Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 59–66

CHARACTERIZATIONS OF BALAYAGES

Sirkka-Liisa Eriksson-Bique

University of Joensuu, Department of Mathematics P.O. Box 111, FIN-80101 Joensuu, Finland

Abstract. We consider balayages in H -cones. Any pair of elements in an H -cone has mixed envelopes formed relative to two partial orders. Our main result characterizes balayages in terms of mixed envelopes. We state an explicit formula of a balayage in an H -cone admitting a certain type of unit.

Introduction

Mappings called balayages are important objects in the theory of harmonic spaces ([4]). An axiomatic counterpart of the cone of positive superharmonic functions on a harmonic space is an H -cone ([3, Section 2]). We consider balayages in H -cones.

Any pair of elements of an H -cone has mixed envelopes, studied by Arsove and Leutwiler ([2]), defined in terms of two partial orders. Using mixed envelopes it is possible to extend the Freudental spectral theorem of vector lattices (or Riesz spaces) for H -cones [1]. In our main theorem (Theorem 2.4) we use mixed envelopes to characterize balayages. This result leads to an explicit formula of a balayage in an H -cone admitting a special unit (Theorem 2.8).

1. Preliminaries

We review the basic concepts.

Definition 1.1. Let E be an ordered vector space and S be a convex cone in E such that S ⊂ E^{+} and E = S − S. The cone S is called an H -cone if it possesses the following properties:

(A_{1}) any upward directed and dominated subset F of S has a least upper bound

in E denoted by ∨F and ∨F ∈ S, (A_{2}) any subset F of S has a greatest lower bound in E denoted by ∧F and

∧F ∈ S, (A_{3}) for any s and t in S the greatest lower bound of the set { u ∈ S | s − t ≤ u }

in E denoted by R(s − t) satisfies R(s − t) ∈ S and s − R(s − t) ∈ S.

1991 Mathematics Subject Classification: Primary 31D05; Secondary 06F20, 46A40.

The equivalence of the preceding definition and the original definition of an H -cone [3, p. 37) is proved in [6].

The partial order ≤ in an H -cone is called the initial order. Another partial order called specific order denoted by ≼ is defined in an H -cone by

s ≼ t if and only if t = s + s′ for some s′ ∈ S.

Proposition 1.2. If S is an H -cone in an ordered vector space E then E is an Archimedean vector lattice with respect to the initial order. Moreover, the set E is a conditionally complete vector lattice with respect to the specific order.

Proof. See [3, Proposition 2.1.1] and [3, Theorem 2.1.5].

Any pair of elements in an H -cone has mixed envelopes introduced by Arsove and Leutwiler in algebraic potential theory ([1, 2]).

Theorem 1.3. Let S be an H -cone. Then for any elements s and t in S there exist a mixed lower envelope

s⌣\t = max{ x ∈ S | x ≼ s, x ≤ t }

and a mixed upper envelope

s⌢/t = min{ x ∈ s | x ≽ s, x ≥ t }

satisfying the equality

s⌣\t + t⌢/s = s + t.

Proof. See [2, Theorem 2.5].

We use the following properties of mixed lower envelopes in H -cones stated in [2, Section 3]:

(1.1)  t ≼ s if and only if s⌣\t = t
(1.2)  t ≤ s if and only if t⌣\s = t
(1.3)  a + s⌣\t = (a + s)⌣\(a + t)
(1.4)  (s + t)⌣\u ≼ s⌣\u + t⌣\u
(1.5)  u⌣\(s + t) ≤ u⌣\s + u⌣\t

Two types of units, defined next, are important in the theory of H -cones.

Definition 1.4. Let S be an H -cone. An element e ∈ S is called a weak unit if s = ∨_{n∈N}(ne) ∧ s for all s ∈ S. An element p ∈ S is called a generator if s = ⋁_{n∈N}(np)⌣\s for all s ∈ S.

Note that a generator is also a weak unit. If the initial order and the specific order coincide, then an element is a weak unit if and only if it is a generator. Moreover, an H -cone possessing a weak unit and a countable order dense set has a generator [6, Theorem 2.8].

2. Characterizations of balayages of H -cones

We are going to state some new characterizations of balayages in terms of mixed lower envelopes. We start by recalling the main concepts.

Definition 2.1. Let S be an H -cone and B be a mapping from S into S. The mapping B is called left order continuous if for any s ∈ S the property

B(s) = ⋁_{t∈F} B(t)

holds for all upward directed subsets F of S such that s = ∨F. The mapping B is called idempotent if B^{2} = B and contractive if B(s) ≤ s for all s ∈ S. Moreover, the mapping B: S → S is called a balayage if it is additive, left order continuous, idempotent and contractive.

A partial ordering in the set of left order continuous additive mappings in S is defined by ψ ≤ φ if ψ(s) ≤ φ(s) for all s ∈ S.

Note that if B: S → S is left order continuous then B is increasing.

The pointwise least upper bound of an arbitrary set of balayages is surprisingly a balayage as proved in [5, Proposition 2.1].

Lemma 2.2. Let S be an H -cone. If B_{i} is a balayage for each i in an index set I then a mapping B: S → S defined by

B(s) = ⋁_{i∈I} B_{i}(s)

is a balayage.

Lemma 2.3. Let S be an H -cone and B: S → S be a balayage. Then the set B(S) is a specifically solid convex cone in S. Moreover, the equality

B(u)⌣\v = B(u)⌣\B(v)

holds for all u and v in S.

Proof. Assume that B: S → S is a balayage. Since B is idempotent and additive we have

B(s) + B(t) = B^{2}(s) + B^{2}(t) = B(B(s) + B(t)).

Hence B(S) is a convex cone. It is also specifically solid. Indeed from w ≼ s for s ∈ B(S) it follows that w + w′ = s = B(s) = B(w) + B(w′). Then the properties B(w) ≤ w and B(w′) ≤ w′ imply that B(w) = w.

Assume that u, v ∈ S. Obviously,

B(u)⌣\v ≥ B(u)⌣\B(v).

To prove the reverse inequality, let t ∈ S be such that t ≼ B(u) and t ≤ v. As B(S) is specifically solid we infer t = B(t). But then the inequality t ≤ v results in t = B(t) ≤ B(v). Consequently B(u)⌣\v ≤ B(u)⌣\B(v) completing the proof.

The preceding lemma gives a characterization of balayages. The first assertion of this result is also proved by Popa in [7].

Theorem 2.4. Let S be an H -cone. Suppose that a mapping ψ: S → S is left order continuous and admits the property

(2.1)  ψ(u)⌣\v = ψ(u)⌣\ψ(v)

for all u and v in S. Then ψ is idempotent, contractive, subadditive and the set ψ(S) is specifically solid. Moreover the mapping B_{ψ}: S → S defined by

(2.2)  B_{ψ}(s) = ⋁ { ∑_{i=1}^{n} ψ(s_{i}) | ∑_{i=1}^{n} s_{i} ≤ s, n ∈ N, s_{i} ∈ S }

is a balayage and therefore satisfies (2.1).

Proof. Assume that ψ: S → S is left order continuous and satisfies (2.1). Using (2.1) we find out that

u ≥ ψ(u)⌣\u = ψ(u)⌣\ψ(u) = ψ(u).

Hence ψ is contractive and so ψ^{2} ≤ ψ. Moreover applying (2.1) twice we obtain

ψ(u) = ψ(u)⌣\ψ(u) = ψ(u)⌣\ψ^{2}(u) ≤ ψ^{2}(u).

Hence ψ is idempotent.

Suppose that w ∈ S and w ≼ ψ(t) for some t ∈ S. Then applying (2.1) we observe that

w = ψ(t)⌣\w = ψ(t)⌣\ψ(w) ≤ ψ(w) ≤ w,

which leads to w = ψ(w) ∈ ψ(S). Thus the set ψ(S) is specifically solid.

The mapping B_{ψ} is well defined, since ∑_{i=1}^{n} ψ(s_{i}) ≤ ∑_{i=1}^{n} s_{i} ≤ s for all s ∈ S and therefore the least upper bound on the right side of the equality (2.1) exists and B_{ψ}(s) ≤ s. To prove additivity assume that u and v are arbitrary elements of S. Then by (1.5) and (2.1) we have

\begin{aligned}ψ(u + v) & = ψ(u + v)⌣\(u + v) ≤ ψ(u + v)⌣\u + ψ(u + v)⌣\v \\ & = ψ(u + v)⌣\ψ(u) + ψ(u + v)⌣\ψ(v) ≤ ψ(u) + ψ(v). \end{aligned}

Hence ψ is subadditive. Applying [3, Proposition 2.2.4] we note that the mapping B_{ψ} is additive and left order continuous. Moreover since the mapping ψ is idempotent and ψ ≤ B_{ψ} we have ψ ≤ B_{ψ}^{2} . As B_{ψ}^{2} is additive and increasing we infer from (2.2) that B_{ψ} ≤ B_{ψ}^{2} ≤ B_{ψ} completing the proof.

Corollary 2.5. If the function ψ_{a}: S → S defined by ψ_{a}(u) = a⌣\u is left order continuous for some a ∈ S then the mapping B_{ψ_{a}} defined by (2.2) is a balayage and B_{ψ_{a}}(s) = ∨_{n∈N}(na)⌣\s for all s ∈ S.

Proof. Assume that the function ψ_{a}: S → S given by ψ_{a}(u) = a⌣\u for some a ∈ S is left order continuous. The preceding theorem asserts that the mapping B_{ψ_{a}} defined by (2.2) is a balayage. Let s be an arbitrary element of S. Assume that s_{i} are elements of S such that ∑_{i=1}^{n} s_{i} ≤ s. Then we infer that

∑_{i=1}^{n} a⌣\s_{i} ≤ (na)⌣\s ≤ n(a⌣\(s/n)) ≤ B_{ψ_{a}}(s).

Hence we conclude B_{ψ_{a}}(s) = ∨_{n∈N}(na)⌣\s.

Corollary 2.6. Let S be an H -cone. Then the mapping B: S → S is a balayage if and only if B is left order continuous, idempotent and the set B(S) is a specifically solid subsemigroup of S.

Proof. The assertion is obvious for a balayage. To prove the converse statement assume that B: S → S satisfies the required conditions. Since the set B(S) is a semigroup and B is idempotent we infer

B(s) ≥ B(∑_{i=1}^{n} B(s_{i})) = ∑_{i=1}^{n} B(s_{i})

for all s_{i} ∈ S and s ∈ S with ∑_{i=1}^{n} s_{i} ≤ s. Hence we only have to verify the equality (2.1) for B. Let u and v be elements in S. Suppose that w ≼ B(u) and w ≤ v. Since B(S) is specifically solid we have w = B(t) for some t ∈ S. As B is idempotent and increasing we obtain w = B(w) ≤ B(v). Hence the fact that w is an arbitrary element such that w ≼ B(u) and w ≤ v implies B(u)⌣\v ≤ B(u)⌣\B(v). Thus noting that B(v) ≤ v we conclude

B(u)⌣\v ≤ B(u)⌣\B(v) ≤ B(u)⌣\v

completing the proof.

Proposition 2.7. Let S be an H -cone and u in S. If the mapping B: S → S defined by

B(s) = ⋁_{n∈N}(nu)⌣\s

is left order continuous then it is a balayage.

Proof. Applying (1.4) and (1.5) we note that

(nu)⌣\x + (mu)⌣\y ≤ (m + n)u⌣\(x + y) ≤ (m + n)u⌣\x + (m + n)u⌣\y

and so B is additive. From (1.1) and (1.2) it easily follows that B is also idempotent and contractive. Hence B(S) is specifically a solid subsemigroup of S. By the preceding corollary B is a balayage.

Our main theorem gives a new presentation of a balayage. Moreover it shows how the value of a balayage at a point is obtained from its value at a generator.

Theorem 2.8. Let S be an H -cone possessing a generator p and B a mapping from S into S. Then B is a balayage if and only if B is left order continuous and

(2.3)  B(x) = ⋁_{n∈N}(nB(p))⌣\x

for all x ∈ S.

Proof. Assume first that B: S → S is a balayage. Let p be a generator in S and x ∈ S. Since B is order continuous we have

B(x) = B( ⋁_{n∈N}(np)⌣\x) = ⋁_{n∈N} B((np)⌣\x).

Hence by Lemma 2.3 we conclude

B(x) = ⋁_{n∈N} B((np)⌣\x) ≤ ⋁_{n∈N} B(np)⌣\x = ⋁_{n∈N}(nB(p))⌣\B(x) ≤ B(x),

which implies

B(x) = ⋁_{n∈N}(nB(p))⌣\x.

To prove the converse assume that B: S → S is left order continuous and

B(x) = ⋁_{n∈N}(nB(p))⌣\x.

Since by (1.5) the relation

\begin{aligned}(nB(p))⌣\x + (mB(p))⌣\y & ≤ ((m + n)B(p))⌣\(x + y) \\ & ≤ ((m + n)B(p))⌣\x + ((m + n)B(p))⌣\y \end{aligned}

holds for all n, m ∈ N the mapping B is additive. Moreover B is idempotent since

\begin{aligned}B^{2}(x) & = ⋁_{n∈N}(nB(p))⌣\B(x) = ⋁_{n∈N}(nB(p))⌣\( ⋁_{n∈N}(nB(p))⌣\x) \\ & = ⋁_{n∈N}(nB(p)⌣\x). \end{aligned}

Hence B is a balayage.

Corollary 2.9. Let S be an H -cone possessing a generator p. If B_{1} and B_{2} are balayages from S into S such that B_{1}(p) ≤ B_{2}(p) then B_{1} ≤ B_{2} .

Proof. Assume that B_{1}: S → S and B_{2}: S → S are balayages with B_{1}(p) ≤ B_{2}(p). Lemma 2.2 assures us that the mapping B: S → S defined by B(s) = B_{1}(s) ∨ B_{2}(s) is a balayage and B(p) = B_{2}(p). Hence by the theorem above B = B_{2} establishing the result.

The mapping B satisfying (2.3) is left order continuous if and only if B is left order continuous at B(p). Generally the following result holds.

Theorem 2.10. Let S be an H -cone and u be an element of S. Then B: S → S defined by

(2.4)  B(s) = ⋁_{n∈N}(nu)⌣\s

is a balayage if and only if u = ∨_{f∈F} B(f) for all upward directed families F with u = ∨F.

Proof. For a balayage B satisfying (2.4) the assertion is clear. To verify the converse assume that B defined by (2.4) satisfies the required condition. Let x ∈ S and F be an upward directed family with x = ∨F. Suppose first that x ≼ u. As the family u − x + F is upward directed with ∨(u − x + F) = u, we have by (1.3)

x + u − x = u = ⋁_{f∈F} B(u − x + f) = u − x + ⋁_{f∈F} B(f).

Hence the mapping B is left order continuous at x with x ≼ nu for any n ∈ N.

Assume next that x is arbitrary. Since (nu)⌣\x ≼ nu we obtain B((nu)⌣\x) = (nu)⌣\x and therefore the relation

(nu)⌣\x = ⋁_{f∈F} B(((nu)⌣\x) ∧ f) ≤ ⋁_{f∈F} B(f) ≤ B(x)

holds for n ∈ N. Consequently B is left order continuous.

A characterization of balayages in terms of their images follows from Theorem 2.8 and Theorem 2.10.

Theorem 2.11. Let S be an H -cone possessing a generator and T be a specifically solid subset of S. Then there exists a balayage B → T such that B(S) = T if and only if T admits a generator u satisfying

u = ⋁_{{n∈N \atop f∈F}}(nu)⌣\f

for all upward directed families F with ∨F = u.

66 Sirkka-Liisa Eriksson-Bique

An H -cone S is a positive cone of the Dedekind complete vector lattice with respect to the specific order by Proposition 1.2. Hence balayages with respect to the specific order are just specific band projections [8, Theorem 2.10]. In addition they have the following simple characterization.

Theorem 2.12. Let S be an H -cone in which the specific and initial order coincide. Then B: S → S is a balayage if and only if there exists a subset F of S such that

(2.5)  B(s) = ⋁_{{n∈N \atop t∈F}}(nt) ∧ s

for all s ∈ S.

Proof. Assume that a mapping B: S → S satisfies (2.5) for some subset F of S. Since by Proposition 1.2 the cone S is a positive cone of a Dedekind complete vector lattice, the mapping s ↦ t ∧ s is left order continuous for all t ∈ F. Hence Lemma 2.2 and Corollary 2.5 imply that B is a balayage.

Conversely, assume that B is a balayage. Applying Theorem 2.4 we find out that

B(t) ∧ s = B(t) ∧ B(s)

for all s and t in S. Thus we conclude

⋁_{t∈S} B(t) ∧ s = B(s)

for all s ∈ S and the condition (2.4) holds for F = B(S).

References

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[2] Arsove, M., and H. Leutwiler: Algebraic potential theory. - Mem. Amer. Math. Soc. 226, 1980.

[3] Boboc, N., Gh. Bucur, and A. Cornea: Order and convexity in potential theory: H - cones. - Lecture Notes in Mathematics 853. Springer-Verlag, Berlin–Heidelberg–New York, 1981.

[4] Constantinescu, C., and A. Cornea: Potential theory on harmonic spaces. - Springer-Verlag, Berlin–Heidelberg–New York, 1972.

[5] Cornea, A., and H. Höllein: Morphisms and contractions on standard H -cones. - Hokkaido Math. J. Special Issue 10, 1981, 157–212.

[6] Eriksson-Bique, S.-L.: Real-valued duals of H -cones. - Math. Scand. (to appear).

[7] Popa, E.: Balayages and morphisms in H -cones. - An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 36, 1990, 203–213.

[8] Schaeffer, H.H.: Banach lattices and positive operators. - Springer-Verlag, Berlin– Heidelberg–New York, 1974.

Received 1 December 1992