Clfs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite sums, with

∞

respect to a speci¬ed topology on X. This holds if A is local, i.e., it contains

17.1

17.1 17. Basic concepts and topological realcompactness 185

any function f such that for each x ∈ X there is some fx ∈ A with f = fx

near x.

Clfcs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite countable

∞

sums.

Interesting algebras are the following, where in this chapter in the notation we shall

generally omit the range space R.

y wC

“

‘

‘‘

Cb

‘‘

™

y wC y wC © Ce

se

∞ ∞ ∞

g

e

¢

Clfcs

ee

j

h

c

h ee

ee g

h

¨

h

p e

s

Py w P 99

y wC y wC © C&

4&

G

x

x C∞

ω ω

&&

f

99

c

( x xx

A

9 &

y wC

ω ω

Cconv

C(X) = C(X, R), the algebra of continuous functions on a topological space X.

It has all the properties from above.

Cb (X) = Cb (X, R), the algebra of bounded continuous functions on a topological

space X. It is only bounded inversion closed and a C ∞ -algebra, in general.

C ∞ (X) = C ∞ (X, R), the algebra of smooth functions on a Fr¨licher space X,

o

see (23.1), or on a smooth manifold X, see section (27). It has all properties

from above, where we may use the c∞ -topology.

C ∞ (E) © C(E), the algebra of smooth and continuous functions on a locally

convex space E. It has all properties from above, where we use the locally

convex topology on E.

∞ ∞

Cc (E) = Cc (E, R), the algebra of smooth functions, all of whose derivatives

are continuous on a locally convex space E. It has all properties from above,

again for the locally convex topology on E.

C ω (X) = C ω (X, R), the algebra of real analytic functions on a real analytic

manifold X. It is only inversion closed.

C ω (E) © C(E), the algebra of real analytic and continuous functions on a locally

convex space E. It is only inversion closed.

ω ω

Cc (E) = Cc (E, R), the algebra of real analytic functions, all of whose derivatives

are continuous on a locally convex space E. It is only inversion closed.

ω ω

Cconv (E) = Cconv (E, R), the algebra of globally convergent power series on a

locally convex space E.

Pf (E) = Polyf (E, R), the algebra of ¬nite type polynomials on a locally convex

space E, i.e. the algebra E Alg generated by E . This is the free commu-

tative algebra generated by the vector space E , see (18.12). It has none of

the properties from above.

17.1

186 Chapter IV. Smoothly realcompact spaces 17.3

P (E) = Poly(E, R), the algebra of polynomials on a locally convex space E, see

(5.15), (5.17), i.e. the homogeneous parts are given by bounded symmetric

multilinear mappings. No property from above holds.

∞ ∞ ∞

Clfcs (E) = Clfcs (E, R), the Clfcs -algebra (see below) generated by E , and hence

also called (E )∞ . Only the Clfs -property does not hold.

∞

lfcs

17.2. Results. For completely regular topological spaces X and A = C(X) the

following holds:

(1) Due to [Hewitt, 1948, p.85 + p.60] & [Shirota, 1952, p.24], see also [Engel-

king, 1989, 3.12.22.g & 3.11.3]. The space X is called realcompact if all

algebra homomorphisms in Hom C(X) are evaluations at points of X, equiv-

alently, if X is a closed subspace of a product of R™s.

(2) Due to [Hewitt, 1948, p.61] & [Katˇtov, 1951, p.82], see also [Engelking,

e

1989, 3.11.4 & 3.11.5]. Hence every closed subspace of a product of realcom-

pact spaces is realcompact.

(3) Due to [Hewitt, 1948, p.85], see also [Engelking, 1989, 3.11.12]. Each Lin-

del¨f space is realcompact.

o

(4) Due to [Katˇtov, 1951, p.82], see also [Engelking, 1989, 5.5.10]. Paracom-

e

pact spaces are realcompact if and only if all closed discrete subspaces are

realcompact.

(5) Due to [Hewitt, 1950, p.170, p.175] & [Mackey, 1944], see also [Engelk-

ing, 1989, 3.11.D.a]. Discrete spaces are realcompact if and only if their

cardinality is non-measurable.

(6) Hence Banach spaces are realcompact if and only if their density (i.e., the

cardinality of a maximal discrete or of a minimal dense subset) or their

cardinality is non-measurable.

(7) [Shirota, 1952], see also [Engelking, 1989, 5.5.10 & 8.5.13.h]. A space of

non-measurable cardinality is realcompact if and only if it admits a complete

uniformity.

(8) Due to [Dieudonn´, 1939,] see also [Engelking, 1989, 8.5.13.a]. A space

e

admits a complete uniformity, i.e. is Dieudonn´ complete, if and only if it

e

is a closed subspace of a product of metrizable spaces

Realcompact spaces where introduced by [Hewitt, 1948, p.85] under the name Q-

compact spaces. The equivalence in (1) is due to [Shirota, 1952, p.24]. The results

(1) and (2) are proved in [Engelking, 1989] for a di¬erent notion of realcompactness,

which was shown to be equivalent to the original one by [Katˇtov, 1951], see also

e

[Engelking, 1989, 3.12.22.g].

17.3. Lemma. [Kriegl, Michor, Schachermayer, 1989, 2.2, 2.3]. Let A be ¯

1-eval-

uating. Then we have a topological embedding

δ : XA ’ prf —¦δ := f,

R,

A

with dense image in the closed subset Hom A ⊆ R. Hence X is A-realcompact

A

if and only if δ has closed image.

17.3

17.6 17. Basic concepts and topological realcompactness 187

Proof. The topology of XA is by de¬nition initial with respect to all f = prf —¦δ,

hence δ is an embedding. Obviously Hom A ⊆ A R is closed. Let • : A ’ R be an

algebra-homomorphism. For f ∈ A consider Zf . If A is 1-evaluating then by (18.8)

for any ¬nite subset F ⊆ A there exists an xF ∈ f ∈F Zf . Thus δ(xF )f = •(f )

for all f ∈ F. If A is only ¯1-evaluating, then we get as in the proof of (18.3) for

every µ > 0 a point xF ∈ X such that |f (xF ) ’ •(f )| < µ for all f ∈ F. Thus δ(xF )

lies in the corresponding neighborhood of (•(f ))f . Thus δ(X) is dense in Hom A.

Now X is A-realcompact if and only if δ has Hom A as image, and hence if and

only if the image of δ is closed.

17.4. Theorem. [Kriegl, Michor, Schachermayer, 1989, 2.4] & [Adam, Bistr¨m, o

Kriegl, 1995, 3.1]. The topology of pointwise convergence on Homω A is realcom-

pact. If XA is not realcompact then there exists an ω-evaluating homomorphism •

which is not evaluating.

Proof. We ¬rst show the weaker statement, that: If XA is not realcompact then

there exists a non-evaluating •, i.e., X is not A-realcompact.

Assume that X is A-realcompact, then A is 1-evaluating and hence by lemma

(17.3) δ : XA ’ A R is a closed embedding. Thus by (17.2.1) the space XA is

realcompact.

Now we give a proof of the stronger statement that Homω (A, R) is realcompact:

Assume that all sets of homomorphisms are endowed with the pointwise topology.

Let M ⊆ 2A be the family of all countable subsets of A containing the unit.

For M ∈ M, consider the topological space Homω M , where M denotes the

subalgebra generated by M . Obviously the family (δf )f ∈M , where δf (•) = •(f ), is

a countable subset of C(Homω M ) that separates the points in Homω M . Hence

Homω M = Hom C(Homω M ) by (18.25), since C(Homω M ) is ω-evaluating by

(18.11), i.e. Homω M is realcompact. Now Homω A is an inverse limit of the spaces

Homω M for M ∈ M. Since Homω M is Hausdor¬, we obtain that Homω A as

a closed subset of a product of realcompact spaces is realcompact by (17.2.2).

Since X is not realcompact in the topology XA , which is that induced from the em-

bedding into Homω A, we have that X = Homω A and the statement is proved.

17.5. Counter-example. [Kriegl, Michor, 1993, 3.6.2]. The locally convex space

∞

R“

count of all points in the product with countable carrier is not C -realcompact, if

“ is uncountable and none-measurable.

Proof. By [Engelking, 1989, 3.10.17 & 3.11.2] the space R“ count is not realcompact,

in fact every c∞ -continuous function on it extends to a continuous function on R“ ,

see the proof of (4.27). Since the projections are smooth, XC ∞ is the product

topology. So the result follows from (17.4).

17.6. Theorem. [Kriegl, Michor, Schachermayer, 1989, 3.2] & [Garrido, G´mez,

o

Jaramillo, 1994, 1.8]. Let X be a realcompact and completely regular topological

space, let A be uniformly dense in C(X) and ¯

1-evaluating.

17.6

188 Chapter IV. Smoothly realcompact spaces 18.1

Then X is A-realcompact. Moreover, if X is A-paracompact then A is uniformly

dense in C(X).

∞

In [Kriegl, Michor, Schachermayer, 1989] it is shown that Clfcs -algebra A is uni-

formly dense in C(X) if and only if A © Cb (X) is uniformly dense in Cb (X). One

may ¬nd also other equivalent conditions there.

Proof. Since A ⊆ C(X) we have that the identity XA ’ X is continuous, and

hence A ⊆ C(XA ) ⊆ C(X). For each of these point-separating algebras we consider

the natural inclusion δ of X into the product of factors R over the algebra, given

by prf —¦δ = f . It is a uniform embedding for the uniformity induced on X by this

algebra and the complete product uniformity on R with basis formed by the sets

Uf,µ := {(u, v) : | prf (u) ’ prf (v)| < µ} with µ > 0.

The condition that A ⊆ C is dense implies that the uniformities generated by

C(X), by C(XA ) and by A coincide and hence we will consider X as a uniform

space endowed with this uniform structure in the sequel. In fact for an arbitrarily

given continuous map f and µ > 0 choose a g ∈ A such that |g ’ f | < µ. Then

{(x, y) : |f (x) ’ f (y)| < µ} ⊆ {(x, y) : |g(x) ’ g(y)| < 3µ}

⊆ {(x, y) : |f (x) ’ f (y)| < 5µ}.

Since XA is realcompact, δC (XA ) = Hom(C(XA )) and hence is closed and so the

uniform structure on X is complete. And similarly also if X is realcompact. Thus,

the image δA (X) is a complete uniform subspace of A R and so it is closed with

respect to the product topology, i.e. X is A-realcompact by (17.3).

17.7. In the case of a locally convex vector space the last result (17.6) can be

slightly generalized to:

Result. [Bistr¨m, Lindstr¨m, 1993b, Thm.6]. For E a realcompact locally convex

o o

vector space, let E ⊆ A ⊆ C(E) be a ω-evaluating C (∞) -algebra which is invari-

ant under translations and homotheties. Moreover, we assume that there exists

a 0-neighborhood U in E such that for each f ∈ C(E) there exists g ∈ A with

supx∈U |f (x) ’ g(x)| < ∞.

Then E is A-realcompact.

18. Evaluation Properties of Homomorphisms

In this section we consider ¬rst properties near the evaluation property at single

functions, then evaluation properties for homomorphisms on countable many func-

tions, and ¬nally direct situations where all homomorphisms are point evaluations.

18.1. Remark. If • in Hom A is 1-evaluating (i.e., •(f ) ∈ f (X) for all f in A),

then • is ¯

1-evaluating.

18.1

18.3 18. Evaluation properties of homomorphisms 189

18.2. Lemma. [Bistr¨m, Bjon, Lindstr¨m, 1991, p.181]. For a topological space

o o

X the following assertions are equivalent:

(1) • is ¯

1-evaluating;

˜x

ˇ

(2) There exists x in the Stone-Cech compacti¬cation βX with •(f ) = f (˜) for

˜

all f ∈ A.

˜ ˇ

Here f denotes the extension of f : X ’ R ’ R∞ to the Stone-Cech-compacti¬-

cation βX with values in the 1-point compacti¬cation R∞ of R.

In [Garrido, G´mez, Jaramillo, 1994, 1.3] it is shown for a subalgebra of Cb (R) that

o

x need not be unique.

˜

Proof. For f ∈ A and µ > 0 let U (f, µ) := {x ∈ X : |•(f ) ’ f (x)| < µ}. Then U :=

{U (f, µ) : f ∈ A, µ > 0} is a ¬lter basis on X. Consider X as embedded into βX and

˜

take an ultra¬lter U on βX that is ¬ner than U. For f := (f1 ’•(f1 ))2 +(f2 ’•(f2 ))2

we have in fact

U (f1 , µ1 ) © U (f2 , µ2 ) ⊇ U (f, min{µ1 , µ2 }2 ).

˜

Let x ∈ βX be the point to which U converges. For an arbitrary function f in A

˜

˜˜ ˜

the ¬lter f (U) converges to •(f ) by construction. But f (U) ≥ f (U) = f (U), so

˜x ˜x ˜

•(f ) = f (˜). The converse is obvious since •(f ) = f (˜) ∈ f (βX) ⊆ f (X) ⊆ R∞ ,

and •(f ) ∈ R.

18.3. Lemma. [Adam, Bistr¨m, Kriegl, 1995, 4.1]. An algebra homomorphism •

o

is ¯

1-evaluating if and only if • extends (uniquely) to an algebra homomorphism on

A∞ , the C ∞ -algebra generated by A.

Proof. For C ∞ -algebras A, we have that

•(h —¦ (f1 , . . . , fn )) = h(•(f1 ), . . . , •(fn ))

for all h ∈ C ∞ (Rn , R) and f1 , . . . , fn in A.

In fact set a := (•(f1 ), . . . , •(fn )) ∈ Rn . Then

1

ha (x) · (xj ’ aj ),

h(x) ’ h(a) = ‚j h(a + t(x ’ a)) dt · (xj ’ aj ) = j

0 j¤n j¤n

1

where ha (x) := 0 ‚j h(a + t(x ’ a))dt. Applying • to this equation composed with

j

the fi one obtains

•(h —¦ (f1 , . . . , fn )) ’ h(•(f1 ), . . . , •(fn )) =

•(ha —¦ (f1 , . . . , fn )) · (•(fj ) ’ •(fj )) = 0.

= j

j¤n

(’) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). By what we have shown

˜

above (1-preserving) algebra homomorphisms are C ∞ -algebra homomorphisms and

hence this is the only candidate for an extension. This map is well de¬ned. Indeed,

18.3

190 Chapter IV. Smoothly realcompact spaces 18.5

let h —¦ (f1 , . . . , fn ) = k —¦ (g1 , . . . , gm ). For each µ > 0 there is a point x ∈ E such

that |•(fi ) ’ fi (x)| < µ for i = 1, ..., n, and |•(gj ) ’ gj (x)| < µ for j = 1, ..., m. In

˜x

fact by (18.2) there is a point x ∈ βX with •(f ) = f (˜) for

˜

n m

(fi ’ •(fi ))2 + (gj ’ •(gj ))2 ,

f :=

i=1 j=1

˜x

and hence •(fi ) = fi (˜) and •(gj ) = gj (˜). Now approximate x by x ∈ X.

˜x ˜

By continuity of h and k we obtain that

h(•(f1 ), . . . , •(fn )) = k(•(f1 ), . . . , •(fm )),

and we therefore have a well de¬ned extension of •. This extension is a homo-

morphism, since for every polynomial θ on Rm (or even for θ ∈ C ∞ (Rm )) and

gi := hi —¦ (f1 , . . . , fni ) ∈ A∞ we have

i i

1 m

•(θ —¦ (g1 , . . . , gm )) = •(θ —¦ (h1 — . . . — hm ) —¦ (f1 , . . . , fnm ))

˜ ˜

1 m

= (θ —¦ (h1 — . . . — hm ))(•(f1 ), . . . , •(fnm ))

1 1 m m

= θ(h1 (•(f1 ), . . . , •(fn1 )), . . . , hm (•(f1 ), . . . , •(fnm ))

= θ(•(g1 ), . . . , •(gm )).

˜ ˜

(⇐) Suppose there is some f ∈ A with •(f ) ∈ f (X). Then we may ¬nd an

/

h ∈ C (R) with h(•(f )) = 1 and carr h © f (X) = …. Since A∞ is a C ∞ -algebra,

∞

we conclude from what we said above that •(h —¦ f ) = h(•(f )) = 1. But since

˜

h —¦ f = 0 we arrive at a contradiction.

18.4. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.2]. If A is bounded

o

inversion closed and • ∈ Hom A then • is ¯

1-evaluating.

Proof. We assume indirectly that there is a function f ∈ A with •(f ) ∈ f (X).

Let µ := inf x∈X |•(f ) ’ f (x)| and g(x) := 1 (•(f ) ’ f (x)). Then g ∈ A, •(g) = 0

µ

1

and |g(x)| = µ |•(f ) ’ f (x)| ≥ 1 for each x ∈ X. Thus 1/g ∈ A. But then

1 = •(g · 1/g) = •(g)•(1/g) = 0 gives a contradiction.

18.5. Lemma. Any C (∞) -algebra is bounded inversion closed.

Moreover, it is stable under composition with smooth locally de¬ned functions, which

contain the closure of the image in its domain of de¬nition.

Proof. Let A be a C ∞ -algebra (resp. C (∞) -algebra), n a natural number (resp.

n = 1), U ⊆ Rn open, h ∈ C ∞ (U, R), f := (f1 , . . . , fn ), with fi ∈ A such that

f (X) ⊆ U , then h —¦ f ∈ A . Indeed, choose ρ ∈ C ∞ (R) with ρ|f (X) = 1 and

supp ρ ⊆ U . Then k := ρ · h is a globally smooth function and h —¦ f = k —¦ f ∈ A.

18.5

18.9 18. Evaluation properties of homomorphisms 191