On the sequential monotone closure of $CD_{\omega}(K)$ spaces

This short note resolves a problem posed by Wickstead regarding the sequential monotone closure of $CD_{\omega}(K)$ spaces, a topic arising from the study of Riesz completions of regular operator spaces between Banach lattices.

The Gist

Here is an explanation of the paper "On the Sequential Monotone Closure of CDω(K) Spaces," translated into simple language with creative analogies.

The Big Picture: Building a "Perfect" House

Imagine you are an architect working with a specific type of building material called Banach Lattices. These are mathematical structures that act like a grid where you can add, subtract, and compare things (like "bigger" or "smaller"), and they have a built-in ruler (a norm) to measure size.

The authors of this paper are investigating a specific construction project: The Sequential Monotone Closure.

Think of your original building material (let's call it Space E) as a rough, unfinished house. It has walls and a roof, but maybe some gaps. The "Sequential Monotone Closure" (let's call it Space Eσm) is the process of trying to "finish" this house by filling in the gaps using a specific rule: You can only fill a gap if you can build a staircase up to it or a ladder down to it using the original materials.

• The Rule: If you have a sequence of materials getting bigger and bigger ($x_1 < x_2 < x_3 \dots$) that seems to approach a limit, that limit must be included in your new, finished house.
• The Goal: The mathematicians wanted to know: "If we start with a perfectly solid, complete building (a Banach lattice), and we use this 'staircase rule' to finish it, will the result still be a solid, complete building?"

The Problem: Wickstead's Question

A mathematician named Wickstead asked a very specific question:

"If we take any perfect Banach lattice and apply this 'staircase finishing' rule, do we always end up with a complete, solid structure? Or does the process sometimes create a house with holes in the floor?"

Most mathematicians suspected the answer was "Yes, it stays solid." The authors of this paper set out to prove that No, it does not.

The Experiment: The "CDω(K)" House

To prove their point, the authors built a specific, tricky test case. They chose a space called CDω(K).

The Analogy of CDω(K):
Imagine a giant canvas (Space $K$) where you can paint pictures (functions).

• The Rule for CDω(K): You are allowed to paint any picture, as long as it looks exactly like a smooth, continuous painting, except for a tiny, countable number of specks (like a few dust motes).
• If you change the painting on 100 points, or even a million points, it's fine. But if you change it on an infinite, uncountable number of points (like changing the color of every single pixel in a specific region), it's not allowed.

This space is a "Banach lattice," meaning it's a solid, complete building to start with.

The Discovery: The "Ghost" Function

The authors asked: "What happens if we apply the 'staircase rule' to this CDω(K) house?"

They constructed a specific mathematical object (a function $f$) that acts like a Ghost.

1. It exists: You can build this function by stacking up a sequence of allowed CDω(K) paintings (a staircase).
2. It looks almost right: This ghost function looks exactly like a "semi-continuous" painting everywhere, except for a huge, uncountable mess of points.
3. The Trap: Because the "mess" is too big (uncountable), this Ghost function does not belong to the original CDω(K) house.
4. The Result: Even though the Ghost function was built using the "staircase rule" from the original house, it doesn't fit inside the "finished" house (the Sequential Monotone Closure) because the closure only allows functions that differ from smooth ones by a small amount.

The Metaphor:
Imagine you are building a wall using bricks. You have a rule: "You can only add a brick if it fits perfectly with the ones below it."
You build a tower. At the very top, you try to place a "Ghost Brick" that is made of smoke. It fits the pattern of the bricks below it (you can see the staircase leading up to it), but because it's made of smoke, it doesn't actually belong in the wall.
The authors proved that for this specific type of house, the "staircase rule" creates a Ghost Brick that the house cannot hold. Therefore, the house has a hole.

The Conclusion: The Answer is "No"

The paper concludes that CDω(K)σm is not complete.

In plain English:

• You can start with a perfect, solid mathematical space.
• You can try to "complete" it by filling in all the gaps reachable by staircases.
• But sometimes, the process creates a new object that is almost in the space but technically falls through the floor.
• Therefore, the "finished" space is actually broken (incomplete).

Why Does This Matter?

This isn't just about abstract math; it has real consequences for how we handle Regular Operators.

Think of Regular Operators as machines that take inputs from one building and produce outputs in another. Mathematicians often need to "complete" these machines to make them work perfectly.

• Wickstead had a theory that if the output building was "almost" complete, the machine would work fine.
• This paper says: "Not necessarily."
• If you use this specific type of building (CDω(K)), the machine might break because the "completion" process leaves a hole.

Summary in One Sentence

The authors proved that if you try to "finish" a specific type of mathematical building by filling in gaps reachable by staircases, you might accidentally create a structure with holes, meaning the process doesn't always result in a perfect, solid space.

Technical Summary

Here is a detailed technical summary of the paper "On the Sequential Monotone Closure of $CD_\omega(K)$ Spaces" by Sukrit Chalana, Denny H. Leung, and Foivos Xanthos.

1. Problem Statement

The paper addresses a specific open question posed by Wickstead in [13] regarding the completeness of the sequential monotone closure of Banach lattices.

• Context: The study arises from the investigation of the Riesz completion of spaces of regular operators between Banach lattices.
• Definitions:
  • Let $E$ be an Archimedean vector lattice with an order completion $E_\delta$.
  • $u(E)$ denotes the set of $\sigma$-upper elements (limits of increasing sequences from $E$).
  • The sequential monotone closure (denoted $E_{\sigma m}$) is defined as the sublattice $u(E) - u(E)$ of $E_\delta$.
• The Question: Wickstead proved that if $E$ is an almost $\sigma$-order complete Banach lattice, then $(E_{\sigma m}, \|\cdot\|_{E_{\sigma m}})$ is complete. He asked whether this completeness holds for any Banach lattice $E$.
• Goal: The authors aim to determine if $(E_{\sigma m}, \|\cdot\|_{E_{\sigma m}})$ is always a complete space for any Banach lattice $E$.

2. Methodology

The authors employ a combination of vector lattice theory, topology, and functional analysis to construct a counterexample.

• Space Selection: They focus on the space $CD_\omega(K)$, defined as the set of bounded functions on a topological space $K$ that differ from a bounded continuous function on at most a countable set:
  $$CD_\omega(K) = \{f \in B(K) \mid \exists g \in C_b(K) : [f \neq g] \text{ is at most countable}\}$$
• Topological Setting: The analysis is conducted on perfect Polish spaces (separable, completely metrizable, no isolated points).
• Characterization of the Closure:
  • They establish that for such spaces, the sequential monotone closure $CD_\omega(K)_{\sigma m}$ can be identified with the set of functions in $B(K)$ that differ from a function in $DBSC(K)$ (differences of bounded semicontinuous functions) on a countable set.
  • They utilize a projection operator $P_c: CD_\omega(K) \to C_b(K)$ which maps a function to its unique continuous representative (up to a countable set).
• Counterexample Construction:
  • They construct a specific function $f \in B_1^1(K)$ (the uniform closure of $DBSC(K)$) using a nested sequence of closed sets $(F_n)$ in a Cantor-like subset of $K$.
  • The function is defined as an alternating series: $f = \sum_{n=1}^\infty \frac{(-1)^n}{n} \mathbf{1}_{F_{n-1} \setminus F_n}$.
  • They prove that for this specific $f$, the set of points where $f$ differs from any $g \in DBSC(K)$ is uncountable.

3. Key Contributions and Results

A. Characterization of $CD_\omega(K)_{\sigma m}$ (Theorem 2.3)

The authors prove that if $K$ is a perfect metrizable Baire space, then:
$$CD_\omega(K)_{\sigma m} = \{f \in B(K) \mid \exists g \in DBSC(K) : [f \neq g] \text{ is at most countable}\}$$
This provides a concrete structural description of the sequential monotone closure for these spaces.

B. The Counterexample (Theorem 2.5 & 2.7)

This is the central result of the paper.

• Theorem 2.5: For any uncountable Polish space $K$, there exists a function $f \in B_1^1(K)$ such that for any $g \in DBSC(K)$, the set $[f \neq g]$ is uncountable.
• Theorem 2.7: Consequently, if $K$ is a perfect Polish space, $CD_\omega(K)_{\sigma m}$ is not uniformly complete.
  • Reasoning: The function $f$ constructed in Theorem 2.5 belongs to the uniform closure of $DBSC(K)$ (and thus is a limit of a Cauchy sequence in the relevant norm), but it does not belong to $CD_\omega(K)_{\sigma m}$ because it cannot be approximated by a $DBSC$ function on a countable set.
  • Implication: This answers Wickstead's question in the negative. The sequential monotone closure of a Banach lattice is not necessarily complete.

C. Relationship with Order Completeness (Propositions 2.8 & 2.9)

• Proposition 2.8: If $E$ is an almost $\sigma$-order complete vector lattice, then $E_{\sigma m}$ is $\sigma$-order complete. This confirms Wickstead's previous positive results for this specific subclass.
• Proposition 2.9: The converse is not true. There exist spaces (specifically $C(K_\infty)$ where $K_\infty$ is the one-point compactification of an uncountable discrete set) where $E_{\sigma m}$ is $\sigma$-order complete, but $E$ is not almost $\sigma$-order complete.

D. Application to Regular Operators (Theorem 2.10)

The paper connects the completeness of $E_{\sigma m}$ to the Riesz completion of spaces of regular operators.

• Theorem 2.10: Let $F$ be a Banach lattice. The following are equivalent:
  1. $F_{\sigma m}$ is uniformly complete.
  2. The Riesz completion of $L_r(c, F)$ (where $c$ is the space of convergent sequences) is uniformly complete.
• This provides a necessary and sufficient condition for the uniform completeness of the Riesz completion of $L_r(c, F)$, improving upon Wickstead's earlier sufficient condition.

4. Significance

• Resolution of an Open Problem: The paper definitively settles Wickstead's question, showing that the property of being a Banach lattice is insufficient to guarantee the uniform completeness of its sequential monotone closure.
• Refinement of Terminology: The authors argue for and adopt the term "sequential monotone closure" ($E_{\sigma m}$) over "sequential order closure" ($E_\sigma$), clarifying the distinction between order limits and monotone limits in this context (Remark 2.6).
• Operator Theory Implications: By linking the completeness of $F_{\sigma m}$ to the Riesz completion of regular operators, the results provide a deeper understanding of the structural properties required for spaces of operators to be well-behaved (complete).
• Counterexample Construction: The construction of the specific function $f$ using nested closed sets in Polish spaces serves as a powerful tool for distinguishing between different classes of function spaces (specifically between uniform closures of semicontinuous functions and sequential monotone closures).

In summary, the paper demonstrates that while sequential monotone closures behave well for "almost $\sigma$-order complete" lattices, they fail to be complete in the general Banach lattice setting, specifically for spaces like $CD_\omega(K)$ over perfect Polish spaces. This failure has direct consequences for the completeness of Riesz completions in operator theory.