Geometry of Spaces of Orthogonally Additive Polynomials on $C(K)$

A real function $f$ on a Banach lattice is said to be orthogonally additive if $f(x+y)=f(x)+f(y)$ whenever $x$ and $y$ are disjoint. Non-linear orthogonally additive functions on function spaces often have useful integral representations — see, for example the papers of Chacon, Friedman and Katz [8, 12, 13], Mizel [21] and Rao [24]. In 1990, Sundaresan [26] initiated the study of orthogonally additive $n$-homogeneous polynomials with particular reference to the spaces $L_{p}[0,1]$ and $\ell_{p}$ for $1\leq p<\infty$. Building on the work of Mizel, he showed that, for every orthogonally additive $n$-homogeneous polynomial $P$ on $L_{p}[0,1]$ with $n\leq p$, there exists a unique function $\xi\in L_{\tilde{p}}$, where $\tilde{p}=p/(p-n)$, such that

for every $x\in L_{p}[0,1]$. When $n>p$, there are no non-zero orthogonally additive $n$-homogeneous polynomials on $L_{p}[0,1]$. He went on to show that the Banach space of orthogonally additive $n$-homogeneous polynomials on $L_{p}[0,1]$ is isometrically isomorphic to $L_{\tilde{p}}$ where the latter space is equipped not with the usual norm, but with the equivalent norm $\|x\|=\max\{\|x^{+}\|_{\tilde{p}},\|x^{-}\|_{\tilde{p}}\}$.

The next significant development was the discovery of an integral representation for orthogonally additive $n$-homogeneous polynomials on $C(K)$ spaces by Pérez and Villanueva [23] and by Benyamini, Lassalle and Llavona [3], who proved a representation of the form

where $\mu$ is a regular Borel signed measure on $K$. The integral representations (1) and (2) have been extended and generalized in various directions in recent years. See, for example, [22, 17, 1, 30].

Orthogonally additive $n$-homogeneous polynomials are also of interest in the study of multilinear operators on Banach lattices and, more generally, on vector lattices. If $E,F$ are vector lattices, an $n$-linear mapping $A\colon E^{n}\to F$ is orthosymmetric if $A(x_{1},\dots,x_{n})=0$ whenever $x_{i}$ and $x_{j}$ are disjoint for some pair of distinct indices $i,j$. Orthosymmetric multilinear mappings are automatically symmetric [4]. In [5], Bu and Buskes prove that an $n$-linear function is orthosymmetric if and only if the associate $n$-homogeneous polynomial is orthogonally additive.

Let $E$ be a Banach lattice. For every positive element $a$ of $E$, we may form the principal ideal

with lattice structure inherited from $E$ and the norm defined by $\|x\|_{a}=\inf\{C>0:|x|\leq Ca\}$. With this norm, $E_{a}$ is a Banach lattice. By virtue of the Kakutani representation theorem [16], the Banach lattice $E_{a}$ is canonically Banach lattice isometrically isomorphic to $C(K)$ for some compact Hausdorff topological space $K$, with $a$ being identified with the unit function on $K$. The Banach lattice structure of $E$ is uniquely determined by its principal ideals. It follows that an analysis of the orthogonally additive $n$-homogeneous polynomials on $C(K)$ is central to an understanding of the behaviour of orthogonally additive $n$-homogeneous polynomials on general Banach lattices.

In this paper, we focus on the geometric properties of the spaces $\mathcal{P\!}_{o}(^{n}{C(K)})$ of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two phenomena that are of particular interest. The first is that there are two natural ways to norm the space $\mathcal{P\!}_{o}(^{n}{E})$. The first is the norm of uniform convergence on the unit ball of $E$, given by $\|P\|_{\infty}=\sup\{|P(x)|:x\in E,\|x\|\leq 1\}$. In this norm, $\mathcal{P\!}_{o}(^{n}{E})$ is a Banach space. Now $\mathcal{P\!}_{o}(^{n}{E})$ also has a lattice structure and so another choice of norm is the regular norm, defined by $\|P\|_{r}=\||P|\|_{\infty}$, where $|P|$ is the absolute value of $P$. In this norm, $\mathcal{P\!}_{o}(^{n}{E})$ is a Banach lattice. The existence of these two norms was first observed by Bu and Buskes [5] and is hinted at in the paper of Sundaresan [26]. These norms are equivalent, but we shall see that they have significantly different geometric properties.

The second phenomenon is the influence of the parity of the degree $n$ on the structure of the space $\mathcal{P\!}_{o}(^{n}{E})$ for the two norms. Bu and Buskes [5] showed that, when $n$ is odd, the supremum and regular norms on $\mathcal{P\!}_{o}(^{n}{E})$ are the same and that they are equivalent when $n$ is even. We sharpen their results, using the strategy of working first on $\mathcal{P\!}_{o}(^{n}{C(K)})$ and then extending to general Banach lattices. The integral representation (2) gives a canonical isomorphism between $\mathcal{P\!}_{o}(^{n}{C(K)})$ and $\mathcal{M}(K)$, the space of regular Borel signed measures on $K$. The regular norm on $\mathcal{P\!}_{o}(^{n}{C(K)})$ corresponds to the usual variation norm on $\mathcal{M}(K)$, but the supremum norm is identified with a different norm on $\mathcal{M}(K)$, given by $\|\mu\|_{0}=\max\{\|\mu^{+}\|_{1},\|\mu^{-}\|_{1}\}$. We show that $(\mathcal{M}(K),\|\cdot\|_{0})$ is isometrically isomorphic to the dual space of $C(K)$, where $C(K)$ is endowed with the norm $\|{x}\|_{d}=\|x^{+}\|_{\infty}+\|x^{-}\|_{\infty}$ and we show that this norm is closely related to the diameter seminorm (see, for example, [6]). We use these identifications to give a complete description of the extreme points of the unit ball of $\mathcal{P\!}_{o}(^{n}{C(K)})$ for both norms, extending the results in [7]. Our starting point is a characterisation of the extreme points in $C(K)$ for the norm $\|\cdot\|_{d}$ and $\mathcal{M}(K)$ for the norm $\|\cdot\|_{0}$. This allows us to prove a Banach-Stone theorem for $(C(K),\|\cdot\|_{d})$.

We finish with a study of the exposed points of the unit ball of the space $\mathcal{P\!}_{o}(^{n}{C(K)})$. The identification of this space with the space of measures $\mathcal{M}(K)$, which is a dual space for both norms, allows us to use the theory of Šmul’yan [28, 29]. Using this machinery, we characterise the weak^∗ exposed and the weak^∗ strongly exposed points of the unit ball.

Let $E$ be a Banach lattice and $n$ a positive integer. A function $P\colon E\to\mathbb{R}$ is called an orthogonally additive $n$-homogeneous polynomial if $P$ is a bounded $n$-homogeneous polynomial with the property that $P(x+y)=P(x)+P(y)$ whenever $x,y\in E$ are disjoint. The space of orthogonally additive $n$-homogeneous polynomials on $E$ is denoted by $\mathcal{P\!}_{o}(^{n}{E})$. It is easy to see that $\mathcal{P\!}_{o}(^{n}{E})$ is a closed subspace of the space $\bigl{(}\mathcal{P}(^{n}{E}),\|\cdot\|_{\infty}\bigr{)}$ of bounded $n$-homogeneous polynomials with the supremum norm. Thus $\mathcal{P\!}_{o}(^{n}{E})$, with this norm, is a Banach space. When $n=1$, this space is simply the dual space $E^{\prime}$, since every bounded linear functional is orthogonally additive.

We have the following integral representation for orthogonally additive $n$-homogeneous polynomials on $C(K)$ spaces, due to Pérez-García and Villanueva [23] and Benyami, Lassalle and Llavona [3] (see also [7]).

In general, there is no guarantee that a Banach lattice supports any non-trivial orthogonally additive polynomials of degree greater than one. Sundaresan [26] showed that there are no non-zero orthogonally additive $n$-homogeneous polynomials on $L_{1}[0,1]$ for $n>1$. In the case of $\ell_{1}$, it is easy to see that an $n$-homogeneous polynomial $P$ is orthogonally additive if and only if there exists a bounded sequence of real numbers, $(a_{j})$, such that

for every $x\in\ell_{1}$, and that $\|P\|_{\infty}=\sup_{j}|a_{j}|$. Thus $\mathcal{P\!}_{o}(^{n}{\ell_{1}})$ is isometrically isomorphic to $\ell_{\infty}$ for every $n$.

To put the results of the previous paragraph in a general context, we recall that a Banach lattice $E$ is an AL-space if the norm is additive on the positive cone: $\|x+y\|=\|x\|+\|y\|$ for all $x,y\geq 0$. The Kakutani representation theorem [15, 18] states that every AL-space $E$ can be decomposed into a disjoint sum of copies of $\ell_{1}$ and $L_{1}$ spaces. Accordingly, $E$ is Banach lattice isometrically isomorphic to a space of the form

In this representation, the unit basis vectors $e_{\gamma}$ in $\ell_{1}(\Gamma)$ are in one-to-one correspondence with the atoms in $E$ of unit norm. We recall that a positive element $x$ of $E$ is said to be an atom if $0\leq y\leq x$ implies that $y$ is a scalar multiple of $x$. We can write the second component in this representation as $L_{1}(\mu)$, where $\mu$ is the product of the Lebesgue measures on the sets $[0,1]^{m_{\alpha}}$. Thus, we see that $E$ can be represented as the disjoint sum $\ell_{1}(\Gamma)\oplus_{1}L_{1}(\mu)$, where the measure $\mu$ is nonatomic.

The Banach lattices $L_{1}(\mu)$, where $\mu$ is nonatomic, do not support any real valued lattice homomorphisms. Our next result indicates that the existence of non-trivial $n$-homogeneous orthogonally additive polynomials on a Banach lattice is closely related to the existence of lattice homomorphisms.

A Banach lattice $E$ is an AM-space if the norm has the property that $x\wedge y=0$ implies $\|x\vee y\|=\max\{\|x\|,\|y\|\}$. In contrast with AL-spaces, there is a good supply of orthogonally additive $n$-homogeneous polynomials on every AM-space. The Kakutani representation theorem for AM-spaces [16] shows that the real valued lattice homomorphisms on an AM-space $E$ separate the points of $E$. It follows that there is a rich supply of orthogonally additive $n$-homogeneous polynomials of every degree on $E$.

We now look at some properties of orthogonally additive polynomials on general Banach lattices. Our starting point is the fact that every orthogonally additive $n$-homogeneous polynomial on a Banach lattice $E$ is regular. This has been shown by Toumi [27, Theorem 1]. One may also argue as follows. Let $P$ be an orthogonally additive $n$-homogeneous polynomial on a Banach lattice $E$. As $E$ is the upwards directed union of its principal ideals, it suffices to show that $P$ is regular on each of them. Since each principal ideal is Banach lattice isometrically isomorphic to a $C(K)$, we can use the integral representation in Theorem 1. Then the Jordan decomposition of the representing measure gives a decomposition of the polynomial into the difference of two positive orthogonally additive $n$-homogeneous polynomials. Therefore $P$ is regular.

Let $P=\widehat{A}$ be a regular $n$-homogeneous polynomial $P$ on $E$. The absolute value of $P$ is given by [5, 19]

for $x\geq 0$, where $\Pi(x)$ denotes the set of partitions of $x$, namely, all finite sets of positive vectors whose sum is $x$. In general, we have

for every $x\in E$ and $|P|$ is the smallest positive $n$-homogeneous polynomial, in the sense of the lattice structure of $\mathcal{P}_{r}(^{n}{E})$, with this property. The space $\mathcal{P}_{r}(^{n}{E})$ is a Banach lattice in the regular norm,

It follows from (5) that $\|P\|_{\infty}\leq\|P\|_{r}$ for every $P\in\mathcal{P}_{r}(^{n}{E})$. In general, these norms are not equivalent.

Now $\mathcal{P\!}_{o}(^{n}{E})$ is complete in the regular norm; indeed, it is even a dual Banach lattice [5, Theorem 5.4]. It follows that the supremum and regular norms are equivalent on this space. Thus, there is a sequence $(C_{n})$ of positive real numbers such that $\|P\|_{r}\leq C_{n}\,\|P\|_{\infty}$ for every $n$ and every $P\in\mathcal{P\!}_{o}(^{n}{E})$. Bu and Buskes [5] show that the two norms are the same for odd values of $n$. For even values of $n$, they show that $C_{n}\leq n^{n}/n!$, the polarization constant. We shall show that, in fact, $C_{n}=2$ for even values of $n$ and that this is sharp. This will follow from estimates we give for the value of $|P|$ at positive points in $E$.

If $\varphi$ is a bounded linear functional on $E$, then [20]

for every $x\geq 0$. It would be suprising if there were such a simple formula for $|P|(x)$ when $P$ is a regular $n$-homogeneous polynomial. As a linear functional, $P$ acts on an $n$-fold symmetric tensor power of $E$ and the set of vectors $y$ satisfying $|y|\leq x$ is now a set of tensors, rather than elements of $E$. However, if $P$ is orthogonally additive, it is possible to establish a relatively simple estimate for the values of $|P|$.

To see that the bound in (6) for even values of $n$ is sharp, consider the example $P(x)=x_{1}^{n}-x_{2}^{n}$ on $\mathbb{R}^{2}$ with any Banach lattice norm. The bound is attained for the vector $x=(1,1)$.

In this section, we study the supremum and regular norms on the spaces of orthogonally additive $n$-homogeneous polynomials on $C(K)$.

The integral representation for orthogonally additive $n$-homogeneous polynomials on $C(K)$ allows us to identify the vector space $\mathcal{P}_{r}(^{n}{C(K)})$ with $\mathcal{M}(K)$, the space of regular Borel signed measures on $K$. The natural norm on $\mathcal{M}(K)\cong C(K)^{\prime}$ is the dual norm. This is the variation norm for measures: $\|\mu\|_{1}=|\mu|(K)$. We shall see that this norm corresponds to the regular norm on the spaces of orthogonally additive $n$-homogeneous polynomials. However, the supremum norm on $\mathcal{P\!}_{o}(^{n}{K})$ corresponds to a different, but equivalent norm on the space of regular Borel signed measures.

The space $\mathcal{P}_{r}(^{n}{C(K)})$ is a Banach lattice with the regular norm, as is the dual Banach lattice $\mathcal{M}(K)$ with the variation norm. We shall see that the lattice structures of these two Banach lattices are the same. We note that the lattice structure of $\mathcal{M}(K)$ as the dual of $C(K)$ is the same as the lattices structure of $\mathcal{M}(K)$ considered as a sublattice of the lattice of Borel signed measures on $K$. In other words, a measure $\mu\in\mathcal{M}(K)$ is positive, in the sense that $\int_{K}f\,d\mu\geq 0$ for every nonnegative $x\in C(K)$, if and only if $\mu(E)\geq 0$ for every Borel subset $E$ of $K$ [25, Theorem 2.18].

We summarize the identifications of the various norms, bearing in mind that the supremum and regular norms coincide for positive polynomials.

We note that the norm $\|\cdot\|_{0}$ is easily seen to be equivalent to the dual (variation) norm on $\mathcal{M}(K)$. In fact, we have

for every $\mu\in\mathcal{M}(K)$.

It will be useful to have an alternative expression for the norm $\|\cdot\|_{0}$ on $\mathcal{M}(K)$. Using the identity $\max\{a,b\}=\frac{1}{2}\bigl{(}|a+b|+|a-b|\bigr{)}$ for non-negative real numbers, we have

Thus, we have

These results clarify the geometric properties of the spaces $\mathcal{P}_{o}(^{n}{C(K)})$; for the regular norm, these spaces are all essentially the same as the dual space $\mathcal{M}(K)$ with the variation norm. The case of $\mathcal{P}_{o}(^{n}{C(K)})$ with the supremum norm and $n$ even is substantially different. To understand this, we must study the extreme point structure of the unit ball of $\mathcal{M}(K)$ for the norm $\|\cdot\|_{0}$.

In this section, we study the extreme points of the unit ball of the space $\mathcal{P\!}_{o}(^{n}{C(K)})$. We begin with the regular norm. We have seen in Proposition 3 that there is an isometric isomorphism

where $\|\cdot\|_{1}$ denotes the variation norm on $\mathcal{M}(K)$, the space of regular Borel signed measures on $K$. Furthermore, when the degree $n$ is odd, the supremum and regular norms on $\mathcal{P\!}_{o}(^{n}{C(K)})$ coincide.

It is a classical result that the extreme points of the unit ball of $\mathcal{M}(K)$ for the variation norm are the measures of the form $\pm\delta_{t}$, where $t\in K$ (see, for example, [11, V.8.6]). The isomorphism between $\mathcal{P\!}_{o}(^{n}{C(K)})$ and $\mathcal{M}(K)$ associates the polynomial $P(x)=x(t)^{n}$ with the measure $\delta_{t}$. Thus, we have