18.6. Lemma. Any inverse closed algebra A is 1-evaluating.

By (18.10) the converse is wrong.

Proof. Let f ∈ A and assume indirectly that Zf = …. Let g := f ’ •(f ). Then

g ∈ A and g(x) = 0 for all x ∈ X, by which 1/g ∈ A since A is inverse-closed. But

then 1 = •(g · 1/g) = •(g)•(1/g) = 0, which is a contradiction.

18.7. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Lem.14] & [Adam, Bi-

o o

str¨m, Kriegl, 1995, 4.2]. For • in Hom A the following statements are equivalent:

o

(1) • is 1-evaluating.

(2) • extends to a unique (1-evaluating) homomorphism on the algebra RA :=

{f /g : f, g ∈ A, 0 ∈ g(X)}.

/

(3) • extends to a unique (1-evaluating) homomorphism on the following C ∞ -

algebra A ∞ constructed from A:

∞

A := {h —¦ (f1 , . . . , fn ) :fi ∈ A, (f1 , . . . , fn )(X) ⊆ U,

U open in some Rn , h ∈ C ∞ (U )}.

Proof. (1) ’ (3) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). Since there

exists by (18.8) an x with •(fi ) = fi (x), we have (•(f1 ), . . . , •(fn )) ∈ U , hence the

right side makes sense. The rest follows in the same way as in the proof of (18.3).

∞

(3) ’ (2) Existence is obvious, since RA ⊆ A , and uniqueness follows from the

de¬nition of RA.

(2) ’ (1) Since RA is inverse-closed, the extension of • to this algebra is 1-

evaluating by (18.6), hence the same is true for • on A.

18.8. Lemma. Every 1-evaluating homomorphism is ¬nitely evaluating.

Proof. Let F be a ¬nite subset of A. De¬ne a function f : X ’ R by

(g ’ •(g))2 .

f :=

g∈F

Then f ∈ A and •(f ) = 0. By assumption there is a point x ∈ X with •(f ) = f (x).

Hence g(x) = •(g) for all g ∈ F.

18.9. Theorem. Automatic boundedness. [Kriegl, Michor, 1993] & [Arias-

de-Reyna, 1988] Every 1-evaluating homomorphism • ∈ Hom A is positive, i.e.,

0 ¤ •(f ) for all 0 ¤ f ∈ A. Moreover we even have •(f ) > 0 for f ∈ A with

f (x) > 0 for all x ∈ X.

Every positive homomorphism • ∈ Hom A is bounded for any convenient algebra

structure on A.

A convenient algebra structure on A is a locally convex topology, which turns A

into a convenient vector space and such that the multiplication A — A ’ A is

bounded, compare (5.21).

18.9

192 Chapter IV. Smoothly realcompact spaces 18.11

Proof. Positivity: Let f1 ¤ f2 . By (17) and (18.8) there exists an x ∈ X such

that •(fi ) = fi (x) for i = 1, 2. Thus •(f1 ) = f1 (x) ¤ f2 (x) = •(f2 ). Note that if

f (x) > 0 for all x, then •(f ) > 0.

Boundedness: Suppose fn is a bounded sequence, but |•(fn )| is unbounded. By

2

replacing fn by fn we may assume that fn ≥ 0 and hence also •(fn ) ≥ 0. Choosing

a subsequence we may even assume that •(fn ) ≥ 2n . Now consider n 21 fn . This

n

series converges Mackey, and since the bornology on A is by assumption complete

the limit is an element f ∈ A. Applying • yields

N N

1 1 1 1

≥

•(f ) = • fn + fn = •(fn ) + • fn

2n 2n 2n 2n

n=0 n=0

n>N n>N

N N

1 1

≥ •(fn ) + 0 = •(fn ),

2n 2n

n=0 n=0

where we used the monotonicity of • applied to n>N 21 fn ≥ 0. Thus the series

n

N

N ’ n=0 21 •(fn ) is bounded and increasing, hence converges, but its summands

n

are bounded by 1 from below. This is a contradiction.

18.10. Lemma. For a locally convex vector space E the algebra Pf (E) is 1-

evaluating.

More on the algebra Pf (E) can be found in (18.27), (18.28), and (18.12).

Proof. Every ¬nite type polynomial p is a polynomial in a ¬nite number of linearly

independent functionals 1 , . . . , n in E . So there is for each i = 1, . . . , n some point

ai ∈ E such that i (ai ) = •( i ) and j (ai ) = 0 for all j = i. Let a = a1 +· · ·+an ∈ E.

Then i (a) = i (ai ) = •( i ) for i = 1, . . . , n hence •(p) = p(a).

Countably Evaluating Homomorphisms

18.11. Theorem. Idea of [Arias-de-Reyna, 1988, proof of thm.8], [Adam, Bis-

∞

tr¨m, Kriegl, 1995, 2.5]. For a topological space X any Clfcs -algebra A ⊆ C(X) is

o

closed under composition with local smooth functions and is ω-evaluating.

Note that this does not apply to C ω .

Proof. We ¬rst show closedness under local smooth functions (and hence in partic-

ular under inversion), i.e. if h ∈ C ∞ (U ), where U ⊆ Rn is open and f := (f1 , . . . , fn )

with fi ∈ A has values in U , then h —¦ f ∈ A .

Consider a smooth partition of unity {hj : j ∈ N} of U , such that supp hj ⊆ U .

Then hj ·h is a smooth function on Rn vanishing outside supp hj . Hence (hj ·h)—¦f ∈

A . Since we have

carr (hj · h) —¦ f ⊆ f ’1 (carr hj ),

18.11

18.12 18. Evaluation properties of homomorphisms 193

the family {carr((hj · h) —¦ f ) : j ∈ N} is locally ¬nite, f is continuous, and since

1 = j∈N hj on U we obtain that h —¦ f = j∈N (hj · h) —¦ f ∈ A .

By (18.6) we have that • is 1-evaluating, hence ¬nitely evaluating by (18.8). We

now show that • is countably evaluating:

For this take a sequence (fn )n in A. Then hn : x ’ (fn (x) ’ •(fn ))2 belongs to

A and •(hn ) = 0. We have to show that there exists an x ∈ X with hn (x) = 0

for all n. Assume that this were not true, i.e. for all x ∈ X there exists an n with

hn (x) > 0. Take h ∈ C ∞ (R, [0, 1]) with carr h = {t : t > 0} and let gn : x ’

h(hn (x)) · h( n ’ h1 (x)) · . . . · h( n ’ hn’1 (x)). Then gn ∈ A and the sum n 21 gn

1 1

n

is locally ¬nite, hence de¬nes a function g ∈ A. Since • is 1-evaluating there exists

for any n an xn ∈ X with hn (xn ) = •(hn ) = 0 and •(gn ) = gn (xn ). Hence

1 1

•(gn ) = gn (xn ) = h(hn (xn )) · h( n ’ h1 (xn )) · . . . · h( n ’ hn’1 (xn )) = 0.

By assumption on the hn and h we have that g > 0. Hence by (18.9) •(g) > 0,

since • is 1-evaluating. Let N be so large that 1/2N < •(g). Again since A is

1-evaluating, there is some a ∈ X such that •(g) = g(a) and •(gj ) = gj (a) for

j ¤ N . Then

1 1 1 1 1

gn (a) ¤ 0 + N

< •(g) = g(a) = gn (a) = •(gn ) +

2N 2n 2n 2n 2

n n>N

n¤N

gives a contradiction.

18.12. Counter-example. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.17]. For

o o

any non-re¬‚exive weakly realcompact locally convex space (and any non-re¬‚exive

Banach space) E the algebra Pf (E) of ¬nite type polynomials is not ω-evaluating.

Moreover, EA is realcompact, but E is not A-realcompact, for A = Pf (E), so that

the converse of the assertion in (17.4) holds only under the additional assumptions

of (17.6).

1

As example we may take E = , which is non-re¬‚exive, but by (18.27) weakly

realcompact.

By (18.10) the algebra Pf (E) is 1-evaluating and hence by (18.7) it has the same

homomorphisms as RPf (E), Pf (E)∞ or even Pf (E) ∞ . So these algebras are not

ω-evaluating for spaces E as above.

Proof. By the universal property (5.10) of Pf (E) we get Hom Pf (E) ∼ (E )— , the

=

space of (not necessarily bounded) linear functionals on E . For weakly realcompact

E by (18.27) we have Homω Pf (E) = E. So if Pf (E) were ω-evaluating then even

E = Hom Pf (E). Any bounded subset of E is obviously Pf -bounding and hence

by (20.2) relatively compact in the weak topology, since EPf (E) = (E, σ(E, E )).

Since E is not semi-re¬‚exive, this is a contradiction, see [Jarchow, 1981, 11.4.1].

If we have a (not necessarily weakly compact) Banach space, we can replace in the

argument above (20.2) by the following version given in [Bistr¨m, 1993, 5.10]: If

o

Homω Pf (E) = Hom Pf (E) then every A-bounding set with complete closed convex

hull is relatively compact in the weak topology.

18.12

194 Chapter IV. Smoothly realcompact spaces 18.15

18.13. Lemma. The Clfs -algebra A∞ generated by an algebra A can be obtained

∞

lfs

in two steps as (A )lfs . Also the Clfcs -algebra A∞ can be obtained in two steps as

∞ ∞

lfcs

∞

(A )lfcs .

Proof. We prove the result only for countable sums, the general case is easier. We

have to show that (A∞ )lfcs is closed under composition with smooth mappings. So

take h ∈ C ∞ (Rn ) and j≥1 fi,j ∈ (A∞ )lfcs for i = 1, . . . , n. We put h0 := 0 and

k k

fn,j ) ∈ A∞ and obtain

hk := h —¦ ( f1,j , . . . ,

j=1 j=1

h—¦( (hk ’ hk’1 ),

f1,j , . . . , fn,j ) =

j≥1 j≥1 k≥1

where the right member is locally ¬nite and hence an element of (A∞ )lfcs .

18.14. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 4.3]. A homomorphism • in

o

Hom A is ω-evaluating if and only if • extends (uniquely) to an algebra homomor-

phism on the Clfcs -algebra A∞ generated by A, which can be obtained in two steps

∞

lfcs

∞

as (A )lfcs (and this extension is ω-evaluating by (18.11)).

Proof. (’) The algebra A∞ is the union of the algebras obtained by a ¬nite

lfcs

iteration of passing to Alf cs and A∞ , where Alfcs := {f : f = n fn , fn ∈

A, the sum is locally ¬nite}. To A∞ it extends by (18.3). It is countably eval-

uating there, since in any f ∈ A∞ only ¬nitely many elements of A are involved.

Remains to show that • can be extended to Alf cs and that this extension is also

countably evaluating.

For a locally ¬nite sum f = k fk we de¬ne •(f ) := k •(fk ). This makes sense,

since there exists an x ∈ X with •(fn ) = fn (x), and since n fn is point ¬nite, we

have that the sum n •(fn ) = n fn (x) is in fact ¬nite. It is well de¬ned, since for

n gn we can choose an x ∈ X with •(fn ) = fn (x) and •(gn ) = gn (x) for

n fn =

all n, and hence n •(fn ) = n fn (x) = n gn (x) = n •(gn ). The extension is

a homomorphism, since for the product for example we have

• fn gk =• fn gk = •(fn gk ) =

n k n,k n,k

= •(fn ) •(gk ) = •(fn ) •(gk ) .

n

n,k k

Remains to show that the extension is countably evaluating. So let f k = n fn be

k

k k

given. By assumption there exists an x such that •(fn ) = fn (x) for all n and all

k. Thus •(f k ) = n •(fn ) = n fn (x) = f k (x) for all k.

k k

(⇐) Since A∞cs is a Clfcs -algebra we conclude from (18.11) that the extension of •

∞

lf

is countably evaluating.

18.15. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.10]. Let • in Hom A

o

j 1

be 1-evaluating, and let fn ∈ A be such that n »n fn ∈ A for all » ∈ and

j ∈ {1, 2}.

18.15

18.17 18. Evaluation properties of homomorphisms 195

Then • is {fn : n ∈ N}-evaluating.

For a convenient algebra structure on A and {fn : n ∈ N} bounded in A the second

condition holds, as used in (18.26).

It would be interesting to know if the assumption for j = 2 can be removed, since

then the application in (18.26) to ¬nite type polynomials would work.

Proof. Choose a positive absolutely summable sequence (»n )n∈N such that the

sequences (»n •(fn ))n∈N and (»n •(fn )2 )n∈N are summable. Then the sum

∞

»j (fj ’ •(fj ))2 ∈ A.

g :=

j=1

If there exists x ∈ X with g(x) = 0, we are done. If not, then consider the (positive)

function

∞

1

»j (fj ’ •(fj ))2 ∈ A.

h :=

2j

j=1

For every n ∈ N there exists xn ∈ X such that •(fj ) = fj (xn ) for all j ¤ n,

•(g) = g(xn ) and •(h) = h(xn ). But then for all n ∈ N we have by (18.9) that

0 < 2n •(h) = 2n’j »j •(fj ’ •(fj ))2 ¤ »j •(fj ’ •(fj ))2 = •(g),

j>n j>n

a contradiction.

18.16. Corollary. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.9]. Let E be a

o o

Banach space and A a 1-evaluating algebra containing P (E). Then for each • ∈

Hom A, each f ∈ A, and each sequence (pn )n∈N in P (E) with uniformly bounded

degree, there exists a ∈ E with •(f ) = f (a) and •(pn ) = pn (a) for all n ∈ N.

Proof. Let (»n )n∈N be a sequence of positive reals such that {»n pn : n ∈ N} is

bounded. Then by (18.15) the set {f, pn } is evaluated.

18.17. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 3.3]. Let (fγ )γ∈“ be a family in

o

A such that γ∈“ zγ fγ is a pointwise convergent sum in A for all z = (zγ ) ∈ ∞ (“)

j

and j = 1, 2. Let |“| be non-measurable, and let • be ω-evaluating.

Then • is {fγ : γ ∈ “}-evaluating.

We will apply this in particular if {fγ : γ ∈ “} is locally ¬nite, and A stable

under locally ¬nite sums. Note that we can always add ¬nitely many f ∈ A to

{fγ : γ ∈ “}.

Again it would be nice to get rid of the assumption for j = 2.

Proof. Let x ∈ X and set zγ := sign(fγ (x)) for all γ ∈ “. Then z = (zγ ) ∈ ∞ (“)

1

and γ∈“ |fγ (x)| = γ∈“ zγ fγ (x) < ∞, i.e. (fγ (x))γ∈“ ∈ (“). Next observe

that (•(fγ ))γ∈“ ∈ c0 (“), since otherwise there exists some µ > 0 and a countable

18.17

196 Chapter IV. Smoothly realcompact spaces 18.18

set Λ ⊆ “ with |•(fγ )| ≥ µ for each γ ∈ Λ. By the countably evaluating property

of • there is a point x ∈ X with |fγ (x)| = |•(f» | ≥ µ for each γ ∈ Λ, violating the

condition (fγ (x))γ∈“ ∈ 1 (“). Since as a vector in c0 (“) it has countable support

and since • is countably evaluating we get even (•(fγ ))γ∈“ ∈ 1 (“). Therefore we

may consider g, de¬ned by

x ’ g(x) := (fγ (x) ’ •(fγ ))2 1

∈

X (“).

γ∈“

This gives a map g — : ∞ 1

(“) ’ A, by

(“) =

g — (z) : x ’ z, g(x) = zγ · (fγ (x) ’ •(fγ ))2 ,

γ∈“

since (•(fγ )γ∈“ ∈ 1 (“). Let ¦ : ∞ (“) ’ R be the linear map ¦ := • —¦ g — :

∞

(“) ’ A ’ R. By the countably evaluating property of •, for any sequence

(zn ) in ∞ (“) there exists an x ∈ X such that ¦(zn ) = •(g — (zn )) = g — (zn )(x) =

zn , g(x) for all n. For non-measurable |“| the weak topology on 1 (“) is realcom-

pact by [Edgar, 1979, p.575]. By (18.19) there exists a point c ∈ 1 (“) such that

¦(z) = z, c for all z ∈ ∞ (“). For each standard unit vector eγ ∈ ∞ (“) we have

0 = ¦(eγ ) = eγ , c = cγ . Hence c = 0 and therefore ¦ = 0. For the constant

vector 1 in ∞ (“), we get 0 = ¦(1) = •(g — (1)). Since • is 1-evaluating there exists

an a ∈ X with •(g — (1)) = g — (1)(a) = 1, g(a) = γ∈“ (fγ (a) ’ •(fγ ))2 , hence

•(fγ ) = fγ (a) for each γ ∈ “.

18.18. Valdivia gives in [Valdivia, 1982] a characterization of the locally convex

spaces which are realcompact in their weak topologies. Let us mention some classes

of spaces that are weakly realcompact:

Result.

(1) All locally convex spaces E with σ(E , E)-separable E .

(2) All weakly Lindel¨f locally convex spaces, and hence in particular all weakly

o

countably determined Banach spaces, see [Vaˇ´k, 1981]. In particular this

sa

applies to c0 (X) for locally compact metrizable X by [Corson, 1961, p.5].

(3) The Banach spaces E with angelic weak— dual unit ball [Edgar, 1979, p.564].

Note that (E — , weak— ) is angelic :” for B ⊆ E — bounded the weak— -closure

is obtained by weak— -convergent sequences in B, i.e. sequentially for the

weak— -topology.

(4) 1 (“) for |“| non-measurable. Furthermore the spaces C[0, 1], ∞ , L∞ [0, 1],

the space JL of [Johnson, Lindenstrauss, 1974] (a short exact sequence

c0 ’ JL ’ 2 (“) exists), the space D[0, 1] or right-continuous functions

having left sided limits, by [Edgar, 1979, p.575] and [Edgar, 1977]. All these

spaces are not weakly Lindel¨f.o

(5) All closed subspaces of products of the spaces listed above.

(6) Not weakly realcompact are C[0, ω1 ] and ∞ [0, 1], the space of bounded

count

functions on [0, 1] with countable support, by [Edgar, 1979].

18.18

18.20 18. Evaluation properties of homomorphisms 197

18.19. Lemma. [Corson, 1961]. If E is a weakly realcompact locally convex space,

then every linear countably evaluating ¦ : E ’ R is given by a point-evaluation

evx on E with x ∈ E.

Proof. Since ¦ : E ’ R is countably evaluating it is linear and F := {ZK : K ⊆

E countable} does not contain the empty set and generates a ¬lter. We claim that

this ¬lter is Cauchy with respect to the uniformity de¬ned by the weakly continuous

real functions on E:

To see this, let f : E ’ R be weakly continuous. For each r ∈ R, let Lr := {x ∈

E : f (x) < r} and similarly Ur := {x ∈ E : f (x) > r}. By [Jarchow, 1981, 8.1.4]

— —

we have that E is σ(E , E )-dense in E . Thus there are open disjoint subsets

—

˜ ˜

Lr and Ur on E having trace Lr and Ur on E (take the complements of the

closures of the complements). Let B ⊆ E be an algebraic basis of E . Then the

— —

map χ : E ’ RB , l ’ (l(x ))x ∈B is a topological isomorphism for σ(E , E ).

By [Bockstein, 1948] there exists a countable subset Kr ⊆ B ⊆ E , such that the

˜ ˜

images under prKr : RB ’ RKr of the open sets Lr and Ur are disjoint. Let

K = r∈Q Kr . For µ > 0 we have that ZK — ZK ⊆ {(x1 , x2 ) : f (x1 ) = f (x2 )} ⊆

{(x1 , x2 ) : |f (x1 ) ’ f (x2 )| < µ}, i.e. the ¬lter generated by F is Cauchy. In fact, let

x1 , x2 ∈ ZK . Then x (x1 ) = •(x ) = x (x2 ) for all x ∈ K. Suppose f (x1 ) = f (x2 ).

Without loss of generality we ¬nd a r ∈ Q with f (x1 ) < r < f (x2 ), i.e. x1 ∈ Lr

and x2 ∈ Ur . But then x (x1 ) = x (x2 ) for all x ∈ Kr ⊆ K gives a contradiction.