UDC 513.88

MATHEMATICS

G. Ya. LOZANOVSKII

ON THE REALIZATION OF SPACES OF REGULAR FUNCTIONALS AND SOME OF ITS APPLICATIONS

(Presented by Academician L. V. Kantorovich on 10 II 1969)

In the theory of vector structures one often uses representations of various spaces in the form of spaces of certain special types, for example, spaces of continuous functions. Such a realization is admitted, in particular, by an arbitrary \(K\)-space (see, for example, \((^2)\), Ch. V). Completely linear functionals on \(K\)-spaces admit an integral representation of the same type as linear continuous functionals in the classical \(L_p\) \((1 < p < \infty)\); this circumstance makes it possible, in the study of such functionals, to apply widely the apparatus of measure and integration theory. The question of the realization of arbitrary regular functionals (and of spaces of such functionals) is more complicated. The purpose of the present note is to construct one method of realizing spaces of regular functionals and to apply it to Banach structures introduced by Calderon \((^3)\). Some results are also given on completely linear functionals in a \(KN\)-space, based on its unit ball.

We shall use the terminology from the theory of \(K\)-spaces (i.e., conditionally complete linear structures) adopted in \((^2)\). Two elements \(x, y\) of a \(K\)-space \(X\) are called disjoint (notation \(xdy\)) if \(|x| \wedge |y| = 0\). A unit \(1\) of a \(K\)-space \(X\) is understood in the weak sense (in Freudenthal’s sense), i.e., \(x \wedge 1 > 0\) for every \(x > 0\). A normal subspace of a \(K\)-space \(X\) is any of its linear subspaces \(X_1\) satisfying the condition: if \(x \in X_1\), \(y \in X\), \(|y| \le |x|\), then \(y \in X_1\). If, in addition, in \(X\) there are no nonzero elements disjoint from all elements of \(X_1\), then one says that \(X_1\) is a fundament in \(X\).

A \(K\)-space \(W\) is called extended if every set of its pairwise disjoint elements is bounded. A bicompact space \(Q\) is called extremal if the closure of every open set in \(Q\) is open-and-closed. For an arbitrary extremal bicompact space \(Q\), the set \(C_\infty(Q)\) of all real continuous functions on \(Q\) that may take the values \(+\infty\) and \(-\infty\) only on nowhere dense sets is an extended \(K\)-space under the natural partial order and algebraic operations (see \((^2)\), Ch. V). Every extended \(K\)-space \(W\) in which a unit \(1\) is fixed is uniquely realized in the form of a space \(C_\infty(Q)\) on a suitable extremal bicompact space \(Q\), if one requires that \(1\) correspond to the function on \(Q\) identically equal to one. Every \(K\)-space \(X\) is a fundament in some extended \(K\)-space \(W\), which is called the maximal extension of the space \(X\) and which we shall denote by \(\mathfrak{M}(X)\).

With every \(K\)-space \(X\) there are associated two spaces of functionals on \(X\): the space \(\widetilde{X}\) of all regular functionals (\((^2)\), p. 267) and the space \(\overline{X}\) of all completely linear functionals (\((^2)\), p. 239), called the Nakano conjugate of \(X\).

A KN-space is a \(K\)-space \(X\) which is simultaneously a normed space in which the norm is monotone, i.e., from \(|x|\le |y|\) it follows that \(\|x\|_X\le \|y\|_X\).

A KB-space is a \(KN\)-space \(X\) in which two additional conditions are satisfied:

(A) If \(x_n\downarrow 0\), then \(\|x_n\|_X\to 0\).

(B) If \(0\le x_n\uparrow\) and \(\lim \|x_n\|_X<\infty\), then there exists \(\sup x_n\in X\).

For an arbitrary \(KN\)-space \(X\), by \(X^*\) we denote its Banach dual. Recall that \(X^*\subset \widehat X\), and, if \(X\) is a Banach space, then \(X^*=\widehat X\).

§ 1. Let \(Q\) be an extremally disconnected bicompact space, \(W=C_\infty(Q)\) the corresponding extended \(K\)-space. For brevity we shall denote \(C(Q)\), i.e., the ordinary space of real finite continuous functions on \(Q\), by \(M\).

Definition 1. Let \(X\) be a normal subspace in \(C_\infty(Q)\), \(f\in \widehat X\), \(u\in X_+\). For any \(x\in M\) put
\[ f_{(u)}(x)=f(xu), \tag{1} \]
where \(xu\) is the product in the sense of multiplication in \(C_\infty(Q)\), see \((^2)\), p. 163. It is clear that \(f_{(u)}\in \widehat M\).

Definition 2. Let \(X\) and \(Y\) be normal subspaces in \(C_\infty(Q)\), \(f\in \widehat X\), \(g\in \widehat Y\). We shall say that \(f\) and \(g\) are disjoint (notation \(fDg\)) if, for any \(u\in X_+\), \(v\in Y_+\), \(f_{(u)}dg_{(v)}\) holds, i.e., \(f_{(u)}g_{(v)}\) are disjoint as elements of the \(K\)-space \(\widehat M\).

We emphasize that one cannot speak about the disjointness of the elements \(f,g\) in the ordinary sense, since they are not elements of one and the same \(K\)-space.

Theorem 1. Let \(X\) be a normal subspace in \(C_\infty(Q)\). Fix a unit \(1_X\) in the space \(\mathfrak M(\widehat X)\) and a unit \(1_M\) in \(\mathfrak M(\widehat M)\). Then there exists a unique pair \((R_X,V_X)\), where \(V_X\) is a component in \(\mathfrak M(\widehat M)\), and \(R_X\) is an isomorphism of the \(K\)-space \(\mathfrak M(\widehat X)\) onto the \(K\)-space \(V_X\), satisfying the conditions:

(1) For any \(f\in \widehat X\), \(g\in \widehat M\),
\[ (fDg)\Longleftrightarrow (R_X f\, dg); \]

(2) \(R_X(1_X)=\operatorname{Pr}_{V_X}1_M\).

Let us note that here \(R_X f\) and \(g\) are elements of one and the same \(K\)-space \(\mathfrak M(\widehat M)\), and one may speak of their disjointness in the ordinary sense.

Definition 3. The operator \(R_X\) introduced in Theorem 1 will be called the canonical realization of the space \(X\).

It is clear that the operator \(R_X\) depends on the choice of the units \(1_X,1_M\) in the spaces \(\mathfrak M(\widehat X)\), \(\mathfrak M(\widehat M)\), respectively.

Theorem 2. Let \(X\) and \(Y\) be normal subspaces in \(C_\infty(Q)\); \(R_X\) and \(R_Y\) the corresponding canonical realizations. Then, for any \(f\in \widehat X\), \(g\in \widehat Y\), and for any choice of the units \(1_M,1_X,1_Y\), one has
\[ (fDg)\Longleftrightarrow (R_X f\, d R_Y g). \]

§ 2. Throughout this section we assume that a unit has been chosen in \(\mathfrak M(\widehat M)\) and that a realization \(\mathfrak M(\widehat M)=C_\infty(Q')\) has been made on a suitable extremally disconnected bicompact space \(Q'\). Let \(X_0,X_1\) be Banach \(KN\)-spaces that are normal subspaces in \(C_\infty(Q)\). Following A. P. Calderón \((^3)\), put, for \(0<s<1\),
\[ X_0^{1-s}X_1^s=\{z\in C_\infty(Q): |z|\le \lambda x_0^{\,1-s}x_1^s,\ \text{where } 0\le x_i\in X_i, \]
\[ \|x_i\|_{X_i}\le 1\ (i=0,1),\ \text{the number } \lambda>0\}, \tag{2} \]
and for \(z\in X_0^{1-s}X_1^s\) we take as \(\|z\|_{X_0^{1-s}X_1^s}\) the infimum of all possible \(\lambda\) in (2). Then \((X_0^{1-s}X_1^s,\|\cdot\|_{X_0^{1-s}X_1^s})\) is a Banach \(KN\)-space.

Let us for the moment choose units arbitrarily in the spaces $\mathfrak M(X_0^*)$, $\mathfrak M(X_1^*)$, $\mathfrak M((X_0^{1-s}X_1^s)^*)$, and identify the spaces themselves $X_0^*$, $X_1^*$, $(X_0^{1-s}X_1^s)^*$ with their images in $C_\infty(Q')$ under the canonical realizations. After this one may consider the Calderón space $(X_0^*)^{1-s}(X_1^*)^s$, constructed from $X_0^*$ and $X_1^*$ in the same way (formula (2)) in which the space $X_0^{1-s}X_1^s$ is constructed from $X_0$ and $X_1$.

Theorem 3. Let the units in the spaces $\mathfrak M(X_0^*)$ and $\mathfrak M(X_1^*)$ be chosen arbitrarily. Then in the space $\mathfrak M((X_0^{1-s}X_1^s)^*)$ one can choose a unit so that, when the corresponding spaces are identified with their images under the canonical realizations, the equality

\[ (X_0^{1-s}X_1^s)^*=(X_0^*)^{1-s}(X_1^*)^s, \tag{3} \]

will hold both as to the stock of elements and as to the norm.

The proof of this theorem is based on results previously obtained by the author $(^4,^5)$.

It follows from Theorem 3 that the Banach conjugates to the family $X_0^{1-s}X_1^s$ $(0<s<1)$ again form a similar family. We emphasize here that in Theorem 3 no additional restrictions are imposed on the Banach $KN$-spaces $X_0$ and $X_1$.

Let us now consider an important special case of Calderón’s construction. Let $X$ be a Banach $KN$-space that is a normal subspace in $C_\infty(Q)$; let $p>1$ be an arbitrary number. Put

\[ X_p=\{x\in C_\infty(Q):\ |x|^p\in X\} \tag{4} \]

and, for $x\in X_p$,

\[ \|x\|_{X_p}=\||x|^p\|_X^{1/p}. \tag{5} \]

It is clear that $X_p=X^{1-s}Y^s$, where $Y=C(Q)$ and $1-s=1/p$.

Theorem 4. a) The formula $(X_p)^*=(X^*)_p$ is valid, where $X^*$ is the Nakano conjugate to the Banach conjugate $X^*$.

b) The Banach conjugate of odd order to $X_p$ is a $KB$-space.

c) If $X$ is not a $KB$-space, then no Banach conjugate of even order to $X_p$ is a $KB$-space.

Theorem 5. Let $\overline X$ be total on $X$ and let the following condition be satisfied: if a directed system $0\le x_\alpha\uparrow$ $(\alpha\in A)$ and $\sup\|x_\alpha\|_X<\infty$, then there exists $\sup x_\alpha\in X$ and $\sup\|x_\alpha\|_X=\|\sup x_\alpha\|_X$.

Then $X_p$ is algebraically and structurally isomorphic and isometric to $(\overline X_p)^*$.

Using Theorem 5, some other results of the author $(^5)$, and the Bishop–Phelps theorem on support functionals $(^1)$, one can prove the following theorem on completely linear functionals in a $KN$-space supporting its unit ball.

Theorem 6. Let $X$ be a Banach $KN$-space satisfying all the conditions of Theorem 5. Then:

a) for any $x\in X$ and any number $\varepsilon>0$ there exist $y\in X$ and $f\in\overline X$ such that $\|x-y\|_X<\varepsilon$, $\|f\|_{X^*}=1$ and $f(y)=\|y\|_X$;

b) for any $f\in\overline X$ and any number $\varepsilon>0$ there exist $g\in\overline X$ and $x\in X$ such that $\|f-g\|_{X^*}<\varepsilon$, $\|x\|_X=1$ and $g(x)=\|g\|_{X^*}$;

c) if $\mathfrak M(X)$ is of countable type, then there exist a weak unit $1$ in $X$ and a functional $f\in\overline X$ such that $\|1\|_X=\|f\|_{X^*}=f(1)=1$.

In conclusion the author expresses gratitude to Prof. B. Z. Vulikh for his attention to the present work.

Received
1 II 1969

REFERENCES

1. E. Bishop, R. R. Phelps, Proc. Symp. in Pure Math., VII, Am. Math. Soc., 1963, p. 393.
2. B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Moscow, 1961.
3. A. P. Calderon, Studia Math., 24, No. 2 (1964).
4. G. Ya. Lozanovskii, DAN, 172, No. 5 (1967).
5. G. Ya. Lozanovskii, Sibirsk. matem. zhurn., 10, No. 3 (1969).