Conditional Riesz Representer

• Conditional Riesz representers are specialized operators that generalize the Riesz representation theorem by capturing conditional expectations and invariant structures via order projections.
• The framework employs Cesàro means, band projections, and conditional Kac formulas to recast recurrence theory in a measure-free, operator-theoretic setting.
• This approach underpins analyses in ergodic theory, finance, and stochastic processes by integrating operator theory with conditional expectation structures in Riesz spaces.

A conditional Riesz representer is a function, operator, or element that, within an abstract operator-theoretic or non-measure-theoretic context, represents the conditional or localized structure of a linear functional, measure, or expectation in analogy with the classical Riesz representation theorem. The concept generalizes the classic formula—where bounded linear functionals are represented via an inner product or integration—by incorporating conditional expectation operators, band projections, or similar structures to capture conditionality, invariance, or information encoded by a sub-σ-algebra, filtration, or an operator subspace. This framework is prominent in the paper of Riesz spaces, conditional expectation preserving systems, ergodic theory, and modern semiparametric estimation.

1. Abstract Formulation in Riesz Spaces

The setting for a conditional Riesz representer is typically a Dedekind complete Riesz space $E$ equipped with a weak order unit $e$ and a strictly positive, order-continuous conditional expectation operator $T$. In this framework, the classical measure and almost-everywhere concepts are supplanted by order projections and band decompositions. A function or operator $f$ is called a conditional Riesz representer if, for a given conditional expectation preserving system or operator $S$ with $T S = T$ and $S e = e$, it “captures” the conditional or invariant component of another object—such as the long-term average or the expected return time—via projections onto the range $R(T)$ or the invariant band.

One central construction is the Cesàro mean of iterates of $S$,

$S^n f := \sum_{k=0}^{n-1} S^k f,$

and its order limit

$L_S f := \lim_{n \to \infty} S^n f,$

where $L_S$ is itself a conditional expectation operator with invariant range. This operator $L_S$ often produces, via band projections, conditional Riesz representers encoding invariance or recurrence.

2. Conditional Poincaré Recurrence and Kac Formula

Ergodic theoretical notions, such as recurrence and return times, are recast in Riesz spaces by replacing measurable sets with components (order projections) of the order unit $e$. For projections $p \leq q$, $p$ is recurrent with respect to $q$ if there exist components $p_n$ such that $p = \sum_{n=1}^\infty p_n$ and $S^n p_n \leq q$ for each $n$. Under bijectivity of $S$, this is equivalent to $p \leq \bigvee_{n=1}^\infty S^{-n}q$.

The abstract Poincaré recurrence theorem asserts that, under these conditions, every component $p$ of a given component $q$ is recurrent with respect to $q$. This generalizes classical recurrence to a measure-free, order-structural setting.

The conditional Kac formula provides an analog of the classical result linking average return times to measure. For a component $p$, define the "non-recurrence" components

$N_p(n) = p \wedge \bigwedge_{j=1}^n (e - S^{-j}p),$

then define the first recurrence time as

$$n_p = \sum_{k=1}^\infty k [N_p(k-1) - N_p(k)] = \sum_{k=0}^\infty N_p(k),$$

in the universal completion of $E$. The key result (Theorem 4.4) shows: $L_S n_p = P_{L_S p} e,$ where $P_{L_S p}$ denotes the band projection onto the band generated by $L_S p$. In the conditionally ergodic case ($L_S = T$),

$T n_p = P_{T p} e,$

so $T n_p$ becomes the conditional Riesz representer of the average recurrence behavior, providing a precise operator-theoretic generalization of Kac’s formula.

3. Connections with Conditional Expectation and Invariance

In this abstract framework, the conditional Riesz representer provides a mechanism for representing conditional expectations and invariant structures. Specifically, for processes in $L^1(\Omega, \mathcal{A}, \mu)$, setting $E = L^1(\Omega, \mathcal{A}, \mu)$ and $T$ as the usual conditional expectation, the operator $L_S$ projects onto the $S$-invariant functions. The equality $T n_p = P_{T p} e$ then reduces to stating that the conditional expectation of the first recurrence time is the projection of the indicator function of the set $A$ (represented by $p = 1_A$) onto the invariant subalgebra—directly generalizing classical results to the operator and order-theoretic context.

Analogously, conditional Riesz representers in this context "capture" the parts of random variables (or more generally, elements of $E$) that are determined by the invariant sub-σ-algebra or operator structure, thus providing functional-analytic analogs of conditional expectations or probability kernels as encountered in measure-theory.

4. Characterization via Band Projections and Averaging

In Riesz spaces, the role of conditional Riesz representer is intimately tied to the structure of band projections and order ideals. If $f$ is an element of $E$, the projection $P_{T f} e$ onto the band generated by $T f$ can be viewed as a conditional indicator or representer, "localizing" $f$ to the conditional information encoded by $T$.

Key formulas in this context include:

• Cesàro mean and limit: $S^n f = \sum_{k=0}^{n-1} S^k f$, $L_S f = \lim_{n\to\infty} S^n f$,
• Non-recurrence components: $N_p(n) = p \wedge \bigwedge_{j=1}^n (e - S^{-j}p)$,
• First recurrence time: $n_p = \sum_{k=1}^\infty k [N_p(k-1) - N_p(k)]$,
• Conditional Kac: $L_S n_p = P_{L_S p} e$, $T n_p = P_{T p} e$ (if ergodic).

These constructions provide order-theoretic analogues of measure-theoretic expectations, essential for the analysis of dynamical, ergodic, or stochastic systems in a measure-free setting.

5. Broader Implications and Applications

The extension of recurrence and invariant structure to Riesz spaces via conditional Riesz representers enables the analysis of abstract dynamical and probabilistic systems beyond the classical context. This approach:

• Generalizes recurrence theory and Kac’s formula to non-pointwise, operator-based frameworks,
• Enables the encapsulation of conditional behavior in abstract settings (e.g., vector lattices, spaces of functions, or martingale and Markov process theory over Riesz spaces),
• Provides foundational tools for developing a measure-free theory of stochastic processes, ergodic theory, and probabilistic inequalities (such as Chernoff bounds and Poisson approximations) via conditional expectations, order convergence, and generating functions.

Applications include studying processes in $L^1(\Omega, \mathcal{A}, \mu)$ where conditional expectations and invariant projections are internalized within the algebraic and order structure, as well as applications in finance (via conditional supremum and arbitrage analysis), the paper of Markov and ergodic processes, and mathematical economics.

6. Synthesis and Key Formulas

The conditional Riesz representer emerges as a unifying construct bridging operator theory, probability, ergodic theory, and functional analysis within Riesz spaces and their generalizations. As a summary, key identities are:

• Representation of recurrence and return time via projections: $L_S n_p = P_{L_S p} e$,
• Relation to ergodicity: $L_S = T$ implies $T n_p = P_{T p} e$,
• Structure of non-recurrence components and their summation encoding recurrence times.

These establish that conditional Riesz representers localize, project, or "average" elements in a Riesz space in a way that precisely reflects the invariant or conditional information dictated by an operator structure, providing the foundation for a broad range of operator-theoretic and probabilistic analysis in non-measure-theoretic settings (Azouzi et al., 2022).