Functional calculus and Minkowski duality on vector lattices Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2009, Vol. 11, No 2, pp. 31-42

UDC 517.98

FUNCTIONAL CALCULUS AND MINKOWSKI DUALITY ON VECTOR LATTICES

To §afak Alpay on his sixtieth birthday A. G. Kusraev

The paper extends homogeneous functional calculus on vector lattices. It is shown that the function of elements of a relatively uniformly complete vector lattice can naturally be defined if the positively homogeneous function is defined on some conic set and is continuous on some closed convex subcone. An interplay between Minkowski duality and homogeneous functional calculus leads to the envelope representation of abstract convex elements generated by the linear hull of a finite collection in a uniformly complete vector lattice.

Mathematics Subject Classification (2000): 46A40, 47A50, 47A60, 47A63, 47B65.

Key words: vector lattices, functional calculus, Minkowski duality, sublinear and superlinear operators, envelope representation.

1. Introduction

For any finite sequence (xi,...,Xn) (N G N) in a relatively uniformly complete vector lattice the expression of the form p(x i,... ) can be correctly defined provided that p is

a positively homogeneous continuous function on RN. The study of such expressions, called homogeneous functional calculus, provides a useful tool in a variety of areas, see [4, 9, 10, 14, 15, 16, 21]. At the same time it is of importance in certain problems to deal with p(x 1,... ,xn) even if p is defined on a conic subset of RN [2, 16, 17]. The first aim of this paper is to extend homogeneous functional calculus on uniformly complete vector lattices.

Let H be a linear (or semilinear) subset of a vector lattice E. The support set dux of x G E with respect to H is the set of all H-minorants of x: dux := {h G H : h ^ x}. The H-convex hull of x G E is defined by cou x := sup{h G H : h G dux}. An element x is called H-convex (abstract convex with respect to H) if cou x = x. Now the problem is to examine abstract convex elements, that is elements which can be represented as upper envelopes of subsets of a given set H of elementary elements. (For this abstract convexity see [13, 20]). The second aim of the paper is the description of H-convex elements in E in the event that H is the linear hull of a finite collection {xi,... ,xn } C E of a vector lattice E. It turns out that under some conditions an element in E is H-convex if and only if it is of the form p(xi,..., xn) for some lower semicontinuous sublinear function p.

Section 2 collects some auxiliary results. In Section 3 the extended homogeneous functional calculus is defined. It is shown that the expression p(xi,...,xn) can naturally be defined in any relatively uniformly complete vector lattice if a positively homogeneous function p is defined on some conic set dom(p) C RN and is continuous on some closed subcone of dom(p). Section 4 contains some examples of computing p(ui,..., Un) whenever ui,..., Un are continuous or measurable vector-valued functions, or p is a Kobb-Duglas type function and Ui := 6(x*, yi) (i = 1,...,N) for some lattice bimorphism b. In Section 5 Minkowski duality is transplanted to vector lattice by means of extended functional calculus.

© 2009 Kusraev A. G.

There are different ways to define homogeneous functional calculus on vector lattices [3, 9, 14, 18]. We follow the approach of G. Buskes, B. de Pagter, and A. van Rooij [3] going back to G. Ya. Lozanovskii [18]. Theorem 2.1 below see in [3, 10, 14, 21]. For the theory of vector lattices and positive operators we refer to the books [1] and [10]. All vector lattices in this paper are real and Archimedean.

2. Auxiliary results

Denote by H (Rn ) the vector lattice of all continuous functions p : RN ^ R which are positively homogeneous (= p(At) = Ap(t) for A ^ 0 and t G RN). Let dt& stands for the kth coordinate function on RN, i. e. dt& : (ti,..., tN) ^ t&.

2.1. Theorem. Let E be a relatively uniformly complete vector lattice. For any p := (xi,..., xn) G EN there exists a unique lattice homomorphism

p: p ^p(p): = p(xi,...,xN) (p G H(RN))

of H(Rn) into E with p(dtfc) = xfc (k := 1,..., N).

If the vector lattice E is universally a-complete (= Dedekind a-complete and laterally a-complete) and has an order unit, then Borel functional calculus is also available on E. Let B(Rn) denotes the vector lattice of all Borel measurable functions p : RN ^ R. The following result can be found in [10, Theorem 8.2.14].

2.2. Theorem. Let E be a universally a-complete vector lattice with a fixed weak order unit 1. For any x := (xi ,...,xn ) G EN there exists a unique sequentially order continuous lattice homomorphism

p : p ^p(p): = p(xi,...,xN) (p G B(Rn))

of B(Rn) into E such that P(1rn) = 1 and p(dt&) = x^ (k := 1,..., N).

Let Heor(RN) denote the vector sublattice of B(Rn) consisting of all positively homogeneous Borel functions p : RN ^ R.

2.3. Theorem. Let E be a universally a-complete vector lattice with an order unit. For any p := (xi,... ,xn) G EN there exists a unique sequentially order continuous lattice homomorphism

p: p ^ p(p) = p(xi,... ,xn) (p G HBor(RN))

of Heor(Rn) into E such that ^p(dtfc) = x& (k := 1,..., N).

< Fix an order unit 1 in E and take p as in Theorem 2.2. Since Heor(RN) is an order a-closed vector sublattice of B(Rn ), the restriction of p onto HBor(RN) is also an order a-continuous lattice homomorphism. If h : Heor(RN) ^ E is another order a-continuous lattice homomorphism with h(dt&) = ^p(dtfc) (k := 1,..., N), then h and p(-) coincide on H(Rn) by Theorem 2.1. Afterwards, we infer that h and p(-) coincide on the whole Heor(RN) due to order a-continuity. >

3. Functional calculus

In this section we define extended homogeneous functional calculus on relatively uniformly complete vector lattices. Everywhere below p:= (xi,...,xn) G EN.

3.1. Consider a finite collection xi,..., xn G E and a vector sublattice L C E. Denote by (xi,...,xn) and Hom(L) respectively the vector sublattice of E generated by {xi,...,xn} and the set of all R-valued lattice homomorphisms on L. Put

[p] := [xi,... ,xN] := {(w(x1 ),..., w(xN)) G RN : w G Hom((xi,... ,xN)^ .

Let e := |xi| + ... + |xN| and Q:= {w G Hom((xi,... ,xN)) : w(e) = 1}. Then e is a strong order unit in (xi,..., xn) and Q separates the points of (xi,..., xn). Moreover, Q may be endowed with a compact Hausdorff topology so that the functions : Q ^ R defined by Pk(w) := w(x&) (k := 1,...,N) are continuous and x ^ p is a lattice isomorphism of (xi,...,xN) into C(Q). Put

Q(xi,..., xN):= {(w(xi),..., w(xN)) G Rn : w G Q} ,

and observe that [xi,...,xn] := cone(Q(xi,...,xn)), where cone(A) is the conic hull of A defined as |J{AA : 0 ^ A G R}. Evidently, Q(xi,..., xn) is a compact subset of RN, since it is the image of the compact set Q under the continuous map w ^ (Pi(w),..., Pn(w)). Therefore, [xi,...,xn] is a compactly generated conic set in RN. (The conic set [xi,...,xn] is closed if 0 G Q(xi,..., xn).) A set C C RN is called conic if AC C C for all A ^ 0 while a convex conic set is referred to as a cone. The reasoning similar to [3, Lemma 3.3] shows that [xi,..., xn ] is uniquely determined by any point separating subset Qo of Hom((xi, ...,xn )). Indeed, if Q0 := {w(e)-iw : 0 = w G Qo}, then Q0 is a dense subset of Q and [xi,..., xn] = cone (cl(Q0(xi,... ,xN))), where Q0(xi,... ,xN) is the set of all (w(xi),... ,w(xN)) G R with w G Q0.

3.2. For a conic set C in RN denote by C C EN the set of all p := (xi,..., xn) G EN with [ p ] C C. Consider a conic set K C C. Let H(C; K) denotes the vector lattice of all positively homogeneous functions p : C ^ R with continuous restriction to K. Fix (xi,..., xn) G C and take p G H(C; [p]). We say that p(xi,..., xn) exists or is well defined in E and write y = p(p) = p(xi,..., xn) if there is an element y G E such that w(y) = p(w(xi),..., w(xn)) for every w G Hom((xi,..., xn, y)). This definition is correct, since for any given (xi,..., xn) G C and p G H(C;[p]) there exists at most one y G E such that y = p(xi,...,xn). It is immediate from the definition that p(Aix,..., Anx) is well defined for any (Ai,..., An) G C and <p(Aix,..., Anx) = p(Ai,..., An)x whenever 0 ^ x G E. The following proposition can be proved as [3, Lemma 3.3].

Assume that L is a vector sublattice of E containing {xi,...,xn,y} and p G H(C; [xi,..., xn]). If w(y) = p(w(xi),..., w(xn)) (w G Q0) for some point separating set Q0 of R-valued lattice homomorphisms on L, then y = p(xi,..., xn).

3.3. Theorem. Let E be a relatively uniformly complete vector lattice and p G EN, p = (xi,..., xn). Assume that C C RN is a conic set and [p] C C. Thenp(p) := <p(xi,..., xn) exists for every p G H (C; [p ]) and the mapping

p: p ^ p(p) = <p(xi,.. .,xn) (p G H(C; [p]))

is a unique lattice homomorphism from H(C;[p]) into E with dtj(xi,...,xn) = xj for j :=1,...,N.

< Let H([ p ]) denotes the vector lattice of all positively homogeneous continuous functions defined on [p]. Then H([p]) is isomorphic to C(Q), where Q := [p] H S and S := {s G Rn : ||s|| := max{|si|,..., |sn|} = 1}. Much the same reasoning as in [3, Proposition 3.6, Theorem 3.7] shows the existence of a unique lattice homomorphism h from H([p ]) into E such that dtj(xi,..., xn) = xj (j := 1,..., N). Denote by p the restriction operator p ^ p|[?] (p G H(C; [ p ])). Then p o h is the required lattice homomorphism. >

Observe that if p, ^ G H(C; [p]) and p(t) ^ ^(t) for all t G [p], then p(xi,..., xn) ^ ?p(xi,... ,xN). Evidently, |p(t)| ^ |||p||| ■ ||t| for all t G [p] with |||p||| := sup{p(t) : t G Q} and

hence

1 p(xi, . . . , xN ) 1 < 1 1 1 p 1 1 1 ( 1 xi 1 v---V 1 xN 1) .

In particular, the kernel ker(p) of p consists of all p G H(C; [p]) vanishing on [p].

3.4. Let K, M, N G N and consider two conic sets C C RN and D C RM. Let x1,...,!n G

E, p := (x1,..., xN), [p] C C, p1,..., pM G H(C; [p]), and denote p := (p1,..., pM) and

y := (y1,..., yN) with yk = <pk(x1, ...,xn ) (k := 1,..., M). Suppose that [ y ] C D, p(C) C D, and p([p]) C [y ]. If I := (^1,..., lK) with ^1,..., lK G H(D; [y ]), then I op,..., lKop G H(C;|pj). Moreover^<p(p) := (Jpifc),..., pM(p)) G EM, l'(y) := (li(y),..., Ik(y)) G EK,

and i) o p(p) := (li o p(p),..., ^k ◦ p(p)) G EK are well defined and

o p)(p) = ^p(<p(p)).

3.5. Theorem. Let C and K are conic sets in RN with K closed and K C C and let

p G H(C; K). Then for every e > 0 there exists a number Re > 0 such that

1 p(p + y) - ^ 1 < e 1 1 1 p 1 1 | + R 1 1 1 y 1 1 1

for any finite collections p = (xi,..., xn) G EN and y = (yi,..., yN) G EN, provided that

p, y G K, p + y G K and |||(u1, ..., Un)||| stands for |u1| V ■ ■ ■ V |un|.

< The proof is a duly modification of arguments from [4, Theorem 7]. Denote Kx : = {(s,t) G K x K : s +1 G K} and define A as the set of all (s, t) G Kx with max{|| s ||, ||1||} — 1 and t(s, t) := |p(s +1) — p(s)| ^ e||s||, where ||s|| := max{|si|,..., |sn|}. Then A is a compact subset of K x K and (s,t) ^ (t(s,t) — e||s||)/||t|| is a continuous function on A, since ||t| = 0 for (s, t) G A. Therefore,

„ ( t (s,t) — e||s|| , .I

R :=supj------------------ : (s,t) G A j < ^

Hence t(s,t) ^ e||s|| + Re||t|| =: a(s,t) for all (s,t) G Kx. Evidently, t G H(Cx,Kx),

a G H(Rn x Rn), and t ^ a on Kx. It remains to observe that (p, y) G Kx and apply 3.3 and the desired inequality follows. >

3.6. Proposition. Let E and F be uniformly complete vector lattices, E0 a uniformly

closed sublattice of E, and h : E0 ^ F a lattice homomorphism. Let C be a conic set in RN, x1,...,xn G E0, and p G H (C; [x1,..., xN ]). Then [h(x1),..., h(xN)] C [x1,..., xN ] and

h(p(xi,..., xn)) = p(h(xi),..., h(xN)).

In particular, if h is the inclusion map E ^ F and xi,...,xn G E, then the element

<p(xi, ...,xn ) relative to F is contained in E and its meaning relative to E is the same.

< Put yi := h(xi) (i := 1,..., N). If w G Hom((y1,..., yN)), then w := w o h belongs to Hom((x1,..., xN)) and (w(y1),..., w(yN)) = (tD(x1),..., w(xn )) G [x1,...,xn ]. Therefore, [y1,..., yN] is contained in [x1,..., xn]. Now, if y = p(yi,..., yN), x = <p(x1,..., xn), and w G Hom((y, y1,..., yN)), then w G Hom((x, x1,..., xN)) and by definition

w(y) = p(w(xi),..., w(xn)) = w(<p(xi,..., xn) = w(h(x)), so that y = h(x). >

Denote Hgor(RN, [p]) := {p G Heor(RN) : sup{|p(s)| : s G S H [p]} < +^}.

3.7. Theorem. Let E be a Dedekind a-complete vector lattice. For p:= (xi,..., xn) in EN there exists a unique sequentially order continuous lattice homomorphism

p: p ^ p(p) = <p(xi, ...,xn ) Sp G Htor(RN, [ p ])) of HSr (Rn, [p ]) into E such that p(dtk) = xk (k := 1,..., N).

< Let E0 be the order ideal in E generated by xi,..., xn. According to 1.3 there exists a unique sequentially order continuous lattice homomorphism p of Heor(RN) into (E0)uo_, a universal a-completion of E0, with p(dtk) = xk (k := 1,...,N). Clearly, the image of

HSr(RN, [ p ]) under p is contained in E0. >

4. Examples

Now, we consider extended functional calculus on some special vector lattices E and for

some special functions p. Everywhere in the section p G H(C; K).

4.1. Proposition. Let Q be a Hausdorff topological space, X a Banach lattice, and Cb(Q, X) the Banach lattice of norm bounded continuous functions from Q to X. Assume that u1,..., uN G Cb(Q, X) and [u1,..., uN] C K. Then [u1(q),..., uN(q)] C K for all q G Q and

p(ui,.. .,Un )(q) = <p(ui(q),.. .,Un (q)) (q G Q).

< Indeed, for q G Q the map p : Cb(Q,X) ^ X defined by p : u ^ u(q) is a lattice homomorphism. Therefore, given ui,... ,Un G C5(Q,X), by Proposition 3.6 we have [p(u1),..., p(uN)] C [u1,..., uN] and p(<p(u1,..., uN)) = <p(p(u1),..., p(uN)) from which the required is immediate. >

4.2. Suppose now that Q is compact and extremally disconnected. Let u : D ^ X be a continuous function defined on a dense subset D C Q. Denote by D the totality of all points in Q at which u has limit and put u(q) := limp^q u(p) for all q G D. Then the set D is comeager in Q and the function u : D ^ X is continuous. Recall that a set is called comeager if its complement is meager. Thus, the function u is the «widest» continuous extension of u i. e., the domain of every continuous extension of u is contained in D and, moreover, u is an extension of every continuous extension of u. The function u is called the maximal extension of u and denoted by ext(u), see [6]. A continuous function u : D ^ X defined on a dense subset D C Q is said to be extended, if ext(u) = u. Note that all extended functions are defined on comeager subsets of Q.

Let Cqo(Q,X) stands for the set of all extended X-valued functions. The totality of all bounded extended functions is denoted by C^(Q,X). Observe that Co(Q,X) can be represented also as the set of cosets of continuous functions u that act from comeager subsets dom(u) C Q into X. Two vector-valued functions u and v are equivalent if u(t) = v(t) whenever t G dom(u) H dom(v).

The set Cqc(Q, X) is endowed, in a natural way, with the structure of a lattice ordered module over the f-algebra CO(Q). Moreover, CO(Q,X) is uniformly complete and for any ui,... ,Un G Cqc(Q,X) the element p(ui,... ,Un) is well defined in CO(Q,X) provided that [ui,..., un] C K.

4.3. Proposition. Let Q be a extremally disconnected conpact space and X a Banach lattice. Let ui,..., un G CO(Q, X) and [ui,..., un] C K. Then there exists a comeager subset Q0 C Q such that Q0 C dom(ui) for all i := 1,...,N, [u1 (q),..., uN (q)] C K for every q G Q0, and p(ui,..., Un) is the maximal extension of the continuous function q ^ p(ui(q),... ,Un(q)) (q G Q0), i. e.

p(ui,.. .,Un )(q) = <p(ui(q),.. .,Un (q)) (q G Q0).

< Put Q' := dom(ui) H ••• H dom(uN) and observe that Q' is comeager. There exists a unique function e G Coo(Q) such that e'(q) := ||ui(q)|| + ■ ■ ■ + ||un(q)|| (q G Q'). Let E be

the order ideal in CO(Q) generated by e and define the sublattice E(X) C CO(Q,X) by

E(X):= {u G Cqc(Q,X) : (3 0 ^ C G R) (Vq G dom(u)) ||u(q)|| ^ Ce(q)}.

In the Boolean algebra of clopen subsets of Q there exists a partition of unity (Q(£)) with XQ(£)e G C(Q) for all £ G S. Put Q^ := Q' H Q^ and Q0 := (J^gS Q^ and observe that Q0 is comeager in Q. Let stands for the band projection in Cqc(Q,X) defined by

: u ^ Xq(^)U. Then (E(X)) C C6(Q,X) and (n^Ui)(q) = Ui(q) (q G Q^; i = 1, ...,N). Finally, given q G Q^, in view of Propositions 3.6 and 4.1 we have [ui(q),..., Un(q)] = [(n^ui)(q),..., (n^un)(q)] C K and

(n^p(ui,..., un))(q)p((n£ui)(q),..., (n^un)(q)) =

= <p(n^ui,.. . ,n^Un)(q) = <p(ui(q),... ,un(q))

and the proof is complete. >

4.4. Let (fi, S, ^) be a measure space with the direct sum property and X be a Banach lattice. Let L°(^, X) := L0(fi, S, ^, X) be the set of all Bochner measurable functions defined almost everywhere on fi with values in X and L0(^, X) := L0(^,X)/ ~ the space of all equivalence classes (of almost everywhere equal) functions from L0(^, X). Then L0(^, X) is a Banach lattice and hence p(ui ,...,Un ) is well defined in L0(^, X) for p G H (C; K) and Ui,...,Un G L0(^, X) with [ui,...,Un ] C K. Denote by u the equivalence class of u G LV,X).

Let L00 (^, X) stand for the part of L0 (^, X) comprising all essentially bounded functions and L°°(^,X):= L°°(^,X)/ ~. Put L°°(^) := L°°(^,R) and L°°(^) := L°°(^,R). Denote by L 00 (^) the part of L 00 (^) consisting of all function defined everywhere on fi. Then L 00 (^) is a vector lattice with point-wise operations and order. Recall that a lattice homomorphism p : L 00(^) ^ L°°(^) is said to be a lifting of L 00 (^) if p(f) G f for every f G L 00(^) and p(l) is the identically one function on fi. (Here 1 is the coset of the identically one function on fi). Clearly, a lifting is a right-inverse of the quotient homomorphism 0 : f ^ f (f G L 00 (^). The space L 00(^) admits a lifting if and only if (fi, S, ^) possesses the direct sum property. If f G L 00 (^), then the function p(f ) is also denoted by p(f).

4.5. Proposition. Let u1,...,un G L0(fi, S,^,F), and [u1,...,un ] C K. Then there exists a measurable set fi0 C fi such that ^(fi \ fi0) = 0, [ui(w),...,Un(w)] C K for all w G fi0, and p(Ui,...,Un) is the equivalence class of the measurable function w ^ <p(u1(w),..., uN(w)) (w G fi0).

< The problem can be reduced to Proposition 4.2 by means of Gutman’s approach to vector-valued measurable functions. Let p be a lifting of L°°(fi, S,^) and t : fi ^ Q be the corresponding canonical embedding of fi into the Stone space Q of the Boolean algebra B(fi, S,^), see [?]. The preimage t-1(V) of any meager set V C Q is measurable and ^-negligible. Moreover t is Borel measurable and v o t is Bochner measurable for every v G Coo(Q,X). Denote by t* the mapping which sends each function v G Co°(Q,X) to the equivalence class of the measurable function v o t . The mapping t * is a linear and order isomorphism of Co°(Q,X) onto L0(fi, S,^,X). If a is the inverse of t*, then [a(u1),..., a(uN)] C K and a<p(u1,..., Un) = <p(a(u1),..., a(uN)) by Proposition

3.6. According to Proposition 4.3 there exists a comeager subset Q0 C Q such that [a(u1)(q),..., a(uN)(q)] C K for all q G Q0 and

p(a(ui),..., a(UN))(q) = p(a(ui)(q),..., a(uN)(q)) (q G Q0).

Clearly, the functions ui := a(Ui) ot and Ui are equivalent and p(Ui,..., Un) is the equivalence class of a(p(ui,..., Un)) o t. Let fi' stands for the set of all w G fi with ui(w) = Ui(w) for all i = 1,..., N. Then fi0 := t-1(Q0) H fi is measurable and ^(fi \ fi0) — 0. Substituting q = t(w) we get [ui (w),..., uN(w)] C K for all w G fi0 and

a<p(Ui,... ,Un)(t(w)) = <p(ui(w),... ,uN(w)) (w G fi0),

which is equivalent to the required statement. >

4.6. A conic set C C Rn is said to be multiplicative if st := (siti,..., SntN) G C for all s := (si,... ,Sn ) G C and t := (ti,..., tN) G C .A function p : C ^ R is called multiplicative if p(st) = p(s)p(t) for all s, t G C.

Take a subset J C {1,...,N} and define R^ as the cone in RN consisting of 0 and (s1,..., Sn) G R+ with si > 0 (i G I). We will write xi ^ 0 (i G I) if [x1,..., xn] C R;^. The general form of a positively homogeneous multiplicative function p : R^ ^ R other that p = 0 is given by

p(ti,...,tN)=0 (ti ■ ... ■ tN = 0),

p(ti, . . . ,tN) = exp(gi(ln ti)) ■ ... ■ exp(gN (ln tN)) (ti ■ ... ■ tN = 0),

where gi,... ,gN are some additive functions in R with ^gi = Jr. If p is continuous at any interior point of RN or bounded on any ball contained in R^, then we get a Kobb-Duglas type function and if, in addition, p is nonnegative, then p(ti,..., tN) = t^1 ■ ... ■ t^N with a1,..., aN G R and ^ N=1 ai = 1.

By definition xi ^ 0 (i G J) implies that p(xi,...,xn) is well defined for every p G H(Rn, [xi,..., xn]). Thus, the expression x^1 ■ ... ■ xNN is well defined in E provided that xk ^ 0 for all k with ak < 0. At the same time p G H(RN) whenever J = 0 and in this case x?1 ■ ... ■ xNN is well defined in E for arbitrary xk ^ 0 and ak ^ 0 (k = 1,..., N).

4.7. Proposition. Let E, F and G be vector lattices with E and F uniformly complete and b : E x F ^ G a lattice bimorphism. Let p := (x1, ...,xn ) G EN, y := (yi,..., yN) G FN, and [p] U [y] C K for some multiplicative closed conic set K C RN. If 0 G H (C, K) is multiplicative on K, then 0(b(xi, yi),..., b(xN, yN)) exists in G and

p (b(xi, yi), . . . , b(xN, yN)) = ^p(xi, . . . , xn ), p(yi, . . . , yN ^ .

< Put u = <p(xi,..., xn) and v = 0(yi,... ,yN). Let E0 and F0 be the vector sublattices in E and F generated by {u,xi,...,xn} and {v,yi,...,yN}, respectively. Let G0 be the order ideal in G generated by b(e, f) where e := |u| + |xi| + ••• + |xn | and f := |v| + |yi| +

+ |yN|. Observe that Hom(G0) separates the points of G0. By [12, Theorem 3.2] every R-valued lattice bimorphism on E0 x F0 is of the form a 0 t : (x, y) ^ a(x)T(y) with a G Hom(E0) and t G Hom(F0). Denote by b0 the restriction of b to E0 x F0. Given an R-valued lattice homomorphism w on G0, we have the representation w o b = a 0 t for some lattice homomorphisms a : E0 ^ R and t : F0 ^ R. Since K is multiplicative, we have

(w(b(xi,yi),... ,w(b(xN,yN))) = (a(xi)T(yi),..., a(xN)t(yN)) = (a(xi),..., a(xN)) ■ (t(yi),..., t(yN)) G K,

and thus [b(xi, yi),..., b(xN, yN] C K. Now, making use of 3.6 and multiplicativity of 0 we deduce

w o b(u, v) = a(p(xi,..., xn))t(p(yi,..., yN))

= 0(a(xi),..., a(xN))0(t(yi),... ,t(yN))

= 0(a(xi)T(yi),..., a(xN)t(yN))

= 0(w o b(xi, yi),..., w o b(xN, yN))

= w o p(b(xi,yi), . . . ,b(xN,yN)),

as required by definition 3.2. >

4.8. In particular, we can take G := F 0 F, the Fremlin tensor product of E and F [5], or E®, the square of E [4], and put b := 0 or b := 0 in 4.7. Thus, under the hypotheses of 4.7 we have

p(xi 0 yi, . . . ,xn 0 yN) = p(xi, . . . ,xn) 0 p(yi, . . . ,yN),

p(xi 0 yi, . . . ,xn 0 yN) = p(xi, . . . ,xn) 0 p(yi, . . . ,yN).

Taking 4.6 into consideration we get the following: If 0 ^ ai,..., aN G R, ai + ■ ■ ■ + aN = 1,

then |x1 0 y1|“1 ■ ... ■ |xN 0 yN|“N exists in E 0 F for all x1,..., xN G E and y1,..., yN G F

and

N / N

n|xi0 yi|ai = (n |xi|ai

i=1 Vi=1

if, in addition, E = F, then we also have

N / N

n|xi0 yi|ai = (n |xi|ai

i=1 Vi=1

4.10. Proposition. Let E be a uniformly complete vector lattice, p := (xi ,...,xn ) G EN,

p := (ni,...,nN) G Orth(E)N, and [p] U [p] C K for some multiplicative closed conic set K C C C Rn. If 0 G H(C, [p]) H H(C, [p]) is multiplicative on K, then 0(n1x1,..., xn)) exists in E and

p(nixi, . . . ,nN xn ) = p(ni, . . . ,nN )( p(xi, . . . ,xn )) .

< The bilinear operator b from E x Orth(E) to E defined by b(x,n) := n(x) is a lattice bimorphism and all we need is to apply Proposition 4.7. >

5. Minkowski duality

The Minkowski duality is the mapping that assigns to a sublinear function its support set or, in other words, its subdifferential (at zero). For any Hausdorff locally convex space X the Minkowski duality is a bijection between the collections of all lower semicontinuous sublinear functions on X and all closed convex subsets of the conjugate space X', see [13, 19]. The extended functional calculus (Theorems 2.3, 3.3, and 3.7) allows to transplant the Minkowski duality to vector lattice setting.

5.1. A function p : RN ^ R U {+^} is called sublinear if it is positively homogeneous, i. e. p(0) = 0 and p(At) = Ap(t) for all A > 0 and t G RN, and subadditive, i. e. p(s +1) ^ p(s) + p(t) for all s, t G Rn. A function ^ : RN ^ R U {-^} is called superlinear if —^ is sublinear. We say that p is lower semicontinuous (^ is upper semicontinuous) if the epigraph epi(p) := {(t, a) G Rn x R : p(t) ^ a} (the hypograph hypo(p) := {(t, a) G RN x R : p(t) ^

( N 0 liter

Vi=i

0 (n iyii“'J ■

a} is a closed subset of RN x R. The effective domain of a sublinear p (superlinear ^) is dom(p) := {t G RN : p(t) < +ro} (dom(^) := {t G RN : ^(t) > —ro}). The subdifferential dp of a sublinear function p and the superdifferential d^ of a superlinear function ^ are defined by

dp:= {t G RN : (s,t) ^ p(s) (s G RN)}, d^:= {t G RN : (s, t) ^ ^(s)(s G RN)},

where s = (s1 ■ ■ ■, Sn), t = (t1 ■ ■ ■, tN), (s, t) := fc=i Sktk, see [11, 19]. Denote by HV(RN, K) and H,(Rn ,K) respectively the sets of all lower semicontinuous sublinear functions p : Rn ^ R U {+ro} and upper semicontinuous superlinear functions ^ : RN ^ R U {—ro} which are finite and continuous on a fixed cone K C RN. Put HV(RN) := HV(RN, {0}) and Ha(Rn) := Ha(Rn, {0}). We shall consider Hv(Rn) and Ha(Rn) as subcones of the vector lattice of Borel measurable functions Heor(RN) with the convention that all infinite values are replaced by zero value.

5.2. Theorem. Let p G HV(RN) and ^ G H^(Rn). Then there exist countable subsets A C dp and B C d^ such that the representations hold:

p(s) = sup{(s,t) : t G A} (s G Rn),

^(s) = inf{(s,t) : t G B} (s G RN)■

< The claim is true for A = dp and B = d^ in any locally convex space X. The sets dp and d^ may be replaced by their countable subsets A and B provided that X is a separable Banach space, say X = RN (see [8, Proposition A.1]). >

5.3. Remark. For this abstract convexity see S. S. Kutateladze.

In this section we deal with the description of H-convex elements in E in the event that H is the linear hull of a finite collection {xi, ■ ■ ■, xn} C E. The following two theorems say that under some conditions an element in E is H-convex if and only if it is of the form p(p) for some lower semicontinuous sublinear functions p.

For A C Rn denote by (A, p) the set of all linear combinations ^^=1 Akxk in E with (A1, ■ ■ ■, An) G A, so that

f N

sup (A, p) := sup £ Akxk : (A1, ■ ■ ■ , AN) G A

k=i

5.4. Theorem. Let E be a a-complete vector lattice with an order unit, xi, ■ ■ ■, xn G E, and p:= (x1, ■ ■ ■ ,xN). Assume that p G HV(Rn), ^ G H,(Rn), and [p] C dom(p) H dom(^). Then p(p) exists in E if and only if (dp, p) is order bounded above, p(^) exists in E if and only if (d^, p) is order bounded below, and the representations hold:

p(p) = sup (dp, p), p(V0 = inf( d^, p) ■

Moreover, p(xi,...,xn ) (^p(xi, ■ ■ ■, xn )) is an order limit of an increasing (decreasing) sequence which is comprised of the finite suprema (infima) of linear combinations of the form N=1 Aixi with (A1, ■ ■ ■ AN) G dp ((A1, ■ ■ ■ AN) G d^).

< Assume that p G HV(RN) and [xi, ■ ■ ■,xn] C dom(p). Let E0 denotes the band in E generated by 1 := |xi| + ■ ■ ■ + |xn| and by 1 and Eg0" stands for the universally a-completion E0. By Theorem 2.3 p(p) always exists in E0 and the required representation holds true in

EQ*0", since p is Borel. In more details, let po vanishes on RN \ dorn(p) and coincides with p on dorn(p). Then po is a Borel function on RN and according to 5.2 we may choose an increasing sequence (pn) of Borel functions such that pn coincides with the finite supremum of linear combinations of the form ^^ A*t* on dorn(p) and (pn) converges point-wise to po. By Theorem 2.3 the sequence (p(pn)) is increasing and order convergent to p(po) = p(p). Now it is clear that (dp, x) is order bounded above in E if and only if x(p) Є Eq. >

5.5. Theorem. Let E be a relatively uniformly complete vector lattice, xi,...,Xn Є E,

and x:= (x1,... ,xN). If p Є H-(RN; [x]) and ф Є H,(RN; [x]), then

p(p) = sup (dp, x), р(ф) = inf (дф, x).

Moreover, p(xi,..., Xn) (^A(xi,..., Xn)) is a relatively uniform limit of an increasing (decreasing) sequence which is comprised of the finite suprema (infima) of linear combinations of

the form N=1 AiXi with A = (A1,... An) Є dp (A Є дф).

< Consider p Є Hv(RN; [xi,...,Xn]) and denote y = p(xi,...,Xn). By Theorem 3.3

vл := Aixi + ... + An xn ^ y

for an arbitrary A:= (Ai,..., An) Є dp. Assume that v Є E is such that v ^ vл for all A Є dp. By the Krems-Kakutani Representation Theorem there is a lattice isomorphism x ^ X of the principal ideal EM generated by u = |xi| + ... + |xn| + |v| onto C(Q) for some compact Hausdorff space Q. Then v, Xi,..., Xn, Vл, and y lie in EM and for any A Є dp the point-wise inequality w(q) ^ гу (q) (q Є Q) is true. By 4.1 and З.б we conclude that

y(q) = p(xi(q),... ,XN(q)) = sup{^(q) : A Є dp} ^ y(q).

Thus we have y ^ v and thereby y = sup{vл : A Є dp}.

Put U := ^л : A Є dp} and denote by Uv the subset of E consisting of the suprema of the finite subsets of U. Then Uv С EM and the set Uv := {г> : v Є Uv} is upward directed in C(Q) and its point-wise supremum equals to у. By Dini Theorem Uv converges to у uniformly and thus Uv is norm convergent to y in Eu. The superlinear case ф Є HЛ(RN; [xi,..., Xn]) is considered in a similar way. >

5.6. In some situation it is important to know wether the function is the upper or lower envelope of a family of increasing linear functionals. Suppose that RN is preordered by a cone K С Rn, i.e. s ^ t means that s — t Є K. The dual cone of positive linear functionals is denoted by K*. A function ф : RN ^ R U {±ro} is called increasing (with respect to K) if s ^ t implies ф^) ^ ф(t). A lower semicontinuous sublinear (an upper semicontinuous superlinear) ф is increasing if and only if dф С K* ^ф С K*) and thus ф is an upper envelope of a family of increasing linear functionals (is a lower envelope of a family of increasing linear functionals). If ф is increasing only on dorn^), then this claim is no longer true but under some mild conditions it is still valid for the restriction of ф onto dorn^), see [13, 20].

Proposition. Let p : RN ^ R U {+^} and ф : RN ^ R U {—^} be the same as in Theorem 5.2. Suppose that, in addition, dorn(p) — K = K — dorn(p) and dorn^) — K = K — dorn^). Then the following assertions hold:

(1) p is increasing on dorn(p) if and only if

p(s) = sup{(s, t) : t Є (dp) П K*} (s Є dorn(p));

(2) ф is increasing on dorn^) if and only if

ф^) = inf{(s,t) : t Є ^ф) П K*} (s Є dorn^)).

< Indeed, we may assume RN = dom(p) — K and then the function p* : RN ^ R defined by p*(s) = inf{p(t) : t G dom(p), t ^ s} (s G RN) is increasing and sublinear and coincides with p on dom(p); moreover, dp* = (dp) H K*. Similarly, assuming RN = dom(^) — K, we deduce that the function ^* : RN ^ R defined by ^*(s) = sup{^(t) : t G dom(^), t ^ s} (s G Rn) is increasing and superlinear and agrees with ^ on dom(^); moreover, d^* = (d^) H K*. It remains to observe that p and ^ are increasing if and only if p = p* and ^ = ^*. >

5.7. Corollary. Assume that p is increasing on dom(p), ^ is increasing on dom(^), dom(p) — K = K — dom(p), and dom(^) — K = K — dom(^). If, in addition, the assumptions of either 4.4 or 4.5 are fulfilled, then in 4.4 and 4.5 the sets dp and dp may be replaced by (dp) H K* and (d^) H K*.

5.8. A gauge is a sublinear function p : RN ^ R+ U {+to}. A co-gauge is a superlinear function ^ : Rn ^ R+ U {—to}. The lower polar function p° of a gauge p and the upper polar function ^° of a co-gauge ^ are defined by

p°(t):=inf{A ^ 0: (Vs G RN) (s,t) ^ Ap(s)} (t G RN),

^°(t) := sup{A ^ 0 : (Vs G RN) (s, t) ^ A^(s)} (t G RN)

(with the conventions sup0 = —to, inf 0 = +to, and 0(+to) = 0(—to) = 0). Thus, p° is a gauge and ^° is a co-gauge. Observe also that the inequalities hold:

(s, t) ^ p(s)p°(t) (s G dom(p), t G dom(p°)),

(s, t) ^ ^(s)^°(t) (s G dom(^), t G dom(^°)).

Denote p°° := (p°)° and ^°° := (^°)°.

5.9. Bipolar Theorem. Let p be a gauge and ^ be a co-gauge. Then p°° = p if and only if p is lower semicontinuous and ^°° = ^ if and only if ^ is upper semicontinuous.

< See [19]. >

5.10. The lower polar function p° and the upper polar function ^o can be also calculate

by

p°(t) = sup (s,t) = sup{(s,t) : s G Rn, p(s) ^ 1} (t G RN)

seRN p(s)

(with the conventions a/0 = +to for a > 0 and a/0 = 0 for a ^ 0) and

^°(t) = inf (s’t) = inf |(s,t) : s G RN, ^(s) ^ 1 or ^(s) =0} (t G RN)

(with the conventions a/0 = +to for a ^ 0 and a/0 = —to for a < 0).

Denote by Gv(Rn ,K) and GA(RN,K) respectively the sets of all lower semicontinuous gauges p : Rn ^ R+ U {+to} and upper semicontinuous co-gauges ^ : RN ^ R+ U {—to} which are finite and continuous on a fixed cone K C RN. Put Gv(Rn) := Gv(Rn, {0}) and GA(RN) := GA(RN, {0}). Observe that Gv(Rn) C Hv(Rn) and GA(RN) C HA(RN).

5.11. Corollary. Assume that either the assumptions of 5.4 are fulfilled and, in addition,

p G Gv(Rn) and ^ G Ga(Rn), or the assumptions of 5.5 are fulfilled and additionally p G Gv(Rn; [x]) and ^ G HA(RN; [x]). Then in 5.4 and 5.5 the sets dp and dp may be replaced

by {t G Rn : p°(t) ^ 1} and {t G RN : ^°(t) ^ 1}, respectively.

< It is immediate from the Bipolar Theorem and the above definitions, since obviously dp = {t G Rn : p°(t) ^ 1} and, d^ = {t G RN : ^°(t) ^ 1}. >

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Received April 12, 2009.

Anatoly G. Kusraev South Mathematical Institute Vladikavkaz Science Center of the RAS, Director Russia, 362027, Vladikavkaz, Markus street, 22 E-mail: [email protected]