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Séminaire Dubreil.
Algèbre et théorie
des nombres
L ÁSSLÓ F UCHS
Riesz vector spaces and Riesz algebras
Séminaire Dubreil. Algèbre et théorie des nombres, tome 19, no 2 (1965-1966), exp. no 2324, p. 1-9
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Seminaire DUBREIL-PISOT
(Algèbre et Theorie des Nombres)
19e annee, 1965/66, nO 23-24
23-01
20 et 23 mai
1966
RIESZ VECTOR SPACES AND RIESZ ALGEBRAS
by Lássló FUCHS
1. Introduction.
In
1940, F. RIESZ investigated the bounded linear functionals
spaces S, and showed that they form a vector lattice whenever
possess the following interpolation property.
(A) Riesz interpolation property. -If f ,
in
S
such
h
S
such that
E
Clearly,
(choose
e.
that
if
S
g.
h =
is
f1
(1)
a
polynomial
(2)
v
(0 , 1) ,
not
f
if and
1)
in the unit interval
only
if
f(x) ~
0
for all
x E
where
some
put
0
f >
is the
for
so-
ordering).
coefficients, defined everywhere
in
(0 , D,
pointwise ordering.
positive integer k, the k-times differentiable
under the pointwise ordering.
a
real functions in
The real analytic functions in
(5)
The real
(6)
The real-valued continuous functions in the unit interval
(0, 1) ,
under the
(0,
the
trigonometric functions
compact Hausdorff
(this ordering will be
(7)
w9
(0, 1) (this
(4)
on a
there is
interpolation property
but there exist a number of important
A
lattice-ordered and have the Riesz interpolation
The rational functions with real
For
and j = 1 , 2 , then
g 2 ~,
f~
are
polynomials
pointwise
under the
(3)
1 , 2
functions
are
g~
For instance :
The real
called
i =
is assumed to
S
lattice then it has the Riesz
a
function spaces which
property.
for
gj
real function
on
space)
if
called the
f > 0
in
has the
(1)-(5)
ordering
(or
f(x)
inequalities).
ipeaning
ordering by strict
The function spaces mentioned in
pointwise ordering.
under the
>
0
is
pointwise.
more
generally,
for all
x
ordering by strict ine-
qualities.
(8)
perty
The cartesian product of function spaces with the Riesz
has again the same property.
interpolation
p ro-
As
further example of
a
property,
lation
reals)
the
over
ing,
iy
x +
we
partially
x +
x >
means
>
iy
These examples whow that it
0
(considered
night
0 ,
as
a
interpo-
vector space
in another order-
or
reals).
y are
be of
y >
and
0
x >
means
(x,
0
ordered vector space with the Riesz
complex numbers
may mention the
where
0
>
a
interest to investigate
some
systemati-
cally the partially ordered vector spaces with the Riesz interpolation property.
Our present aim is to give a brief survey of the results 9 some of the theorems
can be formulated more generally for partially ordered groups. For the proofs, we
refer to
2. Bas i c
By
our
definitions.
vector space
a
is said to be
such that
~
papers
V,
V
in
b
shall
implies
the field
over
mean one
ordered if it is
partially
a
we
partially
c $ b
a +
+
R
ordered under
for all
c
V
of real numbers.
a
binary relation
~a ~ ~b
and
V
c E
positive real X . The set P of all a E V , satisfying a ~ 0 , is
called the positivity domain of Vy and the elements of P are called positive.
b is equivalent to
P completely determines the partial order of V since a
for all
b-» aE P .
such that
a ~
and
c
ordered. If
a, b E
V
b ~
is
(union)
be
u,
is directed if and
V
a ~ b
V , either
lattice under its
a
b,y
a v b
of the elements
a, b
lattice. We shall write
1.
c.
V ,
there exists
c
some
E
V
is the
only if every element
positive elements.
difference of two
If for all
a, b E
if, to each pair
is called directed
V
a A
or b fi
V , then
in
a
partial order,
V
even
if
vector
a
(intersection)
might exist
which
totally
then it is called
for the g. 1. b.
E
is
V
and the
V
is not
lattice-ordered. A directed partially ordered vector space with the Riesz inter-
polation property
will be called
polation property
(A)
is
a
equivalent
Riesz vector space. Note that the Riesz interto either
(B) Decomposition property. - If
bi ~ 0 , then there exist al ’ a2 E V
(c)
b.
J
such
of
Refinement
property. - If
a
V , then there exist elements
that
0
a
b
+
b
such that a =
+
a~
=
c..~0
~-J
b
following conditions :
of the
one
+
b
in
V
for
a
al + a2
for
(i
bi
E
V ,
where
positive elements
=1 ~ 2 ; j= 1 ,
a. ,
2)
A
a , b
E
dered
(i.
W,
it contains
e.
necessarily
has
a
V
E
x
partially ordered vector space
and a ~ x ~ b imply x E W . If
of
W
subspace
o-ideal is defined
A"
convex.
as
o-ideals of
THEOREM 1. - The
Riesz vector space
a
V/W
W
subspace
becomes
THEOREM 2. I
If
V/I
and
A vector space
is
V
homomorphism that
isomorphism
o-isomorphism.
an
distributive sub-
a
positive
vector
coset
a
I
is
a +
(w
a + w
its
mod
V
E
is defined
W
W) .
V , then
o-ideal of
an
convex
are Riesz vector spaces.
A’ sector space
be
a
Riesz vector space and
a
form
ordered vector space if
partially
o-simple.
ordered vector space
partially
a
whenever it contains
positive
both
a
of
V
V .
lattice in the lattice of all subspaces of
The factor space
V
or-
then it is
subspace. If
convex
if
convex
trivially
),
b
a
o-ideals other than the trivial ones, it is called
no
to be
directed
a
is
subspace
a
such that
a,y b
two elements
no
is said to be
V
positivity
preserves
that is order
is called
an
o-homomorphism.
preserving in both directions is said to
3. Algebraic theory.
In order to
first
type
In
investigate the
have to introduce the
we
a
ordered vector space
partially
a A
b
whenever
other go 1. b.
or
simplest and,at
the
same
algebraic point of view,
time,the most important
of Riesz vector spaces : the antilattices.
g. 1. b.
no
Riesz vector spaces from the
a n
b
existence of
lattice is
a
exist~ io
is called
b ,
=
v
a
a ~ b
b
implies
b
a
Ab
antilattice. Because of
b
a v
=
a
Riesz vector space in which
a, b
have
a . A Riesz vector space
the existence of
e.
an
or
V,y the elements
in
= b
or
in
a v
an
A
always
A,
implies
(-
in which
a n
b
=
a
b) ,
the
thus
an
anti-
g. 1. b. and 1.
u,
b.
b
=
-
a A -
antilattice,
only the trivial
a
exist.
It is easy to conclude that if
have a g.
Examples
§ 1. Also,
an
1.
b.,
say
for antilattices
all
totally
a
finite number of elements of
then
... A
a == b ,
are
abundant ;9
see
ordered vector spaces
antilattice which is lattice-ordered is
The following is the structure theorem
on
a. = b
our
are
a
for
examples
an
antilattice
some ’ o
(~,~~,~~(2~, (4)-(7)
in
antilattices. On the other hand,
totally
ordered vector space.
Riesz vector spaces. It shows that the
antilattices play a similar role in the theory of Riesz vector spaces
as
the to-
in the
tally ordered vector spaces
theory of vector lattices.
(a E I)
ces
A
sian
product
fl
and
such that
A
~il
notation n
Here the
if,
considered as § 0
that
means
and
onto
an
(...,
element
,
of antilatti-
subspace of the carte-
a
u,
b. too.
...)
aa ,
for every
0
only if,
family
a
preserves g. 1. be and 1.
cp
-
V
of
o-isomorphism y
an
there exists
V
THEOREM 3. - To every Riesz vector space
a E
E
~!
is to be
A
Cc
I .
More information about the structure of antilattices may be obtained from the
following two results ,
THEOREM 4. - If
Thus
antilattice is
an
THEOREM 5. - Assume that
extension of
embedded
for
If
is
A
in
o-isomorphically
an
H
a
a
means
that ... ,
C
...~
of
A
is
product ~ B~
cartesian
b03B2 ,
n C
>
0
is defined
0 . Then
of
if,
totally
and
A
can
be
ordered vector
only if,
bo
b03B2
>
0
all 03B2 .
A
THEOREM
is
as
in theorem 5 and is in addition
to the real
6* - If
is
o-isomorphic to
the
ordering
is
numbers, and
is an
A
a
we
o-smmple, then
all the
B
are
then
A
get :
o-simple antilattice
such that
vector space of real-ralued functions
(1 C = 0 9
on some
set 0 where
given by strict inequalities.
Note that by choosing suitable
4.
trivially ordered
antilattice such that the intersection
j) c
B
o-isomorphic
ding
maximal
trivially ordered vector space by
a
of all maximal trivially ordered subspaces
Here
a
totally ordered.
is
an
is
ordered vector spaceo
totally
spaces
C
antilattice and
an
A/C
A, then
of
subspace
is
A
to elements of
A
isomorphisms
B- ~ R ,
the functions correspon-
will be bounded.
Topological questions.
The continuous functions
with the uniform
topology.
on a
compact Hausdorff space Q
are
usually furnished
This
topology
the
one
may be
regarded
as
obtained by
ordering the continuous functions by strict inequalities and then taking as a
subbase of neighbourhoods at 0 the open intervals (- f , f) with functions
f > 0 . In accordance with this remark,
follows.
we
introduce the
open-interval topology
as
V
Let
topology
as
0 . This
rise to
gives
We
open-interval topology.
the
as
take the open inter-
V ,
in
neighbourhoods at
subbase of open
which will be referred to
V
on
a
0
u >
Riesz vector space. For all
a
(- u , u)
vals
a
be
have :
open-interval topology
THEOREM 7. - For the
of
a
the fol-
V
Riesz vector space
lowing hold :
(i)
It is Hausdorff
ordered
vially
(ii)
(iii)
V
is
Hausdorff, then V
is
C
subspaces
If it is
of
is non-discrete in the
V
the intersection
if, and only if,
of all maximal tri-
n C
0 ;
topological
group in this
topology ;
only if, it is
and
open-interval topology if,
antilattice ;
an
(iv)
If
is not non-discrete then every dense
V
subspace
V
of
is
an
antilat-
tice.
Notice that the
hypothesis
of theorem 5 is thus
equivalent
to
simple topolo-
a
gical condition.
Let
be
A
gy. We know
antilattice that is
an
by
(ii)
that
A
Hausdorff space in the
a
is then
a
topological
topological completion A*
is again a topological group
..By
the structure of its
gical
groups,
space
over
Cauchy
then
If
Air
R . If
is constructed
which
of
means
is,
in
particular, the antilattice
inequalities,
(0, 1) ,
continuous functions in
This
is, however,
in §
1.
not true in
approximation antilattice,
general,
a , b
E
A
and
if x
E
A
satisfies
u >
0
in
x
if the
A
A*
A
as
a , b
then
following
for
A
c
(0, 1)
by
condition is satisfied
+ u .
such that
It is easy to
in order that
A*
be
c
see
a
case.
(1)
example
our
approximation property,
A
under
the vector space of all
vector lattice in this
a , c e
a
ordered vector space.
is shown for instance
has the
x
a
just
call
we
positive elements,
polynomials in
is
is
A , there exists
proximation property is necessary
THEOREM 8. - Let
A*
then
thus
We shall say that the antilattice
of all
of
vector
a
if
Cauchy nets, and
topolo-
on
is, in addition,
positive whenever it contains a net consisting
readily checks that A* is likewise a partially
the ordering by strict
an
by
may ask for
so we
standard results
net
one
A
A*
group, and
open-interval topolo-
o
or
is
A
Given
a ,
b , and
that the ap-
vector lattice.
be an antilattice which is Hausdorff in its open interval
"
topology.
Then :
(i) If
0), then
(ii)
is metrizable
A
it has
e.
is
countable system of
a
A*
topological completion
its
A*
(i.
is
The canonical map of
is not necessarily
A
A~’
into
is of
o-isomorphism.
an
of
lations, but the image
A~~
in
A
Riesz vector space ;
a
if,
vector lattice if,and only
a
has the
A
course
contains
more
approximation property.
continuous
a
A
embedding
The
neighbourhoods at
~ A~’
isomorphism,
but it
preserves order
positive elements
in
re-
general,
namely, the positivity domain of the image of A in A~~ is the least closed set
P containing the image of the positivity domain P of A . For instance, if A
is the vector space of polynomials in
(0 , 1) under the ordering by strict inequalities, then in A* a polynomial f will be positive if it can be approximated
by polynomials positive in A, i. e. in the image of A in A~~ the ordering will
be
pointwise.
Let
be
L
a
L
vector lattice. We say that
can
be
the antilat-
approximated by
A, if :
tice
(a) A
is
a
A*
completion
The
L ( A
partially ordered vector
positivity domain P by its clusure P ~.
by replacing its
A
space obtained from
(b)
subspace of
vector
of
is
A
a
denotes the
vector lattice
containing
L
as
vector
a
sublattice. :
THEOREM
mating
L
approximating
there exists
It is
5.
an
we
whi ch
our
groups, but also
to say,
we
also
vector lattice. There exists
L , and every dense subspace
Conversely,
if
are
B
has
R x A
this is its
a
proper dense
topological
Ay which
vector spaces in the
A
that the
antilattice
subspace
then
A
=
L .
B.
THEOREM 10. - An antilattice
it is
are
not
only topological
is
mapping
is continuous
topology if,and only if,
an
open-interval topology. That
~a
(where R carries its
open-interval topology). It is easy to prove
into
is
approxi-
to pological vector spaces.
(~. y a) 2014~
of
A
of
is dense and which satisfies
B
attention to antilattices
assume
B
A
antilattice
y
A
cases
an
an antilattice approximating L ,
is
in which
A
in which
problem
open
turn
=
a
antilattice
an
Antilattices
Next
A
such that
L
be
L
9. - Let
A
is
a
usual
:
topological vector
o-simple.
topology ; actually,
space in its
open-interval
For the sake of
brevity,
shall call
we
antilattice
an
A
that is
a
topological
vector space topological.
Let
us
fix
u)
U =
(-
u ,
for
an
arbitrary
(which
element
some
0
the unit ball of
a E
A ,
we
~...~ ,
Recall that
topological antilattice
This is justified in view
in
functional),
topological antilattice
abstract
of the fact
that,
if
set
and the dual space of
an
A,y and call
a
A .
is the well-known Ninkowski
THEOREM 11. - A
norm
u >
A
is
then
we
obtain :
normed vector space under the
A
is
an
abstract Lebesgue
Lebesgue space is
a
a
space.
normed vector lattice which is
a
real
Banach space such that
The
following
result
gives
a
fairly good characterisation of topological antilat-
tices.
THEOREM 12. - For every
topological antilattice
A,y there exists
a
topological
~-isomorphism ~ of A,y with a subspace of all real-valued continuous functions
on a compact Hausdorff space 0 , where the functions are ordered by strict
inequalities. Moreover, 03A6
Here Q
can
so
is the space of all maximal
nished with the Gelfand
The
be chosen
following theorem
THEOREM 13. -
property, then
If
it is
A
as
to be
an
trivially
isometry
ordered
as
well.
wubspaces
of
A, fur-
topology.
of Stone-Weierstrass
is
type
may be mentioned.
topological antilattice satisfying the approximation
o-isomorphic and isometric to a dense subspace of the space
a
of all real continuous functions
compact Hausdorff space Q , ordered by
strict inequalities, and the topological completion A* of A is the vector lattice
6.
C(Q)
on a
of all continuous functions on 03A9
under the pointwise ordering.
Riesz algebras.
An
algebra
partially
over
the real numbers is called
ordered vector space with the
partially ordered algebra if is a
property that its possitivity domain is
a
closed under algebra
tion that
The last
multiplication.
inequalities
may be
is
property
from the left and from the
multiplied
to the condi-
equivalent
right by posi-
tive elements.
ordered
partially
A
tor space is
Riesz vector space. Our
a
A
partially
algebra.
X
then
Here
identity
some
have meant
we
ideal. As
a
o-ideal of
an
an
o-ideal of
an
x E
Riesz
are
is said to be
x A
algebra
z
X
or
=
0 .
X
a
obtains that
one
1
underlying
as
an
antilattice-
an
x A
and
convex
vec-
algebras.
o-ideal in
an
embedding such that if
an
satisf ies
corollary to theorem 14,
o-isomorphic with
7.
by
(1)-(9) in §
be embedded
then either
0
z >
E X ,
3
There is
e .
x , y > 0
with
x, y e Y
sope
x E
with
Y
can
if its
algebra
antilattice,
an
X
algebra
Riesz
a
examples
algebra, that is
ordered
THEOREM 14. - Every Riesz
algebra
is called
X
algebra
y
=
if,
directed
Riesz
a
0
e.
for
g.,
algebra
antilattice-algebra
is
antilattice-algebra with identity.
an
Antilattice-al ebras.
If
introduce the
we
multiplication
need not be continuous in this
criterion for the
Call the
u >
each
continuity
partially
0 in X ,
9
ordered
of
We
are
m-hounded if every
X
v >
some
ring
going to give
0
in
X
a E
a
an
X
satisfies : to
such that
an
Note that if the additive group of
is
topology.
then
algebras,
ring multiplication.
algebra
there is
in antilattice
antilattice-algebra X that is a Hausdorff
topology multiplication is continuous if, and only if,
THEOREM 15. - In
interval
open-interval topology
antilattice-algebra
is
space in the open-
X
is
m-bounded.
o-simple,
then it
m-bounded.
For
Let
not
m-bounded
us
turn
antilattice-algebras
finally
to
in the
o-simple.
is
a
which
8 holds.
topological algebras, i. e.
open-interval topology, but at the same time toare
spaces.
THEOREM 16. - An antilattice
topology
analogue of theorem
antilattice-algebras
only topological rings
pological vector
the
algebra
topological algebra if,
X
and
which is Hausdorff in its
open-interval
only if its underlying vertor space is
23-09
The main result is the following
THEOREM 17. - Let
is
X
be
an
Hausdorff space in the
a
then it is
space,
representation theorem.
antilattice
open-interval topology. If
o-isomorphic
to
an
Q
where the functions
In this case,
X
is a normed
an
isometry as
Note that
but
our
If
X
we
is
and
o-simple
are
algebra,
ordered
and the
assume
as
on a
a
it
vector
compact
by strict inequalities.
o-isomorphism
may be assumed to
well.
have not
a
priori supposed that X
was
associative
or
commutative,
result indicates that it must be both associative and commutative.
is
as
in theorem
Banach algebra which is
a
then its
topological completion is
Riesz algebra too. If, in addition, X
17,
proximation property, then
its
completion
on Q
are
ordered
where the functions
If X
X
e
algebra of continuous functions
Hausdorff space
be
algebra with identity
is the real
it is easy to
see
polynomials in
that Q
will be
is the
a
commutative
enjoys the
ap-
algebra of all continuous functions
pointwise.
(0 , 1) ,
ordered
homeomorphic to
by strict inequalities, then
(0 , 1) .
p
REFERENCES
[1]
[2]
FUCHS
[3]
FUCHS (Lássló). - On partially ordered vector spaces with the Riesz
interpolation property, Publ. Math. Debrecen, t. 12, 1965, p. 335-343.
FUCHS (Lássló). - Riesz rings, Math.
Annalen, t. 166, 1966, p. 24-33.
[4]
FUCHS
(Lássló). (Lássló). -
Riesz groups, Ann. Sc. Norm.
Sup. Pisa, t. 19, 1965,
p. 1-34.
Approximation of lattice-ordered groups, Ann. Univ. Scient.
Budapest, t. 8, 1965, p. 187-203.