On Banach space valued extensions
from split faces.
Tage Bai Andersen
This work was in part supported by the Danish Science
Council~ J. No. 511-630.
- 1 ....
The aim of this note is to prove the following theorem:
Suppose a
F
of a
is a continuous affine map from a closed split face
compact convex set
K with values in a Banach space
enjoying the approximation property.
Suppose also that
p
is a
strictly positive lower semi-continuous concave function on
l!a(k) II
such that
_::
p(k)
for all
continuous affine extension
l!a(k) II
_::
p(k)
for all
k
to
~
in
k
in
F
K into
Then
B
a
B
K
admits a
such that
K ..
We shall use the :rrethods of tensor products of compact convex
sets as developed by Semadeni
[12]~
Lazar [9], Namioka and Phelps
[10] and Behrends and Wittstock [6] to reduce the problem to the
case
B = JR. , and in this case the result follows from the work/
of Alfsen and Hirsberg [3] and the present author [4].
The author wants to thank E. Alfsen for helpful comments.
We shall be concerned with compact convex sets
in locally convex spaces
E1
and
E2
K1
respectively.
and
By
A(Ki)
K.l
we shall denote the continuous real affine functions on
i
= 1,2
•
We let
BA(K 1 x K2 )
biaffine functions on
and that
BA(K 1 x K2 )
K1 X K2 •
We observe that
we define the projective tensor product of
logy.
Then
K1 ® K2
morphic embedding
arise) from
BA(K 1 x K2 )
~
K1 x K2 •
K1
and
E BA(K 1 x K2 )
As usual
K2
~
K1
®
K2 ,
equipped with the w*-topo-
is a compact convex set, and we have a homeo-
wK 1 x K2
K1 x K2
for
be the Banach space of continuous
separates points of
to be the state space of
K2
into
(called
w , when no confusion can
K1 0 K2
defined by the following
- 2 -
We notice that
w is a biaffine map.
It was proved in [10;
Prop. 1.3, Th. 2.3] and [6; Satz 1.1.3] that
oe(K 1 ® K2 ) =
w(oeK 1 x oeK 2 ) , where in general we denote the extreme points of
a convex set
For
a
K by
in
oeK •
A(K 1 )
biaffine function
and
a ®b
A(K 1 ) ® A(K 2 )
We let
b
A(K2 )
in
we define the continuous
by
be the real vector space
n
I
A(K 1 ) ® A(K 2 ) = [ l: a.® b.
"1J.
J.
J.=
a. EA(K 1 ), b. EA(K 2 )}
J.
J.
A(K 1 )
which is a copy of the algebraic tensor product of
A(K 2 ) .
A(K 1 ) ®e A(K 2 )
We denote by
A(K 1) ® A(K 2 )
in
the uniform closure of
BA(K 1 x K2 ) •
We recall that a Banach space
B
is said to have the approxi-
mation property if for each compact convex subset
each
e > 0
T(B)
is finite dimensional and such that
x E C
there is a continuous linear map
It is proved in
Following Lazar [9] we define
Let
P.
J.
and
A(K 1 )
T1a
=a
T2b
=
A(K 2 )
into
and
BA(K 1
® 11
all
a E A(K 1 )
11 ® b
all
b E A(K 2 )
be the adjoint map of
T.J.
of
~
B
!JTx- xll < e
BA(K 1 x K2 )
T1
C
T: B
[10; Lem. 2.5] that if
has the approximation property then
embeddings of
and
for
T2
X
B
and
such that
for all
A(K 1 ) (or A(K 2 ))
= A(K 1 )
®e A(K 2 ).
as the natural
K2)
9
i = 1 '2
i.e.
.
- 3 -
Then
and continuous map of
K1 ® K2
onto
P.~
is an affine
Ki ( = state space of
A(Ki)) , and
The first part of the following proposition was proved by
Lazar in the case where
K1
and
K2
are simplexes, but the
proof l1olds in general.
The last part was proved by Lazar in the
simplex case by means of the Stone-Weierstrass Theorem for simplexes.
Proposition 1.
vex sets
(i)
Let
and
K1
Then
F
F1
K2
and
F2
resp.
be closed faces of compact con-
Let
F
is a closed face in
= P1 1 (F 1 ) n P2 1 (F 2 )
K1 ®
K2
and
F = co(w(F 1 x F 2 ))
(ii)
If
A(F 1 )
F 1 ® F2
Proof:
A(F 2 )
has the approximation property then
is affinely homeomorphic to
P.~
Since
Pi 1 (Fi)
or
F •
is continuous and affine it is immediate that
is a closed face of
K1 0 K 2
and hence
9
F
is a closed
face.
Now let
and hence
rem:
p
co(a(F 1
p = w(k 1 ,k 2 )
E
P1 1 (F 1 )
X F 2 ))
Conversely, let
Hence
w(F 1 x F 2 )
E
n P2 1 (F 2 )
Then
•
P.p
~
= k.
~
E F.~ ,
=F.
By the Krein Milman Thea-
Since
F
_s F •
p E oeF
x.~ E
oe K . •
~
Then
is a closed face we get
P.p
= x.
~
~
belongs to
- 4 -
F.~
by the definition of
F
p E w (F 1 x :b' 2 ) ~ and again
Hence
F S co(w(F 1 x F2 ))
by the Krein Milman Theorem
~
and
(i) is
proved.
Now we shall prove
(ii) under the assumption that
has the approximation property.
affine map
T: F 1
( Tc:p) ( b ) =
Then
= 1 '2
1
X
.
We shall define a continuous
K1 ® K2
~
by
F ) ' c:p E F 1
F2 ' b E BA ( K1 X K2 )
®
2
is compact and convex in
T(F 1 ® F2)
then
cp
= WF
X
1
F (x1 ,x2)
2
?
.
K1 ® K2
where
If
x.~ E oe F.~
'
But then
(Tc:p)(b)
Hence
IF
cp ( b
cp E ae(F1 ® F2)
i
F2
®
A(F 1 )
= b(x 1 ,x 2 ) = WK
1
X
Tc:p = wK x K (x 1 ,x 2 ) E co(wK
1
2
X
1
K (F 1
w E oeF
X
2
Krein Milman Theorem we conclude that
Conversely, if
bE BA(K 1 X K2 ) •
K (x 1 ,x 2 ) (b), all
2
then as
T(F 1
F
F2 )) = F •
By the
F2 ) c F •
®
is a closed face, we
get by Milman's theorem
If
= wK x K (x1 ,x2)' x. E oeFi, then wF 1 x F2 ( x1 'x2) E o e (F1
W
1
and as above
~
2
W= T(wF xF (x1,x2))
1
2
and
so
F :: T(F 1 ® F2)'
we get
We proceed to show that
if
BA ( K1
X
K2) IF
A( K2 ) IF
A( K1 )
®
BA(F 1
X F2 )
and
1
X
1
X
€
F
F
2
2
property, we have that
exist
is dense in
Since
By the Klein Milman Theorem
is injective.
T
BA(F 1 x F2 )
BA(F 1
A(F 1 )
X
F 2 ).
This is the case
We show that
Hence let
c E
has the approximation
A(F 1 ) ®8 A(F 2 ) = BA(F 1 ® F 2 ), so there
a 1 , ••• ,an E A(F 1 ), b 1 , ••• ,bn E A(F 2 )
lie - i~1ai
®
bi\!p1
F2) '
is surjective.
T
is dense in
> 0 .
.
®
X
F2 <
~
•
such that
- 5 -
Now
A(Ki) lF. is dense in A(FiL so we can choose
1.
b~ E A(K 2 ),
i = 1 , ••• , n
such that
1.
n
II . L: 1a.1. ®b.1. - . nL: 1a.1.' ® b 1.~ llp 1 X F2 < 2e: •
1.=
1.=
n
Then
!lc -
L: a~ ® b 1~ lip
i=1 1.
1
xF
2
< e: , and the claim follows.
The next step is to prove that
split face of
Ki
for
K1 ® K2
provided
i = 1,2 , and f.ex.
a.I E A(K1 L
1.
Fi
A(F 1 )
co(w(F 1 x F 2 ))
is a closed
is a closed split face of
has the approximation property.
We shall remind the reader of the following definitions and
facts:
If
F
complementary
from
F
is a closed face of a compact convex
a-face
F'
is the union of all faces disjoint
It is always true that
a split face if
F'
K , then the
K = co(F U F')
is a face and each point in
F
is called
K~(F U
F') can
be decomposed uniquely as convex combination of a point in
a point in
F' •
F and
It follows from a slight modification of the
proof of [2; Th. 3.5] that a closed face is a split face if and
only if each non-negative
an
u.s.c. affine function on
u.s. c. affine extension iD
K , which is equal to
F
admits
0
on
F' •
This characterization is sometimes inconvenient because of the "nonsymmetric" properties of the affine functions involved.
Using the
above characterization we shall give a new one involving the space
As(K)
which is the smallest uniformly closed subspace of the
affiJle
bounded functions on K containing the bounded u.s.c./functions.
This space has been used f.ex. by Krause [8] and Behrends and
Wittstock [6] in simplex theory and by Combes [7] in
theory.
C*-algebra
We shall state some of the lmovm properties of
- 6 Lemma 2.
(i)
If
a E As(K)
and
(ii)
If
a E As(K)
then
(iii)
If
a E As(K)
then
Sketch of proof:
K
and
K •
(i)
9
s < t
If
on
s
oeK
a > 0
on
then
oeK
a > 0
on
K •
lla!iK = l!aJio K •
e
a satisfies the barycentric calculus.
and
t
are
u.s.c. affine functions on
it follows by [5; Lem. 1] that
s < t
on
Hence (i) follows by a limit argument.
Now (ii) follows by
since on
Hence the same in-
oeK: - l!a!1 0 K _:: a_:: l!a\! 0 K •
e
equality holds on
lity is trivial.
K
e
and so
9
l!ai!K .:5. l!all 0 K' The converse inequae
Finally (iii) follows from Lebesgue's theorem
on dominated converger.ce 9 since the barycentric calculus holds
for (differences of)
u.s.c. bounded affine functions, cf. [1;
Cor. I,1.4].
Proposition 3.
Then
F
Let
F
be a closed face of a compact convex set K.
is a split face if and only if each
As(F)+, A(F)
A(F)+ 9 A(F;K), A(F;K)+)
9
a=
such that
0
on
F' •
a E As(F)
has an extension
(or
a E As(K)
If such an extension exists then it
is unique.
Proof:
oe K
The uniqueness statement follows from Lemma 2 (ii), since
c F U F'
•
Assume
F
-
is a split face and let
u.s.c. affine and non-negative
affine extension
lows if
a
functions on
a
with
a=
a
0
a E As(F) •
has as noted above an
on
F' .
In general there are
bn' en
and non-negative, an = bn - en , such that
a
is
u.s.c.
Hence the result fol-
is the difference of two non-negative
K .
If
u.s.c. affine
u.s.c. affine
l!an- alJF n::coo .
We
- 7 -
use Lemma 2 (ii) and the fact that
oeK c F U F'
to conclude
that
!lan - am!! = lla
' n - a m1! 0 e K = lian - a m11::.ue F = !Ian - a m!IF
[an
Hence
Jf
is Cauchy in
will be an extension of a
As (K) .
with
a= 0
Conversely~ assume that each
a E As(K)
such that
~
x = AY + ( 1 - A) z
=0
a
where
A = ~(x)
and since
and hence
F'
F'
~
Now the uniqueness of
a = lim an E As (K)
on
F'
a E A(F;K)+
Let
z E F'
A is uniquely
/\_1()
= XF
0
separates points of
on
y E F
Then
has an extension
x E K ""- ( F U F ' )
and
"'
xF
determined~
is affine 9
is a face 9 cf. [2; Prop. 1.1
F9F1
Then
0 < A < 1 •
~
Cor.1.2].
components is easy 9 since
A(F;K)+
F .
The following lemma can be derived from [6; Formula (1)
p.263 9 Satz 2.1.3].
For the readers convenience we
9
shall give
a proof.
Lemma 4.
Let
b E As(K 2 ) .
by
a ® b
Proof:
9
K1
and
K2
be compact convex sets and a E As (K 1 ) 9
Then there is a function
c E As(K 1
®
K2 ) , denoted
such that
First we shall consider the case where
non-negative
a
and
[a } c
a.
bS \i b ~ pointwise.
A(K 1 )+ ~ [b 13 } ::; A(K 2 )+ such that aa. "!>~a 9
Then [aa. ® b 13 } is a decreasing net in BA(K 1 x K2 )+
that
c(~)
u.s.c. affine function
= inf
a.~l3
~(a
a.
® b~)
1-'
are
Then there exist nets
u.s.c. and affine.
fore there is an
b
9
all
c
~
on
9
and there-
K1 ® K2
E K1 ® K2 .
such
- 8 -
If
where
ai
is
u.s.c. non-negative and affine
u.s. c. non- negative and affine
a(x 1 )b(x 2 )
on
on
K2 , then
K1 , bi
(x 1 ,x 2 ) -·
is linear combination of four terms of the kind con-
sidered in the first part of the proof, and we can choose
the corresponding linear combination of elements from
If
a E As(K 1 ) , bE As(K 2 )
a'
b'
of the type(*) . such that
!Ia -
a~l\K 1
n '
n
<
Then for all
c
= lim
as
As (K 1 ® K2 ).
<.!n '
such that
all
7
(x 1 ,x 2 ) E K1 x K2 •
(x 1 ,x 2 ) E oeK 2
From this it follows that
[en}
n ·' K2
·
a~(x 1 )b~(x 2 )
en (w(x 1 ,x 2 ))
hence
1\b- b'll
~ and en E As (K 1 ® K2 )
cn(w(x 1 ,x 2 )) =
c
are arbitrary then we can find
'
(**)
is
is Cauchy on
I
< -;. +
n
1
[cnlo (K ®K )}
e
K1 ® K2
en E As(K 1 ® K2 ) .
~ ( \!allK
1
is Cauchy, and
2
by Lemma 2 (ii).
Let
Then it is obvious from (**) that
c
satjsfies the requirement.
and
K2
be compact convex sets, and
F1
closed faces of
K1
and
F
F 2 ))
in
Theorem 5.
and
F2
the face
(i)
If
Let
K1
co(w(F 1
F
X
is a split face of
split faces of
(ii) If
either
K1
A(F 1 )
and
or
K2
respectively.
Let
be
Then the following holds
K1 ® K2
then
F1
and
F2
are
K2 .
A(F 2 )
has the approximation property,
- 9 and
F1
and
F2
are split faces of
is a split face of
Proof:
a > 0
on
oeF = w(oeF 1 x oeF 2 )
F1
i.e.
'
a!p
a ® 11
fix
is
((a® 11) • Xp)
!\
is
aeF 2
.:=
w(oeF1 x oeF 2 )
Since
g(x 2 )
and
u.s.c. affine, while
w(J4 x F 2 )
(a•xp 1 )/\
g(x 2 ) _::: a·xp 1 , and since
We know
K1 •
K1 ® K2 , since
F .
Now we
F1
g(x 2 ) = 0
agree on
is
that
on
oeK 1 , and
oeF1
g(x 2 )
is
u.s.c. concave it follows from
g(x 2 ) ~ (a·xp 1 )/\ •
(a•xp 1 )'"
a·xp 1 , we have
it
g(x 2 ) : x _. ( (a ® 11 ) • Xp)/\(w(x,x2))
F' , we have that
(a·xp 1 )/\
such that
and hence on
On
K1 •
Bauers principle (5; Lem.1]
cave majorant of
F
As no-
By Proposition 3
u.s. c. and affine on
u.s.c. and affine on
since
a E A(K 1 )
is affine on
Then the function
•
K2 , then
is a split face.
Let
(a· Xp 1 )/\
is non-negative on
x2 E
.
F
E A(F 1 ;K 1 )+
1
will suffice to show that
that
and
K1 ® K2 •
To prove (i) we assume that
ted before
K1
is the smallest
Moreover
u.s.c. con-
g(x 2 ) ~ (a•xp 1 )/\ , and (i)
follows.
To prove (ii) we shall assume that
faces, and that
sition 3
extension
A(F 1 )
F2
are split
has the approximation property.
we have to show that if
a E As(K 1
and
F1
® K2 )
a E A(F)+
such that
a=0
then
on
a
F' .
By Propoadmits an
Now
a o (wK x K l F XF ) belongs to BA(F 1 X F 2 ) = A(F 1 ) ®8 A(F 2 ) •
1
2
1 2
If e > 0 is arbitrary we can choose a 1 , ••• ,an E A(F 1 ) and
b 1 , ••• ,bn E A(F 2 )
such that
n
L: a. ® b. !IF
II a o wK x K
1
2
such that
'"" = a.
a.
l
l
on
i=1
F1
l
and
l
1
a.l = o
xF
2
<
e •
,..,_
on
F1 , while
b.
l
= b.l
- 10 ,...
and
=0
on
n
,...
By Lemma 4
L: a. ® b . E As ( K1 ® K2 ) and on w( F 1 x F 2 ) it
i=1 ~
~
n
n ,....,
,...
equals
L: 1a. ® b; , while
I: a. 0 b. = 0 on oe(K 1 ® K2 )'\...oeF.
i= ~
i=1 ~
l
b.
~
...L.
As
As (K 1 ® K2 )
is complete in
n
,...
I: a. ® b.
of
is obtained at w(F 1
i=1 ~
~
to the existence of a E As(K 1 ® K2 )
w(F 1 x F 2 )
and
9
a= 0
on
a = a
on
Now let
x E F
and represent
w(F 1 x F 2 ) •
Since
F
a
11 0
and
e
X F2 )
9
(K ® K )
1
a = 0
x
®
on
and the norm
2
this argument leads
such that
oeF' = oe(K 1
to show that
on
II
a = a
K2 ) 'F •
FI
on
It remains
•
by a probability measure
satisfies the barycentric calculus we
get
a(x) =
and so
,...
a = a
sK1®K2 ad11 = w(Fs
on
To show that
b > 0
on
K1 ® K2
ad11 = Ja di-J. = a(x)
F
1 X F2 )
F •
,...._
a = 0
on
and
b > a
and by Lemma 2 (iL b > a"'
-
we let
F'
on
b E A(K 1 0 K2 )
Then
F •
K1 0 K2 •
on
For
b > "'
a
p
with
on C!e(K ®K 2 ) 9
E K1 ® K2
we
have
Since
(a· xF)''' = 0
on
FI
'
we get
a = 0
on
Fi
"
and the proof
is complete.
Remark:
It is easy to see from Lemma 4 that the embedding of the
product of two parallel faces
givesrise to a parallel face
F1
F
and
F2
without the assumption of the
presence of the approximation property in
1\
1\
1\
XF = xF 1 ® xF 2
is affine.
in the sense of [11]
A(F 1 ) .
In fact,
- 11 -
Theorem 6.
set
Let
K •
Let
property.
F
B
Let
function on
be a closed split face of a compact convex
be a real Banach space having the approximation
be a concave
p
K •
Let
a : F ... B
l.s.c. strictly positive real
be an affine continuous map such
that
II a ( k) II
Then
a: K
a
... B
.:S p ( k ) ,
such that
Let
C
is normed by
9
all
be the unit ball of
1\(x,r)ll = IJx!l + !r! •
(x,r) ... (·)(x) + r
A(C) •
k E F .
has an extension to a continuous affine map
lla(k) II < p(k)
Proof:
all
Hence if
k E K •
B*
with w*-topology.
It was observed in
is an isometric isomorphism of
B
[10] that
B xJR
has the approximation property then
We define a biaffine continuous function
b(x,x*) = x*(a(x)) ,
all
x E F ,
b
B xE
on
onto
A(C) has.
F xC
by
x* E C
By Proposition 1 (ii) there is an affine homeomorphism between
F
®
C
and
co(wKxC(FxC)) defined by
T(p)(d)=p(diFxC)
Since
b
for
dEBA(KxC).
is naturally a continuous affine function on F ® C
there is a continuous affine function
b1
on
9
all
co ( wK x C( F x C))
such that
b 1 (T WF
Moreover
l.s.c,. on
we have
K
X
0 (x,x*)) = x*(a(x))
p ... p(P 1 (p))
®c.
For
p = wK X 0 (x,x*)
( x, x*) E F
X
C •
is concave, strictly positive and
p E oe(co(wKxC(FxC)))=wKxC(oeFxoeC)
with
(x,x-*) E oeF X oeC
and hence
- 12 -
Since
Ib 1 ( p) I
p -
is concave and
that
Ib 1 I _::
is convex and continuous and
l.s.c., it followsfrom Bauers principle [5;Lem.1]
p o P1
co (wK x 0 ( F x C) )
on
•
Now it follows from Theorem 5 that
split face of
K® C •
co(wK
By [1; Th.II.6.12]
c
extends
b1
X
0 (F
X
C))
c E A(K ® C)
such
and
all
p E K0 C .
(Actually, it follows from [1; Cor.I.5.2] that a concave
function on a compact convex set is
sense of [3].
is a
and [3; Th.2.2 and
Th.4.5] it follows that there is a function
that
p ..... p(P1 ( p))
l.s.c.
A(K)-superharmonic in the
Moreover it should be remarked that the th0orems
2.2 and 4.5 of [3] are stated for complex spaces, but the proofs
hold almost unchanged for the real case.)
Now we can define a continuous affine map
Then for
c 1 : K ..... A(C)
by
k EK
l!c 1 (k) II = sup llc(w(k,x*))!l.::sup p(P 1 (w(k,x*)) = p(k)
x* EO
By composing the isometry
the canonical projection
S
from
Q
we get an affine continuous map
between
BXJR
A(C)
to
and
B
a(= Q o So c 1 )
k EK •
Moreover, for
x*(a(k)) =
x7~(
= c(w(k,x*)) =
(Q
o
s
o
k E F ,
of
x* E C
c 1 ) (k)) = c 1 (k) (x*)
b 1 (ill(k~x*))
= x*(a(k))
with
which has norm 1 9
such that
for all
BxE
K into
B
-
Hence for
k E F: i(k)
13 -
= a(k)
.
Q.E.D.
Corollary.
set
K •
property.
Let
Let
Let
F
be a closed split face of a compact convex
B be a real Banach space having the approximation
a : F _, B be a continuous affine map.
admits an extension to a continuous affine function
such that
max !la(k) II = max !la(k) II
k EF
Remark:
k EK
if
a
i : K _, B
•
Conclusions similar to those of Theorem 6 and the Corel-
lary hold with no assumptions on
A(F)
Then
B , if instead we know that
has the approximation property.
K is a simplex.
This is f.ex. the case,
- 14 References
[1]
E.M. Alfsen, Compact convex sets and bom~dary integrals,
Ergebnisse der Mathematik, Springer Verlag, Germany,1971.
[2]
E.M. Alfsen and T.B. Andersen, Split faces of compact convex
sets, Proc. London Math. Soc. 21 (1970), 415-442.
[3]
E.M. Alfsen and B. Hirsberg, On dominated extensions in linear
subspaces of C(Di.!.L (to appear in Pacific Journal of Math.)
[4]
T.B. Andersen, On dominated extension of continuous affine
functions on split faces, (to appear in Math. Scand.)
[5]
H. Bauer, Kennzeichnung kompakter Simplexe mit abgeschlossener Extremalpunktmenge 9 Archiv der Mathematjk 14 (1963),
415-421.
[6]
E. Behrends and G. Wittstock, Tensorprodukte kompakter konvexer Mengen, Inventiones Math. 10 (1970), 251-266.
[7]
F. Combes, Quelques propri~tes des
94 (1970)' 165-192.
[8]
U. Krause, Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa, Math.Ann. 184 (1970), 275-296.
[9]
A Lazar, Affine products of simplexes, Math.Scand. 22 (1968),
165-175.
C*-alg~bres,
Bull.Sc.Math.
[10] I. Namioka and R.R. Phelps, Tensor products of compact convex
sets, Pacific J. Math. 31 (1969), 469-480.
~1]
M. Rogalski, Topologies faciales dans les convexes compacts,
calcul fonctionnelet decomposition spectrale dans le
centre d'un espace A(X), Seminaire Choquet 1969-70,No.4.
[12] Z. Semadeni, Categorical methods in convexity, Proc. Colloq.
on Convexity, Copenhagen 1965, 281-307.
University of Oslo
Oslo, Norway
and
University of Arhus
Arhus, Denmark