Products and projective limits of function spaces Miroslav Kaˇcena

Products and projective limits of function spaces

Miroslav Kaˇcena

Abstract. We introduce a notion of a product and projective limit of function spaces.

We show that the Choquet boundary of the product space is the product of Choquet boundaries. Next we show that the product of simplicial spaces is simplicial. We also show that the maximal measures on the product space are exactly those with maximal projections. We show similar characterizations of the Choquet boundary and the space of maximal measures for the projective limit of function spaces under some additional assumptions and we prove that the projective limit of simplicial spaces is simplicial.

Keywords: Choquet theory, function space, product, projective limit, simplicial space Classification: Primary 46A55; Secondary 26B25, 46A32

1. Introduction

Let{Xi}_i∈I be a family of Choquet simplexes. We can construct a compact convex setX as the state space of the space of all continuous multiaffine functions on Q

i∈IXi. It has been shown in [6] and [16] that X itself is a simplex with extreme points being the evaluation functionals at the points (xi)_i∈I ∈Q

i∈IXi

withxi ∈extXifor everyi∈I. Generalizations to products of arbitrary compact convex sets followed (see [11], [18]). Characterization of maximal measures on the product of two compact convex sets, as the measures whose every ‘projection’ is a maximal measure, appeared later in [3] and [2].

In Section 3 we transfer these results to the context of function spaces. We first introduce a notion of a product of function spaces with several special products.

We compare these products and prove appropriate associative laws. Then we show that the Choquet boundary of a product space is the product of Choquet boundaries. We prove that the product is simplicial if and only if every of the original spaces is simplicial. Finally we show that maximal measures on the product of arbitrary many spaces are exactly those with maximal projections.

In Section 4 we transfer known results from [6] and [13] on projective limits of compact convex sets to function spaces. We use Grossman’s definition of the projective limit of function spaces from [10] and prove that the projective limit of simplicial spaces is simplicial. We also derive similar characterizations of the Choquet boundary and maximal measures as in the case of product of function spaces.

2. Preliminaries

Let K be a compact Hausdorff space. We denote by C(K) the space of all continuous functions on K, by M⁺(K) the set of all positive Radon measures on K and by M¹(K) the set of all probability Radon measures on K. Let εx

stand for the Dirac measure atx∈K. We say that a linear subspaceHofC(K) is afunction space, if it contains 1K(the function identically 1 onK) and separates the points ofK. LetMx(H) be the set of allH-representing measures forx∈K, i.e.,

Mx(H) :={µ∈ M¹(K) :h(x) = Z

K

h dµ for every h∈ H}.

The set Ch_HK := {x∈ K : Mx(H) = {εx}} is called the Choquet boundary ofH. It is a Gδ-set if K is metrizable (see [1, Corollary I.5.17]). We denote by

∇_HK theSilov boundaryˇ ofH(see [1, p. 50] for definition) and we remark that

∇_HK is equal to the closure of Ch_H K (see [1, Theorem I.5.15] for the proof).

A non-empty closed setE⊂Kis calledH-extremal, if sptµ⊂Efor everyx∈E and µ ∈ M_x(H). Finally, for every x∈ K we denote Fx(H) := S{sptµ: µ ∈ M_x(H)}.

We define the spaceA^c(H) of all continuousH-affine functions as the space of all continuous functions onKsatisfying the following formula:

f(x) = Z

K

f dµ for each x∈K and µ∈ Mx(H).

ClearlyA^c(H) is a uniformly closed function space withMx(H) =Mx(A^c(H)) for everyx∈K.

Here we recall main examples of function spaces:

(a) Convex case - LetX be a compact convex subset of a locally convex space and letHbe the linear spaceA(X) of all continuous affine functions onX. The Choquet boundary is the set extX of all extreme points ofX.

(b) Harmonic case - LetU be a bounded open subset of the Euclidean space Rⁿand let the corresponding function spaceH(U) be the family of all con- tinuous functions onU which are harmonic onU. The Choquet boundary coincides with the set∂_regU of all regular points.

An upper bounded Borel functionf is calledH-convex iff(x)≤µ(f) for any x∈Kandµ∈ Mx(H). LetK^c(H) denote the family of all continuousH-convex functions onK. Notice that the spaceK^c(H)− K^c(H) is uniformly dense inC(K) due to the lattice version of the Stone-Weierstrass theorem.

The convex coneK^c(H) determines a partial ordering_H (called theChoquet ordering) on the spaceM⁺(K):

µ_Hν if µ(f)≤ν(f) for each f ∈ K^c(H).

(If the spaceHis obvious, we simply writeµν.)

We remark thatµν if and only if µ(f)≤ν(f) for every f ∈ W(H), where W(H) is the smallest family of functions containingHand closed with respect to taking supremum of finite families.

For any measureµ∈ M⁺(K) there exists a maximal measureν with µν.

In particular, for every x∈ K there exists a maximal H-representing measure.

This is the content of the Choquet-Bishop-de Leeuw theorem [1, Theorem I.5.19].

If K is metrizable, then a measure µ ∈ M⁺(K) is maximal if and only if µ(K\Ch_H K) = 0. In nonmetrizable spaces every maximal measureµsatisfies µ(G) = 0 for anyGδ-set disjoint from Ch_H K(see [1, Proposition I.5.22]).

Theorem 2.1. Letµ∈ M⁺(K). Then the following assertions are equivalent:

(i) µis maximal,

(ii) there exists a setS ⊂ C(K)separating points of K such that every func- tion fromS is constant onFx(H)forµ-a.e. x∈K,

(iii) every function fromC(K)is constant onFx(H)forµ-a.e. x∈K.

Proof: See [2, Proposition 2].

Proposition 2.2. Let(K^′,G)be a function space andρ:K→K^′ a continuous mapping such thatF_ρ(x)(G)⊂ρ(Fx(H))for everyx∈Ch_H K. Then the image measureρµis a maximal measure onK^′ for every maximal measureµonK.

Proof: See [2, Corollary 3].

If for everyx∈Kthe maximalH-representing measure is uniquely determined, we say thatH issimplicial. In the convex case it is equivalent to say thatX is a Choquet simplex. We denote the unique maximal measure representingx∈K byδx.

We say that H has the weak Riesz interpolation property (W.R.I.P.), if for everya₁, a₂, b₁, b₂ ∈ Hsuch thatai< bj,i, j= 1,2, there existsc∈ Hsuch that ai< c < bj,i, j= 1,2. It can be shown that His simplicial if and only if A^c(H) has W.R.I.P. (see [1, Corollary II.3.11] or [4, Theorem 3.3]).

For a functionf :K→Rwe define theupper envelope f^∗ as f^∗(x) := inf{h(x) :h≥f, h∈ H}, x∈K,

and thelower envelope asf∗:=−(−f)^∗. We denoteHb:={f ∈ C(K) :f∗=f^∗}.

It is true that A^c(H) = H. By [1, Proposition I.5.9 and Corollary I.5.10], web have:

Proposition 2.3. Letµ∈ M⁺(K). Then the following statements are equiva- lent:

(i) µis maximal,

(ii) µ(f) =µ(f^∗)for everyf ∈ C(K), (iii) µ(k) =µ(k^∗)for everyk∈ K^c(H).

Corollary 2.4. Letx∈K. Then the following statements are equivalent:

(i) x∈Ch_H K,

(ii) f(x) =f^∗(x)for everyf ∈ C(K), (iii) k(x) =k^∗(x)for everyk∈ K^c(H).

If f and g are functions on K, we writef ∨g for their pointwise maximum andf∧g for minimum.

Now we introduce a notation concerning cartesian products: Let{E_i}_i∈I be a family of topological spaces and letE:=Q

i∈IE_i be their cartesian product with the usual topology. We use the conventionQ

i∈∅E_i:={∅}.

LetJ ⊂I. The natural projection fromEontoQ

i∈JE_i is denoted byπ_J. Let A⊂E andz ∈Q

i∈I\JE_i. We denote byπ_J^z(A) the set{x∈Q

i∈JE_i: (x, z)∈ A}.

We use a similar notation for functions. Let f : E → Rand y ∈ Q

i∈I\JE_i. Thenπ_J^y(f) :Q

i∈JEi →Ris defined as

π_J^y(f)(x) :=f(x, y), x∈Y

i∈J

E_i.

In casef is independent ony, we use notationπJ(f).

Finally, forf1:E1 →Rand f2:E2 →Rwe definef1⊗f2:E1×E2→Rby (f₁⊗f₂)(x, y) =f₁(x)f₂(y), x∈E₁, y∈E₂.

We conclude this section with known results on products of Radon measures:

Let{(Ki,Si, µi)}_i∈I be a family of compact Hausdorff spaces with Radon prob- ability measures. There exists a unique product measure µ on Q

i∈IKi with µ(Q

i∈IEi) = Q

i∈Iµi(Ei), whenever Ei ∈ S_i for each i ∈ I and Ei 6= Ki for finitely manyi∈I(see [12, Chapter VI, Theorem 5.3]). By [8, Theorem 417Q],µ can be uniquely extended to a Radon measureN

i∈Iµi. We call this measure the Radon product measure. Radon products satisfy associative law (see [8, Theo- rem 417J]) and we can also use Fubini’s theorem (see [8, Theorem 417H]). Finally we remark that if two Radon measures coincide on the cylinder sets Q

i∈IE_i, where E_i ⊂ K_i is Borel for each i ∈ I and E_i 6= K_i for finitely many i ∈ I, then they are equal (see [12, Chapter I, Proposition 5.3] and the proof of [8, Corollary 417F]).

3. Products of function spaces 3.1 Definitions and relations.

Definition 3.1. Let{(K_i,H_i)}_i∈I be a family of function spaces and letK :=

Q

i∈IK_i. We define

(a) algebraic tensor product J

i∈IHi as the linear span of the set

{h₁⊗. . .⊗hn⊗1^Q_{Ki:i∈I\{i1,... ,in}}:h_k∈ H_ik, i_k∈I,1≤k≤n, n∈N}, (b) injective tensor product N

i∈IH_i as the closure ofJ

i∈IH_i, (c) multiaffine product by

⊠

i∈I

Hi:={f ∈ C(K) :π^y_j(f)∈ Hj for allj∈Iandy∈ Y

i∈I\{j}

Ki}.

We say that a function spaceHonKis aproduct of function spacesH_i,i∈I,

if K

i∈I

H_i ⊂ H ⊂

⊠

i∈I

A^c(H_i).

In caseI is an empty set, we put all products to be equal{∅}.

Remark 3.2. It can be shown, that H₁ ⊙ H₂ is really the ‘algebraic tensor product’, and ifH₁ andH₂ are closed, i.e., Banach spaces, thenH₁⊗ H₂ is their

‘weak (injective) tensor product’ (see [19, 20.5.5]). IfH_i=A(Xi) for some com- pact convex setsX_i,i∈I, then

⊠

i∈IH_iis the space of all continuous multiaffine functions onK.

Example 3.3. LetU₁ ⊂R^m, U₂ ⊂Rⁿ, be bounded open sets. We take H_i :=

H(U_i),i= 1,2 (see Example (b) in Section 2). IfHis a product of H_i, i= 1,2, then H ⊂H(U₁×U₂). Indeed, choose h∈ H ⊂ A^c(H₁)⊠A^c(H₂) =H(U₁)⊠ H(U₂). Then we have

∆h(x₁, x₂) = ∆π^x₁²(h)(x₁) + ∆π₂^x1(h)(x₂) = 0, x₁∈U₁, x₂∈U₂. However, even the largest product does not have to contain all harmonic functions on the cartesian product. Consider Ui := (0,1) ⊂R, i = 1,2. Then H(Ui) = A(U_i), i = 1,2. So every product consists only of biaffine functions. Now take f(x, y) :=x²−y² forx, y∈[0,1]. Clearly,f is harmonic, but not biaffine.

Proposition 3.4. The following assertions hold.

(i) J

i∈IH_i⊂

⊠

i∈IH_i.

(ii) If all Hi are closed, then J

i∈IHi ⊂ N

i∈IHi ⊂

⊠

i∈IHi. Moreover,

⊠

i∈IHi is closed.

(iii) If H_j is not closed for some j ∈ I, then J

i∈IH_i ( N

i∈IH_i and N

i∈IH_i6⊂

⊠

i∈IH_i.

Proof: Statement (i) and the first inclusion in (ii) are trivial. Since (i) holds, the second inclusion in (ii) will be proved if we show that

⊠

i∈IH_i is closed. So let {fn}_n∈N ⊂

⊠

i∈IH_i be such that fn ⇉ f ∈ C(K). Further, let j ∈ I and y ∈Q

i∈I\{j}Ki. Thenπ^y_j(fn)⇉π^y_j(f), and sinceπ_j^y(fn)∈ Hj for each nand H_j is closed, we haveπ_j^y(f)∈ H_j. Thusf ∈

⊠

i∈IH_i.

Using previous inclusions, it suffices to findf ∈(N

i∈IH_i)\(

⊠

i∈IH_i) to prove (iii). Letj ∈Ibe such thatH_jis not closed and putK^′:=Q

i∈I\{j}Ki. There are functions{hn}_n∈N⊂ Hj such thathn⇉h /∈ Hj. Then alsohn⊗1_K^′ ⇉h⊗1_K^′. Since hn⊗1_K^′ ∈ J

i∈IH_i for every n∈ N, we have h⊗1_K^′ ∈ N

i∈IH_i. But πj(h⊗1_K^′) =h /∈ H_j, thereforeh⊗1_K^′ ∈/

⊠

i∈IH_i. Remark 3.5. Using previous proposition, we can see that all products defined in Definition 3.1 are indeed function spaces, since they are linear spaces and contain algebraic tensor product, which contains constants and separates points.

In the rest of this subsection we will show that the two inclusions in Proposi- tion 3.4(ii) may be proper.

Example 3.6. LetKi:= [0,1]⊂R,H_i:=C(K_i),i= 1,2, and denoteK:=K1× K₂. The functions ofH₁⊙ H₂ are of the formPn

j=1f₁^j⊗f₂^j, wheref_i^j ∈ C(K_i), i= 1,2,j = 1, . . . , n, n∈N. SinceH₁⊙ H₂ contains all polynomials, we have H₁⊗ H₂=C(K). HoweverH₁⊙ H₂(C(K), as can be seen by considering the functionf(x, y) :=e^xy,x∈K₁,y∈K₂.

This example also shows that algebraic tensor product of closed function spaces does not have to be closed.

Definition 3.7. A Banach spaceE is said to have the approximation property, if, for every compact set C ⊂ E and every ε > 0, there is a continuous linear operatorT :E→Eof finite rank so thatkT x−xk< εfor everyx∈C.

(We refer the reader to [14, Chapter 7] for more information on the approxi- mation property.)

Theorem 3.8(Namioka-Phelps). The following statements are equivalent.

(i) For every two compact convex subsetsX₁, X₂ of locally convex Hausdorff spaces isA(X1)⊗A(X2) =A(X1)⊠A(X2).

(ii) Every Banach space has the approximation property.

Proof: See [18, Theorem 2.4 and the subsequent remark].

Using Theorem 3.8 and Enflo’s counterexample [7] of a Banach space not having the approximation property, we may state the following:

Corollary 3.9. There exist compact convex setsX1 andX2 such that A(X₁)⊗A(X₂)(A(X₁)⊠A(X₂).

3.2 Associative laws. In order to be able to use products defined above effec- tively, we need to establish ‘associative laws’ for them.

Definition 3.10. We say, that{Jγ}_γ∈Γ is apartition of a setI, ifS

γ∈ΓJγ =I andJα∩J_β =∅ for everyα, β∈Γ such thatα6=β.

To the end of this subsection, let{(K_i,H_i)}_i∈I be a family of function spaces and {Jγ}_γ∈Γ a partition of I. In the following, we naturally identify spaces C(Q

i∈IK_i) andC(Q

γ∈Γ(Q

i∈JγK_i)).

Proposition 3.11. The following assertions hold:

(i) J

i∈IH_i=J

γ∈Γ(J

i∈JγH_i), (ii) A^c(J

i∈IH_i) =A^c(J

γ∈Γ(J

i∈JγH_i)).

Proof: To prove (i), it clearly suffices to show, that the generating functions of both spaces are the same. Functionf is a generating function ofJ

i∈IH_i, if f =h¹₁⊗. . .⊗h^m₁¹⊗. . .⊗h¹_n⊗. . .⊗h^m_nⁿ⊗1^Q_{K

i:i∈I\{i¹1,... ,i^mnn }}, for someh^l_k∈ H_il

k,i^l_k∈Jγk,l= 1, . . . , m_k,k= 1, . . . , n. Since f_k:=h¹_k⊗. . .⊗h^m_k^k⊗1^Q_{K

i:i∈J_γk\{i¹_k,... ,i^mk_k }}∈ K

i∈J_γk

H_i for each k= 1, . . . , n,

we have

f =f1⊗. . .⊗fn⊗1^Q_{{Ki:i∈I\(Jγ1∪...∪Jγn)}}, which is a generating function ofJ

γ∈Γ(J

i∈JγH_i). Reverting the proof we obtain the converse inclusion.

Assertion (ii) follows from (i) and the fact thatA^c(H) =H.b Proposition 3.12. The following assertions hold:

(i) N

i∈IHi=N

γ∈Γ(N

i∈JγHi), (ii) A^c(N

i∈IH_i) =A^c(N

γ∈Γ(N

i∈JγH_i)).

Proof: Using Proposition 3.11, we have O

i∈I

H_i=K

i∈I

H_i =K

γ∈Γ

K

i∈Jγ

H_i

⊂K

γ∈Γ

O

i∈Jγ

H_i

=O

γ∈Γ

O

i∈Jγ

H_i .

For the converse inclusion, it suffices to proveJ

γ∈Γ(N

i∈JγH_i)⊂N

i∈IH_i, since the latter space is closed. Let f be a generating function of J

γ∈Γ(N

i∈JγH_i).

We can write

f =f₁⊗. . .⊗fn⊗1^Q_{{Ki:i∈I\(Jγ1∪...∪Jγn)}},

where f_i ∈ N

j∈J_γiH_j, i = 1, . . . , n. We may assume thatf_i >0, i= 1, . . . , n (otherwise we write fi = (kfik+ 1)−(kfik+ 1−fi) and use distributive law).

DenoteM := max_{i=1,... ,n}kf_ik. Now choose 0< ε <1 so thatfi> ε,i= 1, . . . , n.

For eachfi we can findhi∈J

j∈J_γiH_j such that fi−ε < hi< fi. We define h:=h₁⊗. . .⊗hn⊗1^Q_{{Ki:i∈I\(Jγ1∪...∪Jγn)}} ∈K

i∈I

Hi,

(we used Proposition 3.11) and compute kf −hk= sup

x1∈^Q_i∈Jγ1Ki

. . . sup

xn∈^Q_i∈JγnKi

Yn

i=1

fi(xi)− Yn

i=1

hi(xi)

< sup

x1∈^Q_i∈Jγ1Ki

. . . sup

xn∈^Q_i∈JγnKi

Yn

i=1

fi(xi)− Yn

i=1

(fi(xi)−ε)

= sup

x1∈^Q_i∈Jγ1Ki

. . . sup

xn∈^Q_i∈JγnKi

ε Xn

k=1

(−1)^k−1ε^k−1 X

|α|=n−k n−kY

i=1

fαi(xαi)

≤ε Xn

k=1

X

|α|=n−k n−kY

i=1

kfαik

≤ε

n−1X

k=0

n k

M^k . Sinceεis arbitrary, we conclude thatf ∈N

i∈IH_i.

Assertion (ii) follows from (i) and the fact thatA^c(H) =H.b Proposition 3.13. The following assertions hold:

(i)

⊠

i∈IH_i=

⊠

γ∈Γ(

⊠

i∈JγH_i),

(ii) A^c(

⊠

i∈IHi) =A^c(

⊠

γ∈Γ(

⊠

i∈JγHi)).

Proof: Letf ∈

⊠

i∈IH_i. Pickγ₀∈Γ andk^′∈Q

i∈I\Jγ0K_i. We want to prove that π^k_J^′

γ0(f) ∈

⊠

i∈Jγ0Hi, i.e., that π^k_j^′′(π_J^k′

γ0(f)) ∈ Hj for every j ∈ Jγ0 and k^′′∈Q

i∈Jγ0\{j}K_i. But this is true, sinceπ^k_j^′′(π^k_J^′

γ0(f)) =π_j^{(k′,k′′)}(f)∈ H_j. Conversely, let f ∈

⊠

γ∈Γ(

⊠

i∈JγH_i). Pick j ∈ I and k ∈ Q

i∈I\{j}K_i. Then j∈Jγ0 for someγ₀∈Γ. Using the assumption, we have

π_j^k(f) =π_j^{πJγ0\{j}(k)}(π^π_Jγ0^I\Jγ0(k)(f))∈ H_j.

Assertion (ii) follows from (i) and the fact thatA^c(H) =H.b From now on, we consider (K,H) to be a product of (Ki,H_i), i ∈ I, unless said otherwise.

3.3 Representing measures.

Notation 3.14. LetJ ⊂I. We denote byH_J the space of all functions fromH depending on coordinates fromJ, i.e.,

H_J :={h∈ H:x, y ∈K, π_J(x) =π_J(y)⇒h(x) =h(y)},

and letH_f be the space of all functions fromHdepending on a finite number of coordinates, i.e.,

H_f :={h∈ H:∃J⊂I finite, so that h∈ H_J}.

Observation 3.15. Using the above notation, we observe:

(a) I₁⊂I₂ ⊂I, h∈ H_I1 ⇒ h∈ H_I2, (b) h∈ H_J ⇔ h=πJ(h)⊗1^Q_{Ki:i∈I\J},

(c) µ∈ M⁺(K), h∈ H_J ⇒ µ(h) = (π_Jµ)(π_J(h)), (d) H_f is a product of H_i,i∈I.

Proposition 3.16. Let us assume either (a) H ⊂N

i∈IH_i, or (b) H=

⊠

i∈IH_i. ThenH_f is dense inH.

Proof: Assuming (a), conclusion is trivial, since J

i∈IH_i ⊂ H_f. Assuming (b), we can use the same technique as in the proof of [16, Theorem 3.1] or [6,

Lemma 4].

Corollary 3.17. C_f(K)is dense inC(K).

Proof: Notice that C(K) =

⊠

i∈IC(K_i) and use Proposition 3.16(b).

Example 3.18. The conclusion of Proposition 3.16 does not have to be true for all products. Suppose we havef ∈(

⊠

i∈IH_i)\(N

i∈IH_i), which does not depend on finitely many coordinates. LetHbe the linear span ofJ

i∈IH_i∪ {f}. Then H_f =J

i∈IH_i, butf /∈ H_f.

Now we construct such a functionf. Let (K_i,H_i) := (X_i, A(X_i)),i= 1,2, be as in Corollary 3.9. Then there isf₁ ∈(H₁⊠H₂)\(H₁⊗ H₂). This function is not constant with respect to any of the two coordinates, sincef₁ ∈ H/ ₁⊙ H₂. SetH_2n+1:=H₁,H_2n+2:=H₂,n∈N, and letf_n+1:=f₁ be the function from (H_2n+1⊠H_2n+2)\(H_2n+1⊗ H_2n+2) for everyn∈N. Set

f :=

X∞ n=1

2⁻ⁿ⁺¹fn⊗1^Q_{Ki:i∈N\{2n−1,2n}}.

Obviously,f does not depend on finite number of coordinates and f ∈

⊠

i∈NH_i since this space is closed. Alsof /∈N

i∈NH_i. Indeed, if we suppose the contrary, then

f ∈O

i∈N

H_i = (H₁⊗ H₂)⊗( O∞ i=3

H_i)⊂(H₁⊗ H₂)⊠( O∞ i=3

H_i).

Thus, for y∈Q_∞

i=3Ki isπ^y_{1,2}(f)∈ H₁⊗ H₂. Butπ_{1,2}^y (f) =f1+c, where c is a constant, which is a contradiction, sincef1∈ H/ ₁⊗ H₂.

Definition 3.19. Let (K,H) be a product of (K_i,H_i), i ∈ I. For J ⊂ I we define the projection ofHby

πJ(H) :={f ∈ C(Y

i∈J

Ki) :f⊗1^Q_{Ki:i∈I\J}∈ H}.

Observation 3.20. The following assertions hold:

(a) π_J(H)is a product ofHi,i∈J, (b) π_J(J

i∈IHi) =J

i∈JHi, (c) π_J(N

i∈IHi) =N

i∈JHi, (d) π_J(

⊠

i∈IHi) =

⊠

i∈JHi.

Proposition 3.21. Letx∈K,µ∈ Mx(H)andJ ⊂I. Then πJµ∈ M_πJ(x)(πJ(H)).

Proof: LethJ ∈πJ(H) and defineh:=hJ⊗1^Q_{Ki:i∈I\J}. Thenh∈ Hand hJ(πJ(x)) =h(x) =µ(h) = (πJµ)(hJ).

Proposition 3.22. Let x = (xi)_i∈I ∈ K and µi ∈ Mxi(Hi) for every i ∈ I.

Thenµ:=N

i∈Iµi∈ Mx(H).

Proof: It suffices to prove the assertion forH=

⊠

i∈IA^c(H_i).

(1) First, let|I|=n∈N. Chooseh∈ H. By Fubini’s theorem, µ(h) =

Z

K

h dµ= Z

K1

. . . Z

Kn

h(y₁, . . . , yn)dµn(yn). . . dµ₁(y₁).

Since the functionyn7→h(y1, . . . , yn) is inA^c(H_n) andµn∈ M_xn(H_n), we have Z

Kn

h(y1, . . . , yn−1, yn)dµn(yn) =h(y1, . . . , yn−1, xn)

for every (y₁, . . . , y_n−1) ∈ Q_n−1

i=1 K_i. Using induction, we can see that µ(h) = h(x₁, . . . , xn) =h(x). Thereforeµ∈ Mx(H).

(2) Now, let I be an arbitrary index set. Choose h ∈ H and ε > 0. By Proposition 3.16(b), there isg∈ H_J for some finiteJ ⊂Iso that

kg−hk<ε 2. Using the first part of the proof, we write

µ(g) = O

i∈J

µi

(π_J(g)) =π_J(g)(π_J(x)) =g(x).

Let us estimate

|µ(h)−h(x)| ≤ |µ(h)−µ(g)|+|µ(g)−g(x)|+|g(x)−h(x)|< ε.

Sinceεis arbitrary,µ(h) =h(x). Henceµ∈ Mx(H).

Notation 3.23. Let Ai ⊂ M¹(Ki) for every i ∈ I. We denote N

i∈IAi :=

{N

i∈Iµi :µi∈Ai, i∈I}.

Example 3.24. If|I|= 2, Proposition 3.22 yields the inclusion co^w∗(M_x1(H₁)⊗ M_x2(H₂))⊂ M_x(H), x= (x1, x2)∈K.

Now we show that the inclusion may be proper.

LetK_i :={r_i, s_i, t_i}, H_i := {f ∈ C(K_i) : f(s_i) = ¹₂(f(r_i) +f(t_i))}, i= 1,2.

Then Msi(H_i) = co{εsi,^εri+ε₂ ^ti}. Suppose (K,H) is a product of these two spaces. Denote

C:= co

εs1 ⊗εs2, εs1⊗εr2+εt2

2 , εr1 +εt1

2 ⊗εs2, εr1+εt1

2 ⊗εr2+εt2

2

.

We see that co^w∗(Ms1(H₁)⊗ Ms2(H₂)) =C. Define µ:=ε_(s1,t2)

2 +ε_(r1,r2)

4 +ε_(t1,r2)

4 .

Obviouslyµ∈ M_(s1,s2)(H). For everyx∈K\ {(s₁, t₂),(r₁, r₂),(t₁, r₂)}we have µ({x}) = 0. However, ifµwere an element ofC, then at least one of the points (s₁, s₂), (s₁, r₂), (r₁, s₂), (r₁, t₂) would have a non-zero measure.

Example 3.25. Letx∈K. DenoteM^π_x(H) the set of allµ∈ M¹(K) such that π_i(µ)∈ M_πi(x)(H_i) for everyi∈I. Proposition 3.21 yields

M_x(H)⊂ M^π_x(H).

Once again, we show that the inclusion may be proper.

Let (K_i,H_i),i= 1,2, be as in Example 3.24. Consider µ:= ε_(r1,r2)

2 +ε_(t1,t2)

2 .

We see that π_i(µ) = ^ε₂^ri + ^ε₂^ti ∈ Msi(H_i), i = 1,2. Thus µ ∈ M^π_(s

1,s2)(H).

Howeverµ /∈ M_(s1,s2)(H). Indeed, takefi∈ Hi such that fi(ri) = 0,fi(si) = 1, f_i(t_i) = 2, for i= 1,2. Definef :=f₁⊗f₂. Thenf ∈ H, but

f(s₁, s₂) = 16= 2 =µ(f).

Question 3.26. Is there a way to characterizeMx(H) byM_πi(x)(H_i),i∈I?

Proposition 3.27. Letx= (x_i)_i∈I∈K. ThenFx(H) =Q

i∈IFxi(H_i).

Proof: First we showFx(H)⊂Q

i∈IFxi(H_i). For each µ∈ M_x(H) andi ∈I we have πi(sptµ) = sptπiµ and since, by Proposition 3.21,πiµ∈ Mx_i(H_i), we getπi(sptµ)⊂Fx_i(H_i). Thereforeπi(Fx(H))⊂Fx_i(H_i) for everyi∈I.

Conversely, letµi∈ Mx_i(H_i) for everyi∈I. Proposition 3.22 yieldsN

i∈Iµi∈ Mx(H) and thusQ

i∈Isptµ_i= sptN

i∈Iµ_i⊂Fx(H).

3.4 H-affine functions.

Proposition 3.28. A^c(H)⊂

⊠

i∈IA^c(H_i).

Proof: Choosef ∈ A^c(H),j ∈I and y = (yi)∈Q

i∈I\{j}Ki. We prove that f_j :=π^y_j(f)∈ A^c(H_j). Letx_j ∈K_j andµ_j ∈ Mxj(H_j). Define x:= (x_j, y) and µ:=µj⊗(N

i∈I\{j}εyi). According to Proposition 3.22,µ∈ Mx(H), so we have fj(xj) =f(x) =µ(f) =µj(fj).

Hencef_j∈ A^c(H_j).

Lemma 3.29. Let|I|= 2. ThenA^c(H₁)⊗ A^c(H₂)⊂ A^c(H).

Proof: Considera₁∈ A^c(H₁),a₂∈ A^c(H₂). We show thata₁⊗a₂∈ A^c(H) by using the characterizationA^c(H) =H.b

First suppose thata₁, a₂≥0. Choosex= (x₁, x₂)∈Kandε >0. Findδ >0 so that

δ(a₁(x₁) +a₂(x₂) +δ)< ε.

Sincea^∗_i =a_i, i= 1,2, there are h₁ ∈ H₁, h₁ ≥a₁ and h₂ ∈ H₂, h₂ ≥a₂ such that

h₁(x₁)< a₁(x₁) +δ and h₂(x₂)< a₂(x₂) +δ.

Obviouslyh₁⊗h₂∈ H,h₁⊗h₂ ≥a₁⊗a₂ and

a₁(x₁)a₂(x₂)≤h₁(x₁)h₂(x₂)<(a₁(x₁) +δ)(a₂(x₂) +δ)

=a₁(x₁)a₂(x₂) +δ(a₁(x₁) +a₂(x₂) +δ)< a₁(x₁)a₂(x₂) +ε.

Thus (a1⊗a2)^∗=a1⊗a2.

Now supposea1≥0 anda2is arbitrary. Thena2+ka₂k ≥0. Sincef 7→f^∗ is a sublinear functional onC(K) and (a₁⊗c)^∗=a1⊗cfor every constant function conK₂, we get

a₁⊗a₂≤(a₁⊗a₂)^∗= (a₁⊗(a₂+ka₂k − ka₂k))^∗

= (a₁⊗(a₂+ka₂k)−a₁⊗ ka₂k)^∗

≤(a1⊗(a2+ka₂k))^∗+ (a1⊗(−ka₂k))^∗

=a1⊗(a2+ka₂k) + (a₁⊗(−ka₂k)) =a1⊗a2. For the lower envelope we have

(a₁⊗a₂)∗ =−(a₁⊗(−a₂))^∗ =−(a₁⊗(−a₂)) =a₁⊗a₂. Thusa₁⊗a₂∈Hb=A^c(H).

Finally, leta₁, a₂ be arbitrary. Then

a₁⊗a₂= (a₁+ka₁k)⊗a₂− ka₁k ⊗a₂∈ A^c(H).

SinceA^c(H) is a closed linear space, the conclusion follows.

Proposition 3.30. N

i∈IA^c(H_i)⊂ A^c(H).

Proof: It suffices to prove J

i∈IA^c(H_i) ⊂ A^c(H), since the latter space is closed.

(1) Assume first, that |I| = n ∈ N and the assertion holds for|I| = n−1.

Using the assumption, previous Lemma 3.29 and the associative law, we get Kn

i=1

A^c(H_i) =

n−1K

i=1

A^c(H_i)

⊙ A^c(H_n)⊂ A^c

n−1K

i=1

H_i

⊙ A^c(H_n)

⊂ A^c (

n−1K

i=1

H_i)⊙ Hn

=A^c Kn

i=1

H_i

⊂ A^c(H).

(2) Now, let I be an arbitrary index set. Choose f ∈ J

i∈IA^c(H_i). Then there is a finite J ⊂ I such that f depends only on coordinates from J. So, according to the first part of the proof,π_J(f)∈J

i∈JA^c(H_i)⊂ A^c(J

i∈JH_i).

Sincef =π_J(f)⊗1^Q_{Ki:i∈I\J}, we have f ∈ A^c K

i∈J

Hi

⊙ A^c K

i∈I\J

Hi

⊂ A^c (K

i∈J

Hi)⊙(K

i∈I\J

Hi)

=A^c K

i∈I

H_i

⊂ A^c(H).

Corollary 3.31. A^c(H)is a product of bothHi,i∈I, andA^c(Hi),i∈I.

Proof: From Proposition 3.30 we have K

i∈I

Hi ⊂K

i∈I

A^c(Hi)⊂ A^c(H), and from Proposition 3.28

A^c(H)⊂

⊠

i∈I

A^c(H_i) =

⊠

i∈I

A^c(A^c(H_i)).

Proposition 3.32. If A^c(H)⊂

⊠

i∈IH_i, then H_i=A^c(H_i)for everyi∈I.

Proof: Choosei∈I. We prove thatA^c(Hi)⊂ Hi. Pickfi∈ A^c(Hi) and define f := fi⊗1^Q_{{Kj:j∈I\{i}}}. Choose x = (xj)_j∈I ∈ K and µ ∈ Mx(H). From Proposition 3.21 we haveµi:=πiµ∈ Mxi(Hi), which implies

f(x) =fi(xi) =µi(fi) =µ(f).

Thusf ∈ A^c(H)⊂

⊠

i∈IH_i, sof_i=π_i(f)∈ H_i. Proposition 3.33. Let H =

⊠

i∈IH_i. Then H = A^c(H)if and only if H_i = A^c(H_i)for everyi∈I.

Proof: If H = A^c(H), we use Proposition 3.32. Conversely, from Proposi- tion 3.28 we have

H ⊂ A^c(H)⊂

⊠

i∈I

A^c(Hi) =

⊠

i∈I

Hi=H.

Corollary 3.34. There are function spacesH₁ andH₂ such that A^c(H₁)⊗ A^c(H₂)(A^c(H₁⊠H₂).

Proof: By Corollary 3.9, there areH₁ and H₂ such that H₁⊗ H₂ (H₁⊠H₂ and H_i =A^c(H_i), i = 1,2. Proposition 3.33 implies H₁⊠H₂ =A^c(H₁⊠H₂).

Thus

A^c(H₁)⊗ A^c(H₂) =H₁⊗ H₂ (H₁⊠H₂=A^c(H₁⊠H₂).

Example 3.35. Example 3.6 shows there are function spaces such that A^c(H₁)⊙ A^c(H₂)(A^c(H₁⊙ H₂).

Question 3.36. IsN

i∈IA^c(H_i) =A^c(J

i∈IH_i)?

Question 3.37. IsA^c(J

i∈IH_i) =A^c(

⊠

i∈IH_i)?

Question 3.38. IsA^c(

⊠

i∈IHi) =

⊠

i∈IA^c(Hi)?

3.5 H-extremal sets.

Proposition 3.39. LetE⊂Kbe anH-extremal set. LetJ ⊂I,y∈Q

i∈I\JK_i, and letG be a product of H_i,i∈J. Thenπ_J^y(E)is either empty or aG-extremal set.

Proof: SupposeE^y :=π_J^y(E) is non-empty and notG-extremal. Then there is x∈E^y andµ_J ∈ Mx(G) so that sptµ_J 6⊂ E^y. According to Proposition 3.21, µ_i:=π_iµ_J ∈ M_πi(x)(H_i) for everyi∈J. Since sptµ_J ⊂sptN

i∈Jµ_i, we can see that sptN

i∈Jµi 6⊂E^y. Define µ:= O

i∈J

µi

⊗ O

i∈I\J

ε_πi(y) .

Hence, we have (x, y) ∈ E and by Proposition 3.22 also µ ∈ M_(x,y)(H). But

sptµ6⊂E, which is a contradiction.

The next two propositions are generalizations of Proposition 4.1 and Theo- rem 4.2 from [16] to function spaces:

Proposition 3.40. LetE ⊂K be anH-extremal set. Let∅ 6=J ⊂I and letG be a product ofHi, i∈J. Thenπ_J(E)is a G-extremal set.

Proof: Let x ∈ π_J(E) and µ ∈ Mx(G). Then there is y ∈ Q

i∈I\JK_i such that (x, y)∈E, i.e.,x∈π^y_J(E). By Proposition 3.39,π^y_J(E) is aG-extremal set,

therefore sptµ⊂π^y_J(E)⊂πJ(E).

Proposition 3.41. LetE_i ⊂ K_i be an H_i-extremal set for everyi ∈ I. Then E:=Q

i∈IE_i is anH-extremal set.

Proof: Obviously,E is a closed set.

(1) Assume |I|= 2. Suppose there is x= (x₁, x₂) ∈E and µ∈ Mx(H) so thatµ(K\E)>0. Denoteµ₁:=π₁µandµ₂:=π₂µ. Since

K\E= ((K₁\E₁)×K₂)∪(K₁×(K₂\E₂)), we have

0< µ(K\E)≤µ₁(K₁\E₁) +µ₂(K₂\E₂).

We may assumeµ₁(K₁\E₁)>0. By Proposition 3.21,µ₁∈ M_x1(H₁). But this is a contradiction, becausex1∈E1.

We proceed similarly for arbitrary finite products.

(2) Now, letI be infinite. Suppose there is x= (xi)_i∈I∈E andµ∈ Mx(H) so that µ(K\E)>0. Then there is some g ∈ C(K) such that g = 0 onE and µ(g)>0. Chooseε >0. According to Corollary 3.17, there isf ∈ C_J(K), where J ⊂ I is finite and kg−fk < ε. Thenπ_Jµ∈ M_πJ(x)(π_J(H)) and by the first part of the proof,π_J(x) is an element of theπ_J(H)-extremal setE_J :=Q

i∈JE_i. Thus sptπ_Jµ⊂E_J and|π_J(f)|< εonE_J. Therefore

|µ(f)|=|(π_Jµ)(π_J(f))| ≤ Z

E_J

|π_J(f)|d(π_Jµ)+

Z

(^Q_i∈JK_i)\EJ

|π_J(f)|d(π_Jµ)< ε.

Hence we get

0<|µ(g)| ≤ |µ(g)−µ(f)|+|µ(f)|<2ε,

which is a contradiction, sinceεis arbitrary.

Using previous results, we can derive the main theorem of this subsection (cf.

also [6, Lemma 5], [16, Theorem 3.2] and [10, Lemma 5.11]):

Theorem 3.42. Ch_HK=Q

i∈ICh_Hi Ki.

Proof: Follows immediately from Propositions 3.40 and 3.41.

Corollary 3.43. ∇_HK=Q

i∈I∇_HiKi. Proof: Using Theorem 3.42 we can write

∇_HK= Ch_HK=Y

i∈I

Ch_Hi K_i=Y

i∈I

Ch_Hi K_i=Y

i∈I

∇_HiK_i.

Remarks 3.44. As has been shown by Grossman [9], the characterizations of Choquet and ˇSilov boundary hold also for the spaceH₁+H₂ defined by

H₁+H₂:={h₁+h2:h1∈ H₁, h2∈ H₂}, where [h1+h2](x, y) =h1(x) +h2(y), (x, y)∈K1×K2.

It is clear thatH₁+H₂ does not have to be a product, since the inclusionH₁⊙ H₂ ⊂ H₁+H₂ does not have to hold.

Versions of Theorem 3.42 for various tensor products of compact convex sets have been proved by I. Namioka and R.R. Phelps in [18].

Example 3.45. In Example 3.3 we have shown that the space of all harmonic functions on a cartesian product does not have to be a product of harmonic spaces.

Moreover, it is not even possible to extend the notion of a product so that the product of harmonic spaces would be a harmonic space and Theorem 3.42 would still hold. Indeed, consider the sets from Example 3.3 and denoteU :=U1×U2. Then

Ch_H(U) U =∂regU=∂U 6={0,1} × {0,1}= Ch_H(U1) U₁×Ch_H(U2) U₂.

3.6 Approximation in product spaces. In the following, we will need some results on approximation of functions in simplicial spaces. So we first state here results that are adaptation of Section 2 from [17].

Definition 3.46. Let (K,H) be a function space. A collection of nonnegative functions{ψ_j}^m_j=1⊂ His called apartition of unity onK, ifP_m

j=1ψ_j = 1_K. Lemma 3.47. Let(K,H) be a simplicial function space. Let{f_i}ⁿ_i=1⊂ A^c(H) andε >0. Suppose that{φ_j}^m_j=1 are nonnegative functions defined on Ch_HK, {k_l}^m_l=1⊂Ch_H Kand {α_ij : 1≤i≤n,1≤j≤m}are real numbers such that

(i) Pm

j=1φj = 1,

(ii) φ_j(k_l) =δ_jl, 1≤j, l≤m, (iii) |fi(k)−Pm

j=1αijφj(k)| ≤ε, k∈Ch_HK,1≤i≤n.

Then there exists a partition of unity{ψ_j}^m_j=1⊂ A^c(H)such that (iv) ψj(k_l) =δ_jl, 1≤j, l≤m,

(v) |f_i(k)−P_m

j=1α_ijψ_j(k)| ≤ε, k∈K,1≤i≤n.

Proof: See [17, Corollary 2.2].

The proof of the next lemma is based on the proof of [17, Lemma 2.4]:

Lemma 3.48. Let (K₁,H₁) and (K₂,H₂) be two function spaces, where H₁ is simplicial. Suppose that {f_i}ⁿ_i=1 ⊂ A^c(H₁)⊠H₂ and ε > 0. Then there is a partition of unity {ψ_j}^m_j=1 ⊂ A^c(H₁), {k_l}^m_l=1 ⊂ Ch_H1 K₁ and {y_ij} ⊂ H₂, 1≤i≤n,1≤j≤m, so that

(i) ψj(kl) =δjl, 1≤j,l≤m, (ii) kf_i−Pm

j=1ψ_j⊗y_ijk< ε, 1≤i≤n.

Proof: Denote byHⁿ₂ then-tuple cartesian product of H₂ with the maximum norm, i.e.,

kyk_max= max

1≤i≤nkπ_i(y)k for all y∈ Hⁿ₂,

whereπ_i is the projection to the i-th coordinate. We denote by Br(x) the open ball with centerxand radiusr >0.

Letf be a function fromK₁ toHⁿ₂ defined by

f(k) := (π₂^k(f₁), . . . , π^k₂(fn)), k∈K₁.

Sinceπi◦f is a continuous function for everyi= 1, . . . , n(we use the fact that C(K₁×K₂) is isometric toC(K₁,C(K₂))), f is also a continuous function onK₁.

For eachy∈ Hⁿ₂ set

(1) Uy:=n

k∈K₁:ky−f(k)kmax< ε 3 o

.

The family {Uy}_y∈Hⁿ₂ is an open covering of K₁. Let Uy1, . . . , Uyp be a finite subcovering. Define

Vyj :=Uyj∩Ch_H1 K₁, 1≤j≤p.

Without loss of generality we may assume that there ism≤psuch that{Vyl}^m_l=1 is an open covering of Ch_H1 K₁and for everyl∈ {1, . . . , m}there existsk_l∈Vyl

such thatk_l∈/ Vyj forj6=l, 1≤j ≤m.

Denote

C:={y₁, . . . , yp} −co(y1, . . . , ym), D:=C+B^ε

3(0).

Choosei∈ {1, . . . , n}. SinceC is a compact subset ofHⁿ₂, alsoπi(C) is a com- pact subset ofH₂. By Arzel`a-Ascoli’s theorem, the setπi(C) is equicontinuous.

Therefore, for each ξ ∈ K2 we can find its open neighbourhood Wξ such that oscW_ξ h < ^ε₃ for everyh∈πi(C). From the open covering{W_ξ}_ξ∈K2 we choose a finite subcovering{W_ξir}^q_r=1ⁱ . For everyh∈πi(C) there is x_h ∈K₂ such that