~FACE
The A d v a n c e d Study Institute on "Foundations of Quantum M e c h a n i c s and O r d e r e d Linear Spaces" has been held at Marburg, Germany,
Federal R e p u b l i c of
from M a r c h 26th to April 6th 1973.
M a t h e m a t i c i a n s and p h y s i c i s t s p a r t i c i p a t e d in the meeting.
The lec-
tures of the Institute were i n t e n d e d to prepare a common basis for discussions between mathematicians
and p h y s i c i s t s and for future re-
search on foundations of q u a n t u m m e c h a n i c s by ordered linear spaces. A series of lectures
("Course")
p r o v i d e d a coherent i n t r o d u c t i o n into
the field of o r d e r e d normed vector spaces and their a p p l i c a t i o n to the foundation of q u a n t u m mechanics. special m a t h e m a t i c a l
Additional
lectures treated
and p h y s i c a l topics, w h i c h were in m o r e or less
close c o n n e c t i o n to the lectures of the course.
The present volume contains the notes of the lectures revised by the authors.
The s p o n s o r s h i p of the S c i e n t i f i c Affairs D i v i s i o n of the N o r t h Atlantic Treaty O r g a n i z a t i o n ,
of the Stiftung V o l k s w a g e n w e r k and the
U n i v e r s i t y of M a r b u r g is g r a t e f u l l y acknowledged.
A.Hartk~mper
H.Neumann
C O N T E N T S
Introduction
I
•
g
J
U
D
Q
•
Q
~
U
t
D
O
•
O
g
•
U
O
g
Q
6
COURSE Mathematics: H.H.
Schaefer
J. M a n g o l d R.J.
Orderings
of V e c t o r
Spaces
4
and
Nagel
Duality
of Cones
Unit
in L o c a l l y
R.J.
Nagel
Order
A.J.
Ellis
Minimal
Decompositions
and Base N o r m
Spaces
A. G o u l l e t
de Rugy
Simplex
A. G o u l l e t
de Rugy
Representation
A.J.
. . . . . . . . .
Ellis
Order
Ideals
W. Wils
Order
Bounded
G. W i t t s t o c k
Ordered
E. S t ~ r m e r
Positive
Spaces
Spaces
, °
.......
in Base N o r m e d
11 23
Spaces.
30
. . . . . . . . . . . . . . .
33
of B a n a c h
41
in O r d e r e d
Normed
Convex
Banach
Operators Tensor
L i n e a r Maps
Lattices
......
Spaces
and Central
Products
.....
47
Measures
54
. . . . . . . .
67
of ~ - A l g e b r a s
.....
85
Physics: A.
Hartk~mper
H. N e u m a n n
Axiomatics Procedures
of P r e p a r i n g and M e a s u r i n g . . . . . . . . . . . . . . . .
107
The S t r u c t u r e of O r d e r e d B a n a c h Spaces in Axiomatic Quantum Mechanics. . . . . . . . .
116
G. L u d w i g
Measuring
122
G,M.
M o d e l s of the M e a s u r i n g Process and of Macro Theories . . . . . . . . . . . . . .
C.M.
Prosperi
Edwards
K. Kraus
D.J. C.H.
Foulis and Randall
The Centre
and P r e p a r i n g
Processes
of a P h y s i c a l
O p e r a t i o n s and Effects F o r m u l a t i o n of Q u a n t u m
......
System .......
in the H i l b e r t Space Theory . . . . . . . .
The E m p i r i c a l Logic A p p r o a c ~ to the Physical Sciences . . . . . . . . . . . . . .
,.
I~ 199
206
230
VI
SPECIAL
TOPICS
The S t r u c t u r e
M. D r i e s c h n e r
Suggestions V. Gorini E.C.G.
stical The
U. Krause
L.A.
Physics
and D y n a m i c a l
Operators
Maps
of C o n v e x
Quantum Mechanics
250
of Stati-
Sets
260
in
..........
269
Lu-
and
G. R a m e l l a
Reduced
M. M u g u r - S c h a c h t e r
The Q u a n t u m
Dynamics
in Q u a n t u m M e c h a n i c s . . . .
Mechanical
Hilbert
lism and the Q u a n t u m M e c h a n i c a l Space of the O u t c o m e s R.J.
......
. . . . . . . . . . . . . .
Inner O r t h o g o n a l i t y
Axiomatic
giato
for a U n i f i e d
Irreversibility
and
Sudarshan
L. Lanz,
of Q u a n t u m Mechanics:
Nagel
Mean Ergodic in O r d e r e d
H. N e u m a n n
Banach
Spaces
The R e p r e s e n t a t i o n Quantum Mechanics
E. P r u g o v e ~ k i
Extended rac's
. . .
. . . . . . . . . .
of C l a s s i c a l
Systems
Space F o r m u l a t i o n
to A b s t r a c t
Projections
H.H.
Sc h a e E e r
The ~ i l o v B o u n d a r y
of Di-
Scattering
an A p p l i c a t i o n
Lattices
of a C o n v e x
A Radon-Nikodym-Theorem
Cone
for O p e r a t o r s
to S p e c t ra l
316
and its Appli-
Stationary
on O r t h o m o d u l a r
309
in
. . . . . . . . . . . . . . . . . . .
R0ttimann
288
Ideals
. . . . . . . . . . . . .
G.T.
M. Wolf f
Probability
and Invariant
Bra and Ket F o r m a l i s m
cations Theory
Hilbert
Space Forma-
of M e a s u r e m e n t s
Semigroups
281
Theory
322
....
384
....
342
with
......
345
INTRODUCTION
Since the p h y s i c a l c o n t r i b u t i o n s of the course are not so closely related to e a c h other as the m a t h e m a t i c a l ones it seems useful to give some i n t r o d u c t o r y remarks c o n c e r n i n g the p h y s i c a l topics.
Since the early b e g i n n i n g of w o r k w i t h q u a n t u m m e c h a n i c s p h y s i c i s t s felt u n c o m f o r t a b l e p o s t u l a t i n g the H i l b e r t space structure ad hoc. From this r e s u l t e d attempts to deduce the Hilbert space structure by an axiomatic
foundation of q u a n t u m mechanics.
A further,
far m o r e am-
bitious aim of m o s t of these attempts is to find structures of physical theories
i n c l u d i n g m o r e general t h e o r i e s than q u a n t u m mechanics.
An a x i o m a t i c foundation of not only m a t h e m a t i c a l c h a r a c t e r u s u a l l y starts w i t h w h a t could be called a pretheory,
d e s c r i b i n g the p h y s i c a l
notions and situations on w h i c h the final t h e o r y is based. This pretheory supplies the usual m a t h e m a t i c a l
structures w i t h an
additional
structure and s i m u l t a n e o u s l y yields a p a r t i c u l a r i n t e r p r e t a t i o n of the final theory.
The attempts of an axiomatic foundation of q u a n t u m m e c h a n i c s can be c l a s s i f i e d by the basic notions w i t h w h i c h the p r e t h e o r i e s cerned and by the m a t h e m a t i c a l
are con-
apparatus used in the sequel.
In contrast to the p o s s i b i l i t y of d i r e c t l y p o s t u l a t i n g p r o p e r t i e s of microsystems,
the authors of this volume start from the m a c r o s c o p i c
e x p e r i m e n t a l situation.
The vector space structure enters the t h e o r y either in an early stage by e m b e d d i n g the basic s t a t i s t i c a l d e s c r i p t i o n into a dual pair of vector spaces,
or in a later stage via a linear space of o r t h o a d d i -
tive real v a l u e d functions on a "logic".
In a c c o r d a n c e w i t h the title of this v o l u m e very little w i l l be found c o n c e r n i n g the b r o a d field of lattice t h e o r e t i c a l approaches to an
axiomatic
foundation
of quantum mechanics.
Several articles of this volume contribute
to the discussion
of the
relation between quantum mechanics a n d the classical theories croscopic bodies.
On one hand the latter seem intimately
with quantum mechanics sical theories position, parata.
of many particles.
to a certain
Finally,
the general
of the m e a s u r i n g process
formulation,
K = ~ W ~ Lh(H)
system is described
operators
of the system.
operators
are called properties
However,
in H is the set of observables
onto closed subspaces
of projection
operators
ponding to various different operators
of H. The outcomes
interpretations,
or events or decision
the analysis
suring process
of the functioning
with outcomes
L = ~ F £ Lh(H) O and i, tr(W.F)
/ O ~ F ~ ~ I
projection
effects.
pro-
of the
and, corresoperators
The projection
orthomodular
lattice.
and the statistics
O and 1 and the analysis
and of operations
of
the m e a s u r e m e n t
of compatible
are O and 1 only,
form a complete orthocomplemented
stic measurements
cone
trace class operators:
can be related to the m e a s u r e m e n t
jection operators measurement
as the
K is the base of the positive
By means of the spectral decomposition,
of observables
denotes the
~IT ~tr = tr(( T ~ T) 1/2) < ~
The set of self-adjoint the system.
in an infinite-di-
is considered
of the base normed Banach space B of hermitean
B = { T ~ L h(H) /
of
of H the set
/ W ~ O, tr W = 1 I
set of ensembles
this volume,
in terms of ordered vector
complex Hilbert space H. If Lh(H)
set of bounded Hermitean
appears
and short descrip-
used throughout
frame of quantum mechanics separable,
ap-
of the theory.
A quantum m e c h a n i c a l
mensional,
as a sup-
with m a c r o s c o p i c
it might be useful to give an elementary
tion of von Neumann's spaces.
the m e a s u r e m e n t
In this sense the description
as a problem of consistency
On the other hand the clas-
extent enter quantum mechanics
via axioms concerning
of ma-
connected
of reali-
of the mea-
suggest that the whole set
describes m e a s u r e m e n t s
being the p r o b a b i l i t y
for the outcome
with outcomes 1 in the en-
semble
W.
An element F E L is called effect,
simple o b s e r v a b l e or test. Lh(H)
can be r e g a r d e d as the dual B a n a c h space B' of the space B of herm i t e a n trace class operators/ form on B x
B'. B l = Lh(H)
tr(TA) b e i n g the canonical b i l i n e a r
is e q u i p p e d w i t h an order unit norm, w h e r e
the unit o p e r a t o r ~ is the order unit of B'. L is the order interval [0, ~]
of B', and the set of p r o j e c t i o n o p e r a t o r s is the set of ex-
treme points of L.
ORDERINGS
OF V E C T O R
H.H. Mathematisches
Schaefer
Institut
der Universit~t
THbingen,
What
is m e a n t
this
concept
to g i v e study
an o r d e r e d
in a n a l y s i s
a first
proofs,
-C = { O~
~C
C C
is c a l l e d
ordering,
semi-ordering),
symmetric
binary
relation ~
invariant
under
translations
if x ~ y i m p l i e s O ~ ~
~
set E +
~.
space,
and what
following
is t h e r e l e v a n c e
informal
discussion,
a few typical
examples.
we must
the
Let E denote
C + C c C and
C ~
and
T0bingen
Germany
refer
For
reader
of
we try
a closer
to the
at t h e e n d of t h e p a p e r .
Orderin@s.
satisfying
? In the
detailed
given
i. V e c t o r
vector
introduction
including
references
which
by
SPACES
a proper
i.e.,
(E,~)
:= ~ x ~ E: x ~ O I
space
cone.
of E b y v i r t u e
" x ~ y iff y - x
cone
of r a t i o
vector
called
cone C ~ E defines (cf.
and
for all x , y
an o r d e r e d
~ C"
a cone
[ SI]
a
antiif it is
~ O;
that
is,
e E and space,
and the
the positive (vector)
, Chap.
C ~ E C for
(or p a r t i a l
transitive,
maps
~ x ~y
is a p r o p e r
each proper
a subset
a vector orderin~
and homothetic
of E. C o n v e r s e l y ,
~;
a cone;
An ordering
on E is c a l l e d
is c a l l e d
over
is c a l l e d
a reflexive,
x + z ~ y + z and
The pair
of
a vector (~ ~0)
V,
cone
ordering Exerc.
1-3).
Examples. i. L e t E b e E',
let K b e a
linear H,
hull
and
:=
A(K)
E
!
on K
{ x e E: o f all
uniform
~(E',E)-compact
O ~
on K, w e
functions
locally
H.
space with
subset
of E'
Considering
can
( ~A]
, C h a p . II,
§2),
a n d E = A(K)
ordered
!
cone E+
to e a c h o t h e r
in t h e
sense ~
functions
:=
is t h e
in t h e
respect
space
to the
complete.
and these
cones
t h a t x' ~ E +' ( r e s p e c t i v e l y ,
O for all x e E +
continuous
continuous
b y the c o n e
on K w i t h
L) ~ K,
dual
hyperplane
K ~ . E is d e n s e
if E is s e q u e n t i a l l y
by the
<xrx' ~
s u c h t h a t E'
~(E',E)-closed
a space of affine
~ O for all x' &
continuous
topological
e a c h x ~ E as an a f f i n e ,
E with
is o r d e r e d
to
in a
identify
• x,x'>
affine
norm,
equivalent
convex
of K a n d K is c o n t a i n e d
suppose
function
E+
a Hausdorff
are d u a l
x ~ E+)
Crespectiveiy,
x' &
is E~).
2. Let X denote continuous norm;
a
(Hausdorff)
functions
X-~ ~
the cone C(X)+
natural
ordering
precisely with
~n
3. Let
of X.
is a B a n a c h
If X is finite the o r d e r e d
its usual
The B a n a c h
spaces
functions
LP(/~)
(i~ p ~ + ~ )
that the B a n a c h
the
and contains
can be i d e n t i f i e d
ordering. space
(equivalence
by d e f i n i n g
an f such that
dual of LP(/~)
the s u p r e m u m
defines
space C(X)
measure
of
space of all
under
(hence discrete)
Banach
~-finite
are o r d e r e d
to be ~ 0 iff it c o n t a i n s Recall
The v e c t o r
space C(X)
(coordinatewise)
(X,~ ,/--) be a t o t a l l y
measurable
space.
:= { f: f ( t ) ~ 0 for all t ~ X ~
n elements, under
compact
(see
[DS]
classes
an e q u i v a l e n c e
f(t) ~
of) class
O everywhere
(i~ p ~ + ~ )
can be
).
on X.
identified
/
w i t h Lq(/~),
Let E d e n o t e
where
p-i + q-i = i.
an o r d e r e d
vector
space,
of E. F is o r d e r e d by the cone F+ q: E --)E/F denotes defined being
iff the cone q(E+)
an order
(Notice that E is;
the q u o t i e n t
that
ideal
If E,F
are o r d e r e d consider
proper
cone,
vector
it defines
it is s u f f i c i e n t
F =
~
spaces
and L is a vector
:= { T ~ L: T(E+)
the n a t u r a l
of dual
Examples
i, 3 above.
An e l e m e n t
pairs
x £ E+ ~
called e x t r e m a l multiple
ordering;
the
[J]
of o r d e r e d
{ O~
, where
if O ~ y ~
of x. We point
Let E be an o r d e r e d linear
vector
positively
depends
generated
ideal J is an ideal
mal
ideal.
is e q u i v a l e n t implies
generated
space
to F
z e F. even
if
of linear maps
functionals
forms
is
on E, and the
called
the order dual
spaces
are c o n t a i n e d
vector
space,
of E. in
is
that y is a scalar
fact:
such that E = E+-E+.
in E+ * whenever ideal
L+ is a
E = E+ - E+ and let
space E ~ of all linear
important
on the fact that
If is
(For L+ to be a proper
E is an o r d e r e d
space
generated
The proof
of L.
x, y e E implies
out this
of E/F
~ F+ ~ ; w h e n e v e r
Suppose
linear
vector
form f on E is e x t r e m a l
maximal,
order
E: - E ~ of E ~ is u s u a l l y
Example s
in turn
subspace
imply J = J+ - J+.)
that E = E+ - E+.)
in its s t a n d a r d
subspace
ordering
z ~ E, and x ~ z ~ y
not
o r d e r e d by the cone E~ of p o s i t i v e vector
which
a vector
ordering).
ideal J need not be p o s i t i v e l y
the cone L+
cone
(induced
a natural
is proper,
is, E = E+ - E+ does
E -~F,
map,
in E: x,y ~ F,
an order
and let F denote
:= F ~ E+
A positive
its kernel
f-l(o)
is a
in E.
the a n n i h i l a t o r
in E ~, and that E/J ~
jo C ~
E ~ of a p o s i t i v e l y iff J is a m a x i -
An ordered
vector
ever nx 6y
for some
vector space
space
1-3
2. V e c t o r if t h e xv y
E+
is s e q u e n t i a l l y
lattices.
theory
§i] or
(1)
closed
dual
if x & O
when-
o f an o r d e r e d
a topological
t h e n E is A r c h i m e d e a n
exist
[$2,
for each pair lattices
Chap. II,
+ x~y
vector
(see
Ixl
= x v(-x)
satisfy
Ixl = x + + x-.
E is c a l l e d
of e l e m e n t s
we
§§i-4]
and the
refer
. We
x,y e
to [J,
recall
a vector
lattice
least upper E.
For
Chap. I I ]
the basic
,
in p a r t i c u l a r
bound
[Sl, that
= x + y
x = x + - x- b y
element
space
:= inf { x , y I
f o r a l l x , y ~ E; d e f i n i n g
we obtain
The order
if E is a l s o
vector
xAy
of vector
xvy
holds
An ordered
lower bound
:= sup ~ x , y J
algebraic
moreover,
(ordered)
above).
greatest
Chap. V,
y ~ E a n d all n ~ ~ .
is A r c h i m e d e a n ;
in w h i c h
Examples
s p a c e E is c a l l e d A r c h i m e d e a n
x + = xvO
letting
is c a l l e d
and x- =
(-x) v 0 for x £
E,
(i),
so E = E +
The
y = O in the modulus
Moreover,
and
of x ~ E,
the m o d u l u s
function
and
- E+.
found
to
x~-~ |xl s a t i s f i e s
the relations
(2)
I x + Yl
for all x , y & E, An
important
x,y
A ~ :=
o f E.
to the
like-named
space of set while
[3)
by
l~xl
=
I~I
[ xl
m
vector if
spaces
Ixl m
lattices,
lattices space
shows
theory;
certain
is
lyl = O;
I xl = O for all x e A I , w h e r e
in H i l b e r t
A is a n y
parallels
it is m a i n l y
useful
however,
are g i v e n
1 is a v e c t o r
lattice
ordered
ortho~onal
above
lattice
(Examples
only
simplex;
iff K is a B a u e r
2,3).
The ordered
if K is a p a r t i c u l a r [A,
simplex
Chap. II, (l.c.)
§3]
type
),
provided
E
complete.
that
to E h a v i n g
is d e f i n e d
in l a t t i c e
iff K is a C h o q u e t
E is a v e c t o r
,
(lattice)
in v e c t o r
lattices
of E x a m p l e
is s e q u e n t i a l l y
valent
vector
(precisely,
The property
lyl
concept
of v e c t o r
E'
E:
Orthogonality
for A r c h i m e d e a n Examples
available
are c a l l e d
[ y 6
subset
lyl
6 ~.
concept
orthogonality: we write
~
-~ Ix~ +
the
space
E' of E x a m p l e
the decomposition
1 be
the relation
[ O~x + y ]
=
[Otx ]
a vector
(or i n t e r p o l a t i o n )
+
[O~y]
lattice
is e q u i -
property,
which
holding w ~
for
E:
all x , y ~
O&w
conversely; satisfies
{z]of
E+;
E.
as usual,
Every
if E is a p o s i t i v e l y (3),
its o r d e r
dual
If E is a v e c t o r
lattice,
in its n a t u r a l
finite
infima
positively ideal
and
(briefly,
A vector
lattice
bounded
(i.e.,
sup A,
is c a l l e d
order
Example
3 are o r d e r
compact
space
An
ideal
implies band
sum, Pj
J C E
and
the
(called
famous Let
J A
B1 v
:= B 1 + B 2
projections P1 A
P2
:= P1
For proofs
of t h e s e
considered
in this
nating has
manner
a weak
u ~ E+
such
tation,
order
u ~ the
interval
induced
by E.
[O,e]
a
The
ideal.
is o r d e r
a least
is o r d e r
, then
upper
spaces
bound
of
complete
A & J and
iff the
sup A = x 6 E
J is c a l l e d
with with
lattice.
If a
a projection
of E to be a d i r e c t
kernel J).
We
J ~ is a p r o j e c t i o n can n o w
Each band
B1 A
algebra P2
state
the
under
[LZ]
or
B 2 := B 1 ~
that
= O
E,
family to the
(x e E)
P2
the m a p p i n g
of all b a n d
(in p a r t i c u l a r , B ~ - ~ P B. algebras
in a p a r t i c u l a r l y
u;
implies
and
operations
. The B o o l e a n
in a d d i t i o n unit)
B2
illumi-
to b e i n g
order
that
an e l e m e n t
x = O.
is, For
this
complete,
represen-
result.
lattice, form
the
respect
IS2]
in E is a
of E is a B o o l e a n
:= P1 + P2 - P1
to ~
see
with
can be r e p r e s e n t e d
following
L e t E be any v e c t o r order
2)
of all b a n d s
(or F r e u d e n t h a l I x~
has
decomposition
operations
isomorphic
assume
J be
a lattice
A which
complete.
that
E--~J
vector
to the
theorem
that
is c a l l e d
subset
(BI,B 2 ~ ~ ). A c c o r d i n g l y ,
unit
that
we n e e d
this
family ~
results,
if we
such
associated
' P2 and P1 v and
preserving
theorem.
~ is a B o o l e a n
commutative),
and
for E / J to be
E-*E/J
for e a c h A ~ E, A ~ is a band.
endomorphism
respect
then q:
sufficient
interval)
E = J + J~
complete
with
B2
that
and the
algebra
space
A of E, A ~ is a l a t t i c e
(Example
lattice)
projection
band,
not
disconnected.
J~ = {O I forces
an o r d e r
projection
and
or D e d e k i n d
a band;
decomposition
of E,
an ideal
order
interval
(3), b u t
vector
(§i) w i t h
non-empty
C(X)
(E any v e c t o r is c a l l e d
the b a n d
E be
in some
complete;
associated
Riesz
each
complete
the p r o p e r t y
because
subset
X is e x t r e m a l l y
x e J,
J has
band,
each
that
contained
subspace
Such
For
E such
ordered
ordering
ideal.
the o r d e r
satisfies
lattice.
it is n e c e s s a r y
order
ideal).
generated
J a vector
suprema,
generated
denotes
lattice
is a v e c t o r
a vector
lattice
[ O,z]
vector
and
let e & E+.
a Boolean
algebra
The under
extreme the
points
ordering
of the
In fact,
it is n o t d i f f i c u l t
x ~
(e-x)
the
lattice
[O,e] The
= O,
and t h a t
announced Let E be
In fact,
rated
PI(E) ~ hence,
3. O r d e r
Unit
are d u a l
Let such
(Pl,P2e
to e a c h vector
the
; since
boundary
of
possessing
a weak
order
on E t h e n
projection
Moreover,
~
Norms.
P~-~Up
spaces
other.
We
algebra
= Pu A
(u-Pu)
. Conversely,
with
u is e q u i v a l e n t is an o r d e r
the b a n d
if
gene-
with isomorphism
and
algebras.
We c o n s i d e r
vector
(IE-P)u
of [ O , u ]
associated
P1 u ) P 2
) ; therefore,
O = Pu A
point
unit
which,
suppose
very briefly
as the
two p a r t i c u l a r
following
throughout
that
lectures
E be
will
show,
an A r c h i m e d e a n
space.
e be an o r d e r that
lattice
Pu is an e x t r e m e
of B o o l e a n
and B a s e
of o r d e r e d
ordered
projection
x = Pu.
an i s o m o r p h i s m
types
vector
= O and P is the b a n d
P2(E)
of [ O , e ]
the e x t r e m e
:= Pu is an i s o m o r p h i s m of the B o o l e a n P a l g e b r a of e x t r e m e p o i n t s of [ O , u ] .
therefore,
then
a sublattice
iff
P~-~u
if P is a b a n d
by x,
form
is e x t r e m e
follows.
complete
the B o o l e a n
x /~ (u-x)
elements
x e [O,e]
in E are d i s t r i b u t i v e ,
is as
an o r d e r
and O & P u ~ u ;
that
algebra.
result
u. The m a p p i n g onto
these
operations
is a B o o l e a n
to show
unit
ideal
of E,
that
generated
is,
let t h e r e
by e e q u a l s
E
exist
(so t h a t
an e l e m e n t E =
%2
e ~
E+
n [-e,e]
n 6 the M i n k o w s k i
functional
Pe(X)
is a n o r m (E,p e) each
on E c a l l e d
are p r e c i s e l y
increasing
(and the m o s t A(K)
(§i,
isomorphic
= inf
Pe-Cauchy
order
If such
where
Pe of
[-e,e]
~. 6 6R: -2, e £ x
unit
norm;
units
sequence
example
i).
to C(X),
~
an o r d e r the
general)
Example
(gauge)
the
of E.
has
, defined
_~ le ~ ,
interior
(E,Pe)
a least
points
of E+
in
is a B a n a c h
space
iff
upper
bound.
of an o r d e r - u n i t - n o r m e d a space
X is the
is a v e c t o r
extreme
by
lattice
boundary
A typical
Banach
of K
space
then
is
it is
(Kakutani-
Krein). Dually, for e a c h
if E = E + - E + x £ E+,
x & f Cx[B; the
then
infimum
(z ~ E) ; the
and t h e r e
there
exists
B is c a l l e d
is t a k e n
over
exists
a convex
a unique
number
a base
all p a i r s
set J = qB-l~oj
of E+,
Let
Cx,y) ~
is an o r d e r
subset
such
O for w h i c h
qBCz[
= inf
E+ x E+
ideal
B of E+
f(x)~
f(x+y)
such that
that
where
z = x-y
of E, and qB d e f i n e s
a
);
norm ~B on E/J w h i c h is called a base norm.
The d i s t i n g u i s h i n g
of qB and ~B is additivity
Cbut not the most general)
on E+. A typical
example of an ordered vector the space E' of Example
i.
space whose positive
property
cone has a base,
(Here the base K is compact,
is
and J = { O~
A lattice ordered base normed Banach space is isomorphic
to LI(/~),
/~ is a suitably chosen Radon measure on some locally compact
.) where
space
(Kakutani). In conclusion,
let us consider
the following example which illuminates
the entire preceding discussion. sets,
Let X be a set, ~
a base normed vector lattice with base B = { f ~ its dual L ~ (/~) is an o r d e r - u n i t - n o r m e d e (the constant-one and L ~ ( ~ )
ristic
functions
(and L ~ )
fixed m e a s u r a b l e
~
of
~O,e]
are the
(equivalence
of the functions vanishing
characte-
a.e.
(2~) outside
some
is multiplica-
function of S. Thus the
of the last result of §2 is the mapping PS~-~ ~ S . In parti-
is isomorphic with
On the other hand,
~/N,
the Boolean
(see
algebra of m e a s u r a b l e
sets. the order unit normed space L ~ (/~) can be identified
with C(K), where K is the Stone r e p r e s e n t a t i o n Z/N
classes of)
sets S & ~_ ; each projection band in L 1
set S, and the associated band projection
sets modulo / ~ - n u l l
algebra
lattice with order unit /
: f ~--)~S f, with the c h a r a c t e r i s t i c
isomorphism cular,
is
f f d/~ = i~ , and
function on X). e is a weak order unit of L I ( ~ ) ,
of m e a s u r a b l e
consists
PS
vector
0 :
of sub-
Then LI(/~)
can be identified with the ideal of L 1 generated by e.
The extreme points
tion,
a ~-algebra
and let/~ denote a finite positive measure on ~
tV]
,
space of the Boolean
IS 2] ).
References CA]
Alfsen,
E.M.,
Compact Convex Sets and Boundary Springer-Verlag,
~DS]
Dunford,
N. and J.T. VoI.I.
[J]
Jameson,
Schwartz,
Berlin-Heidelberg-New
York 1971.
Linear Operators.
Interscience
Publ.
4 th print,
New York 1962.
G., Ordered Linear Spaces~ Springer Lecture Notes No.
[LZ]
Integrals.
Luxemburg~
W.A,J.
and A.C.
North-Holland
Zaanen~
141,
19~0.
Riesz Spaces
I,
Publ. Co,~ A m s t e r d a m - L o n d o n
1921.
10
[p]
Peressini,
A.L., Ordered Topological
Vector Spaces.
Harper and Row, New York-Evanston-London
tSll
Schaefer,
H.H., Topological
Vector Spaces.
Springer-Verlag, IS 2 ]
Schaefer,
Springer-Verlag
Iv]
Vulikh,
B.Z.,
Introduction Spaces.
3 rd print.
Berlin-Heidelberg-New
H.H., Banach Lattices
1967
York 1971
and Positive Operators.
(in preparation).
to the Theory of Partially Ordered
(Engl.Transl.)
Wolters-Noordhoff t Groningen
1967.
D U A L I T Y OF CONES IN L O C A L L Y C O N V E X SPACES
J~rgen Mangold
and
Rainer J. Nagel
F a c h b e r e i c h M a t h e m a t i k der U n i v e r s i t ~ t T ~ b i n g e n
TObingen,
If a cone cone
C'
:=
C
Germany
in a locally c o n v e x space
Ix' ~ E ' : ~ x , x ' > ~ o
for all
E
is "big"
x~C
~
, its dual
is "small"
and vice versa. This b e h a v i o r will be e x p r e s s e d m o r e p r e c i s e l y in section 2 by the d u a l i t y t h e o r e m for normal and strict ~ - c o n e s to
S c h a e f e r
In normed vector spaces the n o r m a l i t y resp. can be m e a s u r e d by n u m e r i c a l constants. Again, t h e o r e m is v a l i d
due
([4~, V.3).
(section 3; see
strictness of a cone
a strong d u a l i t y
[2]). In section 4 we discuss a
p r o p e r t y of o r d e r e d B a n a c h spaces w h i c h is m o t i v a t e d by the theory of B a n a c h lattices
(see [i~).
W h i l e all the results are m o r e or less well known, we prove the m a i n t h e o r e m s by a new m e t h o d ~3] in the normed case): ties of polars
(already used by
K u n g - F u
N g
By a c o n s e q u e n t use of some basic proper-
(see section i), all proofs become simple and m e c h a -
nical computations. In general we follow the t e r m i n o l o g y of ~4] and refer to and
~
for a d d i t i o n a l i n f o r m a t i o n and h i s t o r i c a l comments.
[2~
12
1.
Computation
For
a locally
rules
for p o l a r s
convex
space
and the d u a l i t y
<E,E'>
. If
the p o l a r
is d e f i n e d
(resp.
of
M
: <x,x'>
~ 1
for all
x~M~
: ~x,x'>
~ 1
for all
x'~M ~
properties
in
[4~
N
The
be c o n v e x
E
(M + N) °
D
1/2
(M ° n N ° )
D
1/2
(M ° + N ° )
(3)
(M + N) °
C
M°N
(4)
if
1. For
[4~,
E
=
MqN
N) °
=
c--o ( M ° v
a convex
theorem) additional
with
the
=
subset
in
[4~,
(4)
which
of v e r t e x a detailed (5)
vector
from
(3) and
will
be n e e d e d
0
such
proof
of the
we c o n s i d e r
E
containing
Mo + N °
, its
IV.I.5,
I)
closure
with (see
is the
<E,E'>
[4],
corollary
if
is an i n t e r i o r detailed
space
with
same
and
IV.3.1).
2, and t h e
(U - C)) °
(if not
3:
fact
=
ball
N
U
,
for
be a c o n v e x is
C'
see a g a i n
stated)
closed
and
rules
C
cone
co((C' N
otherwise
are M
[3]).
unit
Let
rules,
N
of
see
closed
. The d u a l
following
and
computation
in s e c t i o n 0 £ C
M
point
proof,
(4) f u r t h e r
that
((U + C) ~
of
C
consistent
is s a t i s f i e d
; for a m o r e
is a n o r m e d
E')
to prove:
.
or if the o r i g i n II.l.l
N° )
~(E,E')-closure
from
(M N N) °
assumption
(resp.
, then
topologies
(4) f o l l o w s
can d e d u c e
E'
some
N°
M ~ N
convex
identical
3. The
E'),
).
the b i p o l a r
we need
are e a s y
(M ~ N) °
(M ~ N) °
i) On
of
(2)
(trivially)
For
subsets
inclusions
(i)
2. R u l e
cone
(especially
. F o r our p u r p o s e s
following
locally
polars,
of p o l a r s
, IV.I
of
rules: and
Remarks:
If
(resp.
Ix 6 E
(M ~
we
E
~x'6E'
the origin.
(use
of
E'
by
:=
M
that
is a s u b s e t
space
:=
computation
hence
M
its d u a l
SO
The b a s i c
for all
, we consider
MO
are p r o v e d
Let
E
U O) v
the
=
[3]
(-C) ° . :
-(C' N U°))
2)
~(E',E)-
topology. 2) S i n c e
C' ~
is closed.
D°
is
~(E',E)-compact,
co((C' N
U O) ~
-(C' ~
U°))
13
(6)
if
C
is closed:
((u ° + c') n
2. D u a l i t y
In t h i s space
section,
with
vertex
in
E
we
~x'~
E'
of
DEFINITION:
Let
(i)
[A~
:=
(A + C) N
(ii)
~A[
:=
co
For [A'~
subsets and
Example:
of
E'
Take
E
: l~.2 ,
unit
.
locally
convex
be a c o n v e x cone
cone
of
is =
(-C) O
the pointedness,
.
of
E
.
We
the C-saturated
-(An
C))
and the
dual
~ (A,~)
call
hull
the C - c o n v e x
cone
in an a n a l o g u e
C =
ball
-(c n u))
for all x ~ C ~
be a s u b s e t
are d e f i n e d
A be t h e e u c l i d e a n
C
us to g a u g e
(A - C)
((A ~ C) ~
A'
~A' 5
A
is a real
~ o
C'
n u)~
spaces
. Let
help
and
~((c
. The dual
: <x,x'>
C
E
~
O E C
will
=
convex
that
base
that
definition
the bluntness
(2.1)
assume
such
:=
following
resp.
in l o c a l l ~
O-neighborhood
O C'
The
of c o n e s
(u ° - c')) °
:
A
C'
of
kernel
the
A
3!
of
A
.
sets
way.
~. O,
"2 >iO
. Let
:
(A) 4 ) \\~\
¢\ \" C ' © \ \ \ 2 1
7." L <\ x\O.,yl\J t, ,\"
k ,d
• \
"\ ,"07
I 3)
If
C
is a p r o p e r
we have 4)
(A)
:=
cone
~A~ = k 3 ~ x6A
x,y £A ~-x,x] ;
(and h e n c e
induces
~x,y~ ,where see
(4.2)
.
an o r d e r i n g
Cx,y~
Iz~E
on
E ),
x
14
The family topolog y
:= I /-U] : U&'O[ I
[~I
on E w h i c h
is the coarser
Hence we can r e s t r i c t the f o l l o w i n g
(2.2) i_~f
the
C
t__oo ~
1. E v e r y normal
and the closure further
elementary
U .
cone
C
[2] , p. 88
is normal
IIxll , IIYll ~
Examples:
Then
and
(since See
t_~o ~
E
)
is Hausdorff)
[4~, V . 3 . 1
, uses the term vector
there there
x ~ z ~ y
be a vector
is a n o r m e d E
for
'self-allied
space
E
with
is or, 1 K ~ £ ~ is ~ ,
wedge'.
closed
unit
such that
1 ~ ~ ~ ~ such that
implies
llz(( 9 ~
cone in a locally
space
of b o u n d e d
).
convex v e c t o r
vector
real-valued
ordering
space o r d e r e d
lattice
continous
t~ ~
and
is an ordered
derivative.
Ilfll :=
Banach
space
If(t)I
on
cone.
functions
Define
sup t ~
functions
and the sup-norm.
by a normal
be the space of all r e a l - v a l u e d
a bounded
for all E
(with respect
is normal.
, e n d o w e d w i t h the c a n o n i c a l
3. Let having
iff
1. The p o s i t i v e
E
X E
1
cone
([4], V.7.1).
2. Let a set
is.
to have
properties.
( or equivalently:
is normal
C
C
.
3. Let C be a cone in a n o r m e d
[U] C ~ U
convex
the cone
by r e q u i r i n g
cone is p r o p e r
of a normal
2. J a m e s o n
ball
'blunter' C
is a normal
i_~s e q u i v a l e n t
Remarks:
the of
a locally
property.
DEFINITION:
[~]
'width'
defines
C
on
:= I f ¢ E : f(t) ~
+
sup t~e
, but its p o s i t i v e
cone
If' (t)I
C
is not
normal.
Recall
that a family
saturated
if
members,
(i)
(ii)
members
and
it contains (iii)
the union of each Let
~
the
]~[
elements
following
~
~I
the closed
subfamily
are as
'small' ensures
:
S ~ ~ ]
E
convex
C
C
circled
subsets
is
is
of each of its
of
is a s u b f a m i l y
as the cone that
of
of each of its
hull of
(see [4], p. 81).
family of b o u n d e d
~ IS[
subsets
subsets
all scalar m u l t i p l e s
:=
property
of b o u n d e d
arbitrary
it contains
finite
be a s a t u r a t e d
Obviously, whose
~
it contains
'small'.
is not too
E of Hence
'pointed'.
o
15
(2.3)
DEFINITION:
C
is a strict
f u n d a m e n t a l subfamily of
Remarks: E'
i. If
~'
~
~-cone
if
]~;
is a
5)
is a family of
~ ( E ' , E ) - b o u n d e d subsets of
, an a n a l o g o u s d e f i n i t i o n applies to the dual cone 2. If
i.e.,
E =
~S:
S &~-i
, every strict
4. Let ball
C
is generating,
'~
-decomposition'.
be a cone in a normed vector space w i t h closed unit
U . Take for E . Then
C
such that
U ~
that each
z ~E
and
~-cone
E = C - C .
3. J a m e s o n [2], p. 98, uses the term
of
C'.
~
the family
is a strict oL~U[
~
~
of all b o u n d e d subsets
-cone
iff
(or equivalently:
can be w r i t t e n as
there is
there is
1 -~ ~ E
1 ~ o( ~ fR
z = x - y , where
such
x,yE C
l~x~ + llYlf ~ ~llzII) •
Examples: lattice
E
subsets of
1. The p o s i t i v e cone in a locally convex vector is a strict E )
~-cone
(for
~
the family of all b o u n d e d
(see [4~, V.7.2).
2. Any closed and g e n e r a t i n g cone in an o r d e r e d Banach space is a strict
~-cone
(see
(3.3)
, or
[4~, V.3.5,
corollary).
3. F u r t h e r examples m a y be d e d u c e d from the subsequent d u a l i t y theorems.
We are now able to formulate and prove S c h a e f e r ' s d u a l i t y t h e o r e m for normal and strict spaces
(see [4~, V.3.3).
s a t u r a t e d family ~(E',E)-bounded S° :
t o p o l o g y on
on E
<E,E'>
saturated family
is a E'
-cones in locally convex
of bounded 6) subsets c o v e r i n g
subsets c o v e r i n g
S £ ~ i
Y-topology
duality
~
~
To this end we recall that for every (rasp. of
O - n e i g h b o r h o o d base for the so-called
(rasp.
(rasp. on
E ). In particular: A locally convex E' ) is c o n s i s t e n t w i t h the given
if and only if it is the E'
compact subsets of
E'
covering
compact subsets of
E ). For details and further results see ~ ,
(rasp.
, of
~ - t o p o l o g y for a
, covering
III.3
~
E
E' ) , the family of polars
~(E',E)-relatively E , of
C(E,E')-relatively
, IV.I.5 and IV.3 .
5) This means that each m e m b e r of m e m b e r of ] ~ [ . 6) i.e. ~(E,E')-bounded
~
is c o n t a i n e d in some
(see [4~, IV.3.2,
c o r o l l a r y 2)
16
(2.4) THEOREM: a cone in (i) Let
E
~
Let
with dual cone
subsets
covering
is a strict
on
E . ~
E'
in
E'
space and let
C
be
.
~(E',E)-relatively
compact
is normal
~-topology
:
~ -cone
covering
is a strict
C'
family of
be a saturated
subsets on
be ~ locally convex
be a saturated
C' (ii)Let
E
iff
C
family of
for the
~-(E,E')-relatively
compact
E : ~
-cone
iff
C' is normal
for the
~-topology
Ew .
Remark:
For greater
in its greatest
symmetry,
possible
we did not state the above theorem
generality.
See
[4~, V.3 for additional
results. Proof: cations of
By using the rules
are proved easily.
(i) - (4) of section
1 , all impli-
As an example we give an explicit
proof
(ii) :
' ~
' : Choose
logy on
E'
a
O-neighborhood
. We have to show,
[ SI° ~ C S °
is
S ~ ]1/4 SI[ = A f o r t i o r i , we have
co
by
S° ~ (i)
by
(2)
D
4(SI~ ~
-
S1 ~ ~
such that the
S ~ ~
S c
~ -topology
c
Applying
s °°
rule
~
on
[Sl°]°
=
SI~7))
such that .
~s1° ]
we have to show,
]Sl[
(4) and
and convex,
-(1/4
(S I N (-7))°)
(compare E'
that there exists
(2.3)).
If
C'
=
[Sl° ~ ~
(3) yields
convex,
S°
((Sl° + c') n
(Sl°
Sle
is normal
, there is a circled,
compact SI~ ~ such that Taking polars we get s
, circled
Si~7)°
2((SI~ ~)° N
' : For
such that
.
D (Sl° - c') n (sl° + c') ' ~
S1 6 ~
~-topo-
(2.2)).
((1/4 S I ~ ) u
S ~ 1/4 (SI• 7 - SI~7) By taking polars we get
, for the
that there is a
(compare definition
By assumption,there
S° , S 6 ~
-
c')) °
for
~qE,E')-
17
S
Since
~
c--o ((SI ° + C') ° U
c
c-~ ( (Sl°°f3 (-C)°°) k2 (Sl°°t~ C °O ) )
S1 °° = S 1
S
c
and
(SI ° - C') °)
C °o = ~
c-~ (-(Slt3 ~) v
we get finally
(SI(~))
=
w h e r e the last e q u a l i t y holds since
3.
co
((SIr% ~) ~
SI~ ~
is
-(SIr~))
,
~(E,E')-compact.
N u m e r i c a l dualit~ theorems for cones in normed vector spaces
The previous duality t h e o r e m can be c o n s i d e r a b l y s t r e n g t h e n e d for normal and strict
~
-cones in normed vector spaces
family of all b o u n d e d subsets).
symmetry b e t w e e n normal and strict ~ E
E B'
and
( ~
the
Not only will we obtain complete -cones in the normed spaces
(notice that the norm topology
n e c e s s a r i l y c o n s i s t e n t w i t h the duality
B(E',E)
<E,E'>
on
E'
is not
, hence t h e o r e m
(2.4) does not apply), but we also can introduce a n u m e r i c a l constant w h i c h m e a s u r e s the n o r m a l i t y resp. the strictness of the cone,
and w h i c h is p r e s e r v e d under duality.
already in
(2.2), remark 3, and in
these properties,
using the t e r m i n o l o g y of J a m e s o n ~ ] ,
(3.1) DEFINITION: unit ball
U
Let
and let
C
(i)
C
is ~ - n o r m a l
(ii)
C
i_~s ~ - ~ e n e r a t i n ~
Remarks: 2. C some
This constant a p p e a r e d
(2.3), remark 4. We restate
E
be a normed vector space with closed
be a cone in
if
[U~ ~ if
i. C is normal
is a strict
~-cone
iff
U
E .
oqU ~
C is iff
for
~ ~U[
~6~ for
R-normal
C is
. ~ ~
for some
~ £ ~ .
o<-generating for
~ 6 R .
Examples:
1. The p o s i t i v e cone in an order unit space is
1-normal and 2-generating. 2. ~he p o s i t i v e cone in a base norm space is 2-normal and 1-generating. 3. The p o s i t i v e cone in a B a n a c h lattice is 2-normal and 2-generating.
3.6 .
18
(3.2) a cone (i)
THEOREM:
in C
(ii)
E
. Let
is
E'
iff
E
is a B a n a c h
is
o~-normal
In fact,
we have
(5) a n d
With
(5)
(ii)
Similarly
most
(U°+C ')
The
~
contained
in
contained
Let
x
in the
(AuB)
C
co
of
(3.3)
b2
(AvB)
that
w
this
(a)
E = C - C
C
~
-(C'~
U°))
space,
bI of
and
= let
interior
~u[ A, B
of
to
. be closed,
co
point
of
co
(A v B )
Let
(AVB)
:= 2x - a I = x + co
( A v B)
this
in the
( A ~ B)
is
. Therefore
procedure
interior
of
A
and
.There
exists
(x - a I) is
for co
we h a v e
bI
gives
(Av~B) x =
~
an
such
are b o u n d ~
and c o m p l e t e ,
.
E
be a B a n a c h
E' = C'
space
with
closed
cone
- C'
(b)
C
is a normal,
strict
q-cone.
(C)
C'
is a n o r m a l ,
strict
~
(d)
C
is
and
Z-normal
that
2-n an
equivalent: and
.
if
is e q u i v a l e n t
-(unc))
. The
. Since
COROLLARY: are
(ii)
.
. Repeating
x ~ co
following
using in
if and o n l y
B
The
[ > o.
(6):
where
co
for all
implication
(U-C)
we get
6 n we conclude
b__ee
-(u~c))
E
that
and
~
x = 1/2 a I + 1/4 a 2 + 1/4 b 2 . F i n a l l y a
C
computation
( ( C ' ~ U O) ~
~,U °
~
implies
interior
a2 ~
for o n e
(U+C)
be an i n t e r i o r
such
(a I + b l)
let
is closed:
is an e a s y
the aid of
be a B a n a c h
( A v B)
x = i/2 co
that
c
lemma
subsets
(A~B)
C
(~+£)-~eneratin@
i. O n l y
co ( ( u n c )
E
co
if
i__ss
ot-Iu O
c
Let
Proof: co
if
(U°-C ')
bounded
and
e x t r a work:
following
LEMMA:
convex,
space
topology.
of the p r o o f
c-~ ( ( c ~ u )
(~+~)-i u
and
C
we have with
~
norm
vector
i__ss ~ - ~ e n e r a t i n ~ .
space iff
it is c l e a r
if and o n l y
~-lu
the C'
(6) of s e c t i o n
to do some
(i)
be a n o r m e d
have
~-normal
If
rules
aI £
E
C'
Proof: the
Let
-conein
~-generating
E Bi
I
for some
~,
B
~
1 .
C
.
19
Proof: after and
The
(3.1),
equivalence
a n d the
of
above
(b) a n d
theorem
(d)
follows
implies
the
f r o m the
remarks
equivalence
of
(d)
(c).
'(d)
~--~
ing
in
(a)':
C
is
B-generating
EB' . A f o r t i o r i ,
'(a)
=~
(b)':
that
E =
borhood. some
M
~/n~
nM
By the
C
:=
and
]U~ . E
above
in a B a n a c h
s h o w that,
under
E B' , h e n c e
in
4.
ordered
assumptions
C is n o r m a l
Regularly
In t h i s
the
ordered
section
we
by a c l o s e d
(4.1)
positive
cone
C . If
attain
lattice,
such
~)xll -~ m a x If
For
This
E
that
C
E
1-normal,
C
y
:=
y
=
that
and ,
we get
the
( ~Ixl()-i (x + - x-)
(llxII)-l(liX+il +
lJx-ll) is not
that
C
=
is
for
one
can ~-cone
C
~-normal of
~
normed
S
S
cannot
vector
~-normal E
~+S
is a B a n a c h
exists
llx(~ =
yields
the
o~+B > 2
IIx-fl
space
and
.
Then
, and
=
and
and
and
x 6 E
lJXllI +
~x2[(= 1
contradiction
.
is o ( - n o r m a l ,
~x+II
space
1 ).
is
for
_x2 _z x L Xl
x -~ x +
shows
is
that
1-generating.
lattice
-x- -~ -i (lixll) x
O-neigh.
generating
vector
small
in a d d i t i o n , bound
(llXltI , fix21()< 1 . T h e r e f o r e , is a B a n a c h
such
C
out
cone
lower
is
such
~U[
is a s t r i c t
be an o r d e r e d
> 2 . If,
best
~
Similarly,
how
(which
x = xI - x2 , o ~ xI , x2 6 C
is a l s o
x ~E
~+S
is the
Assume
that c
then
2~
Proof:
It t u r n s minimum
Let
is a [U
a closed
is a n o r m e d
to know,
2 . If its p o s i t i v e
~-generating,
If
their
M
and
spaces
E
PROPOSITION:
dim E ~
vector
that
simultaneously.
cones.
that
C'
assume
can be c h o s e n
with
(a),
~-generat-
E
normed
it is i n t e r e s t i n g
is
circled
hence
again
Z -cone.
of
in
S-generating,
simultaneously
space,
see t h a t
is a s t r i c t
C'
convex,
it f o l l o w s we
, and
are g e n e r a t i n g
is a B a i r e
lemma,
space
C'
E
is closed,
£ > o . As a c o n s e q u e n c e ,
cone
in
=
there 1
and
exists ~IxLl~_o~ .
decomposition
and h e n c e 2
(~Jxll)-I
So-generating
>/
2/o<. for
B ° < 2/~.
and h e n c e
20
+
B
for
~
~
~ +
=
2/~
~
. The m i n i m u m
. The
example
in
of the (2.1)
right
shows
side
that
is a t t a i n e d
this
bound
can be
attained.
The
above
proposition
the
search
The
following
(4.2) with
for a m o r e
symmetric
results
unit
property
can be f o u n d
DEFINITION:
closed
(and the e x a m p l e
Let
ball
U
E
in
in
(2.1))
for c o n e s
~i~
or
be an o r d e r e d
motivates in B a n a c h
~
, p.l15
normed
vector
spaces
.
space
. i
(i)
(u)
(ii)
E
:=
is r e g u l a r l y
(U)
~
Remarks: Every
3.
E
If
( use
E
°n
C
If
P = ~
(U) O
implies
,
U
=
and
-x-~ y-~ x
llx[~ < 1 , t h e r e
space with
(U °)
exists
y~ E
closed
to the
cone
closure
C of
, (U)
.
%
: y
x
for all
P := i z £ C : x'
p(z)
= O
: the
existence
x ~ U ~C > 1
first
of
x' £ E ' y£U
~
for all
y£P.
1
p ( x + y) x,y & C
,
>~
p(x)
A ~IR+
.
y'~
separation
for all
E
. Since
p
is s u p e r l i n e a r
is convex.
and w e get
-~ 1
i.e.
,
is e q u a l
all
ordered.
that
& 1
P M ~
the
.
x,y ~ E
x@E
.
(tJx uC x )°
, take
now
iff
define
p(x)
7)
[U~
Banach
iff
:= sup{
Certainly,
C-
£ >o
(3.2)).
(U)°
p(x)
k_/x,~uO~-X',X' ~
IIyI~ < 1 .
oOCX
y' ~
for all
is r e g u l a r l y
for
and
It is c l e a r
t/x
7)
, and
:=
if
ordered
ordered
LEMMA:
Proof:
Assume
(U)
is an o r d e r e d
lemma
(4.3)
For
C
lattice
-y-~ x ~ y
is r e g u l a r l y
(u°)
(i+~) (U)
]U[
l~Yt( -~ llxll
that
4.
~
Banach
;
ordered
is r e g u l a r l y
implies
E
U
I.
2.
such
q/x~u[-x,x~
[-x',x'~ theorem
separating
U
. ([4~,
and
II.9.1) P
, i.e.
, and
+ p(y)
and
p(Ax)
= Ap(x)
for
21
F r o m the f i r s t i n e q u a l i t y second inequality p(y) If
~
p(y)
hence
if
= o , take
n
follows
x ' ~ U ° . Take
y 6 C . The
implies p(y)
z &P
+
~ o .
. Then
~ 1
(ny + z ) ~ P
for all
n E ~
,
~ o
and
F r o m b o t h c a s e s we c o n c l u d e Z hence
& P(Y) ~ < y , x ' >
y' ~
The
~x',x'~
following
C
theorem
, which
expresses
(U)
and
(U °)
(see
~,
3.6.7 a n d 3.6.8)
easily
and e x p r e s s
Banach
spaces.
(U)
(ii)
(U°) ~
~
Proof: (ii)
E
o(U °
iff
U
c
From
U
(U) O
=
' •
_t
Since
Again 2-n x n
6
(~+£) (U)
in
(U)
~
U
Proof: o r d e r e d.
If
a new,
U
closed
.
~( (UO) (~+£) (U)
for all
£ > o .
lemma. w e get by t a k i n g p o l a r s
(4+£) (U) (U °)
. Because
~
(U O)
uOO (U) (3.2)
=
this
is v a l i d
for
.
U .
for any s e q u e n c e
~Xn~
that the i n t e r i o r
of
C- (U)
,
(U)
~ > o .
Every ordered
Banach
cone and g e n e r a t i n g
lent n o r m u n d e r w h i c h
defines
space w i t h
~
for all
positive
follow
ordered
. Afortiori,
(4.5) C O R O L L A R Y : generating
corollaries
t a k i n g p o l a r s w e get
~
one c an s h o w as in l e m m a is c o n t a i n e d
of
~U°[
by the a b o v e
c~U O
c<(u °)° ~
and
of r e g u l a r l y
U . Let ~ g R c
behaviour
~
be an o r d e r e d B a n a c h unit b a l l UO
~
' ~>
subsequent
iff
o , we c o n c l u d e
(U)
. The
the symmetric to t h a t of
~ U
' :
(0(+~)U° ~>
Let
(i) is t r i v i a l
'~
all
is s i m i l a r
c o n e and c l o s e d
(i)
y~ C ,
the m a i n p r o p e r t i e s
(4.4) T H E O R E M : positive
for all (U O)
E
is the
space
E
with closed
d u al c o n e has an e q u i v a -
is r e g u l a r l y o r d e r e d .
(original)
but e q u i v a l e n t
norm,
u n i t ball,
then
for w h i c h
E
(U)
is r e g u l a r l y
22
(4.6) COROLLARY: iff
An ordered Banach space is regularly ordered
its dual is regularly ordered.
references: Davis, E.B.:
The structure and ideal theory of the predual
of a Banach lattice.
Trans. Amer. Math. Soc.
13!, 544-555
(1968). Jameson, Math.
G.:
141, Berlin-Heidelberg-New
Ng, Kung-Fu: Scand.
Ordered Linear Spaces.
On a computation
26, 14-16
Schaefer,
H.H.:
Lecture Notes in
York: Springer
197o.
rule for polars. Math.
(197o). Topological
Berlin-Heidelberg-New
Vector Spaces,
York: Springer
1971.
3 rd print.
ORDER
UNIT AND BASE NORM
Rainer
Fachbereich
J. N a g e l
Mathematik
der Universit~t
T~bingen,
The dual behavior senatation and base
theorems
norm
spaces
vector-lattice
case,
In s e c t i o n
1 we
of
(AL)-spaces
known
represent
every
space
A(K)
. The dual
of a n o r d e r
basic
properties
are
theory
in s e c t i o n We use be 3.9
found
of o r d e r
stated
and the Kakutani
see:
[6], V.8)
generalizations
see that
whose
duality
(e.g.
are the natural and we will
T~bingen
Germany
(AM)- a n d
are w e l l
SPACES
order unit
similar
space
in s e c t i o n
unit and base
norm
space
is a b a s e
will
are valid. as a
norm
2 . Finally,
spaces
unit
to the non-
results
unit Banach
repre-
. Order
the
space, complete
be developed
3 . the terminology
in
A 1 f s e n
of
S c h a e f e r
[i~,
II.l
or
[6];
the results
J a m e s o n
[2],
can
3.7 -
.
i.
Let having
l)
Order
E
unit
be an ordered
an o r d e r
u
space§
unit
is a n o r d e r
vector
u I).
unit
If
if
space E
(over R)
with
is A r c h i m e d e a n ,
E
=
l ! k ~ / n [-u,u~
positive
cone E+
24
is a norm on
E , for w h i c h the unit ball
p o s i t i v e cone
(1.1)
are closed
DEFINITION:
order unit space)
E+
u
(see
U
~],
=
~u,u]
3.7.2)
and the
.
An A r c h i m e d e a n o r d e r e d vector space
is called an order unit space
E with
(order unit B a n a c h
if it is endowed w i t h the order unit norm
Pu
(and is com-
plete). Remarks: normal,
i. The p o s i t i v e ~cone
E+
2 - g e n e r a t i n g and regular
in an order unit space is
(see
2. An o r d e r e d Banach space is i s o m o r p h i c vector space) is normal,
closed and has interior points
Examples:
I. Let
~
be a
ordered by the p o s i t i v e cone order unit Banach space 2. Let
K
(as ordered t o p o l o g i c a l
to an order unit Banach space
~+
iff
the p o s i t i v e cone
([6], V.6.2,
C~-algebra
subspace of all s e l f - a d j o i n t elements
l-
[3] for the definitions).
c o r o l l a r y i).
(with algebraic unit).
~s :=
1=
The
Ix6(~ : x = x~
~x 6Gt :
x = yy'~
is an
(see [5]).
be a compact convex subset of a locally convex vector
space. The space
A(K)
of all continuous affine functions on
K
is
an order unit B a n a c h space for the natural positive cone and the sup-norm. We will see in
(1.3)
can be r e p r e s e n t e d as an
A(K)
3. Let
E
be an a r b i t r a r y Banach space. Take
geneous h y p e r p l a n e in E+
:=
~(x,A) : ~ix|i ~
ding norm E
, that every order unit B a n a c h space
Pu
E
and w r i t e and set
E
= F ~R u
=
F
a closed homo-
. Define a cone
(o,i)
.
is a new, but e q u i v a l e n t norm on
The c o r r e s p o n -
E , w h i c h makes
an order unit Banach space. F
/ //~ , ~ ~1~
all Banach spaces.
E+
I i j i
,
I
This example shows that the class
J
i
l
>
I
f order unit spaces
"contains"
25
(1.2)
DEFINITION: K
Let :=
is c a l l e d the s t a t e
Remarks: unit b a l l 2.
1.
a unique
x 6E
The s e c o n d
~u,x'>
The set
= 1
E .
~(E',E)-compact
U°
A6 R+
is p o s i t i v e condition
!) . S i n c e
4.
is a
an__~d
is a b a s e of the d u a l c o n e
there exists
base
s p a c e of
K
be an o r d e r unit space.
convex
s u b s e t of the d u a l
U° .
K
3.
(E,u,p u)
~ x ' ~ E': o ~ x'
=
iff
:
(see £3~,
~ o
3.2)
K . x'~ K :
for all
x ' e E+ ~' (K
xgE+
=
UO
=
is a
(-E$) O
from the fact that i.e.
,
o ~ x ' g E+
for all
implies
follows E+'
cone
!
for e a c h
Ax'g
> o
this
This
1-generating
%2-(U°~E$))
<x,x'> <x,x'>
is closed,
co(K~-K)
b a s e of the
such that
implies
E+
E+' , i.e.
K
co((U°~
is a
E$) U
. %
5.
UxI{
immediate 6.
=
sup~<x,x'>~
from
E+'
(for the n o r m t o p o l o g y
THEOREM:
der i s o m o r p h i c on the s t a t e
Proof: defines
is r e g u l a r
on
E'
to the s p a c e K
of
a continuous
A(K)
affine
s~ace
>
<x,x'>
~(E',E)-closed
on
!
to
x ' ~ E'
that
~
f o r m on
E'
a
=
. Decompose ,
a subnet
converge.
But =
o
a
functions
on
, i.e.
E'
,
x'~ K 3
and
extended
to a
to an e l e m e n t
of
a6A(K)
. For the
to
of
E .
E'
is
~(E',E)-continuity >
theorem
5 to
is
(~6], IV.6.4)
and
E ):
can c h o o s e
.
and it r e m a i n s
c a n be l i n e a r l y
form
Uo
£ [o,i]
affine
x eE
a & A(K)
(use the K r e i n - S m u l i a n
~
[3]
is n o r m and or-
K . By r e m a r k
and n o r m i s o m o r p h i s m
linear
of
from
E
for
t h a t the l i n e a r e x t e n s i o n
linear
the c o m p l e t n e s s
Jx~ ~
is
1-generating
follows
it suffices to show that
Let
: This
and
of all c o n t i n u o u s
function
is an o r d e r
~(E',E)-continuous
of
) . This a g a i n
x
, that every element
a well defined
x&E
E .
The evaluation
First observe
, 2-normal
Every order unit Banach
space
this evaluation show
for all
4 .
The d u a l cone
(1.3)
: X'eK~
co(K~-K) x~
X~l
,
~x'~
= x'~2
~
~X'~l ~
K
such that
is c o n t i n u o u s
for all
be a net c o n v e r g i n g
on
implies
K
-
(i - ~ ) x ' ~ 2
. Since ~ ,
K
such
is c o m p a c t ,
~X'~l I
and l i n e a r on
=
o
weakly
and
we Ix'~21
UO . Hence
, qed.
26
Remark: of
Let
be the
K . The K r e i n - M i l m a n
order
isomorphic
C(X)
iff
see
[6],
in this
2.
E
Base
is a v e c t o r or the
Let
norm
above
normed
E
for some
that
strictly B
is l i n e a r l y
(2.1)
has
~x~E+
to
de
R u g y
on
E
a base
Ix ~E+:
K
:=
:
K
with
class
generating
positive
cone
= l}
form 2)
~~ +
An ordered
If
E
f
on
E
.
If
:
x ~ ~B
vector
spac e
B = co(Kv-K)
is a b a s e
norm
E+
f ~E$
is c l o s e d such t h a t
positive
a norm
we h a v e
the n o r m
cone
for all
of
in a b a s e ~ >I
E norm
PB
cone on
K
=
3.8.3). and
is a d d i t i v e space
on
f(x) E+
is r e g u l a r
~Ix~
=
. ,
2-normal
(see [3])
f
is s t r i c t l y
positive
if
B
is l i n e a r l y
bounded
(linearly
a n d closed)
(/2~, ~fl( = 1
3)
.
defines
space,
2)
E
whose
.
, or:
(bounded
E
space.
iff
exists
M-generating
that
norm
,,x~ = i~
x 6 E+
of
(as
a new
, i.e.
f(x)
linear
inf
such
The positive
bounded
space
.
is c l o s e d
There
for all
unit
to d e f i n e
b o u n d e d 3) , t h e n
I.
origin
and
co(K ~-K)
DEFINITION:
K
of an o r d e r
space
a base
positive :=
Remarks:
and
norm
is e q u a l
G o u 1 1 e t
are n o w u s e d
vector
=
a base
4.
is also
, which
spaces:
E+
is c a l l e d
3.
A.
of t h e d u a l
vector
PB(X) is a n o r m
2.
E
C(X)
points
(K a k u t a n i's theorem;
of
remarks)
be an o r d e r e d
K
=
lattice
that
of
extreme
spaces
in the
. Assume
has
implies
subspace
lecture
of the p r o p e r t i e s
of o r d e r e d
E
theorem
of the
volume).
Some
E+
~(E',E)-closure
to a c l o s e d
V.8.5
collected
E+
X
f(x) >
segment
o
for all
compact)
for e v e r y
if line
o ~ x ~ E+ B nL
.
is a
L through
the
27
5.
If
equal to 6. on
B
is linearly compact 3)
B . In this case
If
K
E , then
Examples:
E+
is
, the closed unit ball
U
is
1-generating.
is compact for some locally c o n v e x H a u s d o r f f t o p o l o g y (E,PB)
i.
is a B a n a c h space
Every
(AL)-space
([i], II.l.12).
, hence every
LI(x,/~)
is a base
norm space. 2.
The dual of an o r d e r unit space is a base n o r m space by the
remarks in 1.2 . 3.
The s e l f - a d j o i n t linear forms on a
norm space. Moreover, predual) 4.
of a
C ~ - a l g e b r a form a base
the s e l f - a d j o i n t normal linear forms
(i.e. the
W ~ - a l g e b r a form a base n o r m space.
On every Banach space one can define an o r d e r i n g and an
e q u i v a l e n t norm under w h i c h it becomes a base norm space
(compare
i.I, e x a m p l e 3).
(2.2) pROPOSITION:
The dual of a base norm space is an order unit
B a n a c h space.
Proof: which
K
We have only to show that the linear form =
{x&E+
:
f(x) = i~ ,
f g E +'
is an order unit in
that it d e t e r m i n e s the dual n o r m on
E'
E'
for and
. This is clear from the
f o l l o w i n g computation:
llx'~
3.
=
sup~l<x,x'>l
:
xeB~
=
sup ~ l<x,x'>[
:
x ~ K~
=
i n f { ~E ~+
-A
&<x,x'>
=
inf~Aea+ :
-A<x,f>
=
inf I l e ~+
x' e A ~ f , f ]
=
pf(x')
:
:
~
~
~ ~x,x'>~
for all
A<x,f>
for all
x6~
.
D u a l i t y O f order unit and base norm spaces
(3.1)
THEOREM:
Let
E
be an ordered B a n a c h space w i t h closed
a n d g e n e r a t i n g p o s i t i v e con_~e. (i)
E
is an order unit space
iff
E'
is a base norm space w i t h
~ ( E ' , E ) - c g m p a g t base. (ii)
E
is a base norm s p a c e
i ff
E'
is an order unit spage.
28
Pro0f: and
in (i)
f
Two
(2.2)
. It r e m a i n s
"~"
:
be the
=
Ix'6
- K)
the unit
"~=="
:
is
UO
=
defines
K
the n o r m
= of
is a b a s e
norm
Remark: duality
The
proof,
in
Let
for
unit
[3~,
=
3.2
Hence .
to
in
to f
As ~f,f]
E'
is
in
closed
co(Kv-K)
space
this
E
norm
in
we
and
retrieve
(AL)-spaces
(i) no
, while
a theorem
the d u a l
cone
E+
functional
unit
ball
cone
E+
sp@ce
V.8.4).
if one d r o p s
dual
is not
of
co
an o r d e r
K u n g - F u
unit
N g
(ii)
each
is an o r d e r order
the
is
e1 space.
[4]
,
Banach
space
with
closed
and
:
iff
E+
i_~s 1 - n o r m a l
and the
open
upwards.
norm
is it true,
E
E
situation.
be an o r d e r e d
be a b a s e
(i)
E
classical
Problems
assumptions
of
of
consequently,
the
holds
The
co
of
(~6],
longer
~(E',E)-compact:
is d i r e c t e d
E
, one
.
, the positive
the
~E',E)
(2.2)
, so t h a t
the M i n k o w s k i
U
lattice,
(AM)- a n d
be
p0sitive
E
is e q u a l
,
B
a vector
we g i v e
Let
is a b a s e
let
I
norm
THEOREM: ~eneratiDg
and
Since
, we get
K
is a b a s e
4.
.
E+'
is e q u a l
f 6E+
~ > 1 . Hence,
equivalence
that
with
ball
E
E
deals
E'
on
and
E
(-E+~U))
of
. = o I
order
By
, remarks,
space.
theorems
(3.2)
which
in
[-f,f~.
U) V
E+
For
assumption
Without
U
base
~(E',E)-closed.
be the
(1.2)
following:
}
IV.6.4)
:
is a b a s e
which
f(x')
it is
in
for w h i c h
= 1
and
for all
co((E+~
E'
: f(x')
f
=
on
[63,
proved
~E',E)-compact
1
ball
Let
~-generating B
,
(again by
that
ball
is
llX'll &
already
to s h o w the
be the
t x'EE+ '
I/2(K
unit
K
were
functional
E':
continuous
(ii)
Let
linear
K Since
shows
implications
Banach
space.
Under
which
additional
that ideal
in
E''
intervall
in
E
? is w e a k l y
compact
?
uni X
,
29 (iii)
E
is the order ideal of all order continuous linear forms
on some order complete order unit space ? While none of these properties holds in the general case (use 2.1, example 4), they are true if if
E' is isomorphic
part of a
E
is a vector lattice
([6], V.8)
or
(as an ordered Banach space) to the self-adjoint
C*-algebra.
references
[l~
Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Berlin-Heidelberg-New York: Springer 1971.
[2]
Jameson, G.: Ordered Linear Spaces. Berlin-Heidelberg-New York: Springer 197o.
[3]
Mangold, J. - Nagel, R.J.: Duality of cones in locally convex spaces. See the previous article.
[4]
Ng, Kung-Fu: The duality of partially ordered Banach spaces. Proc. London Math. Soc. 19, 269-288 (1969).
[5~
Sakai, S.: CW-algebras and W~-algebras. New York: Springer 1972.
[6]
Schaefer, H.H.: Topological Vector Spaces, Heidelberg-New York: Springer 1971.
Berlin-Heidelberg3 rd print. Berlin-
MINIMAL DECOMPOSITIONS
IN BASZ NORMED SPACES
A.J. Ellis Department
Let cone
of Pure Mathematics,
E
If
x
College
of Swansea,
Wales
be a base normed Banach space with a closed positive
E + and base
a positive
University
K.
Therefore,
decomposition
each
x = y - z
has such a decomposition
x~
E
has, for each
such that
for
~ = 0
~ ~ 0,
~y[~ + ~z[~ ~ (I +E)I~xU.
then that decomposition
is called minimal. If
E
then if
is a vector lattice,
x = y - z,
y, z ~ 0 ,
i.e. if
K
we have
I[Yll + llzll ~ ~[x+ll + llx-ll = llxll.
is a Choquet
y ~ x +, z ~ x-,
In this case each
x
simplex, so that
has a minimal
decomposition. Example
(L. Asimow).
and
x E E
let
E
such that
x
does not have a minimal
have the base norm induced by
I
E=
There exists a base normed Banach
eCo:Xo
I
,
] , =__ x
the base norm is equivalent for all
decomposition:
,
Xo
E (1+E)co(K U -K)
E
where
x I +x 2
Xi
this example
K
space
g~
(0,I~,-~, 0,0, ...).
to the Co-nOrm
0 , but
E
In
and
does not belong to
co(K U-K). For each strictly obtain a base base norm, x
f
on
Kf = i x ~ E+: f(x) = I]
the Minkowski
an element which
positive
x ~ E
functional
we consider
has a minimal
of
E , for
i.e. E+,
f ~ (E:) °,
and an equivalent
co(Kf v -Kf)
the problem
we
of finding
for
E.
Given
a base norm for
decomposition.
Since if x = y - z and f ~ E we have f(y) + f(z) = 2 f ( y ) - f ( x ) , it is easy to show that x = y - z is a minimal decomposition of x relative f(y)
to
f ~ (El) °
= infif(u):
u@
E+~
if and only if (E++x)]
.
y @ E+~
(E+ + x)
and
It is clear then that any
$I
f ~ (El)°
which is a support functional
rise to a minimal decomposition
of
for
x.
E+ ~
(E+ + x)
gives
We have (cf. [ 4]):
Theorem I. The strictly positive support functionals for E+ ~ (E+ + x) are dense in ( E ) o , and hence also in E*+. Proof. If f ~ (El)° then f is bounded below on E+ ~ (E+ ÷ x) and so, by a theorem of Bishop and Phelps [2], f can be approximated in the norm topology~ by support functionals for E+ ~ (E+ + x) . Since f ~ (E+) ° in (E+) °
it can be approximated by support functionals
lying
Although there are many support functionals for E+ ~ (E+ + x) there may be only one support point, in fact if x + exists then x + will be the unique support point for all f E (E+) °. Conversely, only finitely many support points of E+ ~ (E+ + x) exist ( E + )O)
(for
then
x+
if
exists.
Ne now consider the existence of unique minimal decompositions. An element x will have a unique minimal decomposition x = y - z relative to
f ~ (El)°
if and only if
f
supports
E+ ~
(E+ + x)
at the exposed point y. If E+ ~ (E+ + x) is locally norm-compact, e.g. if E is finite-dimensional, then a result of Klee [7] shows that that
E+ A y E
(E+ + x)
always has an exposed,point
must be supported by some
y;
it can be shown
f E (E+) °.
will be said to have the unique minimal decomposition property
(u.m.d.p.) if every x ~ E has a unique minimal decomposition relative to the given base K . This property of K can be interpreted [4] as an intersection property reminiscent of that defining Choquet simplexes:Theorem 2. E has the u.m.d.p, for K if and only if co(K u -K) is closed and, for each x ~ E, K A ( x + K ) is either empty, a singleton, or contains a set of the form y + k K , for some y ~ E
k>o. R3
The AL-spaces have the u.m.d.p, for all bases, but then so does with a circular cone. An important example of a space E which
possesses
the u.m.d.p,
for the given base is the space of Hermitian
functionals on a B -algebra A with identity; here K is just the usual state-space for A, and E = limK. This property of l i n k does not distinguish B -algebras amongst unital complex Banach algebras. In fact, if A is a Dirichlet algebra on a compact Hausdorff space ~ then the state-space K is precisely the probability Radon measures on.0_, so that l i n k is an AL-sp~ce.
32
Let let
A
be a complex unital Banach algebra with identity
K = [#@
A* : #(e) = I =~ll~
be the state-space of
e
A.
and
Then,
using a result of Bohnenblust and Karlin (cf.[3]) and a construction of AsimowL1~, the map 0 : A ~ A ( Z ) = A(co(K V -iK)), ~a(z) = r e z ~ ) a ~ A, z @ Z, A(Z), on Z
is a topological real-linear isomorphism of
onto
the Banach space of all continuous real-valued affine functions with the supremum norm. The dual space of
base
A
Z.
A(Z)
is the base normed space
Using the Vidav-Palmer theorem (cf. [3 D
lin Z,
with
the following
characterization of B*-algebras can be obtained [5]. Theorem 3If A is a complex unital Banach algebra then B -algebra if and only if lin Z ~ A(Z) ) has the u.m.d.p.
A
is a
If lin Z is an AL-space then K is a simplex and it is well known that this implies that the B*-algebra A is commutative. A related result is the following [5~Theorem 4.
If
A
is a complex unital Banach algebra such that
is isometrically isomorphic to a complex L1-space then commutative B -algebra.
A
A*
is a
Theorem 4 was first proved, for the case of a function algebra, by Hirsberg and Lazar [6~. References 1.
L. ASINOW,
'Decomposable compact convex sets and peak sets for function spaces', Proc. Amer. Math. Soc. 25(I)(1970)
2.
75-9. E. BISHOP and R.R. PHELPS, 'The support functionals of a convex set', Proc. Symp. Pure Mathematics VII(Convexity),
3.
F.F. BONSALL and J. DUNCAN,
4.
normed spaces and of elements of normed algebras'~ Cambridge 1971. A.J. ELLIS, 'Minimal decompositions in partially ordered normed
5.
A.J. ELLIS,
Amer. Math. Soc. (1963), 27-35.
6. 7.
'Numerical ranges of operators on
vector spaces', Proc.Camb.Phil.Soc. 64(1968),989-1000 'Some applications of convexity theory to Banach algebras', (submitted for publication). B. HIRSBERG and A.J. LAZAR, 'Complex Lindenstrauss spaces with V.L. ELEE,
extreme points', (to appear). 'Extremal structure of convex sets II', Math. Z. 69(1958),
90-104.
SIMPLEX SPACES
Alain GOULLET de RUGY Equipe d'Analyse • Universit~
de Paris VI
O. INTRODUCTION. The aim of this lecture is to make a survey of the theory of integral representation
on compact convex sets and its connexions with the theory of ordered
Banach spaces. i
As I am short of time and as there now exists the book of ALFSEN covering matters,
I shall not give full proofs.
lying ideas of the theory. Discussion
these
I shall simply try to give the major underof references
is rel=ga~ed
to the last
section.
I. KREIN-MILMAN
THEOREM AND BAUER'S MAXIMUM PRINCIPLE.
1.0. NOTATIONS. By a compact convex set, I shall always mean a compact convex set of a Hausdorff locally convex real topological vector
space (HLCRTVS).
Let X be a compact convex set in some HLCRTVS E. Denote A (X) the space of affine C
continuous
functions on X and Qc(X)
(resp. u.s.c.)
(resp. Qs(X))
the space of convex continuous
functions on X.
An extreme point x ~ X
is, by definition,
a point such that (X v Ix})
is still
convex. Denote E(X) the set of extreme points of X. 1.1. THEOREM.-(BAUER's
maximum principle).
Let f in Qs(X). Then f attains its maximum on E(X). PROOF
(sketch of).
Define a stable set of X to be a non empty closed subset S of X such that : x, y ~ X ,
Vt~]0,1[,
tx + (l-t) y 6 S ~ x , y ~ S .
Prove that the family of stable sets of X is downwards minimal
inductive and that the
stable sets are just the one point sets ~x] with x in E(X) using the
following fact which relate stable sets and convex functions -
If S is a stable set, f ~ Q s ( X )
stable set.
and r = m a x , ( s )
:
: s ~ S], then S ~ F - I ( r )
is a
34
1.2. COROLLARY
(KREIN-MILMAN
Theorem).
Let A be a closed subset of X. Then, the following are equivalent
:
(a) X is equal to the closed convex hull c'-~6"~(A) of A ; (b) A D E ( X )
;
(c) A has the property
: ~f~A
(X) : f ~ O
on A
>f~O
on X ;
C
(d) A has the property
: Vf ~Qs(X)
: f attains its maximum on A.
PROOF. By 1 . 1 .
(b)"
~(d) and it is clear that (d)
a Hahn-Banach argument and (a)
~ (c). The assertion
(c)
~(a)
i,
~ (b) follows from :
1.3. LEMMA. In a compact convex set X each extreme point x is strongly extreme in the. sense that : For any neighbourhood slice [ f < r ~ X
V of x in X there exists f ~ E '
contains x and is contained
and r ~ R
such that the
in V.
PROOF. See H.H. SCHAEFER'lecture
: The Silov boundary of a cone.
2. LINKS BETWEEN COMPACT CONVEX SETS AND ORDERED SPACES. 2.0. DEFINITIONS. Let V be an order unit normed
space. A state of V is a linear functional
such that : x(V+)C R_+ and x(e) = l where e denotes
x on V
the order unit which define the
norm in V. Clearly one can replace the assumption x(e) = 1 by ~xU = I. In particular, every state of V is continuous.
The set X of all states of V is called the
state space of V. Then, X is a non-empty of the dual V' of V. For x ~ X and a ~ V
convex and ~ (V',V)-compaet
convex subset
let ~(x) = x(a). The map a ~ - ~ ,
the KADISON map. It carries V into A (X). The properties
is called
of this map are surmnarized
C
in the following NAGEL'lecture
theorem,
the proof of which reduces to the bipolar
theorem.
(See
: Order unit and base norm spaces).
2.1. THEOREM. Let V be an order unit normed space and let X be the state space of V. Then the map a ~ - ~
from V into A (X) is a bipositive
linear isometry of V onto a dense sub-
C
space of the order unit Banach space A (X) and ~(x) = 1 for all x m X, where e is C
the order unit of V. The map a ~-->~ is onto A (X) if an only if the normed space V C
is complete. If we apply Kadison map to a space A (X) for some compact convex set X, do we C
obtain the same space at the end? The answer is given by the following. 2.2. PROPOSITION. Let X be a non-empty compact convex set and for every x ~ X the evaluation map defined by e(x)f = f(x) for any f ~Ac(X).
let us denote by e(x) Then the map x~-~ e(x)
35
is an affine homeomorphism
of X onto the state space of A (X). c
2.3. COROLLARY. Let X be a non-empty compact convex set and @ ~ ( X ) .
Then, there exists a point
x in X, denoted r(8) and called the resultant or barycenter @(f) = f(r(8))
of 8 such that :
(Vf~Ac(X)).
PROOF. By restriction,
a probability measure defines a state of A (X). Conclusion c
follows from 2.2. 3. CHOQUET'S THEOREM OF INTEGRAL REPRESENTATION. 3.0. NOTATIONS. In the sequel X will denote a fixed compact cnnvex set. We shall identify X with the state space of A (X) (see 2.2.). e = [ y £ A c ( X ) ' ; Y~O t.
In particul~r,
X is the base of the cone
Consequently, to each 8~q~l,+(X) we can a s s o c i a t e a p o i n t x ~
denoted r ( 8 ) and
called the resultant of 8, such that : 8(f) = r(8)(f)
(Vf~A
c
(X)).
3.1. FINITE DIMENSION. Let X be of finite dimension.
We then have the famous CARATHEODORY's
which says that : For every x ~ X ,
there exist x],...,x n ~ E ( X )
theorem
and r 1,...,r n ~ ]O,1[
such that : ~" i
r. = | i
In other words,
and
x = ~ i
if d(y) denotes
r.x.. 11
the Dirac measure at the point y of X, x is the
resultant of the discrete measure
~_ i
rid(x i) which is concentrated
on E(X).
Does the same hold for general convexes with the word Radon probability measure instead of discrete measure?
The answer is very difficult
even in a as simple case
as when E(X) is denumbrable. 3.2. THEOREM. Suppose that X is a metrizable denumbrable
intersection
bability measure concentrated PROOF.(sketch
compact convex set. Then E(X) is a G~ set i.e. a
of open sets, and every x K X
is the barycenter
of a pro-
on E(X).
of).
First step. The key idea is to introduce a relation on ~%+(X) which says that a measure is "closer"
to the extreme points than another.
This is the following
:
36
8~@'"
.~@(f) 4@(f')
for any f£Qc(X).
As Qc(X) is total in c(x), this relation is an order and as Qc(X) f% Qc(X) = A c(X) two comparable measures have the same resultant in X. For each x ~ X, denotes by M
x
the set of those O ~ ~]%+l(x) with r(O) = x. One shows easely that : (i)
E(X) = [ x K X
(ii) The order
; M x = [d(x)}}
;
K is inductive.
Second step. Express that the maximal measures are in fact close to E(X). The key notion is the following : to each f ~ Q c ( X ) = inf [ g ~ - Q c ( X )
associate :
; g~f}.
The function ~ is concave and upper semi-continuous. And ~ = f on E(X). The main result of that step is : 3.3. LEMMA. A measure 0 is maximal if, and only if, O(f) = O(~) for any f
Qc(X). Consequent-
ly, O is maximal if, and only if, O is concentrated on each of the G~ B E = [~ = f]
sets
for any f GQc(X).
Third step. The last step is the remark that if X is metrizable, there exists a convex continuous function f such that Bf = E(X). It suffices to consider a sequence (fn) of affines functions total in A (X) s.t. c
0$f
$ 2 -n and to consider f = ~ f 2 n" n The third step proves that E(X) is a G~ , thus a Borel set. And, if x ~ X , consider n
a maximal measure O which majorizes d(x), then 8 has resultant x and is concentrated on E(X) by steps 2 and 3. 3.4. EXAMPLE. Let A be the closed unit disk and f a continuous function on A harmonic in the interior. The classical Poisson formula :
f(z) =
l 2 7r
JO 2 ~
f(cos(O), sin(e)) l-jzJ2 ~lelO-z J
de
can be interpreted in terms of maximal measures on a convenient compact convex set. It is the same for the Bernstein's theorem which says that if f is a completely monotone function on R, i.e. a C ~
function s.t. ( - l ) k f k ~ o for all k £ N ,
exists a Radon measure O on [ O , + ~ [ , positive, such that :
f (x) =
e-kXdo(k)
there
37
4. UNICITY. The problem we are interested in now is the question of the unicity of the maximal measure associated to a point of the compact convex set X. If we look at some X in R 2 we easily see that the only convex sets bearing this unicity property are the triangles. This unicity property has many different, at first surprising, expressions which I shall state in the following theorem. In the sequel E will denote an order unit space, X the state space of E and ~ the cone generated by X. We say that E is a simplex space if E satisfies the Riesz's interpolation property : YUl, u2, Vl, v 2 there exists a w g E
such that such that :
u i ~ v j (i,j = 1,2), ui~w~v
j (i,j = 1,2).
We shall say that X is a simplex if every x in X is the barycenter of a unique maximal measure. 4.1. EXAMPLE. Let ~ b e
any open set in R 2 with compact closure. Let A ( ~ )
continuous functions on ~
, harmonic in
~
, then A ( ~ )
be the space of
is a simplex space.
4.2. THEOREM. The following statements are equivalent : (a) X is a simplex ; (b) E is a simplex space ; (c) E' is lattice (when ordered by the positive cone ~) ; (d) Edward's separation property : For any f, -g in Qs(X) with f sg, there exists h~A
c
(X) with f ~ h 4 g .
If X is metrizable, these four statements are equivalent to the following : (e) The weak Dirichlet's problem : For every compact subset K of E(X) and every fGC(K)
there exists an ~ A
(X) (= E) such that : c
IK = f
and
l~f~IK = l[~n
PROOF. (a)
>(c).
By (ii) of 3.2, the set of maximal measures M(X) is a cone hereditary in
~+(X),
thus lattice for its own order. By (a), the resultant map is a linear b i j e c t i o n from M(X) onto ~. The latter is thus lattice for its own order and so E' is lattice. (c) T
>(b).
Take ul, u2, Vl, v 2 g E
(= Ac(X)) such that u i ~ v j
(i,j = 1,2), and consider
38
f = sup(ul,u 2) and g = inf(vl,v2). By the Riesz decomposition property, it is easy to see that ~ ¢ ~
(where ~ = -(-g)^). Thus, by Hahn-Banach, there exists w ~ A
(X) C
s.t. ~ 6 w $ ~ .
To replace
The same argument holds to prove (b) r (d)~
>(e).
~(a).
(Even in the non metrizable case).
Consider K and f as in (e). For sake of simplicity assume O ~ f $I. Define fl,f2 on X by the following conditions : f = fl = f2 on K ; fl = 1 = I-f2 elsewhere. We have f]' -f2 ~ Q s (X) and f2 $ f] thus, by (d) there exists h ~ A
(X) such that C
f2 S h ~fl" In particular, O ~ h $I and h = f on K. (e)
> (a). (In the metrizable case).
Suppose 0 and O' are two maximal probability measures with same barycenter. Sustracting 0 ^ 0 ' given
and normalizing we can suppose O and O' disjoint. Thus, for a
E > O, there exists two disjoint compact subsets of E(X), say K I and K 2 such
that : O(K l) ~ l - ~ and O'(K 2) $ g. Take f ~ A
(X) with 0 ~ f 6 1
f = ] on K| and
C
f = O on K 2. By the barycenter formula : O(f) = f(r(@))~ I-E
;
O'(f) = f(r(O')) ~ ~ , but f(r(O)) = f(r(O')), a contradiction if
I
E ~ ~.
5. BAUER SIMPLEXES. We are now going to characterize the lattice spaces among simplex spaces. 5.1. THEOREM. The following are equivalent : (a) X is a Bauer simplex i.e. a simplex with E(X) closed ; (b) Solution of the Dirichlet's problem : Any bounded continuous function on E(X) extends to an affine continuous function on X ; (e) E is lattice ; (d) There exists a compact space T such that X is affinely homeomorphic to the compact convex set
~(T)
;
(e) There exists a compact space T and a bipositive linear isometry from E onto
C (T). PROOF. Note that (e) (a)
~,(d) as (d) is the dual statement of (e).
~(b).
comes form 4.2.(e).
(b)=
> (c).
D e n o t e Cb(E(X)) t h e s p a c e o f bounded c o n t i n u o u s f u n c t i o n s
on E(X). By B a u e r ' s
39
maximum principle,
the map f ~-P f~E(X) is a linear bipositive
onto Cb(E(X))._ _ __ As the latter is lattice,
(c)
isometry form At(X)
so is At(X) and E.
~ (a).
If E is lattice, E satisfies 4.2. Furthermore,
one has : : x(avb)
E(X) = [xEX
the Riesz interpolation property and X is a simplex by
= max(x(a),x(b)),
(¥a,b~ E)}
so, E(X) is closed. (a) and (b) clearly give (e) with T = E(X) and (d) ~(T)
~ (a) comes from the fact that
is a Bauer simplex.
6. FUNCTION SPACES. It happens very often that an order unit space is given under the form of a function space
: A function space on a compact space T is a closed separating
space F of C(T) containing - A ~ilov
the constants.
The following notions
sub-
are considered
:
set of F is a closed subset S of T such that :
sup If(z)~ = sup If(t)[ s~S t~T
(Vf~F)
- If F admits a smallest ~ilov set, this set is called the ~ilov boundary of F. - The Choquet boundary of F is the set of all t ~ T such that the following
is
true : If e ~ ( T )
is such that e(f) = f(t) for any f ~ F ,
It is easy to interpret the map t ~--~e(t)
then e = d(t).
these notions in terms of compact convex sets. Consider
from T into the state space X of F, where e(t) is the evaluation
at t. By 1.2, e(T) DE(X)
and by (i) of 3.3, one has that :
6.1. PROPOSITION. The Choquet boundary Furthermore,
is the inverse image of E(X) by the evaluation map.
by 1.2. :
6.2. THEOREM. (a) A closed subset S of T is a ~ilov set for F if and only if S contains
the
Choquet boundary. (b) F admits a ~ilov boundary which is the closure of the Choquet boundary. By 3.3, we have : 6.3. THEOREM. If T is metrizable, continuous
the Choquet boundary of F is a G s
set of T and, for any
linear positive functional L on F there is at least one probability mea-
sure e on T concentrated L(f) = @(f)
on the Choquet boundary
for all
f ~F.
such that :
40
This measure @ is unique if F satisfies the Riesz interpolation property. In that case, one can solve the weak Dirichlet's problem : - For eveny compact K in the Choquet boundary and every f ~ C(K), there exists a norm preserving extension of f in F. 7. NOTES. The book of ALFSEN [I] is the most comprehensive work on compact convex sets and its scholarly notes give
precise references for the research of sources as well as
for further reading. The following numbers correspond to the sections in the text : I. For ].1., 1.2. and ].3., see [2], 11,§7. 2. 2.1. is due to Kadison, see p. 74-75 of Ill° For 2.3., see [1], 1,§2. 3. The first proof of 3.2. is due to Choquet. The present proof is due to Choquet and Meyer, see [1]. The idea of the order in the first step goes back to Bishop de Leeuw and in the present form to Mokobodzki,see [1], 1,§4. For the examples 3.4., see [3], §31. 4. Theorem 4.2. is due to Choquet for the equivalences (a), (c) and (e) ; the others are due to Edwards, see [I], 11,§3. 5. Due to Bauer, see[l], 11,§4. 6. See [I], 1,§5.
BIBLIOGRAPHY
l] E.M. ALFSEN. Compact convex sets and boundary integrals. Springer-Verlag, Berlin ]971 2] N. BOURBAKI. Espaces vectoriels topologiques. Chap. I e t
II. Hermann, Paris 1966, 2gme ~d. (ASI 1189)
3] G. CHOQUET. Lectures on Analysis. Vol. II, W.A. Bengamin Inc, New-York, [3
1969
G. CHOQUET et P.A. MEYER. Existence et unicit~ des representations int~grales dans les convexes compacts quelconques.' Ann. Inst. Fourier (Grenoble) 13, p. 139-154, 1963.
+
++ + ++
REPRESENTATION
OF BANACH LATTICES
Alain GOULLET de RUGY
Equipe d'Analyse ,
Universit~ de Paris VI
O. INTRODUCTION. Recall that a Banach lattice is a couple
(V,V+) where V is a Banach space and V+
a cone in V defining the order of V and for which V is a lattice space. The norm and the order are related by the following axiom : Vx,y£V
: Ix14 l y [ ~ l l x ~ l
6 UY~.
This axiom implies that the lattice operations continuous.
Consequently,
: x,y~---) x V y
; Co(T),T locally compact
: Represent V as a concrete
; LP(T)
; ip, . . . .
rich theory, where a lot has been done in the past twenty yearssand myself to some of the most significant Two kinds of representation functions on some topological The kind of representation extreme generators
: Representation
space with or without
by mean$of
in the cone P(V) of positive functionals
on V i.e. on the abun-
Without any restriction
on V, there won't
and we shall only have Davies's representation
(of. 2.4.) of V by real continuous
functions on some compact space, with
possible infinite values on some rare subset. On the contrary, such as when V is an M-space, we shall have representation continuous
functions
on some non-compact
abundance of real lattice homomorphisms. space is not a handicap.
continuous
infinite values.
we shall obtain will depend upon the abundance of
exist any real lattice homomorphism
valued,
This is a very I shall restrict
results.
will appear
dance of real lattice homomorphisms.
theorem
are
the cone V+ is closed.
The problem we are concerned with is the following space : C(T), T compact
or x ^ y
cases,
theorem by real, finite-
topological
The non-compactness
On the contrary,
in particular
space, due to the of the representation
its structure expresses precise features
of the Banach lattice V. References
to sources and complements
are relagated
to the end of this paper.
I. CASE OF FINITE VALUED FUNCTIONS. l.]. NOTATIONS. V will be a fixed Banach lattice
; V] denotes its unit ball ; V' is topological
dual ; V 1' its dual unit ball ; P(V) the positive elements
in V' and P I ( V ) = P ( V ) ~ V I.
42
An extreme generator of P(V) is, by definition, a generator D of P(V) such that (P(V)TD) is convex. If P(V) has a base B, D is extreme if, and only if, D ~ B extreme point of B. P(V)
g
is an
will denote the union of the extreme generators of P(V)
e|(V)g = P ( V ) g ~ e l(v). Recall that L E P ( V ) L(avb)
g
if, and only if, L is a lattice homomorphism i.e. :
= max(L(a),L(b)),
(¥a,b~V).
Thus P(V)g is closed in P(V). In particular PI(V)g is compact. 1.2. EXAMPLES. If V = C(T), the space of continuous real functions on a compact topological space T, P(V) =
@~.+(V) the cone of positive Radon measures on T ; if V = Co(T) , the
space of continuous real functions on some locally compact topological space T vanishing at infinity, P(V) =
~(T)
the cone of positive bounded Radon measures on
T. In both cases, P(V)g consists of the ponctual measures rd(t) where r~R_+ and d(t) is the Dirae measure at the point t ~T. If V = LP(x,@), where I S p < + ~
, and
@ a positive Radon measure on some locally compact topological space, P(V) = Lq(x,@)+ where q is the conjugate number of p and P(V)
is made of the g functions with support reduced to a point of X of @-measure non null. The first theorem we state is simply a restatement of Bauer's theorem : 1.3. THEOREM. If V is an order unit Banach lattice space, there exists a compact topological space T and a bipositive linear isometry of V onto C(T). Let us now consider a more general case : 1.4. DEFINITION. We say that a Banach lattic~ V is an M-space if the following is true : ~la vbl~ = max(UaU,Jlbll)
for all
a,b~V+.
The main interest of such spaces V is given by the following result which expresses the abundance of extreme generators : 1.5. LEMMA. If V is an M-space, then PI(V) is a ca___p_pof P(V) i.e. the complement of P](V) in P(V) is convex. In particular, E(PI(V))CP(V)g. From this, one gets Kakutani's theorem in a slightly modified version : 1.6. THEOREM. Let V be an M-space. To each v ~ V
associate the homogeneous function ~ on P(V)
defined by : : L ~-~ L(v)
for all
L~P(V)g.
g
43
Then the map v ~--P~ is a bipositive Ho(P(V)g)
of continuous
homogeneous
logy of uniform convergence The representation
linear isometry of V onto the space
real functions on P(V)g endowed with the topo-
on Pl(V)g.
theorem bear some interesting properties
Pl. To the supremum in V correspond
:
the upper envelope in the function space.
P2. Every element of P(V) is represented
by some measure on the underlying
topologi
cal space (here P(V)g). In order to understand ordered Vx~V,
the next property
linear space V is a positively Vy~J
: OSx~y
>xGJ,
let us recall that an ideal in an
generated
subspace J of V such that
:
An ideal J is said to be dense if every positive
element of V is the supremum of a net of positive elements of J. P3. To every closed ideal J of V correspond topological
a closed set Sj of the underlying
space such that :
J = Iv : ~ = 0 on Sj~. All "good" representations
must possess these properties.
added, also very important, which is not verified
A fourth one can be
in theorem 1.6.
P4. The image of V in the function space is an ideal of continuous This last property is extremely is "small".
It will be possible
strong
: it implies that the representation
to get a representation
certain M-spaces by "cutting off" P(V)
space
satisfying Pl to P4 for
in the following sense
g
functions.
:
1.7. DEFINITION. A positive element e of a Banach lattice V is a topological unit if the closed ideal generated by e is the whole space. It is not difficult B
= [L~P(V)
to prove that if e is a topological unit, then, the set
; l(e) = I~, is a (non-compact)
e generator of P(V) at a point different
base of P(V) i.e. B meets each ' e from O. Every separable Banach lattice has
a topological unit. ].8. THEOREM. Let V be an M-space with topological unit e. Let us denote Te = Be ~ P ( V ) g the restriction
of the dual norm to T . To each v ~ V e function ~ on T defined by : e : L :
:L(v)
for all
L~T
e
tisfying
continuous
functions
linear isometry of V onto the space ~ ( T
that is the continuous
:
¥ ~ > O,
~K
compact C T e s.t.
and
the continuous
.
Then, the map v ~--~ ~ is a bipositive of ~ -dominated
associate
jf[ • g ~
out of K
functions
f on T
c
sa-
e)
44
endowed with the norm :
Uf~
= inf [r ; ~f~ 6 r ~ ]
Furthermore,
this representation
. satisfies property PI to P4.
The proof is too long to be summarized fact that the cone P(V) is a bir~ticul~
in a few lines. It rests mainly on the cone, the theory of which I have developed
in [l]. Note that property P4 is clear. PI is a consequence ments of T following
of the fact that ele-
commute with the supremum. Property P2 can be made more precise by the e : P(V) can be identified with the positive Radon measures on the Cech-
compactification ~ of T concentrated on T (which is a K~) which integrate e e e ' Also, a stronger version of P3 holds : Closed ideals of V are in bijection, in a natural way, with the closed sets of T a . To end this section, M-spaces
let us caracterize
the spaces of the form C (T) among o
:
1.9. PROPOSITION. Let V he an M-space.
Then, there exists a bipositive
linear isometry of V onto
some space Co(T) where T is a locally compact topological the dual norm, when restricted
to (P(V)g~[O])
space if, and only if,
is continuous.
2. INFINITE VALUED FUNCTIONS. We shall first treat the case when V is order complete where results of algebraic nature are available.
Then we shall treat the case when V is not order complete
where results are known only when V has a topological unit. 2.1. THEOREM. Suppose that F is an order complete vector space. Then, compact topological
space T and a linear bipositive
ideal of the space C~(T) of continuous
there exists a s t o n i a n
isomorphism from F onto a dense
functions from T into [ - ~ , + ~ ]
, finite on
a dense subset of T. Recall that a s t o n i a n
compact topological
that the closure of every open set is open.
space T is, by definition,
a space such
We shall apply this theorem to repre-
sent L-spaces. 2.2. DEFINITION. We say a Banach lattice V is an L-space if the given norm is additive on V+. 2.3. THEOREM. Suppose that V is an L-space.
Then, there exists a locally compact topological
space T, a positive Radon measure 8 on T and a bipositive onto the space LI(T,8).
linear isometry from V
45
PROOF
(sketch of).
First note that V is order complete and by 2.]. can be represented ideal J G C ~ ( T )
for some stonian T. Using the abundance
exists a dense open set T' of T such that J contains nuous real functions positive
arguments
X
the space
in V, there
~(T')
of conti-
in T' with compact support as a dense ideal. Denote by L the
linear functional
tion of L to
by a dense
of projections
on J which coincide with the L-norm on J+. The restric-
(T') defines a positive Radon measure e on T' and some convergence
show that L and e still coincide on J which almost ends the proof.
Let us now consider
the second case. The best result is the following
:
2.4. THEOREM. Let F be a Banach lattice with topological unit e. Then, there exists a compact topological continuous
space T and a bipositive
linear bij~ction from F onto an ideal C F of
functions on T with values in [ - ~ , + ~ [
, finite on a dense subset of T,
such that C F contains C(T) as a dense ideal. Furthermore, the set of Radon measures PROOF
F' can be identified with
on T which integrate every function in C F.
(idea of).
Consider
the ideal J
generated by e. With e as order unit, it is an order unit e Banach space. Thus there exists a bipositive linear isometry from J onto some C(T) e with T compact. In fact, this isometry extends to a bijection from F onto some space of continuous
functions on T as described
in the theorem.
3. NOTES. 3.1. I have said nothing about the uniqueness in each of the representations
studied.
it is unique within an homeomorphism. change of norm respecting required
of the associated
topological
space
Simply note that, except for theorem 2.3.,
Furthermore,
this space is invariant by a
the locally convex space and the additional properties
to the initial norm in every statement.
3.2. All the theorems about representation to locally convex lattices.
of Banach lattices stated here generalize
For example let us restate theorem 2.3.
:
3.3. THEOREM. Let F be a locally convex L-space, topological vector which F is lattice, semi-norms Then,
that is a locally convex Hausdorff
such that the topology of F can be defined by a family P of
additive on F+ and such that : p(~x]) = p(x) for all x ~ F
there exists a locally compact topological
= (@i)(i~ I) of positive Radon measures phism from F onto the space L I ( ~ ) i~l,
complete
space together with a closed cone F+ defining the order, for
and p ~ P .
space T, a family
on T and a bipositive
of the @i-integrable
functions
endowed with the topology associated with the semi-norms
linear isomoron T, for all
: f ~
ei(~f~).
46
3.4. SOURCES AND REFERENCES. The numbers below refer to sections in the paper. I. Theorem 1.3. goes back to Kakutani ([4]). Theorems 1.6., 1.8. and proposition 1.9. are found in Goullet de Rugy [2], corollaires 1.31., 3.18.
and
proposition 2.31. 2. For theorem 2.1. and sources, see the Chapter 7 of [6]. This book of Luxemburg and Zaanen is so complete that it becomes confusing. So it might be look at Vulikh [8], theorem V.4.2..
better
to
Theorem 2.3. is due to Kakutani, see [5].
Theorem 2.4. is due to E.B. Davies. See theorem IO of [3]. For another proof, see [7~, theorem 1.
BIBLIOGRAPHIE
[U
A. GOULLET de RUGY. La th~orie des cSnes bir~ticulgs. Ann. Inst. Fourier (Grenoble), 2 1 (4), 1-64, 1971
[2j
A. GOULLET de RUGY. La structure id~ale des M-espaces. J. Math. Pures et Appl. 51, 331-373, 1972
[3]
E.B. DAVIES. The Choquet theory and representation of ordered Banach spaces. Illinois J. Math., 13, 176-187, 1969
[4]
S. KAKUTANI. Concrete representation of abstract M-spaces. Ann. of Math. 42, 994-1024, 1941
[5]
S. KAKUTANI. Concrete representation of abstract L-spaces and the mean ergodic theorem. Ann. of Math. 42, 523-537, 1941
[6]
W.A.J. LUXEMBURG and A.C. ZAANEN. Riesz spaces. Vol. I, North Holland, Amsterdam, London 1971
[7]
H.H. SCHAEFER. On the representation of Banach lattices by continuous numerical functions. Math. Z. 125, 215-232, 1972
[8]
B.Z. VULIKH. Introduction to the theory of partially ordered spaces. Moscow 1961 (English translation, Groningen 1967)
ORDER IDEALS IN ORDERED BANACH SPACES A.J. Ellis Department
Let of E
E
of Pure Mathematics,
E
be an ordered Banach
is closed,
If
y ~ I, I+
i.e. if
generates
I
then
6
E
I
= 0,
~
ideal, and
However
I
x&
I
ideal in
Let E
and important,
to study
and their annihilators
is an order ideal whenever
may be an order ideal without E = R3
E+
[(x,y,z):
subspace
of
positive-generation,
I
w~
-Wn + ~ Y n
I
E
I
I
is
being an
z ~ 0, x 2 + y2 ~
z21
which intersects
is described
E .
is perfect,
I , Yn ~ Z n ~ E x ~w n + ~Zn,
For extensive and Nagel
be an order ideal in
if and only if
sequencesl
E+ a
in the following
such that for each
generalizations
Then
I°
is an order
i.e. for each
x ~ I
~IYn~ ~ I , I~ z n ~ ~ I
n.
of this result
see Jameson ~16]
[17~.
For the remainder space, with base an ideal in I
I
is also an
[12].
Theorem I.
if
implies E+
The precise property which I must satisfy,
kind of approximate
and
x
of
.
I°
is any two-dimensional
in an extreme ray. result
I°
for example when
0 ~ y~
E+
I
subset of
E , with the dual ordering,
between order ideals
: f(x)
cone
subspace
and
is an extremal
Wales.
is called an ideal.
It is easy to verify that an ideal.
A linear
space and so it is natural,
the relationship =
E+
of Swansea,
i.e. the positive
x ~ I , y ~ E
I+ = I ~
The Banach dual space ordered Banach
space,
College
normal and generating.
is called an order ideal if
that
I°
University
E
B
of these notes let
and closed unit ball
if and only if
is closed then so is
I = linF
E
be a base normed Banach
co(B U -B). for some face
F , but the converse
Then F
of
I
is B ;
is much more subtle.
48
In fact if f ~ Ab(F)
F
is closed then
linF
is closed if and only if each
has an extension belonging to
Ab(B).
Here we denote by
Ab(B) the Banach space of all bounded affine real-valued functions on B ; this space is readily identified with E*. If
K
is a compact convex set then
space of the ordered Banach space K
we write
F~=
Ab(K)
A(K).
If
is the second dual F
~f & A(K): f(x) = 0~ ~ x ~ F~,
(F~) ~ = ~ x @ K: f(x) = O, ~ f ~ F±}. closed face of B we write F ~ = ~ f ¢
is a closed face of and
Similarly if Ab(B): f ( x ) =
F is a normO, ~ x ~ F } ,
and (F~)~ = ~ x ~ B: f(x) = O, ~ f @ FI}. It is often of importance to know that F = (FI) ~ or F = (FA)~; this is always the case if F is finite-dimensional. due to J.D. Pryce. Example 1.
Let
G = tf e E: f on
[0,I~.
However,
E = L2[0,1],
o,
fll
Then, if
we have the following example
F = ~f e E: 0 ~ f ~ I } ,
I}
and let h @ E+ be essentially unbounded K = co(F u (G + h)), K is weakly compact and
F is a closed face of K, since all elements of F are essentially bounded. If ~ F ~ then, since G - G is a neighbourhood of 0 in E and since l i n F is dense in E~ it follows that ~ = 0. Therefore (Fl)~ = K ~ F , and a f o r t i o r i (Fl)± = K . The bipolar theorem shows that if F is a closed face of K (or a norm-closed face of B) then F = (Fi)~ (F = (F&)~) if and only if F = K ~ L (F = B ~ L ) where L is the w -closed (norm-closed) linear hull of F ; these conditions are certainly satisfied if L is w -closed (norm-closed). and D.A. Edwards [ 1 0 ~ . Theorem 2.
If
F
The following result is due to Alfsen
is a closed face of
K,
then the following state-
ments are equivalent: (i) lin F is norm-closed; (ii) lin F w -closed; (iii) ~ a constant M such that each f E A(F) extension g ~ A(K) with IlgU ~ Mllfll. If these statements hold then A(K)/F~ is Archimedean ordered.
[21
A(K)+IF = A(F) +
Precisely analogous results hold for the space exception of (ii)) .
is has an
if and only if
Ab(B)
(with the
An ideal I in A(K) such that A(K)/I is Archimedean ordered is called an Archimedean ideal; if, in addition, I ° is positively generated then I is called a strongly Archimedean ideal. Since an
49
Archimedean ideal
I
satisfies
I = (Ii)&
the bipolar theorem shows
that
I ~ is strongly Archimedean if and only if the conditions
(i)-
(iii) of Theorem 2 hold for
F = I ~.
Analogous definitions
and results apply in the case of ideals in
Ab(B).
A closed face F of K (or of B ) is called semi-exposed if for each x ~ K k F (BkF) ~ f ~ A(K) + (Ab(B) +) with f(x) > 0 while
f(y) = 0
of
then
x
for all
F
y & F .
If
is called exposed.
f
can be chosen independently
Clearly a face is semi-exposed
if and only if it is the intersection of a family of exposed faces. It is not difficult to show that a semi-exposed face
F
of
K
is
exposed if and only if it is a G&-set, which is always the case when K
is metrizable.
G&-set;
Of course every norm-closed face of
B
however not every semi-exposed face is exposed,
is a
as the
following example shows.
Example 2. Let Y = ~f: [o,1]~R: with the natural ordering, and let y* Then Ab(B) is isomorphic to ×
R
functional in
0.
follows that
Y 0
supports is an
B
at
Ilfll = s u p { I f ( t ) l
: o
B = ~f e Y: f >
0, llfll~
and each positive linear
0
is not an
Since
Y
= Y+ - Y+
Ab(B)-semi-exposed face of
strictly positive linear functional in hence
Y
B .
B.
The following result gives dual characterizations K
and
B .
it
However, no
exists (cf. [10]) and
Ab(B)-exposed face of
and semi-exposed faces of
,
I}.
of exposed
For a proof of parts (i)
and (ii) see ~ 3 ~ Theorem 3. only if f & F~
(i)
and
6 > 0 (ii)
only if
For a closed face
F = (F~)/-
and ~
~
F
of
an element
~f,i)
> 0
The closed face
F = (F&) ~- and, given
with F
f&
of F&,
K,
F
h ~ F~ f~ K
is exposed if and such that for each
lh +~
.
is semi-exposed if and
[>
0
~g&
F~
with
f~g+£. (iii) only if £ > 0
The closed face
F = (F~)~
~ k>O
and
and
F
~ g ~ FA +
h e Ab(B)
with
of
B
is
such that
Ab(B)-exposed if and ~ f ~ F~,
~h(x)~ -~ I
and
f~
x~
B,
~g +gh.
(iv) The closed face F of B is Ab(B)-semi-exposed if and only if F = (F~)/. and given f ~ F j-, x ~ B , £ > 0 ~ g E h ~ xb(B) with ~h(x)~ ~_ 1 and f ~ g + g h . It is possible for
F
to be an exposed face of
being even a perfect ideal in
K
without
F ~+,
F~
Ab(K), as the following example shows.
N
Example 3.
Let
I, n=q
Then to
K
is ~'(~l,Co)-Compact and
co ~ R
and
~x
R
A(K),
respectively.
Ab(K) 0
is
are norm-isomorphic A(K)-exposed in
oweve to perfect ideal in
Ab(B)
sno a
Ab(K).
In order to get a duality between faces of in
K
we need to define an order ideal
and order ideals
f ~ i,
x ~ B,
1
-g + £ h I ~ f ~ g + ~ h 2 .
It is then
true that
I&
B
w -perfect.
F
exists a disjoint face 0 ~ k ~ 1 .
If
is split. B
if and only if
of a convex set F!
of
has a unique decomposition If
K B
is split.
~ I,
to be
~hi(x)~ ~
A face
~g
Ab(B)
such that
is a face of
0
B in
w -perfect if given
and
~•
I
C C
I
h 1, h 2 E Ab(B)
is
is said to be split if there such that every point
of
C
[1~ every closed face of
K
x = ~y + (1-~)z,
is a simplex then
y ~ F,
x
z ~ F t,
is a simplex then [ 7 ~ every norm-closed face of
This latter result requires the completeness of the
base normed space
E , as the following example shows. o:
F = ~f~
B:
(x)dx
0
.
The base norm for
E
induced by
B
is
0 the relative
Ll[0,1~-norm,
and
F
is a closed face of
B
which is
not split. If
F
is a split face of
there exists an
f ~ Ab(B) Moreover,
F l = f-I(1).
exists an
f ~ Ab(B)
Therefore
F
F
and
F
and
B
if
g ~ Ab(F)
such that
F!
with complementary face
such that f = g
A face
L-ideal in
E
F
and on
while
h ~ Ab(F ~) F,
f = h
F!
then
F = f-q(0), then there on
F~ .
are norm-closed and it is easy to check that
are strongly Archimedean,
existing.
0 ~ f ~ 1
of
B
with norm-preserving extensions
is split if and only if
in the sense of Alfsen and Effros [5~,
linF
is an
and it follows
[5, If, 1.q3~ that the intersection of an arbitrary family of split faces of
B
is split, and that the closed convex hull of an arbitrary
family of split faces of
B
is split.
The situation for closed split faces of For example, if
K
K
is rather different.
denotes the probability measures on
[0,1~
then
each extreme point ~x is split but its complementary face is dense in K ; in this example there are, of course, far more norm-closed
51 split faces of
K
than closed split faces.
In the next theorem we
sum up some of the results of Alfsen and Andersen ~4, 6~ concerning closed split faces of Theorem ~.
Every closed split face of
and norm-preserving faces of
K.
K
extensions exist.
K
is strongly Archimedean,
The family
~
of closed split
is closed under arbitrary intersections and the convex
hull of finite unions.
The sets
F ~ ~K,
closed sets for a facial t o p o l o g y ~ K and only if
K
for
F ~
, are the
; this topology is Hausdorff if
is a Bauer simplex.
It is not generally true that the closed convex hull of an arbitrary family of closed split faces of
K
is split, as the follow-
ing example of A. Gleit shows.
Example 5.
Let
A(K) = ~ f e C[-1,11: f(O) = ~ ( f ( - 1 )
I n = ~f ~ A(K): f(1) = 0}. space
A(K)
so that
I n&
= n=l
In
is an ideal in the simplex
is a split face of
If
= 0,Vn
K
However
,
whic
is not
n=1
positively generated, K
Each
+ f(1))l,
so
c-oU i# n=1
is not split (cf. [11~).
is said to satisfy St~rmer's axiom if
split face of
K
whenever each
F~
c-oUF~
is a closed
is a closed split face of
K.
A simplex satisfies St~rmer's axiom if and only if it is a Bauer simplex.
However the state space
satisfies St~rmer's axiom. for any
K
K
of any unital
Alfsen and Andersen [ ~
B-algebra have shown that
which satisfies St#rmer's axiom a hull-kernel topology
may be defined,
and this topology gives, in the case of a unital
B -algebra, precisely the Jacobson topology of the primitive ideal space. Some other relevant results (cf. ~ 3 , # , 1 8 ~ ) a r e the following theorem. Theorem 5and let F
Let K be the state space of a unital B -algebra A , be a closed face of K . Then the following statements
are equivalent:
(i)
F
is a split face of
Archimedean ideal in A(K); (iii) closed two-sided ideal in A .
K
contained in
Fi
K;
(ii)
F~
is an
is the self-adjoint part of a
Chu has shown that, for K as in Theorem 5, every closed face of is semi-exposed. Moreover he has proved the following result
[8, 9].
Theorem 6.
If
K
is the state-space of a unital B -algebra then
the following statements are equivalent: (ii)
A(K)
either with
is an anti-lattice,
f ~ g F, G
or
i.e.
g ~ f ; (iii)
(i)
f ^ g
K
A
is a prime algebra;
only exists in
is prime, i.e.
semi-exposed faces implies either
F = K
A(K)
K = co(F~ or
if
G)
G = K.
In connection with Theorem 6 we recall that a unital B -algebra is commutative if and only if Let let
K
A
is a lattice.
be a function algebra on a compact Hausdorff space-(~,
be the state space of
split faces of of
A(K)
Z
A
and let
Z = co(K ~ -iK) .
The
are also connected with the algebraic structure
A , as the following result shows (cf. [14] and [15]) •
Theorem
7-
Let
F
be a closed face of
is a split face of set for
Z
A ; (ii)
F
if and only if
K .
Then:
F ~ X
is a split face of
Z
is a generalized peak interpolation set for
(i)
co(F v -iF)
is a generalized peak if and only if
F ~ X
A .
References I.
E.M. ALFSEN,
'On the decomposition of a Choquet simplex into a d i r e c t convex sum of complementary faces', Math. Scand. 17(1965) 169-176.
2.
E.M. ALFSEN,
'Facial structure of compact convex sets', Proc.
London Math. S.c. 18 (1968) 385-404. E.M. ALFSEN, 'Com~agt convex sets and boundary integrals', Springer-Verlag, 4.
Berlin, 1971.
E.M. ALFSEN and T.B. ANDERSEN,
'Split faces of compact convex
sets', Proc. London Math. S.c. 21 (1970) 415-442. .
E.M. ALFSEN and E.G. EFFROS,
'Structure in real Banach spaces
I, II', Ann. Math. 96 (1972) 98-173. 6.
T.B. ANDERSEN,
'On dominated extensions of continuous affine
functions on split faces', Math. Scand. 29 (1971)
.
298-306. L. ASIMOW and A.J. ELLIS,
'Facial decomposition of linearly
compact simplexes and separation of functions on cones', Pac. J. Math. 34 (1970) 301-310. 8.
9.
CHU CH0-HO,
'Anti-lattices and prime sets', Math. Scand. 31
CHU CH0-HO,
(1972) 151-165. 'Prime faces in C -algebras', (to appear).
J. London Math. S.c.
53
10.
D.A. EDWARDS,
11.
locally compact cone into a Banach dual space endowed with the vague topology', Proc. London Math. Soc. 14 (1964) 399-414. E.G. EFFROS, 'Structure in simplexes', Acta Math. 117 (1967)
'On the homeomorphic affine embedding of a
12.
I03-121. A.J. ELLIS, 'Perfect order ideals', J. London Math. Soc. 40
(1965) 288-294. 13.
A.J. ELLIS,
14.
annihilators', Math. Ann. 184 (1969) 19-24. A.J. ELLIS, 'On split faces and function algebras', Math. Ann.
'On faces of compact convex sets and their
17.
195 (1972) 159-166. ' M-ideals in complex function spaces and algebras', Israel J. Math. 12(1972) 133-146. GRAHAM JAMESON, 'Ordered linear spaces', Lecture Notes in Mathematics, No.141, Springer-Verlag, Berlin, 1970. R.J. NAGEL, 'Ideals in ordered locally convex spaces', Math.
18.
E. ST~RMER,
15. 16.
B. HIRSBERG,
Scand. 29 (1971) 259-271. 'On partially ordered vector spaces and their duals with applications to simplexes ~nd C -algebras', Proc. London Math. Soc. 18 (1968) 245-265.
ORDER BOUNDED
OPERATORS
AND CENTRAL
MEASURES
W. Wils
i.
Introduction
Attemp%~ to use the setting O.S.)
ics are very old. ed too w e a k
The g e n e r a l
to prove
it became n e c e s s a r y
Considerable
In due time
C* -algebras
apparent Attempts
notions
central
over to a more general
spaces,
however,
which
one wanted.
classes
of P.O.S.
for the class
that many
and Riesz-spaces.
some of the p o w e r f u l
Spaces
analogies
were made
to b u i l d
class of P.O.S., In these
of C* -algebras,
decomposition
seem-
Therefore
the field
of Riesz-spaces.
could be made bea unified
which
lectures
(P.
of Q u a n t u m Mechan-
took place on the one h a n d w i t h i n
a substantial
quotient-algebras,
theory
i n c l u d e d both
I want
to show how
as two sided ideals,
theory
and factors
carry
setting.
DecoMposition
Mathematicians objects
often
by w r i t i n g
Definition:
attempt
E+ = Zi
to simplify
the study of c o m p l i c a t e d
them as sums of simple
A splitting
is a family of subspaces
Hence
special
and on the other h a n d
it b e c a m e
cover
foundation
theory of these
to c o n s i d e r
the two fields.
which would
2.
O r d e r e d Vectors
the kind of results
development
of C* -algebras,
tween
of P a r t i a l l y
in the theory of the m a t h e m a t i c a l
of a P.O.S. {Ei~
components.
(E,E+),
i of E such
Thus:
(with E = E+-E+),
that E = ~ i E i
and
(Ei~ E+)"
every e l e m e n t
k e E can be w r i t t e n
finitely m a n y ki, k i e Ei, k . e E. ~ E +. l 1
and m o r e o v e r
in a unique way
as a sum of
if k e E + then every
55
The subspaces E. w h i c h appear in a s p l i t t i n g of E are called splitl
subspaces of E and the E i ~
P r o p o s i t i o n l:
E + are called s p l i t - f a c e s of E +
There is a one-one c o r r e s p o n d e n c e b e t w e e n split-
subspaces of E and the range spaces of linear maps P:E--gE with the p r o p e r t y 0 ~ Pk ~ k for all k ~ E + and p2 = p.
Any two such
maps P commut4 and hence the set of s p l i t - f a c e s of E + is a Boolean algebra.
The above p r o p o s i t i o n means
firstly that there is an operator-
c h a r a c t e r i z a t i o n of s p l i t - s u b s p a c e s i n t e r s e c t i o n s of s p l i t - s u b s p a c e s (faces).
and secondly that sums and
(faces)
are again s p l i t - s u b s p a c e s
Thus it is always p o s s i b l e to find a r e f i n e m e n t of any two
splittings.
Later on, we shall ask:
Does there exist a finest
s p l i t t i n g of E ?
In this context we introduce a l o c a l i z a t i o n of the notion of splitting as follows. Cp =
~>0
Let k e E + and F k =
0 <_ h < k~,
IF k and Vp = Cp - Cp.
Definition:
Two subspacesE i and E 2 are called disjoint,
E~ ~
~EI,E2]
E2, if
is a s p l i t t i n g of E 1 + E 2.
and H of E + are called disjoint, disjoint.
notation
Two split-faces G
n o t a t i o n G ~ H, if G-G and H-H are
Two e l e m e n t s p, p' e E + are said to be disjoint, n o t a t i o n
p ~ p', if Cp ~ Cp,. is a splitting of V
And p = ~i Pi is a s p l i t t i n g of p if
i
. P
As long as we consider d i s j o i n t elements it is as if we are dealing w i t h a lattice.
We have:
Suppose g, h E E +, g i h and o < k < g + h.
i)
Then there are unique kl, k 2 £ E +, w i t h k = ~ + k 2, ~
~ g,
k 2 .< h and. if k' . < k, k' . < g then k' < k I. ii)
If in addition k ~ g' w i t h g' ~ h then k ~ g.
In many cases a splitting in direct summands does not exist. operator-characterization
of s p l i t - s u b s p a c e s
The
allows a g e n e r a l i z a t i o n
w h i c h can be used to d e s c r i b e smooth decompositions.
58
Definition.
The set of o r d e r - b o u n d e d operators Lob
(E), of a p o s i -
tively g e n e r a t e d P.O,S. E is d e f i n e d to be the set of linear maps T:E-~E such that there existed a I>o w i t h -II < T < + lI. is the identity map on E a n d T < lI means:
Then Lob(E)
is an algebra of operators.
The p o s i t i v e cone Lob(E) + =
T e L ~(E) I T > o] is closed for m u l t i p l i c a t i o n . for L~b (E) is also an o r d e r - u n i t for Lob(E). E is Archimedean,
There I
for all k e E + , Tk < Ik.
The a l g e b r a - u n i t
If the o r d e r i n g on
i.e., k, feE and k ! a f for all a~o implies k ~ o,
then the o r d e r i n g of Lob~(E) is A r c h i m e d e a n too.
It is w e l l known
that an o r d e r e d algebra w h i c h is A r c h i m e d e a n is i s o m o r p h i c to an algebra of functions and hence,
in particular,
is commutative.
If all the spaces V
with p e E + as their order unit and e q u i p p e d P with the c o r r e s p o n d i n g n o r m are complete, then Lob(E) is complete in its o r d e r - u n i t norm.
Every T e Lob(E)
leaves the spaces Vp in-
variant.
The use of Lob(E)
in d i s c o v e r n i n g remnants of lattice structure in E
is i l l u s t r a t e d in the next proposition.
P r o p o s i t i o n 2: i.
Let k be an order unit for E.
ii.
Suppose S, T e Lob(E)
Then the map Lob(E)gT-->Tk e E
is bipositive. are such that sup
(S,T)
e Lob(E).
f ~ E are such that Sk, Tk < f then sup
(S,T)
k < f.
If k,
This closes the e l e m e n t a r y theory of sets of order b o u n d e d operators. There are two main directions for further development.
In both cases
E is given the additional structure of an ordered Banach space and duality theory is used.
The first d e v e l o p m e n t is a further elabo-
ration on closed split faces, ideals in A, q u o t i e n t spaces and ration e x t e n s i o n theorems. in A. Ellis's
Several aspect of this theory have been treated
lectures.
Here I shall add only a few remarks.
The
second line of thought concerns the construction of d e c o m p o s i t i o n s of a p a r t i a l l y ordered space. ures on compact convex sets.
The main tools are r e p r e s e n t i n g measThis topic will be covered in these
lectures. [For a more extensive t r e a t m e n t of the topic treated in § 2, Alfsen [7], Ch. II, or Wils [5] ].
57
3.
Order
Although
the
siderably almost
bounded
theory
exposed
by A l f s e n
classical
spaces.
operators
in this
and E f f r o s
case
[2] w e
of s p e c i a l
An o r d e r - u n i t
section
space,
(A,e)
vector
space
A with
vector
space
in the o r d e r - u n i t
shall
classes
distinguished
extended
consider
only
of p a r t i a l l y
denotes
order
has been
the n o w
ordered
an A r c h i m e d e a n
unit
con-
Banach
ordered
e considered
as a n o r m a l
norm, llall = inf{l > ol-~e~a<_+le ~
for
a ~ A.
A base-norm E + has
space
a base
sidered
that
first
assumes
(A,e)
base
dual
linear
Theorem
The
Moreover tions
Lob(A)B
The
S x,
Edwards
of n o r m a l T--~
converse
and E = A* w i t h
of this dual.
can be This 73,
paper.
identified
We w a n t [i] II,
con-
and
for e a c h
has
1.15].
Lob(E)
on
shown
In this
is a c o m p l e t e
x e E + and
that
T-->T* e L o b ( ~
representation
positive
shown
and
theorem.
the
functionals
space
functional
the m a p L o b ( E ) ~
all
space
f e A +.
linear
The
functional.
linear
combina-
on A are of the
is a g i v e n
order-complete
e E l=l Ifl I= i} interest.
This
A is a C* - a l g e b r a
K with the
II,
operators
norm
form
x e E.
(A,e)
E.g.
[7],
Moreover
[3] h a v e
linear
some
will
E for w h i c h
compact,
and E l l i s
following
is a n o r m a l
K = ~f
with
identification II,
algebra
of c o n s i d e r a b l e
We p r o v i d e
e E*
and
case w h e r e
is also
e
f > e~
for
space
its
bounded
and G e r z o n
x>
the
[See
the
onto.
be
stonean
positive
space
rest
space.
operator
is h y p e r
Then
[2] p r o v e d
isomorphism
Lob(E) ~ S-~<
Let e
1 on K.
set of o r d e r
normed
for Lob(A) map
of E.
the v a l u e
and E f f r o s
is an i s o m e t r i c
space
is r a d i a l l y
= inf { ~ ~ o 1 x E I B}.
is an o r d e r - u n i t
commutative
vector
(Ku-K)
t h a t E is a n o r m - c o m p l e t e
case A l f s e n
I:
a directed
B =conv
I Ixll
is the B a n a c h
E which
denotes
that
in the n o r m
L e t us s u p p o s e A = E*
(E,K)
K such
the r e l a t i v e
setting
unit
throughout
affine
Then A
functions
the paper.
for the
e and E is
weak*-topology.
set of c o n t i n u o u s
be u s e d
is a b a s e - n o r m
is the with
order-unit
on K.
[See A l f s e n
§ i].
to f i n d d i f f e r e n t §7
or Wils
[5]).
representations
for Lob(A).
(See A l f s e n
58
T h e o r e m 2:
The map Lob(A) 9 T-gTe e A is an isometric i s o m o r p h i s m of
the ordered space Lob(A)
Hence Lob(A)
into A.
can be i d e n t i f i e d w i t h a subspace of A.
is called the center of A.
This subspace
For a C* - a l g e b r a A w i t h unit e, Lob(A) e
coincides w i t h the a l g e b r a i c center of A.
The second r e p r e s e n t a t i o n
theorem requires more work.
The center Lob(E)
of E is order c o m p l e t % and the set of weak*
-closed
split-faces of E + is closed under arbitrary i n t e r s e c t i o n and finite sums.
The i n t e r s e c t i o n s of the closed split-faces of E + with the
extreme b o u n d a r y
~ K of K, defines a topology on ~ K the so called e e
facial topology.
T h e o r e m 3:
If x ~ ~e K and T e Lob(A),
constant 1 T
(x) such that 1 T
Lob(A)gT-91TEC f
(x)x=T*x
Let 1T : x-~l T
(x).
The map
(~e K) is a b i p o s i t i v e algebra i s o m o r p h i s m of Lob(A)
onto C f (~eK), the algebra of bounded, on
then there exists a unique
facially continuous
functions
~ K. e
T h e o r e m 4:
For every g e Cf
~I~eK = g.
M o r e o v e r ~ E Lob(A)
b e A with bl~eK = ~l~eK
(~e K) there is a unique ~ ~ A such that e and for every a e A there exists
. al~eK.
Suppose we r e s t r i c t A, v i e w e d as the space of continuous affine functions on K to
~e K.
Then the last theorem tells us that Lob(A)
e
consists of those elements in A I ~ e K with which one can m u l t i p l y other a r b i t r a r y elements in AI~eK and still stay in AI~eK. exist three r e p r e s e n t a t i o n s I.
Hence there
of Lob(A).
as the set of order b o u n d e d operators on A.
2.
as the set of facially continuous functions on ~e K.
3.
as the set of m u l t i p l i e r s w i t h i n AI~eK.
Further d e v e l o p m e n t s of this part of the theory leads to the consideration of the r e s t r i c t i o n of A to a closed split face and the subspace of A c o n s i d e r i n g of those elements w h i c h v a n i s h on such a closed split-face.
The q u o t i e n t of A w i t h respect to the last sub-
space is in a natural way isomorphic w i t h the first space.
The prop-
erties of these spaces are being studied and sharp e x t e n s i o n theorems for continuous affine functions on closed split-faces of K can be
59
given.
The
e e A.
Extensions
theory does n o t d e p e n d
given by W. Habre theories obtained,
4.
Central
We return
to the ideas (A,e)
Thus
more
either
ered.
The d i r e c t
theory,
we a s s o c i a t e
As the s p l i t t i n g s
llkl I =
a tendency
§2.
there
Eil Ikil I, since it be-
spaces
of direct
integral
decom-
to introduce
repre-
in the C h o q u e t - t h e o r y
for e l e m e n t s
of a
similar
difficulties
as here
are encount-
have b e e n w o r k e d
out for the case of C*
integral
involves
set up,
good results offers
spaces.
in the s e p a r a b l e
The results
is p r e s e n t e d
however
for p a r t i a l l y
splitting
finer,
a limit,
k =
on the set of points,
which
are
The
and applies
ordered
spaces.
Z. k. of k 1 l on K, w h i c h r e p r e -
increase
at g.
in the order corres-
is c o n c e n t r a t e d
can no longer be splitted.
is to make
of
less detailed.
is the central m e a s u r e
is that this m e a s u r e
idea in the rest of the p a p e r
case.
proofs
is the p o i n t e v a l u a t i o n
the m e a s u r e s
which
a g r e a t deal
less t e c h n i c a l
g, g e K, of course, become
are,
or one has
measures
The hope
and finer,
to go to zero so that
a theory
With every
and have
to k.
and,because
a measure, Z iIIkil I~ [ki/iikiii),
Here ~
of C h o q u e t
in
a base-norm
of E, then k = E l. k.1 with finer
ordered
approach
let k s Z E +.
k.
and E = A*
a limit.
and gives
Here the second
ki,
as is done
also to n o n - s e p a r a b l e
sense
space,
of E b e c o m e
to develop
Both a p p r o a c h e s
ponding
developed
set, where
use of r e p r e s e n t i n g
sents
w h i c h were
k i has
one has
convex
measure
Thus
on d e c o m p o s i t i o n , an o r d e r - u n i t
for p a r t i a l l y
-algebras.
for com-
[i].
to take
senting m e a s u r e s compact
algebras,
have b e e n
see A l f s e n
and more
impossible
positions
similar
results
references
the splittings
E is a base-norm, comes
Several
of f u n c t i o n
unit
unaltered.
When
in general,
[2] c o n s i d e r e d
spaces.
k e E+ and < E i~ i a s p l i t t i n g
k. e E~. l l
of an order
cones have been
decomposition
stays
Suppose
For more
complete
and Effros
in the context
spaces.
setting w i t h
space,
Alfsen
real B a n a c h
especially
Banach
The
[4].
for g e n e r a l
plex
on the e x i s t e n c e
to the case of w e a k l y
the above h e u r i s t i c
in some The approach
80
more precise,
to find properties of the central measure, and to indi-
cate further p o s s i b i l i t i e s
As before,
for g e E +, C
g and Vg
Cg-Cg
for research.
denotes the smallest face E + w h i c h contain~ g The ordering of E induces an o r d e r i n g on Vg. If
is a positive measure on K, we let ~ ~ : L ~ < a,
~
(~)> =
a d ~, a e A,
~e
(K,~)--gE be the map
(K, ~)
~ ~(~)
is de-
fined as an element in A* = E by the above formula.
The following theorem is more general than is n e c e s s a r y for just central measures, but it p o s s i b l e to find other applications.
T h e o r e m 5.
Let g £ K,
I Igl I = i, and g e W e V
plate linear lattice in the induced ordering.
, w h e r e W is a comg Then, the set of dis-
crete p r o b a b i l i t y measures
Ziai~f. w i t h f i e W / ~ K, Ziei fi = g' is 1 d i r e c t e d in the order of Choquet-Meyer. Let ~ be the s u p r e m u m of this net of m e a s u r e s then that ~ ~
~ is the unique p r o b a b i l i t y measure such
is a lattice i s o m o r p h i s m from L~(K,~)
Various choices for W can be made.
onto W.
Let me indicate two w h i c h for
the case of C* - a l g e b r a coincide, but in general are different. W = Lob(E) tion 2].
g then W is a complete linear sublattice of Vg
If
[Proposi-
This means that only splittings of all of E occur,
and one
obtains a kind of central m e a s u r e s which has not been studied yet. Another choice is to take W = Lob(Vg) p r o p o s i t i o n 2 that W is a lattice,
g.
It follows once more from
and it is not d i f f i c u l t to verify
that W is complete.
Definition: measure ~
For h e K, we denote Lob(Vg)
by Z h.
A probability
~ on K, w h i c h represent h e K, is said to be central iff
maps L ~
T h e o r e m 6:
(K, ~) i s o m o r p h i c a l l y onto the lattice Zhh ~ V h.
For g e
K, there is a unique central measure ~
which
represents g.
The proofs of both theorems 5 and 6 do not contain many new ideas. The next result is much harder to obtain.
It concerns the support
of the central measure.
Definition:
A point k e K is called primary when Lob(Vg)
consists
81
only of m u l t i p l e s m a r y points
of the i d e n t i t y map on V k.
The union of all pri-
in K is denoted by ~ p r K.
In other words:
A p o i n t is primary w h e n it can not be split in two
d i s j o i n t elements.
T h e o r e m 7:
E v e r y g e K, can be r e p r e s e n t e d by a unique central
measure
and
o~
~
~
(0) = O for every B a i r e - s e t
O £ K with
~ p r K = ~.
The Baire sets and the Borel sets coincide in the separable case and it has been shown by J. R. C h r i s t e n s e n that case is u n i v e r s a l l y measurable.
(Kopenhagen)
that ~ pr K in
His proof uses the E f f r o s - B o r e l
structure on the set of closed subsets of K.
In the appendix another
simpler proof is given.
T h e o r e m 8:
(J. P. Reus Christensen).
Let K be a m e t r i z a b l e
convex subset of a locally convex space E.
Then the set,
~
compact
pr
K, of
p r i m a r y points in K, is co-analytic.
C o - a n a l y t i c means that the c o m p l e m e n t of the set is a n a l y t i c and it implies that for every R a d o n - m e a s u r e s n u l - s e t from a Borel set. all Radon-measures.
the set differs
at m o s t by a
Hence c o - a n a l y t i c sets are m e a s u r a b l e
It is unknown under w h a t conditions ~
pr
for
K is a
Borel set.
In v i e w of t h e o r e m 8 it w o u l d be nice to have a simpler proof of t h e o r e m 7 than is available,
When
(A,e)
then Lob(A)
e s p e c i a l l y in the s e p a r a b l e case.
is a C* algebra with unit and K is the state space of A, e coincides w i t h the center of A, and a state f ~ K is
p r i m a r y iff if the r e p r e s e n t a t i o n G.N.S.
construction,
closure of
~f(A)
representations
~f, of A, c o n s t r u c t e d via the
is a primary r e p r e s e n t a t i o n ,
is a factor. ~f and
Two states
Zg are disjoint.
F of K there corresponds
i.e.
the weak
f and g are disjoint iff the To every closed split face
an ideal I £ A such that F= { f ~ K l f ( I ) = ~ o 1
and vice versa.
For lattices, d i s j o i n t n e s s notion of disjointness. b o u n d e d operations
as
i n t r o d u c e d here coincides w i t h the usual
The i n t e r p r e t a t i o n s of split-faces
are s e l f - e v i d e n t in this case.
and order
62
5.
Areas for further r e s e a r c h
The f o l l o w i n g lines of d e v e l o p m e n t have been started in i.
Do there exist i n t r i n s i c c h a r a c t e r i z a t i o n s
2.
As r e m a r k e d earlier, set Lob(A) fore,
for a C* -algebra
[5].
for central m e a s u r e s ?
(A,e) w i t h unit e, the
e coincides w i t h the a l g e b r a i c center of A.
in order to interpret the results on Lob(A)
There-
in a context
of C* -algebras one has to study the centers of C* -algebras. Various s e q u e n t i a l closures for C* -algebras have been considered and E. B. Davies has shown that the center of these enlarged C* -algebras, primary points.
in the separable case, separates d i s j o i n t
It is easy to see that these s e q u e n t i a l clo-
sures also exist when A is an o r d e r - u n i t space. center now ? Answers to these questions
How big is the
can be used in the
formation of a theory of direct integrals of p a r t i a l l y o r d e r e d spaces. 3.
Because closed split-faces have so many important
properties,
it is d e s i r a b l e to develop techniques to handle more general kinds of split-faces.
What is the b e h a v i o u r of central meas-
ures with respect to split-faces? 4.
C o n s i d e r the map K 9 c o r r e s p o n d i n g to g.
g--~g
where
~g is the central measure
In the case of simplices K, where the cen-
tral measure coincides with the unique m a x i m a l measure, above map is w e a k * - m e a s u r a b l e .
the
W h a t are the p r o p e r t i e s of this
map in general?
ad i:
A p r o b a b i l i t y measure
~
on K, is said to be s u b - c e n t r a l
if for every Borel set B ~ K, w i t h 0 <
~
(B) < i, the resultants
of the r e s t r i c t e d measures ~I B and ~K/B are disjoint.
[ ~ B(A)
=
~
(B(] A)
for a Borel set A c K]
Subcentral m e a s u r e s have m a n y - n i c e properties. i)
The subcentral m e a s u r e s r e p r e s e n t i n g a given point g e K, form a complete lattice for the C h o q u e t - M e y e r o r d e r i n g of measures. This lattice is i s o m o r p h i c with the sublattices of Zg.g and the central measure of g is the unique maximal ~ s u b c e n t r a l measure r e p r e s e n t i n g g.
ii)
Suppose
~
and ~ r e p r e s e n t a point g e K, a n d ~ i s
Then there exists a smallest m e a s u r e ~ , ~
subcentral.
w i t h respect to the
63
ordering
of C h o q u e t ~ M e y e r ,
is subcentral, ~ , ~ i s
iii)
which majorizes
~
and ~ .
subcentral.
Let g s K, and ~ the central m e a s u r e
of g.
by every m a x i m a l
represents
-algebras,
the
which
again,
measure
The proof
of this
ization h i n g e s
This
g.
For C*
element
property
in the
measures,
does not hold
examples.
measures,
in the case of separa-
in the o r d e r
representing
a given
of C h o q u e t -
point
and w i t h
points.
last fact is not very
on questions,
largest
are minimal,
all m e a s u r e s
in the set of p r i m a r y
Then ~ is m a j o r i z e d
by all m a x i m a l
from simple
of central
is that they
among
is the
are m a j o r i z e d
a given point.
characterization
support
on K w h i c h
as can be seen
ble C* -algebras, Meyer
which
representing
in general
Another
measure
central
set of m e a s u r e s
If also
difficult
touched upon
but
in the next
its g e n e r a l section,
which
are unsolved.
ad 2:
The
set of all b o u n d e d
ly i d e n t i f i e d s malles t
ments exists
sists
in the center a ~ s Lob(E)
The most
functions
contains
dual A**
A and is closed with
Then A m =
of A**.
But,
(Am) + -
for every
and then T * * E
A conjecture
Lob(A**)
in this
(Am) +.
respect
that
of ele-
there
for all a E A m ,
is that Lob(Am)
elements
to
It is not
T* s Lob(Am) such
connection
to A m of those
con-
T E Lob(A**)
such
A m.
important
question
is, w h e t h e r ,
L o b ( A m ) e is big enough
For C* -algebras
the answer
ad 3:
split
For every
face F' of K.
F'
to s e p a r a t e
face F of K there
is the b i g g e s t
split
Then K =
Conv(F,F') .
Every
affine
extension
to K, w h i c h
vanishes
In p a r t i c u l a r equals
admissable
at least in the s e p a r a b l e states
in K.
if G is a split
face
is a c o m p l i m e n t a r y
face of K, w h i c h
affine
function
let PG be the
the b a r y c e n t r i c
affine
G' of G. calculus
split-
is d i s j o i n t
on F has
on the c o m p l e m e n t a r y
1 on G and 0 on the c o m p l e m e n t
if PG satisfies
disjoint
is yes.
of F.
which
on K, can be n a t u r a l of A. Let A TM be the
in the center of A TM are r e s t r i c t i o n s
of the r e s t r u c t i o n s
that T e ~
case.
which
n o r m over K.
that e l e m e n t s
Ta = a o T.
affine
the second B a n a c h
set in A**,
the s u p r e m u m obvious
with
We
an
face F'
function call G
for central
64
measures,
that is, if a e A and g ~ K, w i t h a s s o c i a t e d central m e a s -
ure ~, then S PG d ~ = Admissable
.
faces have m a n y appealing properties.
sible faces is closed for relative complementation,
The set of admisthat is~ if G and
H are admissable and G c H, then G'/]H, w i t h G' the c o m p l e m e n t of G, is admissible.
The set is closed for m o n o t o n e s e q u e n t i a l limits and
contains the closed split-faces, of those.
their complements
and i n t e r s e c t i o n s
But it is not know w h e t h e r the intersections
faces is again admissible. arate d i s j o i n t states?
of admissible
Are there enough admissible faces to sep-
The answer to these questions
can help in
b u i l d i n g a theory of direct integrals. ad 4:
No comments.
Several other p r o b l e m s have been m e n t i o n e d in 15].
Can one recover
the results of the v o n - N e u m a n - M u r r a y theory of direct integrals of o p e r a t o r algebras, using just central m e a s u r e s
?
What are the geo-
metric c h a r a c t e r i z a t i o n s of the types of a state for C* -algebras (Type I, II, III)?
Do such types or others exist in a setting of
just compact convex sets?
Appendix
T h e o r e m 8:
(J.P. Reus Christensen).
Let K be a compact convex set
in a locally convex space E, such that the r e l a t i v e t o p o l o g y of K is metrizable.
Then the set of primary points of K is the c o m p l e m e n t
of an analytic set.
The notion of primary p o i n t can be defined for every convex set, but w i t h o u t loss of g e n e r a l i t y we may assume that K is the state space of A(K), w h e r e A(K)
is the Banach space of affine continuous
tions on K w i t h the s u p r e m u m norm.
func-
Let the function in A(K) w h i c h
is i d e n t i c a l l y equal to 1 on K be denoted b y e and the B a n a c h dual A(K)* of A(K) by E.
Then K = {f e E
I
I ]fll = < f,e > = 13 .
set K is endowed with the relative o(E,A(K)) §2].
Finally,
i:
~i, chapter 2,
for x e E +, let
F x = [h s E+I h ~ x ] , C x = Definition
topology
The
Ul>o
I F x and V x = C x - C x
Two elements x,y e E + \ O
n o t a t i o n x ~ y, if l.Vx+y = Vy @ Vy direct sum of V x and Vy)
are said to be disjoint,
(i.e., V x + y is the a l g e b r a i c
2. Cx+y = C x + Cy.
65
2.
A point x ~ K is said to be primary
if it cannot be written
as x = y + z with y, x e E + \ 0 and x ~ y. Lemma i.
i:
Two points x, y e E + are disjoint
Fx+y = F x + Fy
Lemma
2:
and
2.
Let T be a Hansdorff
Fx ~ Fy = ~0~ ~ [5] prop.l. space
If R 1 and R 2 are compact
subsets
then ~ (RIt) R2\ R I ~
is a Borel
Proof:
R2)
We may assume
borhoods
R 2 --c R I.
in K, consisting
(RI~R2) = ~ n ~ ( R l ~ 0 n ) \ and z (Rln On ) are closed Theorem
9:
A2
{ (x,y,z)
= T x 0 . Then n n ~(R2~0n)" Clearly, the sets R1 f3 0n and hence the lemma follows. I h _> 0,
then it follows subsets
[~ (AI\A 2) D ~ (BI\ B 2) D ~K x 0~ U The last equality
K, which
The set
subset of K x K.
four sets are all closed
and z (BI\B 2) are Borel
{(x,y) I x ~ y~
I Ih11
in K x K x 2K.
+
x, y ~ K~.
Let ~: (x,y,z)--~(x,y),
Proof of theorem
for the neigh-
Put ~
I x, y e K; z e F x + Fy"I.
B 2 = {(x,y,O)I
too.
space.
(t,k)BT x K~gteT,
be a basis
sets.
I x, y e K; z e F x
B1 = {(x,y,z)
metric
.
subset of T.
Let {0n~n
of closed
Let K = {h e E
The following
and K a separable
of T x K and ~ :
{(x,y) I x,y e K,x ~ y~ is a Borel Proof.
iff
8:
follows
of K x K and consequently {0 x K~]c = {(x,y) i x ~ y~ is Borel from lemma
The continuous
onto the complement
therefore
is analytic.
from lamina 2 that ~ (AI~ A 2)
map
i.
(x,y)-~x + y maps
of the set of primary
points
in
66
References [i]
Alfsen, E.M., Compact convex sets and boundary integrals Berlin, Heidelberg, New York, Springer 1971
[2]
Alfsen E.M., Effros, E., Structure in real Banach spaces Annals of Math.
[3]
96:
98-173,
(1972).
Edwards D.A°, The ideal centre of certain partial ordered Banach spaces
(preprint)
[4]
Habre W., Th~se dutroisie~e
cycle, Paris 1972
[5]
Wils, W., The ideal center of partially ordered Banach spaces Acta.Math.
127:41-77
(1971)
ORDERED
NORMED
Gerd
Fachbereich
TENSOR
Wittstock
Mathematik
der
Saarbr~cken,
Recently in
the
ordered
structures
and
known
if
this
paper
hand,
topological
literature.
the
topologies
is
to b u i l d
to o b t a i n
of normed
are
nice
spaces
regular
ordered
should
be viewed
on
usua~
we
define
by an universal
1.1
Definition.
and w:EXF product there
in
this
phism.
~(x,y)=x
If ~ is
1.2
the
tensor
Example.
tions
is
aim
of
On
the
other
to the
class
B-cone,
respectively
o f E.B.
Davies.
This
on
the
product
paper
subject.
results
of
F,w),
E ®
mapping,
every
linear
the
tensor
injective,
E @F
more
The
self
order
two
real
linear
spaces
linear
space
property:
(E ®
if f o r
our
studied
strict
preliminary
tensor
Much
type.
theory.
restrict
been
possible
product.
general
sense
and
have
F
a real
is c a l l e d
bilinear
the
mapping
mapping
~:E ~F~G
product
is u n i q u e
tensor
~:EXF~G
such
that ~=~'~.
®y(xeE,yEF).
definition
to i d e n t i f y of
F,
an unique
tensor
report
a bilinear
of E a n d
We d e n o t e By
F
Saarlandes
several
of a special
we
the
mapping
The pair
-- E @
exists
the
are
by a normal
as a short
E,F
the
up a more
ordered
des
products
there
spaces
I Notation As
tensor
results,
spaces
Universit~t Germany
In g e n e r a %
factors
PRODUCTS
as
it
is p o s s i b l e
a subspace
of G.
and
Then we
up
to a n
often have
isomor-
convenient a special
model
product. Denote
o n a s e t M.
by ~(M)
the v e c t o r
If E c ~ ( M ) , F c S ( N )
bilinear
mapping
EXF--5(MXN),
function
{ (m,n)~x(m)y(n)},
where
for all
are
space
of
all
subspaces,
(x,y)
is m a p p e d
xeE,yeF,mEM,neN.
real
func-
define
a
onto
the
The
induced
68
linear mapping E @F~(MXN) and
is injective.
x @y:(m,n)~x(m)y(n)
for all meM,neN.
A special case is the usual finite d i m e n s i o n a l
representation
the space B(E,F)
real forms on EXF w i t h the dual (E @ F , B ( E , F ) > . ~(x,y) Since
= (x @ y , ~ >
the a l g e b r a i c
defined
We d e n o t e
and the
of
(E @ F ) *
of a%% b i l i n e a r
and d e n o t e
this d u a l i t y
Thus we w r i t e for x e E , y e F , V g B ( E , F ) .
duals E * c 3 ( E ) , F * c S ( F ) ,
bilinear.
We o b t a i n
corresponding
the m a p p i n g
(xEE,x*EE*,yEF,y*EF*)
x*@y*:(x,y)~(x,x> is
of t e n s o r p r o d u c t s
spaces as matrices.
We i d e n t i f y in a n a t u r a l way
by
We embed E ® F c ~ ( M × N )
i n t e r p r e t x @ y as a f u n c t i o n of two v a r i a b l e s :
an
bi%inear
embedding
mapping
E*@F*c(E @F)*
by
~ * : E * X F * ~ ( E @F)*. 1.3 Example: on M.
Let 5(M;F)
If EcS(M)
EXF-5(M;F),
be the space of all F - v a l u e d f u n c t i o n s
is a l i n e a r subspace,
where
(x,y)
is m a p p e d
we have a b i l i n e a r m a p p i n g
onto
the f u n c t i o n
{m~x(m)y].
The induced l i n e a r m a p p i n g E ®F--5(M;F) is injective. E @FcS(M;F)
We embed
and i n t e r p r e t x ® y as a F - v a l u e d f u n c t i o n
x @y:m~x(m)y.
As a special into
case we o b t a i n from E*c~(E)
the space of all l i n e a r o p e r a t o r s
1.4 Example.
If fEL(EI,E2),
is bilinear.
There
f @g,
from E into F.
gEL(F1,F2),
exists an u n i q u e
the e m b e d d i n g E * @ F c L ( E , F )
then w2" (fXg):EIXFI--E2XF2
l i n e a r mapping,
d e n o t e d by
such that the d i a g r a m m EIXF 1
, E2XF 2 f×g 1~ 2
l~ I EI@F 1
, E2@F 2 f ®g
commutes.
The b i l i n e a r m a p p i n g
(f,g)~f @ g
induces an e m b e d d i n g
L ( E I , E 2) ® L ( F I , F 2 ) c L ( E I ® F 1 , E 2 ® F 2 ) . 1.5 Remark. bounded
Let H I , H 2 be H i l b e r t
%inear operators
spaces,
L b ( H I ) the space of a%%
on HI,K(HI ) the space of a%%
compact
89
Linear trace
operators class).
and
The
N ( H I ) the
trace
K(HI)'
= N(HI) , and
If n o w
ftH1--HI,gtH2--H2
operators,
like
cations
behave
If E,F
troduce
to
x ~y,
x ®f,
only
- or we m a y
are
introduce
identification we have
or o r d e r
identifi-
we m a y
universal ~,~
of
of c o n f u s i o n .
structure,
mappings
special
the
danger
by a restricted
by
or n u c l e a r
meanings
But all
is no
(the
H I ' = H I.
compact
of p o s s i b l e
there
or p o s i t i v e
them
over
x1@x 2 a.s.o..
on E ~ F
bounded
a natural
a lot
norm
operators
more
respectively
manner,
an additional
nuclear
in-
mapping
in d e f i n i t i o n
representations
as
1.2
1.5.
1.7
Definition.
the b i l i n e a r
defined
are
1.8
@
F,m' :E'XF'--(E ®
This
is
a tensor
norm
(cross-norm)
with
Less
norm
equivalent
then
one.
We d e n o t e
xEE,yEF
llx' @ Y'I]C¢, = ilx'll IlY'II
for all
x'EE',y'EF',
U.110~, d e n o t e s solution
of
operators
distance
functional
balls
]]tll ~ dual
the dual the is
corresponding
the
F =
norm.
the R r o j e c t i v e
of
E ®
to for all
The
if
F)'
lix ®Yl]~ = [IX[l HYll
bilinear
unit
11.115 is c a l l e d
bounded
and
(E ®F,II.II~).
where
A norm
mappings
w:EXF~E
The
2 there
or o r d e r i n g s
- take
induces
bounded
in a n a t u r a l
have
norms
property 1.1
f ~g,
of all
= Lb(H1),
are
x,xlx2£H1,YEH
symbols
1.6
form
N(HI)'
space
convex
mapping tensor
hull
of
problem
norm
the
for
bounded
l[.Iln. It is
tensorproduct
the of
the
S E , S Fn = i n f [ ~ -~ ~=I
of E @
F is
llx~[[ IlyvI I : n E N , t the
space
n =E x~®y~} . V=I
Bb(E,F)
of all
bounded
biLinear
forms. I. 9 By a f u n c t i o n cally
embedded
jective is
tensornorm
independent
space
into
of
representation
spaces
of b o u n d e d
11.116 ; F %
FC~b(M,N).
the r e p r e s e n t a t i o n .
EC~b(M),FC~b(N) functions, The
we
induced
, isometri-
obtain
the
injeetive
By E C S b ( E ' ) , F C ~ b ( F '
) we
innorm obtain
70
tltll E
sup{ll:x'CE',llx'l[
:
1.10
Proposition.
If
1.11
Definition.
Let
E+,
E,F
cone
a
be
F+.
We
call
(E ® F , C
),
if
the
canonical
®
F)*
are
~*:E*XF*-(E
a
I1,11= i s
C~cE
®F
a
1,y'CF',Hy'[[
norm
tensor
ordered
~
bilinear
I}.
then
linear
tensor
~
spaces
cone
with
and
mappings
proper
write
w:EXF~E
cones
E ®~F @
=
F and
positive:
x @yEC
for
all
xEE+,yEF+
x*®y*EC*
for
all
xEE*+,vEF*+
and
where
E*+,F*+
1.12
The
tive
bi%inear
and
C~*
solution
of
denote
the
mappings
the
dual
cones.
corresponding
~,
,
universal
~(E+,F+)cG+,
is
the
problem
posi-
for
projective
cone
n C
= co(E P
If
E
+
1.13
,F
are
+
@F +
+
proper
Definition.
) = [ -~ xv@yv:xveE+,yvCF+,ne~]. v=1 cones
Let
E,F
E + = [xCE:(x,E* Let
ECS(M),FcS(N)
he
be
the
special
representation:
C
is
p
a proper
ordered
linear
in~ective
embeddings,
tensor
@F:(t,E*
example
of
rent
from
projective
and
injective
A ~A n m The
by = A
trace
be
the
nm
. An
form
the cone
real A
easy induces
such
that
n+
tensor
the
linear of
the
is
embedding
independent
E~FcS(MXN) of
the
> a O}.
We s h o w a n
n
a
+
It
Example.
A cL(~ n)
is
@F*
C.. l
1.15
ordered
If C
+
cone
Proposition.
Let
spaces,
then
1.14
the
cone.
+ > ~ O} order
induces
C. : [ t E E l
then
-
a duality
a
space
positive
computation
cone
then
tensor
of
C cC cC.. p ~ l
cone,
(An,An>
is
diffe-
cone. complex
semidefinite
shows,
which
that with
the
hermitian
matrices,
matrices.
Then
closure
C cA . p nm+
71
(x,y)
= Tr(xy)
x,yEA
. n
By this
dua%ity ~ p c_An m +
We s h o w now, el,
...
~n(~m).
we
obtain
= (Anm)*
that ~
+
$ A
p
Define
%inear
It f o % % o w s
that
cC p * = C.. 1 if n , m
nm+
,e n r e s p e c t i v e % y
(~,U=1,2)
(A )* = A and C * = C.. p l n + n+
el,
...
~ 2.
,e m be -
operators
then
nm
)*
~ C *. Let p
+
the c a n o n i c a %
cn~n
P~:
(A
basis
respectiveZy
of
:¢m~m
PgU
by
P v ~ : e ~-e , P
:ek~-0(k:~'~).
Then 2
Pv~@Pv~CAnm+ V,~=1 since
it is h e r m i t i a n
and
2
2
( ~ P~@P~a,a> V,~=I The m a p p i n g
= ~ ~V~ 9,~=1
T : L ( C n ) - L ( £ n)
(T®id)AnmCAnm , (T®id)~
(T@id)
P
c~
> 0
(transposed P
n m for a =~___ ~ G ~ k e ~ @ e x. K=I k=1
matrix)
maps
A
into A
n+
and
n+
. But
2 2 2 -~ Pv~®P~)~ = ~ P~v®P~ V,~=l V,~=I
¢Anm
+
since 2
2
~,~=1
V,~l=l
is not p o s i t i v e spring
semidefinite.
2 Re~uZar Let
(E,E+,II.II)
and
F + are
Later
2.1
,
2.2
ordered
(F , F + , l[-ll) b e
closed
normal
o n w e wi%% a s s u m e ,
finiton
This
exampZe
is due
to W.F.
Stine-
[15].
strict that
normed ordered
tensor
products
normed
spaces,
B-cones E and
F are
(see
such
Schaefer
regu%ar
that
E+
[13]).
ordered
(see
normed
tensor
de-
be%ow)
Definition.
We ca%% ( E ®F,C~,II.I[~).... an o r d e r e d
pro-
72
duct
if (i)
C
is a c l o s e d
normal
(ii)
C
is a t e n s o r
cone:
(iii)
llx ®YlI~
strict
E +@ F + cC ~ a n d
llxll IIyll
f o r a%%
IIx'@Y ' [1~, ~ IIx ' II Ily'll (where
C
' is
is
the d u a l
An
ordered
[~].
cone
an
space
(i)
{llyl]
We
x
+ , .- . ' E F '
'CE'
positive
and
linear
+
forms
and
l].II~,
a normal
regular
strict
norm
B-cone
If.IfI in
if a n d
the
sense
only
if
of E . B . D a -
a normed
ordered
space
E regular
ordered
if
and then
+
if I]xll <
:y+_xes+}.
call
is c l o s e d if y + x C E
(ii)
has
equivalent
Definition.
the c o n e
all
'
Take
llXHI = i n f
2.2
x C E + ,y C F +
for
of b o u n d e d
E' +®F' + cC
norm.) normed
it p o s e s s e s vies
the
B-cone
]]xl] ~
I then
llY]i
there
exists
a yCE + such
that
HyH
~
i and
y+__xeE+. 2.3
Definition.
(E @ F , C it
,li.I[~)
is an
2.~
Let
be r e g u l a r
ordered
normed
spaces.
a regular
ordered
normed
tensor
is c a l l e d
ordered
Remark.
E,F
normed
Regular
tensorproduct
ordered
normed
and
spaces
E @
F =
product,
the n o r m
is r e g u l a r .
have
following
the
if
proper-
ties (i)
If E is r e g u l a r
ordered,
then
the
completion
E
is r e g u l a r
ordered. (ii)
The
dual
E'
is r e g u l a r
ordered,
if a n d
only
if E
is r e g u l a r
ordered. (iii)
If E is o r d e r e d
by a c o n e
perties
(ii)
(i)
a regular N o w we 2.5
consider
Lemma. (i) (ii)
and
ordered
always
If E , F
[Ixl[ : sup Hx,]I = sup
are
normed
regular regular
E+,
and
the n o r m
of d e f i n i t i o n
2.2
fulfills
then
space.
ordered ordered
{ (x,x'>:x'CE'+,Hx' [(x , x'>:xCE+,llx[]
normed normed
~ g <
I}
I}
the pro-
(E,~+,II.II)
spaces. spaces,
then
f o r all
xCE+
for all
xEE'
+
is
73
(iii)
I1~11 = sup £11~<x,y)il =xC~+,llxll
(iv)
f o r all b o u n d e d
positive
regular
normed
If
We d e n o t e forms
ordered
~+3EBB(E,F)+
space
~ 1}
mappings
~tEXF
~ G into a
G.
{1~{1 ~ N*[I"
then
by Bb(E,F) + the
~ 1 , yCF÷,llyll
bilinear
cone
of
all
hounded
positive
bilinear
on ExF.
2.6
Proposition.
sor
product
of
If
(E ® F , C
regular
,ll.ll a)
ordered
is
a regular
normed
spaces
ordered
E,F
normed
ten-
then
~ It.L[~
andll.ll~
is a t e n s o r
norm
(see
If ~CC0c ' = (E ® a F ) ' +
Proof.
I]~lla, = s u p
If ~ £ C
then
[<~,t>:teta,,S]tll a <
> sup
I}
[ <~,X ®y>:x£E+,HXII
' t h e n ~ E B b ( E , F ) +. S i n c e
II~ll~,
= inf
1.7)
definition
< I, yEF+,liyll <
II'II~, is a r e g u l a r
[11C011~, : ~0+~CCcc l ]
for
all
norm
~E(E ® F)'
--
By Lemma 5 ( i v )
If tEC
c~
0c
we o b t a i n
II*IL~, > i n f Therefore
I] = II~011
= II*ll = II*11.,
[II~II-m+*EBb(E,F)+]
l l ' l l ~ ~ I]'l117 then by 2.5(iii)
iltll~ = sup £=~cc
, ,II~11~, ~ I}
~up [ x'c~'÷,ilx'll ~ i, y'CF,÷,
IIY'II ~ i]
= tltli~ 2.7
The pro~ective
ordered per
normed
cone.
spaces.
normed
tensor
The projective
product.
cone
Le E , F b e r e g u l a r
Cp = c o ( E +®F + ) i s
a pro-
The functional
L[tlLp = s u p is
ordered
monoton,
{ :~EBb(E,F)÷,
subadditiv
and posit±v
IL®[[ ~ 1] homogeneous
functional
for
all
t~Cp
on C . T h u s P
the
11sllp = inf[IItllp : t ± S e C p ] is
a norm on E ®F.
(E ® f , ~ , i l . i i p ) ~ space
is
is
It
follows
a regular
Bb(E,F) + -
from
ordered
Bb(E,F) + with
the
remark
normed the
norm
4(iii)
tensor
that
product.
E ® F = P
The d u a l
74
[[~[]p, E @
F has
the
: inf
{]l~[[: ~ C B b ( E , F )
following
+]
universal
property:
if ~ : E X F
~ G is a p o s i -
P tire
bounded
then
the
bilinear
induced
mapping
linear
into
mapping
a regular
~:E @
ordered
normed
F ~ G is p o s i t i v e
space
bounded
G,
and
P
I[~H = I[~[I" It has positive
and
and ITr @Ell 2.8
The
rive
a functorial
bounded
f @g:
ordered
(biprojective
injective
dered
tensor
injective
IIslli
EI®pFI--E2®pF2
is p o s i t i v e
bounded
normed
tensor
product.
We
take
the
~ 0
for
_If.IfE
norm.
all
x'EE'+,
is m o n o t o n
y'eF' +]"
on C.. z
We d e f i n e
{IItIl
:t+seC.}._i
E @ . F = (E @ F , C i , l [ . [ I i ) i s a r e g u l a r o r d e r e d n o r m e d t e n s o r 1 It has a functorie% property: if f,g are positive bounded
2.9
then
f ®g
Proposition.
Example.
is p o s i t i v e
If E @
t h e n ~ cC cC. a n d p ~ z
2.10
the or-
norm
= inf
operators,
injec-
cone)
C.l = [ t : < t , x ' @ y ' > The
g:FI~F 2 are
if f:EI--E2,
IIfll llgll
:
injective
cone
then
property:
In
F is a r e g u l a r
]I.1[ z ]I.1[ p
the
bounded
case
of
product. linear
l[f ®Ell = [[fll [lgll.
and
ordered
normed
tensorproduct,
k [I-[[i "
the H i l b e r t
space
b 2 we h a v e
the
sequence
%2@~%2 ~ %2®p%2 -- %2®2L2 ~ %2®iL2 The
cones
~
p and
different 2.11
and not
C. c o i n c i d e b u t z equivalent.
Proposition.
The
(i
an
isometric
(ii
an
order
i)
If ~ E ( E
Proof. ded
linear
duced sion
by
form
the
natural order
embedding
@pF)'+
~i :EXF"--R.
isometric
E @.FEE 1
has Thus
P
FeE ®
normes
embedding P
are
FcF"
all
induces
F" a n d
@.F". I xEE+,
a natural we
~ICBb(E,F")+.
E ®F (] (E ® ( F " ) = (E ® p F ) + P +
order
E ®
= Bb(E,F)+,
F'cF"'
Of c o u r s
corresponding
embedding
~(x,.):y~0(x,y)
embedding
the
%2@a L2 .
then
the p o s i t i v e
extension
to F",
get a n a t u r a l
bi%inear
By d u a l i t y
obtain
we
bouninexten-
75
The
linear
E ®
FeE ® P
extension F"
is a n
mapping
~ ~ ~I
if a n d
only
,F") k 0
It f o l l o w s
that
3. A
dense
in F 'j'
for all
x'EE'
k 0 for all
E @.FcE
®.F"
1
1
tensorie%
y"'
I,
thus
E posesses
the R i e s z
[O,x] This
is
+ [O,y]
equivalent
If y v - x e E + ( ~ , ~ EE+(V,~
=
Theorem.
every
F the
(ii)
1,2).
(i)
If K is a c o m p a c t
Lemma.
that
Then
(u k u' ' n
A regular if a n d
The
lattice
ordered only
decompo-
if
x,yeE
a z,
normed
if the
+
such
dual
the p r o j e c t i v e
that
spaceEhas E'
y -zeE+ the R i e s z
is a l a t t i c e .
decomposition
in C. i n E ® A ( Q ) , i property. set
property,
tensor
lemma
there
> = 8
norm
then
II.ll~ is
for the
the
for
the p r o o f
lattice,
exist
space
order
of
the
, and
there
exists
of all
and
the
the
affine sup-norm.
theorem.
Xl,...,Xn,UCE
~eEu+
E has
u'£(Eu)
, such i
that
+ , ~ = I , ...,m,
>u.11 <
space
with
is
the p o i n t w i s e
then
and
k~
~
linear
Q a square,
then A(K)
on K with
U
~=I
Banach
all
exists
the R i e s z
of Cp in
llx~ -2" <x,u, Proof.
there
E be a Banach
~>0.
[Ix11=
then
Riesz
interpolation p r o p e r t y :
the R i e s z
following Let
u+xCE+__ a n d
the
for
convex
functions the
of
property
C.cE ®F. 1
is d e n s e p decomposition
We need
embedding.
= [O,x+y]
If E h a s
closure
order
property,
property
cone
continous
+
+
decomposition
to
If C
Riesz
, y'eF'
characterisation
= 1,2)
decomposition
injective
+
Therefore
£F'"
is a n
sition
such
than
if
3.2
less
isometry.
F' + is ~ ( F " ' (t,x'®y'>
3.1
norm
P
(ii)
z-x
has
~
"
E u = Ix"
a X>O,ku+xeE+]_
the n o r m
= inf [~>0, X.+xCE+}
for
all
xCE u .
is a
76
E u is an exists ping
(AM)-space.
a compact
f : T -- R n,
By the K a k u t a n i
representation C(T)
set T s u c h t h a t E u
f:t ~
guef(T) s u c h
of t h e o r e m
map-
of u n i t y
IIe-u=lz~EuYu(E )[IRn < n
3.1.
= Eu+,U'Cf-I(EU)C(Eu)'
+
and
properties; (i) A s s u m e
first
that E is a B a n a c h
lattice.
n t
be
= 6XU,
T h e n u~ = y o f E C ( T ) +
the d e s i r e d
Let
a continous
m
0 ( Y~,~__Iy~ = 1, yk(g~)
Proof
there
that
m
they have
Define
( x v ( t ) ) v =n I . In Rn we f i n d a p a r t i t i o n
¥ ~ ( ~ = I , . . . ,m) a n d p o i n t s
f o r all E C f ( T ) .
theorem,
an
there (uX,u'
=
~
x~®y V
element
C 1.cE ®F.
of
exist by
lemma
3.2
Choose
ugE + , such
elements
u eEu+~'
that u _+ x e E + . e(Z u) ! + s u c h
For ~ > 0 that
> = 5k~ a n d m
H % - ~ 1= <xv , u ' u > ~ I I
< n •
Define n m tn = 2 C ~=I ~=I Since
m
(xV'u'~>u~®Yv
E u+ = E + nE u it f o l l o w s
o((Eu)',Eu>
dense
in ~ O
(Eu)'+.
=~-~-1 U u ® Z u =
that
"
the r e s t r i c t i o n
Since
E' + [E u
is
t e E u ® F we o b t a i n
f o r all u ' e ( E u )'+,y'eF' +
Now n
o~=F V=I <x , U ' >< y ~ , y ' >= = (z , y ' > f o r all y ' E F ' + Hence
z CF+ and
t ~ C C p C E ®F.
Now
n m n [[t-t111]IX ~>--~V=II[Xv-~=I(Xv'u'~/>u~II "HY~[I ~- 13~V=I[IYv[I"
Assume
now
a Banaeh
that E has
lattice.
C- = ( E ® F)
p
p
the R i e s z
By p r o p o s i t i o n +
=(E"®
p
F)
+
decomposition 2.11
we h a v e
f'l(E ® F ) = ( E " ® . F ) m
+
property.
Then
E" is
the r e l a t i o n s
n(E ®F)=(E
@.F) =C.. x + i
77
Here ~
is the closure P II.I]p. B u t c o n s i d e r the
The p o l a r theorem (ii)
o f Cp i s
A(Q)
l]'ll~ c o i n s i d e points
o f Q,
extremal
v'
I
affine
and
q5
have - v' 2
rays
Now
choose
form
v'~:v
the +
sitive,
space
= EXEXE.
Let
The
these
is ~ p-
II.ll~ - c l ° s u r e
of c
By t h e p
ql,-..,q 4 b e
functional
tensor
the
norms
extrema%
v'EA(Q)'
rays
bipolar
.
a%% p o s s i b l e
extremal
-
generate
generate
the
cone
= O.
v'k
vl,...,v4EA(Q)
the
cone
, such
extremal
= Yl
t(''v'2)
= Y2
by
vv(q~+3)=O rays
of A(Q)+
and
these
extre
A(Q) + . that
,A(Q)' ) by
t(''V'l)
defined
v~(q~+2)=O,
generate
x,Yl,y2CE
4)
bipolar
relation
a.s.o,
3)
the
the
-- v ( q ~ ) .
v'3
tee ®A(Q)cB(E'
Then t(.,v'
is
p
v~(q~+l)=l,
generate
t(.,v'
and
a 3-dimensional
functions
= ql
mal
is
~
of A(Q)' + and
vv(q~)=l, where
that
: E ®aA(Q)
rays
A(Q)' + . We
The
Bb(E , F)+,
we o b t a i n ,
Since
of C in the ordered projective tensornorm P duality <E ® F , B b ( E , F ) ) = < E ® F , ( E ® F ) ' ) .
O,x
the
~ y l , y 2.
We
define
a bilinear
formulae
= Y2-X.
= t(.,V'l-V'2+v'3)
= Yl-X.
The b i l i n e a r
form
t is
po-
tEC.. i
Assume
for
a moment,
that
tCC
, then P
4
t = ~ z~v v v=l where
zVEE+,
because
the
v
generate
Yl=t(.vl')=zl+z2,y2=z2+z3,Y2-X=Z3+Z 4. we o n l y
have
tCC.=~ , but 1 p
result. For
~ >
0 there
exists
the
cone A(Q)+.
T h e n we o b t a i n
Thus O,x~z2~Yl,y 2.
by an iteration
process
In general
we o b t a i n
the
same
78 4
s
v=l )-~ z~®v~eCp, z~e~+,
=
such that Ilt-sll
< ~12 •
We o b t a i n
UYl-(~1 + ~2 )ll < ~ , IIY2-(z2 +z3 )11< ~ ' II(y2-x Thus there exists a~ u¢~÷, Ilull < ~ -u
~ yl-(Zl+Z2)
-u
< y2-(z2+z
-u
~
We o b t a i n tion we
such that
< u 3)
~ u
(Y2-X)-(z3+z
4)
~; u .
O,x < z 2 < Y l + W , y e + w
find
)- (~ 3÷~4)II <~2"
sequences
where
an,WnE~ + such
w = 3uCE+, that
[]w][ < ~ .
By itera-
[[Wn[[ _L 2 -n,
O,x < a I ~ Y 1 + W l , Y 2 + W l and O,x,an_1-wn_1 It f o l l o w s
< an
<
Y1+wn,Y2+Wn
a n _ 1 + w n-
that
- ( w n _ 1 + w n) < an-an_ I ~ W n _ 1 + w n and
therefore
ll%-an_111< 2 -n÷2 The
limit
z = lim a n C ~
the R i e s z
interpolation
3.3 T h e o r e m . every
If E,F,
complete
interpolation Proof.
lattice
there
the R i e s z
ordered
interpolation
tensor product
have
the R i e s z
and
Then
interpolation (E ® p F ) '
The
space
property,
E ~ F has
that exists
the p r e d u a l
(E @ F)'
is a l a t t i c e
the R i e s z
If ~ I , ~ 2 C ( E
• = inf(~ I , ~ 2 ) C B b ( E , F ) + .
I]¢0110~, = sup[ (t,~>:lltll a < 1, sup{ (t,~O,> V=I , 2
property,
= Bb(E,F) +
E ~ F has P (see Ng [9] a n d P e r e s s i n i [10])
We s h o w n o w then
O,x ~ z < Yl 'Y2"
E has
then
the R i e s z
property.
If E,F,
property.
and
property.
have
regular
Lb(E,F' )+ is a l a t t i c e . Banach
exists
then B b ( E , F ) + =
Bb(E,F)+
is a
interpolation
@aF ) ' +CBb(E,F) +
Since
tCCo~} <
:Iltl]~l ,t¢c a} = infCHmlll il~ell}
79
we
obtain
predua%
that
E ~
A.
~E(E
F has
~
F)'
the
Ordered
and
Riesz
specia%
tensorproducts
normed 4.1
examp%es
and
order
Definition.
space,
if
the
Definition.
med
space,
The
the
norm
unit A
if
A
is d e t e r m i n e d
Examp%e.
of a % % the
sup-nom
(ii) 4.4
The
L[xll where
u'
is a b a s e
is
the
Theorem.
a
E,F
E ~pF
is
(ii)
E ~.F x
is b a s e
base
Proof.
(i)
Since
llx[] =
<x,u'>
I]~]I ~
i
a%%
Let
The
norm
consider
now
u
normed
cone.
space
e%ement
is a b a s e
in
is a n the
order
unit
+
}
for convex
unit
baLL.
u
the
space
A(K)
xCE.
set,
then
the
pointwise
order
if a n d
onZy
if E'
is
an
xCE + , order
normed space
u'£E',v'CF' for
if a n d It
is a d d i t i v e
be
aZL on%y
the
xCE
unit
in
spaces.
E' Then
and ....It-lip = ....l[-]]~,
llt]lp
on C
. P
norm we
if u ' ~ v '
fo%%ows K
+
determing
obtain
that
~ ~CBb(E,F) +
that for
and
space.
space,
a~
normed
is
space.
normed
normed
a~
on K with
normed
a base
base
nor
formu%a
determing be
base
unit.
functions unit
space
normed.
~ C B b ( E , F ) +. Iltllp ~
the
for
norm
Let
order
Then
<x,u,>
(i)
for
is
space. :
order
is a c o m p a c t
order
and
we
positive
normed
If K
affine
normed
resu%ts
the
: X u +--x C E
A(K)'
E
normed
on
[~>O
is a n
dual
Remark.
unit
4.5
(i)
continous
property.
normed
greatest
by
the
spaces.
ordered
is a
Therefore
spaces.
ordered
determing
Ilxll = i n f 4.3
normed
reguZar
is a % a t t i c e .
base
preceding
is a d d i t i v e
there
norm
of
normed
the
regu%ar
norm
4.2
ca%ted
of
F)'
interpo%ation
unit As
(E ~
a%%
t C ~ p"
order
units.
order
80
I f ~CBb(E , F)
ll~II ~ 1 t h e n
,
u'®v'+~CBb(E,F)
+.
Thus B b ( E , F )
= (E ® p F ) '
and II.llp --ll-I%. (ii)
If
tCC. t h e n l
.
lltl] = NtH i E
The norm
= sup[ < t , x ' ® y ' > : x ' E E '
is a d d i t i v e
on
the p o s i t i v e
'CF'
+'Y
cone
+
} =
C.. 1
~.6
Theorem.
order
units
Let
E,F be
u,v.
order
(i) E ~
F is a n o r d e r P unit u ®v and
(ii)
unit
normed
spaces
with
norm
determing
Then normed
space
with
norm
determing
order
E ® . F is a n o r d e r u n i t n o r m e d 1 u n i t u ® v a n d II.]]i = II.]IE .
space
with
norm
determing
order
Proof.
unit
(i)
If ~ E B b ( E , F ) +
libel
=
It f o l l o w s
~(u,v).
then by
~f IItIIp
that u ®v-tC~
%emma
I,
<
2.5(iii)
then ~(u,v)
. So u ® v
is
the
-
we
get
> O.
greatest
element
of
the
P unit
ball.
(ii)
If
Tltll
~
i, x ' ¢ E ' + ,
o ~ llx,NIly,IIit
follows
that
determing 4.7
unit
Theorem.
Let
med
spaces).
If E
E ®
F = E ®.F p x
Proof. order
Iltllp
if.lip = ll.lli.
(ii)
Let
E,F
are
units.
base
unit
Since
~p
NtIIp = i n f { k > O ' l u
4.8
Example.
Consider
the
the
cone
spaces
normed,
u ~
is
the
norm
(or b o t h
order
property,
unit
nor-
then
u'CE'+,v'CF'+
the n o r m
determing
obtain
for
normed = Ci,
we
space
all
spaces,
tC~'p = C.1.
uCE+,vCF+
the n o r m
deter-
obtain
@v+te~_ P = Ci}
v=O with
Therefore
decomposition
lltll i
=
order
'
isomorphic.
= C. w e p i
=
Thus
ruing o r d e r
C
.
normed
order
E,F are
=
lltlli ~
the R i e s z
and
then
]l.IIi = I].]]6.
be base
has
Since
i and
and
E,F
norm
(i) L e t units.
u ®v-tCC
order
y'CF'+
= []tlli
for all
iCE @F.
81
)~ Igv[
¢O
X1+ = [(~v)v=0 As
a Banach
With
the
space
" ~o
we
have
V=I
> o}.
k I = %1'
but
space,
where
we
consider
another
ordering.
norm
~=0 it
is a n
terming
order order
unit unit.
l]<:vL=oll it
is
a base
normed With
the
equivalent
= max( [~o] ,.~" 9=1
normed
space.
(I,0,0,...)
is
the
norm
de-
norm
I:~l)
We need
this
example
for
the next
propo-
sition. 4.9
Proposition.
E an
order
Let
unit
f:E
normed
the
Riesz
(ii) ~
has
lutely
summing.
Proof•
A
f ®id:
E ®£ l 1 ~ F ® u 1 1
linear
~ F be
space
a positive
and
F a base
decomposition
mapping
bounded normed
property,
f is a b s o l u t e l y is b o u n d e d
We
then
summing have
The
mapping
f ~d:E
theorem
4. 7 a n d
the
E @6 11
and
(ii)
The
theorem and
F ®pkl
mapping 4.7
F @i11
and
= F ®
construct
obtain
are
abso-
and that
bounded.
By
E @pl I = E ®.I i I
of
is p o s i t i v e
1 1 we
obtain
and
that
bounded.
E ®i11
By
= E @a 11
= F @~11.
5.
We
@ 11 1 ~ F ® i 1 1
properties
= F @pll
1 1 we
f'
or
11 "
f @id:E the
of
(i) E
%I.
~
® p k I -- F @ p l I is p o s i t i v e
properties
f and
operator,
If
if
l1
•
(i)
linear space.
another
An
approximation
regular
EtF
= E ®.F.
5.1
The
t-product.
Let
II.II~ i s
monotone
on t h e
problem
ordered
product
regular
ordered
EtF.
In
some
cases
1
Ct = [tee The
by
linear
the
space
regular
E,F
be
g~F
: < t , E ' +® ~ ' + > ~ O l .
EtF
= C t - C t is
norm
normed
spaces.
The
norm
cone
ordered
by
the
cone
C t and
normed
82
IItll, It
= inf
£11sll~
11.11~ ~ 2[1.11~.
is
Then
there
exists
Let
a
: s~tCcl}
for
(tn)n=l,lltnil
<
all
tCEIF.
2-n,
be a s e q u e n c e
(Sn)n=l,Sn¢C~,llSnlI <
sequence
2 -n
ElF.
in
such
that
s +t c C n-- n ~" Then t =
tn
s =
n=1 Since
s~tECt,
norm
n=1 is
ll-llt. T h e
5.2
The
two
(AL)-spaces,
page an
t an
element
i-product
approximation
410):
Is
same
where
problem
is w h e t h e r
In
case
two
of
F is
5.3
Definition. unit
5.4
Remark.
dual
5.5
E'
base
E
is a n
5.6
Definition. space
with
simplex
5.7 Theorem. E @
(ii)
E~F
of
of all
A
simplex
the
Riesz
space,
Let E,~
F is
an
in
the
E ~ . F = E~F. In the c a s e l by H.Schaefer ([14]
operator
a%%
case
of
an
we
have
nuklear
E,F
we
a more
ordered
(see
space
space
med
(i)
normed
converges
of
raised
this
spaces show
open
approximately
normed
E is a
unit
We w i l l
the
of
sum
(AM)-space E'~
operators
E
to
F = E'@ from
F : P E into
= ~i"
unit
is a p p r o x i m a t e l y
The
was
In
space
regular
if
is b a s e
Example.
space
An
normed~
the
space.
is w h e t h e r
compact
C~
order
E ~.F = EIF = E ~aF. 1 of a simplex space.
der
the
and
Banach
nuklear~
E'@.F, 1 F. T h e the
E'~
EIF a
problem
positive
F already
of is
problem
the
every
(AL)-space
E$F
normed ball
order
have
by
general
space
in
4.6 the
case
is a p p r o x i m a t e l y
is d i r e c t e d
unit
theorem
result
o__~r-
upwards.
normed,
if a n d
only
if
the
K.F.Ng[~)
compact order
selfadjoint
unit
selfadjoint
space
is
normed nuklear
an
only
if
be approximately
approximately
order
The
on a H i l b e r t dual
is
the
operators.
approximately
decomposition
if a n d
operators space.
order
unit
nor-
property. the
dual
order unit
E'
unit
normed
is a n
(AL)-space.
normed space
spaces.
Then
and
P
Proof.
is a n
(i)
We
approximately show
that
order
(E @
F)'
unit
normed
is a b a s e
space
normed
and
space.
P
Bb(E,F)+, 7>0
then
there
exist
xvEE+,
IIx, ll < 1,y,c~÷,lly~t[
E~F=E
~aF.
I f C01,C~2E
<
1
83
(V=I,2) such that
(~-~)II%11 ~ % ( % , y ~ )
~ I1%11
(v=~,2)
x 3 e £ +, IIx311 < 1, Y3eF+, Ily311 < 1
There e x i s t s
~ith
x3-xe£
+,
y3-y EF+. Then
('-~)(IIm111 (ii)
Denote
K and
L are
11®211) ~
+
( m , + m 2 ) ( x 3 , y 3) ~
b y K : [x'CE' + :l[x'll < w*-eompaet.
If 1
I},
t 1 ,t2gE
~aF,
IIt~ll
1
imately o r d e r u n i t n o r m e d ,
ll~zll.
+
L = [y'EF' + "I[y'll <
I<% ' x ' ® y ' > l < g + 7 IIx'lllly'll Since llx' II -- sup [ <x,x' > x e E ÷ ,
llmlll
for
< 1
all
then
x'eK,y'eL.
llxll < 11, K is w * - c o m p a c t
there
exist
I}.
uIEE+,
a n d E approx-
llUllI < 2- ~ ,
vIEF+,
4
IIv 111 < 2 -
~
such t h a t
I I< gI + Inductively
we
u n eE +
find
(u1®vl,x'@y'}
f o r all
x'EK,y'EL.
, 11%11 < z - T
VneF÷ ' Ilvnll < Z- T
su ch t h a t
n
[ ( t ~ , x ' ® y ' > I < 2 -n + ( ~ ~=I The
series
Ct
E ~
x'eK,y'CL.
U2Vn,
IIsll£ <
1, a n d
F is a n a p p r o x i m a t e
order
s + t v c C t.
unit
normed
space
and
E ~a F
- C t = EtF.
5.8
Theorem.
normed
If
space E ~
p
F = E ~
Since
spaces
the d u a l
rays. rays
In
the
E @
(K,L) the
origin.
x
simplex
F = EtF
as
manner
Thus
E ~
property
of all
O
a
space,
= E %F
F a n d E ~. F are P cones have bases
same
coincide.
approximation BA
'E i s
"F a n
approximate
order
unit
then
Proof.
at
for all
converge
S = ~ n=1 Thus
u~®vV,x'®y'>
in
continous
o
(K,L)
approximately and
[2] we
F = E @.F p i the l a t t e r
biaffine
= BA
are
order
generated
can
show,
= E~F
= E ~F.
space
coincides
functions
unit
by
that
these
Since with
on KXL,
normed
there
extreme extreme
E has the
which
the
space vanish
84
References
I. B e h r e n d s Mengen.
E. und W i t t s t o c k I n v e n t i o n e s math.
2. B e h r e n d s
E. und W i t t s t o c k
tiones math. 1 1 ~ 1 8 8 - 1 9 8 3. D a v i e s E.B.,
4. Davies
10)251-266
G,: T e n s o r p r o d u k t e und S i m p l e x e .
G.F.: T e n s o r products,
Trans.
Amer.
J o u r n a l Math.
ideal Math.
5. F r e m l i n D . H . : T e n s o r p r o d u c t s
544-555
131,
(1972)
spaces.
A.J.: A f f i n e p r o d u c t s
F.)Phelps
Pacific J. Math. Kung-Fu:
of simplexes.
R.R. :Tensor p r o d u c t s
3_!1,469-480
Math.
Scand.
22)
of c o m p a c t convex
sets.
rule for p o l a r s . M a t h .
Scand. 2 6
(1970)
Evanston
and London:
topological
vector
ducts.
Proc.
L o n d o n Math.
12. Popa N. : P r o d u i t et Appl.
I~, 2 3 5 - 2 4 6
topological
t e n s o r pro-
I__9)177-190 (1969)
ordonnes.
Rev.
R o u m a n i Math.
Pures
(1968) v e c t o r spaces.
tensor p r o d u c t s
13, 4 O O - 4 1 5
Stinespring:
Berlin-Heidelberg-
of Banach
lattices.
Israel
(1973)
Positive
Soc. 6, 2 1 1 - 2 1 6
16. W i t t s t o c k
N e w York,
Springer:t971
14 S c h a e f e r H . H . : N o r m e d J. Math.
Soc.
tensoriels
13. S c h a e f e r H.H.; T o p o l o g i c a l N e w York,
spaces.
Harper ~ Row:t967
11. P e r e s s i n i A . L and S h e r b e r t D.R.: O r d e r e d
math.
A n a l y s i s ~,
(1969)
On a c o m p u t a t i o n
10. P e r e s s i n i A.L.: Ordered
Math.
of tensor pro-
J. F u n c t i o n a l
(1968)
8. Namioka.
15. W.F.
of a
(1968)
(1968)
165-175
14-16
Scand.
of a r c h i m e d i a n v e c t o r lattices.
777-798
94,
ducts of p a r t i a l l y o r d e r e d
9. Ng.
Math.
t h e o r y of the p r e d u a % Soc.
6. H u l a n i c k i A. and Phelps R.R. : Some a p p l i c a t i o n s
7. Lazar,
i n f i n i t e pro-
limits of C h o q u e t simplexes.
E.Bo: The s t r u c t u r e and
177-201
Inven.
(1968)
B a n a c h lattice.
Amer.
(1970)
(1970)
Vincent-Smith
ducts and p r o j e c t i v e 2_~2,145- 164
G~ T e n s o r p r o d u k t e k o m p a k t e r k o n v e x e r
{unctions on C * - a l g e b r a s .
Proc.
Amer.
(1955)
G.: C h o q u e t S i m p % e x e und n u k l e a r e Pd~ume.
I_~5)251-258 (1972).
Inventiones
Lecture Notes in Physics Bisher erschienen / Already published Vol. 1: J. C. Erdmann, W~rmeleitung in Kristallen, theoretische Grundlagen und fortgeschrittene experimentelle Methoden. 1969. DM 20,VoL 2: K. Hepp, Th6orie de la renormalisation. 1969. DM 18,Vol. 3: A. Martin, Scattering Theory: Unitarity, Analyticity and Crossing. 1969. DM 16,Vol. 4: G. Ludwig, Deutung des Begriffs physikalische Theorie und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch Haupts~tze des Messens. 1970. DM 28,Vol. 5: M. Schaaf, The Reduction of the Product of Two Irreducible Unitary Representations of the Proper Orthochronous Quantummechanical Poincar6 Group. 1970. DM 16,Vol. 6: Group Representations in Mathematics and Physics. Edited by V. Bargmann. 1970. DM 24,Vol. 7: R. Balescu, J. L. Lebowitz, I. Prigogine, P. R6sibois, Z. W. Salsburg, Lectures in Statistical Physics. 1971. DM 18,Vol. 8: Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics. Edited by M. Holt. 1971. DM 28,Vol. 9: D. W. Robinson, The Thermodynamic Pressure in Quantum Statistical Mechanics. 1971. DM 16,Vol. 10: J. M. Stewart, Non-Equilibrium Relativistic Kinetic Theory. 1971. DM 16,Vol. 11: O. Steinmann, Perturbation Expansions in Axiomatic Field Theory. 1971. DM 16,Vol. 12: Statistical Models and Turbulence. Edited by M. Rosenblatt and C. Van Atta. 1972. DM 28,Vol. 13: M. Ryan, Hamiltonian Cosmology. 1972. DM 18,Vol. 14: Methods of Local and Global Differential Geometry in General Relativity. Edited by D. Farnsworth, J. Fink, J. Porter and A. Thompson. 1972. DM 18,Vol. 15: M. Fierz, Vorlesungen zur Entwicklungsgeschichte der Mechanik. 1972. DM 16,Vol. 16: H.-O. Georgii, Phasen0bergang 1. Art bei Gittergasmodellen. 1972. DM 18,Vol. 17: Strong Interaction Physics. Edited byW. R0hl and A. Vancura. 1973. DM 28,Vol. 18: Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Vol. I. Edited by H. Cabannes and R. Temam. 1973. DM 18,Vol. 19: Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Vol. I1.Edited by H. Cabannes and R. Temam. 1973. DM 26,-
Vol. 20: Statistical Mechanics and Mathematical Problems. Edited by A. Lenard. 1973. DM 22,Vol. 21 : Optimization and Stability Problems in Continuum Mechanics. Edited by P. K. C. Wang. 19?3. DM 16,Vol. 22: Proceedings of the Europhysics Study Conference on Intermediate Processes in Nuclear Reactions. Edited by N. Cindro, P. Kuli~ie and Th. MayerKuckuk. 1973. DM 26,Vol. 23: Nuclear Structure Physics. Proceedings of the Minerva Symposium on Physics. Edited by U. Smilansky, I. Talmi, and H. A. WeidenmLiller. 1973. DM 26,Vol. 24: R. F. Snipes, Statistical Mechanical Theory of the Electrolytic Transport of Non-electrolytes. 1973. DM 20,Vol. 25: Constructive Quantum Field Theory. The 1973 "Ettore Majorana" International School of Mathematical Physics. Edited by G. Velo and A. Wightman. 1973. DM 26,Vol. 26: A. Hubert, Theorie der Dom~nenw~nde in geordneten Medien. 1974. DM 28,-
Vol. 27: R. Kh. Zeytounian, Notes sur les Ecoulements Rotationnels de Fluides Parfaits. 1974. DM 28,Vol. 28: Lectures in Statistical Physics. Edited by W. C. Schieve and J. S. Turner. 1974. DM 24,Vol. 29: Foundations of Quantum Mechanics and Ordered Linear Spaces. Advanced Study Institute Held in Marburg 1973. Edited by A. Hartk~mper and H. Neumann. 1974. DM 26,-
