ON CD_{0}(K)-SPACES

VLADIMIR G. TROITSKY Dedicated to the memory of Yuri Abramovich.

Abstract. We present an elementary proof of the (known) fact that a CD_{0}(K)- space is a Banach lattice and is lattice isometrically isomorphic to a particular C(K̃) for some compact space K̃.

CD_{0}(K)-spaces were introduced by Y. A. Abramovich and A. W. Wickstead in [AW91, AW93b] and further investigated in [AW93a, AE00, Erc04]. It is known [AW93b, AE00] that a CD_{0}(K)-space is a Banach lattice and a unital AM-space. In [Erc04] it was shown that CD_{0}(K) is lattice isometrically isomorphic to C(K × {0, 1}) with K × {0, 1} equipped with a compact Hausdorff topology. In this note we present elementary proofs of these facts.

Throughout these notes, K stands for a compact Hausdorff topological space without isolated points. For x ∈ K, let N_{x} be a base of open neighborhoods of x in K. As usually, for a real-valued function f on K and x_{0} ∈ K we write lim_{x→x_{0}} f(x) = r if for every ε > 0 there exists V ∈ N_{x} such that |f(x) − r| < ε for all x ∈ V \ {x_{0}}. Note that this notation is not vacuous for every x_{0} ∈ K because K has no isolated points.

We denote by C(K) the Banach lattice of all continuous functions on X, equipped sup-norm and point-wise ordering. Denote by c_{0}(K) the set of all real-valued functions f on K such that the set {|f| > ε} = {x ∈ K : |f(x)| > ε} is finite for every ε > 0. Clearly, c_{0}(K) is a vector subspace of ℓ_{∞}(K), the space of all bounded functions on K equipped with sup-norm.

Lemma 1. f ∈ c_{0}(K) iff lim_{x→x_{0}} f(x) = 0 for every x_{0} ∈ K.

Proof. Suppose that f ∈ c_{0}(K). Fix ε > 0. The set {|f| > ε} is finite; since K is Hausdorff there exists V ∈ N_{x_{0}} such that V doesn’t contain any points of this set with the possible exception of x_{0} itself. Thus, |f(x)| ⩽ ε for all x ∈ V \ {x_{0}}. Therefore,

lim_{x→x_{0}} f(x) = 0.

Vladikavkaz Math. J., 6(1) (2004).

1 2 V. G. TROITSKY

Date: March 31, 2004.

Suppose now that lim_{x→x_{0}} f(x) = 0 for every x_{0} ∈ K and assume that the set {|f| > ε} is infinite for some ε > 0. Since K is compact, this set must have an accumulation point x_{0}, which contradicts lim_{x→x_{0}} f(x) = 0. □

Lemma 2. c_{0}(K) is a closed subspace of ℓ_{∞}(K).

Proof. Suppose that a sequence of functions (f_{n}) in c_{0}(K) converges in sup-norm to f ∈ ℓ_{∞}(K). Fix ε > 0, then ∥f_{n} − f∥ < ε/2 for some n. It follows that {|f| > ε} ⊆ \{|f_{n}| > \frac{ε}{2}\}, hence is finite. □

It follows that c_{0}(K) equipped with the sup-norm is a Banach space. Define the space CD_{0}(K) as follows: f ∈ CD_{0}(K) if f = g + h for some g ∈ C(K) and h ∈ c_{0}(K). Equipped with the sup-norm, CD_{0}(K) is a normed space, a subspace of ℓ_{∞}(K). We also equip CD_{0}(K) with the pointwise order. We will see that CD_{0}(K) is a Banach lattice, and, moreover, an AM-space.

Lemma 3. If f ∈ CD_{0}(K), namely, f = g + h for some g ∈ C(K) and h ∈ c_{0}(K), then g(x_{0}) = lim_{x→x_{0}} f(x) for all x_{0} ∈ K.

Proof. By Lemma 1, lim_{x→x_{0}} f(x) = lim_{x→x_{0}} g(x) + lim_{x→x_{0}} h(x) = g(x_{0}) for every x_{0} ∈ K. □

It follows that every f in CD_{0}(K) has a unique decomposition into a continuous and a discrete part. Indeed, suppose that f = g + h = g′ + h′ where g, g′ ∈ C(K) and h, h′ ∈ c_{0}(K), then for every x_{0} ∈ K Lemma 3 implies g(x_{0}) = lim_{x→x_{0}} f(x) = g′(x_{0}). Hence, g = g′ and, therefore, h = h′. In the rest of the paper, for f ∈ CD_{0}(K) we will write f_{c} for the continuous component of f and f_{d} for the discrete component. The uniqueness of the decomposition also implies that (f+g)_{c} = f_{c}+g_{c} and (f+g)_{d} = f_{d}+g_{d} for f, g ∈ CD_{0}(K) because f + g = f_{c} + f_{d} + g_{c} + g_{d} = (f_{c} + g_{c}) + (f_{d} + g_{d}), and f_{c} + g_{c} ∈ C(K) while f_{d} + g_{d} ∈ c_{0}(K).

Proposition 4. If lim_{x→x_{0}} f(x) exists for every x_{0} ∈ K then f ∈ CD_{0}(K). In this case

f_{c}(x_{0}) = lim_{x→x_{0}} f(x).

Proof. For every x_{0} ∈ K, put g(x_{0}) = lim_{x→x_{0}} f(x), and let h = f −g. Then lim_{x→x_{0}} h(x) = 0 for every x_{0} ∈ K, so that h ∈ c_{0}(K) by Lemma 1. It remains to show that g ∈ C(K). Fix x_{0} ∈ K and ε > 0, there exists V ∈ N_{x_{0}} such that |f(x) − g(x_{0})| < ε for all x ∈ V \ {x_{0}}. It follows that for every y ∈ V we have

|g(y) − g(x_{0})| = |lim_{x→y} f(x) − g(x_{0})| < ε.

□

ON CD_{0}(K)-SPACES 3

Combining Lemma 3 and Proposition 4 we get the following result.

Corollary 5. f ∈ CD_{0}(K) if and only if lim_{x→x_{0}} f(x) exists for every x_{0} ∈ K.

Lemma 6. For every f ∈ CD_{0}(K) we have ∥f_{c}∥ ⩽ ∥f∥ ⩽ ∥f_{c}∥ + ∥f_{d}∥.

Proof. The first inequality follows from Proposition 4 while the second inequality is just the triangle inequality. □

Corollary 7. CD_{0}(K) is a Banach space.

Proof. Suppose that a sequence (f_{n}) is Cauchy in CD_{0}(K). It follows from Lemma 6 that the sequence of the continuous parts (f_{n})_{c} is Cauchy, and, therefore, the sequence of discrete parts (f_{n})_{d} is Cauchy. Since C(K) and c_{0}(K) are complete, (f_{n})_{c} converges to some g ∈ C(K) and (f_{n})_{d} converges to some h ∈ c_{0}(K). Hence (f_{n}) converges to g + h, which belongs to CD_{0}(K). □

Next, we show that for this topology CD_{0}(K) is order isometric to C(K × {0, 1}), if the topology on K × {0, 1} is defined as follows. We put discrete topology on K × {1}, that is, we put N_{(x,1)} = {(x, 1)} for each x ∈ K. Then all the points of K × {1} are isolated points of K × {0, 1}. For a point (x, 0) in K × {0} we take the basic open neighborhoods to be of the form Ṽ = (V × {0, 1}) \ {(x, 1)}, where V ∈ N_{x}. One can easily verify that these sets indeed form a base of a Hausdorff topology. From now on we consider K × {0, 1} equipped with this topology.

One can easily see that K × {0} is a closed subspace of K × {0, 1}, and the map x ↦ (x, 0) is a homeomorphism between K and K × {0}. It the future we will often identify K × {0} and K.

Lemma 8. K × {0, 1} is compact.

Proof. Consider an open cover of K × {0, 1}. By replacing each set in the cover by a union of basic open neighborhoods of all the points in the set, we can assume that the cover is formed by basic open neighborhoods. Hence, the cover is of the form

{{(x_{α}, 1)}}_{α∈Λ} ∪ {Ṽ_{γ}}_{γ∈Γ},

where x_{α} ∈ K and V_{γ} ∈ N_{x_{γ}} for some x_{γ} ∈ K. It is easy to see that {V_{γ}}_{γ∈Γ} is an open cover of K, so that there is a finite sub-cover V_{1}, …, V_{n}. But then Ṽ_{1} ∪ ⋯ ∪ Ṽ_{n} only misses finitely many points of K × {0, 1}, so that if we add the corresponding open sets from the original cover then we obtain a finite cover of the entire K × {0, 1}. □

4 V. G. TROITSKY

Theorem 9. CD_{0}(K) is lattice isometrically isomorphic to C(K × {0, 1}). In particular, CD_{0}(K) is an AM-space.

Proof. Define T: CD_{0}(K) → C(K × {0, 1}) via (T f)(x, r) = f_{c}(x) + rf_{d}(x). In other words, T f agrees with f on K × {1} and with f_{c} on K × {0}. It follows immediately that T is an isometry. It is obvious that T f ⩾ 0 implies f ⩾ 0. On the other hand, if f ⩾ 0 then f_{c} ⩾ 0 by Proposition 4.

Observe that T f is indeed a continuous function. Clearly, T f is continuous on K × {1}, as the later set consists of isolated points. Finally, it is left to show that lim_{(x,r)→(x_{0},0)}(T f)(x, r) = (T f)(x_{0}, 0) for every x_{0} ∈ K. Observe that (x, r) → (x_{0}, 0) in K × {0, 1} implies that x → x_{0} in K, so that f_{c}(x) → f_{c}(x_{0}) and f_{d}(x) → 0 by Lemma 1. It follows that (T f)(x, r) = f_{c}(x) + rf_{d}(x) → f_{c}(x_{0}) = (T f)(x_{0}, 0).

Show that T is onto. Let F ∈ C(K ×{0, 1}). For every x ∈ K define f(x) = F (x, 1). Fix x_{0} ∈ K and ε > 0, there exists V ∈ N_{x_{0}} such that |F (x, r) − F (x_{0}, 0)| < ε for all (x, r) ∈ Ṽ. In particular, for every x ∈ V \ {x_{0}} we have |f(x) − F (x_{0}, 0)| = |F (x, 1) − F (x_{0}, 0)| < ε, so that lim_{x→x_{0}} f(x) = F (x_{0}, 0). It follows from Lemma 4 that f ∈ CD_{0}(K) and f_{c}(x) = F (x, 0) for all x ∈ K, so that F = T f. □

References

[AE00] Ş. Alpay and Z. Ercan. CD_{0}(K, E) and CD_{ω}(K, E)-spaces as Banach lattices. Positivity, 4(3):213–225, 2000. Positivity and its applications (Ankara, 1998).

[AW91] Y. A. Abramovich and A. W. Wickstead. Regular operators from and into a small Riesz space. Indag. Math. (N.S.), 2(3):257–274, 1991.

[AW93a] Y. A. Abramovich and A. W. Wickstead. The regularity of order bounded operators into C(K). II. Quart. J. Math. Oxford Ser. (2), 44(175):257–270, 1993.

[AW93b] Y. A. Abramovich and A. W. Wickstead. Remarkable classes of unital AM-spaces. J. Math. Anal. Appl., 180(2):398–411, 1993.

[Erc04] Z. Ercan. A concrete descreption of CD_{0}(K)-spaces as C(X)-spaces and its applications.

Proc. Amer. Math. Soc., 132:1761–1763, 2004. Department of Mathematical and Statistical Sciences, University of Alberta, Ed-

monton, AB, T6G 2G1. Canada.

  E-mail address: vtroitsky@math.ualberta.ca