face
point of
points
K
p,a
a split face
is contained in
of
F
from § 1 that if
FU F ' •
•
is said
such that
pEK
face if there exists a (necessarily unique) face
with
For
Note that if
F
is
F ' , then every extreme
We say that two extreme
are ~~~~~~~~~s~p=l=i~t~f~a~c~e
separated by a
if there exists
K
such that
K
PE F
and
I
aEF
is spectral, then a face
Finally we recall
F
of
A-exposed iff it is projective, and that every subset
K will be
E c K
is
- 33 contained in a smallest projective (or
A-exposed) face
(By [ 2; Prop.10.2] the split faces of
K are precisely those
faces which correspond to central
F(E) .
P-projections.)
We shall need the following elementary result from general
convexity theory.
Proposition 3.1.
set
K
If two extreme points
can be separated by a split face, then
equal to the line segment
are
p,a
[e,a] •
A-exposed points, then
Proof.
1.
of a convex
face((p,a})
If in addition
p
is
and
a
is an A-exposed face.
[p,a]
The statement that
= [p,a]
face((p,a})
, fol-
lows in a straightforward way from the definitions; we leave the
details to the reader.
2.
Assume now that
p
a,b E A be chosen such that
b > 0
that
on
K'-..(a}, (b,a)
p EF
cp( w) E F
and
and
aEF
w(w)E F
( 3.1 )
w
Clearly
c
K"-..[P,a]
F
=
0
and
be a split face such
wEK , let
A(w) E [0,1] ,
c
on
( 1 -A ( w) ) W( w) •
K by writing
( 1 -A ( w) ) ( b , W( w) ) •
is bounded, and it is seen by a straightforward veri-
fication that
on
K""-[P}, (a,p)
Also let
= A ( w) ( a, cp ( w) ) +
( c , w)
(3.2)
A-exposed, and let
be (uniquely) determined by
= A ( w) cp ( w) +
Now define a function
on
For every
•
I
•
are
a
a > 0
=0
I
and
c
and
is affine.
c
=
0
Hence
on
[p,a] •
c EA •
Clearly also
c > 0
This completes the proof.
By a Hilbert ball we· shall understand the closed unit ball
of some real Hilbert space
H •
The dimension of
H
can be
fl
- 34 finite or infinite, and for convenience we also admit the "zero
dimensional Hilbert ball", consisting
Definition.
if face({p,cr})
A convex set
is an
ot
a single point.
K has the Hilbert ball property
A-exposed face affinely isomorphic to a
Hilbert ball for every pair
p,cr
of extreme points of
Note that if the two extreme points
face((p 9 cr})
= face{p} =
p,cr
coincide, then
K has the Hilbert ball pro-
(p} ; so if
K is
perty, then every extreme point of
K •
A-exposed.
Note also that by Proposition 3.1, a convex set
K will
have the Hilbert ball property iff every extreme point is
posed and
face({p,a})
is an
A-ex-
A-exposed face affinely isomorphic
to a Hilbert ball for every pair
p,a
of distinct extreme points
which can not be separated by a split face.
The convex sets to be studied in the sequel will all be
spectral, and then the notions of
"A-exposed face" and "projective
face" will coincide, as noted above.
set
Thus if a spectral convex
K has the Hilbert ball property, then
(3.3)
F([p,a}) = face([p,a})
for every pair
p,cr
of extreme points of
(3.4)
F(p)
=
for every single extreme point
K ; in particular
fp}
p
of
K •
We now proceed to prove that the normal state space of a
JBW-algebra has the Hilbert ball property.
some preliminaries from the theory of
For a
elements
JEW-algebra
a, bE A
A
But first we recall
JB-algebras.
the notion of "compatibility" of two
defined via the spectral duality of
(A, e)
with
~
its predual
35 -
(V,K), will coincide with "operator commutativity"
of the spectral resolutions
[ 3 ; Lem.2 .11].
{e~} , (e~} , as can be seen from
Hence our defini tj,on of center for
equivalent to the
A ( § 1) is
JB-algebra definition of center for
We will use the term
center is trivial.
JB-factor to denote a
A
[ 3 ;§4].
JEW-algebra whose
This definition is in agreement with the
general definition of a factor for spaces in spectral duality
(§ 1), but it is formally different from the definition of a
JB-factor in [ 3 ] , which does not involve the notion of a
algebra.
JEW-
However, it can be seen from [20] that the two defini-
tions are equivalent.
By a minimal idempotent in a
JEW-algebra
A we mean a minimal element of the set of non-zero idempotents
of
A , or equivalently an atom (of. § 1) defined in the spectral
duality of
that a
(A,e)
with its predual
(V,K).
Recall from [ 3]
JB-factor is said to be of type I if it contains a minimal
idempotent (this is in agreement with the general definition given
in§ 1), and that it is said to be of type
r2
if the maximal
cardinality of a set of orthogonal non-zero idempotents is
2 •
The first step towards the Hilbert ball property will be to
prove that every extreme point of the normal state space of a
JEW-algebra
A is
A-exposed.
exceptional
JB-algebra
over the Cayley numbers.
We begin by showing this for the
of all self-adjoint
3 x 3-matrices
This algebra is finite dimensional, so
we do not have to worry about normality for states, and we can
use the term "exposed point" without further specification.
Lemma 3.2.
of
Mg
Every extreme point of the state space
is exposed.
K
- 36 Proof.
Consider the spectral duality of
with
result there exists in
( a Ib)
·
dua 1 cone, ~.e.
A an inner product making
~
( 5 ; Ch.11 ,Satz 3.8]).
isomorphism of
A
Now
A+
I a)
a I-> (
A* = V •
onto
every extreme ray of
for all b E A+
0
-iff
~
(V,K)
By a known
(cf. Proposition 2.1).
where
v+
(A,e)
a self-
A+
a E A+
(cf.
is seen to be an order
Therefore, if we can prove that
is exposed, then every extreme ray of
K will be
will be exposed, and so every extreme point of
exposed.
But an extreme ray of
therefore of the form
Th.12.3].
im+P
AT is a closed face of A+ , and
for some
By Proposition 1.4 of [2] im+P
by finite dimensionality, exposed.
Proposition 3.3.
space
K
P-projection
of a
Proof.
p
A
is
[]
A-exposed.
be an extreme point of
Also let
(cf. Proposition 1.8).
P
F
p
p
(c(p),p) = 1
P-pro-
be the corresponding
= (aE Kj (c(p),a) = 1
by Proposition 2.1.
c(p)
be the corresponding
F
p
K and let
A satisfying
P
c(p) = P e , and let
projective face, i.e.
is semi-exposed, and
This completes the proof.
be the smallest central idempotent of
jection, i.e.
(2;
Every extreme point of the normal state
JEW-algebra
Let
P
Note that
To show that
p
we will relativize to the spectral duality of
is
A-exposed,
(AP,c(p))
and
*
A=
p imPp and Vp =imP p (cf. Proposition 1.4).
Specifically, we will show that {p} is an A -exposed, or equi(V p ,F)
p
where
P
valently a projective, face of
F
p
•
By Proposition 1 .8 (or by [ 3 ; Lem. 5. 5]) Ap is a factor.
or to
Therefore by [3; Th.8.6] A p is either isomorphic to
a Jordan algebra of operators on a Hilbert space (a "JC-algebra").
- 37 .,..
In the former case we are done by Lemma 3.2.
In the latter case
the argument in the proof of Theorem 8.7 in [ 3] shows that the
projective face generated by
p
is minimal.
It then follows
from Proposition 1.10 that this face consist of
(p}
FP , as desired.
is a projective face of
p
only.
Thus,
[]
Next we shall prove a series of lemmas needed for the study
of face([p,cr})
space of a
for two extreme points
with its Eredual
Proof.
(3.7)
be a
(A 1 e}
(V 2 K}
JEW-algebra in SEectral
dualit~
czt
~cf.
By Lemma 6.2 of [ 3 ] there exists
sEA
be
with
such that
us ( u ' vu ' }
( 3. 6 )
where
Let
Pro:eosition 2.1 ). Let UzVE
2
Then there exists sEA with s = e such that
arbitrar~.
=e
of the normal state
JEW-algebra.
Lemma 3.4.
s2
p,cr
u'
=
e-u
= (vu ' v }
'
By the proof of Lemma 6. 3 of [ 3 ] we also have
Usr((u'vu'})
=
r((vu'v}) •
By Lemma 1.2 and equation (1.10):
r((u'vu'}) = (vvu)Au' = (vvu)-u
and
r((vu'v})
=
(u'vv')Av
which completes the proof.
=
(uAv) ' J\ v
=
v-(uAv),
0
The following corollary will not be needed in the sequel,
- 38 -
but seems of interest in its own right.
and
a,b E L ; one says
if for all
x EL
A lattice is
if
(a,b)M
(a, b)
with
a
1\
x
~
b
Topping (22]
be a lattice
there holds
semimodular if the relation
implies
L
is a modular pair (written
~
b
Let
(b,a)M •
(a,b)M)
x = (x v a)
1\
b •
M is symmetric, i.e.
(For background see [4]
or [13]).
has shown that the projection lattice of a von Neu-
mann algebra is semimodular.
We will now generalize this to
JEW-algebras.
The lattice c~ of idempotents (= projective
Corollary 3.5.
units) in a
JEW-algebra
Proof.
A
is semimodular.
1£
We know that
is an orthomodular lattice (cf.
[2; Th.4.5] or (3; Prop.4.9]).
By Corollary 36.14 and Theorem
29.8 of (13], it now suffices to prove that whenever
and
of
u vv = e , u
1.t
taking
1\ v
= 0 , there exists a lattice automorphism
to
u
u,v E ~
and
to
v
u
I
•
If sEA and
e ' then the map Us : a 1-> { sas } sa tisfies us2 = I and it maps idempotents into idempotents; by
( 3 ; Prop. 2. 7] it is order preserving so it is a lattice automorphism of
Now if
U ( e-u) = v
s
and
us v
= u'
1t.
u vv
v
and
e
for some
sEA
u
1\
v = 0 , then Lemma 3.4 gives
with
s2
=e •
Thus
, which completes the proof that
Lemma 3.6.
and
=
If
u
?t
Usu = e- v = v'
is semimodular.(J
is a minimal idempotent of a
is an arbitrary idempotent, then
JEW-algebra
uv v-v is either a
minimal idempotent or zero.
Proof.
By Lemma 3.4 there exists
sEA
with
e ,
- 39 -
such that
of
Us
exchanges
u v v- v
and
v , the latter either equals
since
Us
v
v- u
By minimali ty
Now note that
is a Jordan automorphism (cf. [10; p.60] or use
patents to minimal idempotents.
Lemma 3.7.
JBW-algebra.
uVv
•
or is zero.
Macdonald's Theorem [10; p.41]), then
a
AV
If
such that
Proof.
Let
w
u,v
Us
maps minimal idem-
This proves the Lemma.
be distinct minimal idempotents of
w is an idempotent different from
~
Define
u v v , then
=u
w*
(]
0
and
w is also minimal.
v v- w •
Then
u .!: u v w* ~ u v v ;
we claim one of these relations is actually an equality.
For if
both inequalities are strict, then we also have the strict inequalities
0 < u v w*- u < u v v- u ,
which contradicts the fact that the last term is a minimal idempotent by Lemma 3.6.
Thus, either
u
=
uv w*
In the former case, minimali ty of
w
= uv
u
=
u v v- u ,
= u v v- w* =
u v wl- w*
v- wl
or
forces
u vw*
u
=
= w4f
uv v •
; so
which is minimal by Lemma 3.6.
In the latter case
w
which is again minimal by Lemma 3.6.
Remark.
0
We note for future use that the conclusion of
Lemma 3.7 will hold with the same proof in the more general case
where
u,v
are atoms and
w
is a general projective unit for
- 40 -
an order-unit space
space
(A,e)
in spectral duality with a base-norm
(V,K) , provided that the conclusion of Lemma 3.6 holds.
Lemma 3.8.
idempotent in
Proof.
If
A
A is a
then
JB-factor and
(uA u}
Note first that
is a
(uA u}
u
is a non-zero
JB-factor.
is a JBW-algebra, as can
be seen from [3; Prop. 4.11].
Now suppose that
central in
a
is a non-trivial idempotent which is
(uA u} , and let
b
= u- a
By Lemmas 6 .. 4 and 6.7
•
of [3] there exist non-zero idempotents
s E A with
s2 = e
then since
a 1 , b 1 E (uA u}
such that
But by positivity of the maps
(sa 1 s}
Ut
= b1
and
Ub
a,
.$.
.
a
Let
'
b,
t
=
and
b
.$.
(usu}
.
'
[ 3 ; Prop.2. 7] we
now get
and since
and
a, b, t E {uA u}
are compatible by centrality of
b , this gives (cf.[ 3; Lem.2.11])
a contradiction.
This completes the proof that there is no non-
trivial central idempotent in
Lemma 3.9.
a
a
JB-factor
If
A , then
u,v
A
[uA u} , so
(uA u}
is a JB-factor.
are distinct minimal idempotents in
= { (u V
v )A( u v v)} is a type
I2
------------------------u,v.--~~--~~~~~~~~~-~-~~
JB-factor.
0
- 41 -
Au,v is a JB-factor. Since Au,v
contains the minimal idempotents u and v , it must be of type I.
Proof.
By Lemma 3.8,
Note also that the maximal number of elements in a set of orthogonal idempotents is at least
2 , since
orthogonal and both non-zero.
On
w1 ,w 2 ,w 3
Then
w1 +w 2
2
and
u v v- u
would be a non-minimal idempotent under
states of
A
u,v
u vv '
Thus, there can not be any set of more
orthogonal non-zero idempotents in
Lemma 3.10.
are
other hand, suppose that
were three non-zero orthogonal idempotents in
contrary to Lemma 3.7.
than
t~e
u
If
A
• so
u,v '
A is a JB-factor of type
A are normal and the state space of
A
is
,
u,v
I 2 , then all
A is affinely
isomorphic to a Hilbert ball.
Proof.
Let
A be any
JB-factor of type
I 2 , and note
that by Proposition 7.1 of [3] A will be an (abstract) spin
~,
factor in the sense of Topping (21], i.e.
A
can be equipped
with an inner product
(alb)
in such a way that
is a unit vector and for every pair
of elements of
e
N = [e}
l.
aob
(3.8)
In particular
~Iaiii
= ~(ala)
for
a EN •
!la1! 2
=
l!a 2 1J
which makes ita real Hilbert space
a,b
one has
=
=·
(ajb)e •
I (a! a)/
, so the Hilbert norm
will coincide with the given
JB-algebra norm
!laJI
Clearly also the two norms will coincide on Re •
By Lemmas 3 and 4 of [21] the following inequalities will hold
for a general element
(3.9)
a EA
Therefore
A is linearly homeomorphic to a Hilbert space, and
so is reflexive.
Hence the predual of
A will coincide with its
dual, and all states must be normal.
We will prove that the closed unit ball of the subspace
of
N
provided with the Hilbert norm !\Iaiii , is affinely isomor-
A
phic to the state space of
To this end we first note that
A •
by Lemma 1 of [21 ] a general element
a+ a.e E A , with
a EN
and
a. E E. , is positive iff
(3.10)
a. Z
For every given
on
A
Ill a Ill •
b E N we define a linear functional
pb
by
(3.11)
where
and
a EN
Clearly
a. E R •
is an affine map of
b 1-> Pb
lflbl/l ~ 1 , then for every
a+ a.e > 0
with
N
a EN
into
and
A*
If
a. E JR
we
get by (3.10) and (3.11):
Pb (a+ a. e) Z a - / (a I b) I Z a.
Hence
pb > 0
pb(e) = 1
when
for all
l!lb IJI
.::;
bEN •
the closed unit ball of
Now let
a
= b1-
b2
-1
0 •
By the definition ( 3.11),
Thus,
b.,_> pb
is an affine map of
N into the state space of
be two distinct vectors in
b1,b2
N •
A .
Letting
0 , we have by ( 3. 11)
Pb (a)- Pb (a)
1
2
Hence the map
1 •
-Ill alii 2:
b~>
pb
=
(ajb 1-b 2 )
is
= lllb 1-b 2 !112 -1
0 •
1-1 •
Finally we consider an arbitrary state
p
of
A •
Since
..,. 43 the two norms on
A
coincide on
will have norm at most
Hence there exists
for all
a EN
=
space of
ll!b Ill _s 1
~>
pb
a. E E. •
such that
p (a)
=
(a Ib)
(alb)+ a.
Hence
p
=
pb •
This proves
N
onto the state
maps the unit ball of
[J
A , and we are done.
Theorem 3.11.
=
(alb) +O.p(e)
a E N and
b
with respect to the Hilbert norm.
bE N with
p(a +a.e)
that the map
to N
p
By linearity
~
for arbitrary
1
N , the restriction of
The normal state space of a
JBW-algebra
has the Hilbert ball property.
Proof.
space
K •
A-exposed.
of
Let
A be any
JBW-algebra with normal state
By Proposition 3.3 every extreme point of
We consider an arbitrary pair
p,cr
K
is
of extreme points
K which can not be separated by a split face, and we will
prove that
face({p,cr})
is an
phic to a Hilbert ball.
A-exposed face affinely isomor-
By previous remarks this will prove our
theorem.
Let
R be the
P-projection, and
corresponding to the projective face
and
a •
Since
(p}
and
also be projective faces.
projective units by
u
Therefore
R
?£
F([p,cr})
generated by
is the
Ra
p
A-exposed faces, they will
In the lattice~
v , respectively.
K , we have
F((p,cr})
=
(P}v [cr} •
of projective units, we have
by Proposition 2.1 ,
(3.12)
the projective unit,
We denote the corresponding (minimal)
and
of projective faces of
in the lattice
{cr} are
w
w
P-proj ection corresponding to
R
= Uuvv
=
[(uvv)a(uvv)}.
•
Thus, for
a EA
=
Hence
uvv •
u v v , and
.,.. 44 ...,
Au,v = im R, and we note that A
u,v is a JBWalgebra by [3; Prop.4.11]. By Proposition 1.4 ( im R*, F ( { p , a} ) )
We write
will be the predual of
, u v v) • Thus F({p,cr}) will be
u,v
the normal state space of A
•
u,v
Now, let c(p) be the smallest central projective unit
of
A
such that
projective face
(A
(c{p),p)
G
(Proposition 1.8).
=1
, and recall that the corresponding
is the smallest split face containing
Since the two extreme points
not be separated by a split face, then also
p
a EG •
F({p,cr}) ~ G, and by passage to projective units,
Let
let
Ac(p)
and
p
can
a
It follows that
uvv;:::; c(p) •
Q be the (central) P-projection corresponding to c(p),
=
imQ, and note that
uEAc(p)
(Recall that by [2; Cor.2.12], im Q
genera ted by
vEAc(p).
is the order ideal of
By Lemma 5. 5 of [ 3 ] ,
c ( p)) •
and
Ac ( p)
is a
factor; and since it contains the minimal idempotents
A
JB-
u,v , it
must be of type I.
Au,v = RA = RAc(p) since Au,v -c Ac ( p )·•
Then it £ollows £rom Lemma 3.9 and formula (3.12) that A v is
u,
an I 2 -factor. By Lemma 3.10 the normal state space F((p,cr})
Next we note that
of
Au,v must be a Hilbert ball.
Finally we note that the face generated by any pair of
distinct extreme points of a Hilbert ball, is the entire ball.
Hence
face([p,cr})
tive, hence
= F({p,cr})
•
A-exposed, face of
Thus,
face({p,cr})
is a projec-
K affinely isomorphic to a
0
Hilbert ball.
Corollary
3.12~
The state space of a
JB-algebra has
the Hilbert ball property.
Proof.
The state space of a
JB-algebra is the normal state
- 45 -
space of its enveloping
JBW-algebra (cf. [20]), so Theorem 3.11
0
applies.
We now turn to discuss a second geometric property:
symmetry.
First we define a reflection of a convex set
~
some linear space) to be an affine automorphism
ce 2 = id).
note that such a
has a non-empty set of fixed points
K0
= ( ~ ( p + ~ ( p) ) I
cp(p)
point
To justify the term "reflection",
p E K } , and that for each
Definition.
~(p+ ~(p))
E K0
A convex set
to a convex subset
Lemma 3.13.
V
K0
Let
F
if there exists a reflection of
K0
(i)
(V,K)
is a projective face of
(ii)
K
whose
K
•
K be a convex set embedded in a linear
in such a way that
P-projection on
[p,cp(p)]
•
is a base-norm space in
separating duality with an order unit space
If
the image
K is symmetric with respect
set of fixed points is precisely
space
pEK
is obtained by reflecting the line segment
about its mid-point
(in
K which
of
is involutory (i.e.
cp
K
K and
P
(A, e)
where
A
!:::!
is the corresponding
A , then the following are equivalent:
is symmetric with respect to
co(FU F#)
2P + 2P ' - I > 0 •
If these equivalent conditions hold, then
is the unique reflection of
K with
cp =
co(FU F*)
(2P+ 2P ' -I) *
IK
as its set
of fixed points.
Proof.
Assume first that (i) holds, and let
flection of
K with
"'=
~(~+id)
Define
co(F U F*)
,
~
be a re-
as its set of extreme points.
and note that
V* •
$(K) c K and
$2
= lll
•
- 46 -
Furthermore, for
onto
K
co(FUF,g.)
.
cp
w=
Thus we have
iff
w is an affine retrac-
Thus
By [2 ; Th.3.8]
exactly one affine retraction of
I
* jK.
(P+P)
w(p) = p
it is easily verified that
l\l(K) = co(FU F4/:) •
cp(p) = p' so
tion of
pEK
there exists
K
onto co(FU Fl) , namely
I
*
(P+ P ) jK ,which implies
I
*
= ( 2P + 2P - I) * IK • In particular ( 2P + 2P - I) leaves the
I
invariant.
K of the cone
base
the cone
A+
I
leaves
2P + 2P -I
Hence
invariant, so (ii) is proven.
In addition we have
proven the uniqueness statement of the lemma.
I
Assume next that (ii) holds.
and
2P + 2P -I ~ 0
I
*
(2P + 2P -I)
will leave
Then since
I
(2P+ 2P - I)e = e , the dual map
invariant not only the cone
v+
(2P+2P 1 -I) 2 = I , the map
(2P+2P 1 -I)*jK
of
K.
but also its base
( 2P + 2P t - I) * p =
Furthermore
p
which by [ 2 ; Th. 3 "8] is equivalent to
shows that
K
iff
K •
is a reflection
(P + P ' )*p = p ,
p E co (F U F'*)
is symmetric with respect to
Since
~
This
co(FU Ff) , which
0
completes the proof.
Lemma 3.13 will apply in the special context of spectral
convex sets.
Definition.
A spectral convex set
K is said to be
symmetric if it is symmetric with respect to
projective face
co(FuF 1 ) for every
F , or what is equivalent (by Lemma 3.13), if:
t
2P + 2P - I > 0
(3.13)
Theorem 3.14.
for all P E tfP •
The normal state space of a
JBW-algebra
is symmetric •
. Proof.
Let
A
be a
JBW-algebra with normal state space K.
- 47 We will first verify the following identity valid for any
idempotent
q EA
q1 = e - q :
with
{(q-q 1 )a(q-q')}
(3.14)
By definition,
Applying this to
b
=
q- q
I
=
{(q-:q' )a{q-q 1 )}
(3.15)
=
=
(bab}
=
2(qaq}+ 2(q'aq 1}- a •
Interchanging the roles of
2b o (boa)- b 2 oa
for any
bE A •
2q- e , we find
4(qaq}- 4qoa+ a •
q
and
q
I
, we get
({q-q 1 )a{q-q 1 )} = 4{q 1aq'}- 4q 1 oa+ a •
(3.16)
Addition of (3.15) and (3.16) gives (3.14).
Now suppose . P
position 2.1,
P
is any
P1
= Uq,
P-projection on
= Uq1
A •
Then by Pro-
for some idempotent
qEA.
Thus ( 3.14) gives the following equality valid for each
( 2P + 2P 1 - I) a = ( 2U q + 2U q, - I ) a
By Proposition 2.7 of [ 3] the maps
for all
K ,.
Ub: at-> (bab}
•
are positive
2P + 2P'- I .2! 0 , which proves
b , and so we conclude
symmetry of
= {( q -q' ) a ( q -q' ) }
a EA :
0
Passing from a
JB-algebra to its enveloping
JEW-algebra,
we also obtain:
Corollary 3.15.
The state space of a
JB-algebra is
symmetric.
Remark.
Thus each map
form
Us
=
I
In the proof above
s
2P. + 2P' - I
JB-algebra context is of the
for some element
in the
s
q- q
such that
s2
satisfies
=
e •
In the operator
- 48 -
algebra context the map
unitary
s •
Us
is conjugation by the self-adjoint
Thus symmetry of the state space for a
c*-algebra
can be viewed as a consequence of the fact that conjugation by a
self-adjoint unitary is an involutary
*-automorphism, and thus
induces a reflection of the state space.
~
~·
49 ..,.
Consequences of the ball and symmetry properties.
In this section we will derive as consequences of the ball
and symmetry properties certain properties of the extreme points;
these properties each have a natural physical interpretation, and
they will be useful in the sequel.
In fact, we will show in a
later section that they can replace the ball and symmetry properties as necessary and sufficient conditions for a strongly spectral compact convex set to be the state space of a
JB-algebra.
Our setting in this section will be that of a spectral convex set
v
K which we think of as embedded in a linear space
(V,K)
in such a way that
is a base-norm space in spectral duality with an order unit-spaee (A, e) where A= V* , and we will
use terminology from spectral theory with reference to this duality.
The extreme rays of
treme point. of
K •
v+
are the sets E+p
Thus, to say that a map
preserves extreme rays, means that
treme point for every extreme point
p*
with
(with
p
an ex-
P E fP
)
P* p is a multiple of an exp
of
K •
(Here we allow
P * p = 0) •
the possibility that
Proposition 4.1.
If a spectral convex set K is symmetric
and has the Hilbert ball property, then p * preserves extreme
rays for every
Proof.
Let
and shall, assume
and then
PE
P* p
f!P
•
P E K be an arbitrary extreme point. We can,
I
*
*
(P + P ) p I P; for otherwise 0 ~ P p ~ p ,
would be a multiple of the extreme point
p •
( 2P + 2P 1 - I)* acts as an
( 2P + 2P I - I) * P is an extreme
By the symmetry property, the map
automorphism of
point of
K •
K •
Hence
w=
By the ball property,
isomorphic to a Hilbert ball.
face({P,w})
is affinely
- 50 -
Now let
Define also
liP * p ll
}. =
a
=
I
1-A. =liP '* Pill 0.
0 , and observe that
1\P * PI!- 1 P *P and
T
= liP '* PI! -1 P* P
Then
' * p = A.a +(1-A.)T.
w) = (P+P)
~(p+
Now it follows from the definition of a face that a,r E face({p,w}).
The points
K n imP* and
cr
K n imP
and
I*
not be extended beyond
are contained in disjoint faces
'1'
; therefore the line segment
a
or
within
T
K •
[a, T]
Thus,
a
can
and
T
must be boundary points, and then also extreme points of the ball
f'ace({P,w}) •
It follows that
point of the given convex set
a
=
liP * p 11 -1 P * P is an extreme
0
K , and the proof is complete.
In our next proposition we will introduce two properties
P*
which are consequences of the preservation of extreme rays by
for
PE
§) .
point of
In fact, under the assumption that every extreme
K is exposed, it is not hard to verify that each one
of them is equivalent with the preservation of extreme rays.
However, this will not be needed in the sequel, so we omit the
proof.
Proposition 4.2.
that
(i)
p*
If
K is a spectral convex set such
preserves extreme rays for every
For each atom
uE
U and each vE
PE
qt
PP ,
(u v v')
then:
A
v
is
either an atom or zero.
(ii)
Each
Proof.
P E fP maps atoms to multiples of atoms.
1.)
Let
Q be the
jective fa-ce corresponding to
vE
P-projection and
c11 ,
G the pro-
and recall that
the projective face corresponding to the atom
u E
U .
(u}
By
is
- 51 -
Corollary 1.3
( 4.1 )
By hypothesis
point.
Q*"'u
is either zero or a multiple of an extreme
projective, face of
7! .
wE
fw}
((u} v G#) n G •
is an
A-exposed, hence
!IQ*un- 1 Q*u = w
Therefore
K ..
atom
=
( !lQ*u!l- 1 Q* u
In the latter case
for some
Now (4.1) can be rewritten in the form
By the isomorphism of the lattice
projective units and the lattice ~i
gives the equality
w
= (u v v' ) A v
1f
of
of projective faces, this
, which completes the proof
of ( i).
2.)
Let
PE
ffJ
u EU
and let
= (u v P' e)
r ( Pu)
A
be an atom.
By Lemma 1 .2
Pe ,
and by statement (i) the right term is either zero or an atom.
Therefore
Pu
Corollary 4.3.
p*
I]
must be a multiple of an atom.
Let
K be a spectral convex set such that
preserves extreme rays for every
wEU
are distinct atoms and
is either an atom or else
w
PE
satisfies
=0
or
w
00
Now if
•
w ..::: u v v , then
= u·v v
By (1.10)
(uvv)Av'
w
=
(u v v)
A
v'
mark after Lemma 3. 7.
Remarks.
A •
is
(uvv) -v, and
therefore the conclusion of Lemma 3.6 holds in the lattice ~
projective units in
U
•
Proof. By statement (i) of Proposition 4.2
either an atom or zero.
u,vE
of
Now the Corollary follows from there-
0
As was first observed by Pool in a related context
[16] 1 the statement (i) of Proposition 4.2 is closely related to
~
52 -
the notion of semimodularity of the lattice
1t .
It is well
known that for every orthomodular lattice semimodularity implies
(i), and it can be proved that the converse implication holds if
there are "sufficiently many" atoms, i.e. if every element of the
lattice is l.u.b. of atoms.
Proposition 4.4.
If
(See [17]
and
[13; Th.30.2,Cor.7.7]).
K is a spectral convex set with the
Hilbert ball property, then for every pair of atoms
(4.2)
(u,v)
Proof.
F
= Kn
F
imQ *
By hypothesis
F
=
face((~,v})
for
Proposition 1.4,
QE
fP •
. A
Since
u
and
Thus the restriction of
tions with values in
and
1
at
0
v"'
Being
A-exposed
A-exposed,
A
u,vEF, we can pass to the
u
u,v
~
Qe •
Now by
will determine projective units in
v
the relati vi zed spectral duality of
u
is an
is also projective, therefore of the form
corresponding projective units and obtain
,.
2t :
= (v,u)
face affinely isomorphic to a Hilbert ball.
the face
u,v E
and
v
( im Q, Qe)
to
F
and
( im Q*, F) •
will be affine func-
[0,1] ; the former with the values
1
at
at its antipodal point, and the latter with the values
and zero at its antipodal point.
Denoting by
I
of the ball
angle between
F
the center
TT
and by
u- TT
and
a
the
....
v-
TT
and using some elementary plane
geometry, we obtain
(see fig. 1)
Fig. 1 •
(u,v) = (v,u) = ~(1 + cos a) •
- 53 Note that this_proof is valid also when
Then
""'
n.
180°
--
~nd
=
F
is one-dimensional:
<u, VA) = <v,uA) = 0 •
0
The properties established in Propositions 4.1 and 4.4,
will always be used in connection with the property that all
extreme points of
K are
A-exposed.
It will be convenient to
be able to refer to them by a single name; since they are all
related to the extreme points of
Definition.
K , we give the following:
A spectral convex set
has
K
the~~
state properties if:
(4.3)
Every extreme point of
(4.4)
p*
(4.5)
(u,v)
= (v,u)
u,vE
1.£,..
K
is
A-exposed.
preserves extreme rays for every
P E ,qJ
•
for every pair of atoms
Remarks on the physical interpretation of the pure state
properties.
As in §
2
we may view each
beam of particles with intensity
!IPII =
pE
v+
as representing a
(e,p) , and each
as representing a filter transforming any given beam
new beam
beam.
P*p
PE
pEV+
fP
to a
of intensity at most equal to that of the given
(By a fundamental property of
P-projections, the intensity
remains undiminished only when the filter is neutral to the beam,
cf. [2; §2]).
Property (4.3) states that for an arbitrary beam of particles in a pure state, i.e. for an arbitrary extreme point
of
K , there exists a filter
PE
fP
p
which transforms every in-
coming beam to (a multiple of) the given beam, i.e.
(This may be expressed by saying that the filter
P
P
*< V+ )
=JR.+ p •
"prepares" p).
- 54 Property (4.4) states that each filter
P E []J transforms
pure states to (scalar multiples of) pure states.
(The scalar
factor represents the decrease in intensity, and is strictly less
than 1 unless P* p = p , as noted above).
Property (4.5) is a statement of a symmetric relationship
between the "transition probabilities" connecting pure states.
For a more detailed discussion of these and related properties
see [16], [8 ], [14].
~
1_2.
55 -
Ellipticity.
In this section we will introduce and study a new property
which is relevant in the context of spectral convex sets.
This
property, which we call ellipticity, is closely related to the
notion of "facial homogeneity" defined by Connes [6] in a somewhat different context.
state spaces of
We will prove in this section that the
JB-algebras are elliptic.
The notion of ellip-
ticity will play an important role in the development leading up
to the main results in§ 7, but it is not (as far as we know)
sufficient as an axiom to characterize state spaces of
bras.
JB-alge-
In fact, it is an open problem whether ellipticity alone
will suffice to guarantee that a strongly spectral compact convex
set is the state space of a
JB-algebra.
We begin by a lemma describing the normalized orbits of
one-parameter groups
t 1-> exp t(P-P I ) * p
tions
p
P
and points
Lemma 5.1.
with
Let
A= V* , let
ment of
K
p
such that
determined by
P-projec-
in a given spectral convex set.
(A,e)
be a
P*p
and
(V,K)
be in spectral duality
P-projection on
A
and
p
an ele-
# 0 and P* p t p • Moreover, let
1*
* 1 *
'*..P_...;a;;.;;n;.;;d;....;;;;;l..;;..e_t____.WF = ( P + P ' ) * 1K~
~g_=---.jloo.:Jl'~p..u.I!----:::;..P...~:P;._,z,•-T~=__,~,ll~.,;;.P_&::..p
....
1I-1
_...;P;;__
the unique affine retraction of
If
K
onto
p is not contained in the line segment
goes from
( 5.1)
to + co
- oo
co(FU F#)
[cr,T] , then as
t
the point
exp t(P-P')*p
Pt = (e,exp t(P-Pr)*p) E V
will describe one half of the (unique) ellipse
which has
[2 ; Th.3.8].
(q,T]
~(p)
through
as one diameter and has the conjugate diameter
p
- 56 -
in the direction of the vector
as
t
goes from
~degenerate
-co to +co
ellipse"
Proof.
Ep(p)
p- WF(p)
the point
= [cr,T]
If
•
Pt
p
E (cr,T], then
will describe the
•
We begin by giving an alternate expression for
exp t (P -P ' ) •
Since the two idempotents
P
and
P'
commute,
an expansion of the exponential function gives
where
A = exp t
Let
1-.a.
=I
exp t ( P-P ' )
(5.2)
P - P ' + AP + A-1 P '
for arbitrary
a. = liP * PI!
= liP '* Pll
~
=
(e,P * p) •
= ·(e,P '* p) ,.
t ER •
Then we also have
Using (5.2), we can now express
the denominator of (5.1) by the formula
(5.3)
D
= (APe + A- 1P' e , p )= Aa. + A- 1 ( 1-a.)
and we can express
Pt
,
by the formula
(5.4)
Assuming
Pi [ cr, 'f J ,
system in the plane
we introduce an affine coordinate
[p,cr,T] such that the origin is
and the basis vectors are
Fig. 2.
n
=
~(cr+,.)
- 57 Then
p- •F( p) = 2[a.( 1-a)
.!..
J2
e2
= rr + e 1
'!"
and
= TT -
cr
e1 •
Substitution into ( 5. 4) gives
where
(5.3) we find the alternate expression
Using
then also the equality
for
Pt
=
1- ; 2
~ =
4a.(1-a.)D- 2 •
-1
1 - 2A.a. D
, and
Hence the expression
takes the form
(5.5)
Now as
from
t
through
0
;
for
that
goes from
1
-co through
to
0
to
=
+co , A.
exp t
goes
+ex· , and we see from the expression
that it will go from
+1
through
0
to
- 1 •
This shows
Pt
traces out the orbit described in the statement of the
If
pE [cr,'l"] , then
Lemma.
Pt
= rr+se 1
e: 1
(where
$p(p) = p ; and now we simply obtain
is defined as above).
0
statement of the Lemma also in this case.
Lemma 5.2.
the ellipses
With the assumptionsand notation of Lemma 5.1,
;§p ( p)
are contained in
K
for all
I
p EK
iff
and all
K
is symmetric and
Let
for every
Assume first that
PE
2f>
pEK
t •
be fixed.
and
invariant, and so
all
exp t (P-P )
.a
0
PE
1fJ
and
P E fP
for all
t ER •
Proof.
pEK •
This proves the
Therefore
t ER •
expt(P-P 1 )
Ep( p) _s:
K
for all
PE
fP
and
By Lemma 5.1 , exp t (P- pI ) * p E v+
Hence
exp t(P-P ' ) *
will leave·
exp t (P-P 1 ) > 0
A+
for all
leaves
invariant for
P E
§P and
t E:R •
- 58 For any given
p EK
we consider the 1Teflected" point
( 2P + 2P ' - I) * p = 2 WF ( p) - P E Ep ( p) E K •
Hence
( 2P + 2P r - I)*
leaves
A+
Pt
PE
!}> and
e K = v+ n {w e v 1
is contained in
in
K •
Let
( e, w) = 1J
,
PE
fJJ
and
2P + 2P- I
K is symmetric.
e,xp t(P-P ' )
is symmetric and that
K
tER •
K •
invariant, and so
By definition (3.13),
invariant.
Assume next that
for all
v+
leaves
p EK •
Then
so one half of the ellipse
By symmetry of
~ 0
Ep(P)
K the other half is also
0
As before we will regard any given spectral convex set
as embedded into
(V,K)
in spectral duality with
(A,e)
K
where
A = V * , and we will use terminology from spectral theory with
reference to this duality.
Definition.
ellipses
A spectral convex set
are contained in
Ep( p)
exp t(P-P ' )
Remark.
z
K
0
P E qJ
and
p EK ,
is symmetric and:
all
PE
P/J ,
t ER •
One can also characterize elliptic spectral convex
sets in terms of the geometry of the fibers
F E ~ and
is elliptic if the
K for all
or what is equivalent (by Lemma 5.2), if
(5.6)
K
cr E F ,
T
E Fl •
[cr,r]
where
Such a fiber is said to have elliptic
cross sections if every plane
subspace) through
~F 1 ([cr,r])
M (i.e. two-dimensional affine
meets the fiber in an elliptic disk
(i.e. the convex hull of an ellipse).
Now it can be proved that
a spectral convex set is elliptic iff the fibers have elliptic
cross-sections.
The proof is not difficult; but since we will
not need this result in the sequel, we will only sketch the idea:
- 59 -
Assuming
where
[cr,r]
K
aEF
M n WF 1 ( [a, r ] )
set
elliptic, we consider a plane
'
r E F#
with
CfF.
FE
M through
Then the compact plane
is a union of elliptic disks all with one
diameter, and the direction of the conjugate diameter, in common.
Mn $F 1 ([a,r])
Hence
Assuming that
is itself an elliptic disk.
K has elliptic cross sections, we consider
FE (}j( corresponding to
a,r
let
where
E
PE
tfP •
Let
be defined as in Lemma 5.1.
is the intersection of
[a,r]
E
onto
E
Ep(P) c E
with the plane
has a unique affine re-
(the projection in the direction of the
conjugate d1ameter).
of
We will prove
$F 1 ([a,r])
[p,a,r] • Being an elliptic disk,
traction onto
p E K be arbitrary, and
But
[a,r] •
WF
is seen to be an affine retraction
Therefore the vector
1jrF(p)- p
points in
the direction of the diameter of
E
which is conjugate to
Hence, in the two elliptic disks
E
and
conjugate to
of
[a,r]
and
co(Ep(p))
are colinear.
JBW-algebra
Proof.
A
= Pe
all
K
.=
of a
By Theorem 3.13
K is symmetric so we only have
P-projection
P
on
Pa
=
A •
Let
£ua u} for
By Mac donalds' Theorem [1 o; p.41], or directly from
the definition of the Jordan triple product, we find for
and
K •
is elliptic.
, and recall from Proposition 2.1 that
a EA •
Ep(p) c E
The normal state space
to verify (5.6) for any given
u
is on the periphery
p
p E K , we conclude that
Proposition 5.3.
the diameters
co(Ep(p))
Since
[a,T].
a EA
A. E E+ :
(AP+ (I-P-P 1 )+ x- 1 p']a
By [3; Prop.2.7] the maps
= f(A.~u+).~u')a(X~u+A.-tu')}.
al->
fbab}
are positive for all
bEA.
- 60 ·-:-
Therefore
AP + (I ..,.p- P
I)
+ A
-1
P
I
By formula (5.2), this completes the proof.
Passage to the enveloping
Corollary 5.4.
elliptic.
for all
~ 0
). e::n+ •
0
JBW-algebra gives:
The state space of a
JB-algebra is
- 61 ..,.
1_£.
The type-I factor case.
In this section
K will be a spectral convex set which we
assume is imbedded as the base for a base-norm space
A = v* ~ Ab(K)
(V,K)
., our
in
spectral duality with
(A, e)
is to show that if
is a factor of type I ( cf .. § 1 ) with the
A
where
pure state properties ( cf. § 4), then
is a
purpose
product defined by the functional calculus:
JB-algebra for the
2
2 2
a o b = t ((a+ b ) - a - o ) ~
To prove this, we will eventually show that
A
A
has the key
"interchange property" (2.18), and thus by Theorem 2.6 that
is a
A
JB-algebra for this product.
We will begin by showing that for an arbitrary pair of
atoms
u,ve A
with
uv v = Pe
(PE
fP ) ,
the "relativization"
u,v =imP (cf. Prop.1.4) satisfies (2.18), and so is a JBalgebra. Then it follows from Proposition 5.3 that the normal
A
Au,v is elliptic, and we will use this to show
K is elliptic. Finally, we will use ellipticity of K
state space of
that
together with the pure state properties to show that (2c18) holds
for
A , and thus that
Lemma 6.1 ..
Let
A is a
K be a s12ectral convex set with the 12ure
state Ero12erties 1 and assume
If.
U 1V
i.e.
are atoms in
JB-algebra.
A and
u v v = P 0 e , then
A = Ab~K2
is a factor of tyEe I.
POE qJ corres12onds to
Au,v- = im P 0
is a
uv v
'
JEW-algebra for the
12roduct defined by the relativized functional calculus (i.e. the
functional calculus in the spectral duality of
(imp *0
'
K n imp *0 L cf" PrOE· 1. 4).
Proof.
e0
For brevity we write
= u v v = P0 e
( 6.1)
(Au,v' u vv) yd
•
A0 = Au,v = im P0
We are going to show that for
[P,Q)e 0
=
i
I
[Q ,P ]e 0
•
P,Q
and
~
P0
:
- 62 ·By Proposition 1.4 this will show that the condition (2.18) is
satisfied for every pair of
P-projections on
Theorem 2.6 we will have proved that
A0
A0
is a
,
and so by
JBW-algebra for
the product mentioned above.
Write
both under
A
0
or
e0
•
,
b
= Qe 0
= u v v = P0 e
e0
the elements
= Pe 0
a, b,
e 0 -a~
,
and note that
since
e 0 -b
P,Q
~
P0
a
and
b
are
By Corollary 4.3,
•
will either be atoms or else equal
In the latter case (6.1) is trivially satisfied (both
sides equal zero), so we will assume
a, b, e 0 -a, e 0 -b
are all
atoms.
By Proposition 1.10 the
to
A0
have one-dimensional ranges.
I
P c
Qc
=
P,Q,P ' ,Q '
P-projections
restricted
By ( 1 • 16) we get for
= (c,(e 0 -a)
A
c E A0
)(e 0 ~a)
(c,b)b
Using the equality (4.5) of the pure state properties, we find
[P,Q]e 0
= Pb-
Qa
= (b,a)a- (a,'S)b = (b,a)(a-b)
and
= (e0 -
A
a, ( e 0 -b) ) ( e 0
A
-
b) - ( e 0
-
b, ( e 0 -a) ) ( e 0
-
a)
....
= (e 0 -a,(e 0 -b) )(a-b)
...
= (1- (a,(e 0 -b) ))(a- b)
=
(1-(e 0 -b,a))(a-b)
= (b,a)(a-b)
•
This proves (6.1) and completes the proof.
0
In order to use Lemma 6.1 to prove results for
establish that
A has sufficiently many atoms.
A , we must
In this connec-
:
~
63 -
tion we will need the following general implication:
If
(6.2)
Pa E im+Q
then
where
AE A+
is a weak*-closed face of
A+
Pa E im+Q , then
containing
P- 1 (im+Q)
a , and so must con-
by [2 ; Lem.12.2].
r(a)
Lemma 6.2.
Let
K be a spectral convex set with the pure
A~ Ab(K)
state properties, and assume that
type I •
P,Q E@ ,
P(r(a)) E im+Q •
To verify this, we observe that i.f
tain
and
Then every projective unit in
A
is a factor of
is the
l.u.b. of
an orthogonal family of atoms.
Proof. 1 •
l.u.b.
U0
E
U
We will first verify that
of all atoms in
plete lattice by
.
[2
'
generated by
u0
[ 2 ; Cor • 2 • 12 ] ..
,
u0
so
(Note that
•
1t
is a com-
Cor.12.5], so this l.u.b • exists).
We consider an arbitrary
is compatible with
u
is equal to the
e
•
Let
=
[u0 ]
If
PE @
[u 0 ]
, and we will show that P
be the order-ideal of
im Q where
u0
=
Qe
with
A
Q E (jl)
~
is any finite set of atoms, then
by Proposition 4.2 (ii)
Pu 1 , ••• P~
are multiples o.f atoms;
therefore
By the definition of the map
r(a)v r(b)
=
r(a+b)
for
a~>
a,b E A+ •
r(a)
(see§ 1), we have
Hence by the implication (6.2)
above:
By Proposition 2.11 of [ 2], this shows that
Clearly the family of all
l.u.b.'s
P(u 1 v ••• v ~) ~ u 0
of finite sets of atoms is
•
- 64 -
l ~ u. b. equal to
directed upwards with
u0
by [ 2 ; Lem. 12.1]
;
is also the weak limit of this directed family.
seen that
P
P
P
u0
=e
Since
•
u0
By Proposition 5 .1·
•
u0
•
Since
=
U
A is of type I , it contains at
= Pe with PE 1/J , there exists at least one
u
w such that
tion 4.2
Pv
=u
Pe
w
~
u •
v
If
is any atom, then by Proposi-
is either a non-zero multiple of an atom or
In the former case
~
w
=
[IPvl!- 1 Pv
=
r(Pv)
=0
, so our claim is verified.
plies
where
= Qe
p...LQ, and so
for all atoms
e
v
with
uJ.v
v , then
00 ,
'
VE
then
=e-u
l.u.b.
u
PQe
•
=
In tact, if
0 , which im-
Thus, if
for all atoms
=
Pv
0
v ; since
of atoms, this gives
e
~
e-u,
u = 0 , contrary to assumption.
3.
Again we consider an arbitrary non-zero element
By Zorn's lemma there exists a maximal orthogonal family
of atoms under
we must have
u •
Let
VE ?f., be the
u = v , for otherwise
which would be orthogonal to all
of
= 0.
We will show that the
(cf. [ 2, p.28]).
v~u
has been shown to be the
i.e.
QE
Pv
is an atom satisfying
latter case can not prevail for all atoms
Pv
Therefore
Next we will show that for an arbitrary non-zero element
where
atom
Pv
f!P,
e , as desired.
2.
uE
PE
A is a factor, either
least one atom, so the first possibility is ruled out.
u0
by
;
is compatible with any
and so is by definition central.
u0
u0
is compatible with
Now we have proved that
or
Pq,~
we conclude
of ( 2] this shows that
=0
We have just
maps each element of this family under
weak c6ntinui ty of
u0
u0
(ua.} •
l.u.b.
u- v
UE
n•
(ua.}
of
(ua.J1. • Now
would contain an atom
ua. , contrary to the maximality
This completes the proof.
o·
- 65 Lemma 6.3.
Let
K be a spectral convex set with the pure
A~ Ab(K)
state properties, and assume that
type I •
K is elliptic.
Then
Proof.
is a factor of
By the definition of ellipticity (§ 5) we have to
prove that for any fixed
PE
rJ'J
(6.3)
2P + 2P ' - I 2 0 ,
(6.4)
exp t(P-P ' )
~ 0
for all
tER •
We will first show that each of these maps sends
an arbi-
rlJ., into A+ • Note first that if Pu = 0 then
u E ker+P = im+p' , so P ' u = u , and one easily sees that
( 2P + 2P ' - I ) u ~ 0 and expt(P-P ' )u ~ 0 for all tER. Thus
we assume Pu I 0 • By Proposition 4.2 Pu is a non-zero multrary atom
uE
tiple of an atom
i.e.
u vv
leave
of
=
v E
7t .
Qe , and let
Q E f!lJ
Let
Au,v
=
correspond to
uv v ,
We claim that
imQ.
p
and
p
'
u,v invariant. In fact, since P(u+v) is a multiple
v , this element will belong to the order ideal (u v v] = im Q
A
(cf. [2; Cor.2.12]; therefore we can apply (6.2) to get
P(uv v) = P(r(u+v))E im+Q •
Now it follows from Proposition 2.11 of [ 2 ] that
Hence
P , and then also
(cf. [2 ; Prop.5.1]) •
P( u v v).::::; u v v •
P' , will be compat~ble with
Therefore
P
[2 ; Prop.5.2], and so they leave
A
and
u,v
=
P'
u vv
will commute with Q
im Q invariant, as
claimed.
Now it follows by Proposition 1.4 that
ed to
Au,v
are quasicomplementary
P
and
P'
restrict-
P-projections (with respect
to the "relativized spectral duality").
Since
A
u,v
is known to
- 66 be a
JEW-algebra, its normal state space (i.e,
elliptic.
and that
Kn im Q* )
is
( 2P + 2P' - I ) u E A+ v c A+ ,
u, for all tER •
From this it follows that
exp t(P-P 1 )u EA+
c: A+
u,vClearly the maps in (6,3) and (6.4) are positive also on
orthogonal sums of atoms, and by weak* continuity and by Lemma 12.1
of [2] they are positive on
every
u E 7~
is a
l.u.b.'s of such sums.
By Lemma 6.2
l.u. b. of this type, so the maps in question
are positive on all projective units.
Finally, it follows by
spectral theory and norm continuity, that these maps are positive
on all elements
Notation:
a E A+ , which completes the proof.
Assuming, as before, that
set, we will use the symbol
atoms in
K is a spectral convex
to denote the linear span of all
A~ Ab(K) •
Lemma 6.4.
Let
state properties.
form
Af
0
( I )
on
K be a spectral convex set with the pure
Then there exists a unique symmetric bilinear
Af
(6.5)
for all pairs of atoms
variable separately.
such that
(ulv)
= (u,v)
u,v.
This form is norm continuous in each
PE~
Furthermore, every
into itself, and the restriction of
P
to
Af
will map
Af
is symmetric with
respect to this form, in that
(Pajb)
(6.6)
for all pairs
(ajPb)
a, b E Af •
Proof.
bE Af , say
=
1•
b
=
Consider first an a tom
m
L: 'V·v· where
j=1 J J
uEU
and an element
are atoms.
Then by
- 67 '
equation (4.5) of the pure state properties:
m
,..
m
,..
,..
I: ~-(u,v.) =I: ~.(v.,u) = (b,u) •
j=1 J
J
j=1 J J
m
"
I: ~-(u,v.) depends only on u and b
j=1 J
J
and not on the way in which b is represented. By linearity,
Hence the expression
this result will subsist if we replace
of
Af , say
a
n
=. I: 1>. 1 u.l.
u
by an arbitrary element
where
are atoms.
Thus,
l.=
there is a well defined bilinear form on
Af
given by
m
n
...
m
,.
I: A·~-(u.,v.) =.I: ~J.(a,vJ.).
j=1 i=1 l. J l. J
J=1
(6.7)
(ajb) =I:
Clearly, this form satisfies the requirement (6.5), and
symmetry follows by application of (4_.5) to the terms
(u.
l.
,v.>
J
of (6.7).
2.
For fixed
since each
A
v.
J
bE Af
a 1-> (a Ib)
the map
is norm continuous
occurring at the right side of (6.7) is a contin-
uous linear functional.
Hence the bilinear form
continuous in the second variable.
(
I )
is norm
By symmetry, it is also norm
continuous in the first variable.
3.
Let
P E fP
be fixed.
atoms, hence it leaves
Af
By Proposition 4. 2
invariant.
P
preserves
In order to prove that
<is symmetric with respect to the bilinear form on
Af, we start
by establishing
(6.8)
((I-P)u
I Pv)
for an arbitrary pair of atoms
By Proposition 4.2
Pv
= AW
where
AE JR+
and
P
=
0
u,v •
is a multiple of an atom, say
w
=
Qe
for a
one-dimensional range (cf. Prop.1.10).
P-projection
Now, im+Q
Q with
is the ray
P
- 68 determined by
w , and since
wE im+P
im+Q ~ im+P ,
we must have
which shows that the projective face corresponding to
tained in the projective face corresponding to
By definition,
*A
[ 2; Lem.2.16].
,.
w is the unique element of the projective face
Kn im Q* corresponding to
p
P
Q is con-
Q •
wE K n imP* , and so
Hence
A
A
w= w •
Using this and the definition of the bilinear form on
Af ,
we obtain
((I-P)ujPv)= A((I-P)u!w) =
= A((I-P)u,-w)
= A(u, (I-P) *'"'w) = 0
,
which establishes (6.8).
From (6.8) we conclude that
(Pujv) = (PujPv) = (u!Pv)
for every pair of atoms
u,v •
since by definition
is the linear span of atoms.
Af
Now (6.6) follows by linearity,
[]
We are now ready for the main result of this section.
Theorem 6 .. 5.
Let
K be a spectral convex set with the
A ~ Ab(K)
pure state properties and assume that
type I •
Then
A
is a
is a factor of
JB-algebra for the product
aob = i-((a 2+b 2 )- a 2 -b 2 ) •
Proof.
Let
P,QE
fP
be arbitrary.
By Theorem 2.6 and
Lemma 2.3 we shall be through if we can prove
I
I
[P-P ,Q-Q ]e
=
0 ,
which is equivalent to
(6.9)
( exp t (P -P' , Q-Q' ] ) e
=
e
for all
tE R •
- 69 To prove (6.9) we will show that the maps
Ut
I
=
I
expt[P-P ,Q-Q] with tEJR are affine automorphisms of
+ , then show that they preserve the bilinear form
the cone Af
introduced in Lemma 6.4, and finally use these properties of the
maps
Ut
to show that they leave the order-unit
e
invariant.
We first recall the following known identity valid for arbitrary operators
R,S
on a Banach space:
(6.10)
= exp t
2
for all
[R,S]
(Cf. e.g. [9; p.96 and p.105]
tE JR..
where the relevant series expan-
sions are given in a finite dimensional context which admits a
straightforward generalization to bounded operators on a Banach
space).
By Lemma 6. 3
expt(Q-Q 1 )
send
K
A+
into itself for all
lows by (6.10) that the maps
t
~
0 •
that
Since
Ut
Ut send
Ut
Ut
send
A+
A+
P
P ' interchanged,
and
into itself also for
must be an affine automorphism of
U_t , it follows
A+
for every
From this it follows, in particular, that every
A+
Then it fol-
into itself for all
is invertible with inverse
extreme rays of the cone
exp t (P- P ) ,
tER.
Repeating the same argument with
we conclude that the maps
t< 0 •
I
is elliptic, so the maps
onto extreme rays.
Ut
tEJR.
will map the
By the spectral
theorem [2; Th.6.8] and Proposition 1.10, it follows that the
extreme rays of
atom.
A+
Thus, every
so leave
Af
are precisely the rays
Ut
invariant.
automorphism of the cone
From now on we fix
R+u where
u
is an
will map atoms to multiples of atoms, and
This shows that
A;
for every
t ER •
Ut
acts as an affine
t E JR. •
Note that by Lemma 6. 4 the ope-
..., 70
P- P '
rators
linear form
(
and
Q- Q'
I )
on
~
are symmetric with respect to the bi-
Af , therefore their commutant is "skew".
More specifically, for every pair
I
t
(6.11)
a, b E Af
we obtain by ( 6. 6)
I
t
([P-P ,Q-Q ]ajb) =- (ai[P-P ,Q-Q ]b) •
Writing
T
= [P-P
I
I
,Q-Q ] , we now obtain
and then also
This shows that
ut
preserves the bilinear form
We have previously shown that
Ut
(
I )
on
Af •
maps atoms to multiples
-
of atoms; we now show that atoms are actually mapped to atoms.
Fix an atom
v E t(_t
where
so
A
=
uE
U ;
then for some
is an atom.
1 , showing that
A. E E.+
we have
Utu
is equal to the atom
v •
= A. v
Now
Utu
We now claim that for atoms, spectral orthogonality is equivalent to orthogonality defined in terms of the bilinear form
( I ) .
To verify this, we consider atoms
orthogonal to
v
u,vE
i.e.
u
and
v
If
u
in the spectral sense, then (by [2; (4.6)])
u+v.:;: e, so 0 < (ulv) = (u,v) < (e-v,v) = 0.
(ujv) = 0 , then
1t .
(e-u;v)
=
Conversely, if
1 , so by [ 2; (2.19)]
v.::; e- u ,
are orthogonal in the spectral sense.
From what we have just proved, it follows that
Ut
maps
orthogonal atoms to orthogonal atoms.
We are now ready to show that
Ute
=
there exists an orthogonal family of atoms
e •
(ua}
By Lemma 6.2
with
l.u.b.
is
- 71 -
equal to
fva}
e •
For eavh index
a
we define
is an orthogonal family of atoms.
va
= Utua
•
Then
For every finite set
J
of indices we have (cf. (1.9)):
Now
E u
converges weakly to e
a.EJ a
by weak continuity of Ut we get
Ute
This holds for
tivity of
t E JR
ut
=
lim Ut( l: ua.)
J
a.EJ
=
as well as
t
-t
gives
e
as
J increases [ 2 ; Lem.12 .1];
lim L: va.
J O.EJ
= Ut(U_te)
,
so
~
e •
u_te .S e
,::: Ute •
Hence
,
which by posi-
Ute
=
e •
Since
was arbitrary, this establishes (6.9), and the proof is
0
complete.
The following corollary will not be needed in the sequel,
but may be of some interest in itself.
Corollary 6.6.
A convex set
the normal state space of a
K is affinely isomorphic to
JB -factor of type I iff it satisfies
conditions (i) and (ii) and either one of (iii) and (iii)' below:
(i)
K
contains no proper split faces and has at least one
extreme point
(ii)
(iii)
K
is spectral
K is symmetric and has the Hilbert ball property.
(iii)' K has the pure state properties.
Proof.
The conditions are necessary by Proposition 2.1,
Theorems 3.11 and 3.14, Propositions 4.1 and 4.4, and by the
- 72 -
definition of a factor of type I.
(Recall the connections between
central projections and split faces [2 ; Prop.10.2], and between
atoms and exposed points, cf. Proposition 1.10).
The conditions (i), (ii), (iii)' together will be sufficient
by Theorem 6.5; condition (iii) implies (iii)
I
tions 4.1 and 4.4), which completes the proof.
(again by Proposi[]
- 73 -
U·
Characterization of state spaces of
JB-algebras.
In this section we will combine our previous results and
establish the main theorem characterizing state spaces of
algebras.
As before, every spectral convex set
K
JB-
is assumed to
be imbedded as the base for a base-norm space (V,K) in spectral
duality with (A, e) where A = V* ~ Ab (K) • When K is a compact convex set, we will use the notation
all continuous affine functions on
K •
A(K)
for the space of
The crucial point in the
development will be to prove that under appropriate hypotheses on
a strongly spectral compact convex set
K , the space
A(K)
is a
JB-algebra for the product defined by the functional calculus, i.e.
(7.1)
We will show that the product (7.1) is bilinear,
then
that
A(K)
the fact that
is a
A(K)
JB-algebra.
and
Our procedure will be to use
has a separating set of
"type-I factor re-
By the results of§ 6, the product (7.1) will be
presentations".
bilinear modulo each such representation, and therefore will be
bilinear in general.
The following Lemma will be needed.
Lemma 7.1.
If
PE
fP
and
Proof.
Let
aE
Since
(A,e)
)..<
0
and
A.E R
fA
(V,K)
P
is compatible with
eA.; therefore
(cf. [ 2; Prop.5.2)).
=
be in spectral duality.
A are compatible, then
with all its spectral units
for all
and
I
PeA+ P e
for
P(a 2 )
=
(Pa) 2 •
a , it is compatible
PeA
=
(Pe)A eA. E
Defining
·'I£
fA -- Pe A for
A .2: 0 , we get a spectral family;
this family must be the (unique) spectral resolution for
Pa
- 74 since
By the definition of squares
[J
which proves the lemma.
Theorem 7.2.
If
K is a strongly spectral compact
conve.r~
then the following are equivalent:
(i)
(ii)
(iii)
(iv)
K is symmetric and has the Hilbert ball property
K
has the pure state properties
A(K)
is a
A= Ab(K)
JB-algebra for the product (7.1)
is a
JEW-algebra for the product (7.1).
(i) => (ii) was proven in § 4.
Proof.
To prove (ii) => (iii) we assume the pure state properties.
For a fixed extreme point
projection
P
corresponding to
sition 1. 9 that
AP = imP
tivized spectral duality of
V
p
= imP * and
Fp
c(p) , and we recall from Propo-
(A p ,c(p))
and
(VP,FP)
A+) , and that the extreme points of
is a face of
K).
where
Note that the atoms of
A which are contained in
ly those extreme points of
P-
is a factor with respect to the rela-
F p = K n imP * •
exactly those atoms of
is a face of
pE K we consider the (central)
K which are contained in
are
(since
Ap
F
Ap
are exact-
p
F
p
(since
Therefore the pure state properties in the
relativized duality will follow from those in the given duality
(cf. also Proposition 1.4).
Furthermore,
A
p
contains at least
one atom, namely the atom corresponding to the minimal projective
- 75 -
face
{ p}
Hence
A
that
p
Ap
of
(cf. (4-.3) of the pure state properties).
Fp
JB-algebra for the product (7.1).
is a
By the definition of
the form
Now it follows from Theorem 6.5
is a factor of type I.
Pa
for
the elements of this space are of
Ap
a E A , and it follows from Lemma 7.1 that the
product determined by the functional calculus on
(7.2)
(Pa)o(Pb) = P(aob)
This product in the
JB-algebra
have the following identity for
A
p
for
Ap
is given by
a,bEA.
is bilinear; therefore we
a,b,c EA :
P(ao(b+c)- aob- aoc) =
= (Pa)o(Pb+Pc)- (Pa)o(Pb)- (Pa)o(Pc) = 0.
Applying the state
p = P* p
to the element
ao (b+c)- aob- aoc ,
we now conclude
(7.3)
(a o ( b+ c) - a o b - a o c, p)
From now on we assume that
merely in
A(K)
A= Ab(K) •
Since
K
=0
a,b,c
•
are in
A(K)
and not
is strongly spectral, the space
is closed under squaring, and hence under the product (7.1).
Therefore the element at the left side of (7.3) will also be in
Since the identity (7.3) holds for an arbitrary extreme
A(K) •
point
p
of
K , it now follows from the Krein-Milman theorem
that it holds for every
larly we prove
a 2 o(boa) •
pEK.
Thus, ao(b+c) = aob+aoc.
ao (A. b) = A. aob , and the Jordan identity
a
2
(a ob )o a=
This shows that (7.1) defines a product making
a Jordan algebra, and so by Theorem 2.1 of [3 ],
A(K)
Simi-
A(K)
becomes
JB-algebra for this product.
To prove (iii) => (iv) we assume that
A(K)
is a
JB-alge-
..,. 76 -
bra for the product (7.1).
A(K)** ; Ab(K)
will be a
By [20; Ths.1.2 and 1.4] the bidual
JBW-algebra for the Arens product, and
can be identified with the enveloping algebra
A(K)""' •
Now by
[2 ; Th.12.13] it follows that the Arens product must coincide
with the product (7.1), which proves (iv).
Finally, the implication (iv) => (i)
3.11 and 3.14 since
space of the
K
follows by Theorems
can be identified with the normal state
JBW-algebra
Ab(K)
(cf. [20; Th. 2.3]).
. []
The main theorem is now an easy consequence of preceding
results.
Theorem 7.3.
A compact convex set
K
is affinely and
topologically isomorphic to the state space of a JB-algebra
(with the weak* ~topology) iff K is symmetric and strongly
spectral and has the Hilbert ball property.
Proof.
The conditions are sufficient by the
implica~ion
(i) => (iii) of Theorem 7.2, and they are necessary by Corollaries
2 • 2 , 3 • 12 and 3 • 1 5 •
0
~
77 -
References
1.
E.M. Alfsen, Compact convex sets and boundary integrals.
Ergebnisse der Math. 57, Springer Verlag, Berlin 1971.
2.
E.M. Alfsen and F.W. Shultz, Non-commutative spectral theory
for affine function spaces on convex sets. Memoirs Amer.
Math. Soc., No. 172, Providence R.I., 1976.
3.
E.M. Alfsen, F.W. Shultz and E. St0rmer, A Gelfand-Neumark
theorem for Jordan algebras. Advances in Math. (to appear).
4.
G. Birkhoff, Lattice theory. Amer. Math. Soc. Colloq. Publ.,
Vol. 25, 3rd. edition, Providence, R.I. 1967.
5.
H. Braun und M. Koecher, Jordan Algebren.
Springer Verlag, Berlin 1966.
6.
A. Cannes, Caracterisation des espaces vectoriels ordonn~s
sous-jacents aux algebres de von Neumann.
Ann. Inst. Fourier, Grenobles, 24 (1974), 121-155.
7.
E.B. Davies and J.T. Lewis, An operational approach to
quantum probability.
Communications Math. Phys. 17 (1970), 239-260.
8.
J. Gunson, On the algebraic structure of quantum mechanics.
Communications Math. Phys. 6 (1967), 262-285.
9.
s.
Helgason, Differential geometry and symmetric spaces.
Academic Press, New York 1962.
10.
N. Jacobson, Structure and representations of Jordan algebras.
Amer. Math. Soc. Colloq. Publ. Vol. 39, Providence R.I. 1968.
11.
P. Jordan, J. von Neumann and E. Wigner, On an algebraic
generalization of the quantum mechanical formalism.
Annals of Math. 35 (1934), 29-64.
12.
D. Lowdenslager, On postulates for general quantum mechanics.
Proc. Amer. Math. Soc. 8 (1957), 88-91.
- 78 13.
F. Maeda and s. Maeda, Theory of symmetric lattices.
Grundlehren der math. Wissensch., No. 173, Springer
Verlag, Berlin 1970.
14.
B. Mielnik, Theory of filters.
Math. Phys. 15 (1969), 1-46.
15.
J. von Neumann, On an algebraic generalization of the
quantum mechanical formalism, Part I.
Mat. Sbornik 1 (1936), 415-484.
16.
J.C.T. Pool, Semimodularity and the logic of quantum
mechanics. Communications Math. Phys. 9 (1968), 218-228.
17.
E. Schreiner, Modular pairs in orthomodular lattices.
Pacific J. Math. 19 (1966), 519-528•
18.
I.E. Segal, Postulates for general quantum mechanics.
Annals of Math. 48 (1947), 930-948.
19.
S. Sherman, On Segal's postulates for general quantum
mechanics. Annals of Math. 64 (1956), 593-601.
20.
F.W. Shultz, On normed Jordan algebras which are Banach
dual spaces. Preprint.
21.
D. Topping, An isomorphism invariant for spin factors.
J. Math. and Mech. 15 (1966), 1055-1064.
22.
D. Topping, Assymptoticity and semimodularity in projection
lattices. Pacific J. Math. 20 (1967), 317-325.
23.
w.
Communications
Wils, The ideal center of a partially ordered vector space.
Acta Math. 127 (1971), 41-79.