A bang-bang principle for the conditional expectation vector measure

Representation of Conditional Expectations in Gaussian Analysis on Sequence Spaces

From a given nuclear triplet we construct a nuclear triplet of sequence spaces and introduce a correlated Gaussian measure via the Bochner-Minlos theorem. Considering special types of correlation operators on such sequence spaces, certain conditional expectations can be given in an explicit way.

Orlicz空间中条件期望的连续性

Continuity of conditional expectation in Orlicz spaces

本文研究Orlicz空间中条件期望的连续性，给出了σ-子代数序列使条件期望$L^φ$-收敛的充要条件，推广了$L^p$空间的结果。

A disjointness type property of conditional expectation operators

We give a characterization of conditional expectation operators through a disjointness type property similar to band preserving operators. We say that the operator $T:X\to X$ on a Banach lattice $X$ is semi band preserving if and only if for all $f, g \in X$, $f \perp Tg$ implies that $Tf \perp Tg$. We prove that when $X$ is a purely atomic Banach lattice, then an operator $T$ on $X$ is a weighted conditional expectation operator if and only if $T$ is semi band preserving.

On Convergence of Conditional Expectation Operators

Given an operator $T:U_X(Σ)\to Y$ or ${T:U(Σ)\to Y$, one may consider the net of conditional expectation operators $(T_π)$ directed by refinement of the partitions $π$. It has been shown previously that $(T_π)$ does not always converge to $T$. This paper gives several conditions under which this convergence does occur, including complete characterizations when $X={\bold R}$ or when $X\sp *$ has the Radon-Nikodým property.

Conditional expectations and interpolation of linear operators on ordered ideals between $L^1(0,1)$ and $L^\infty(0,1)$

The monograph contains the detailed exposition of the results obtained by the author during the last several years. In particular it contains an improvement of the well known Calderon - Ryff interpolation theorem and description of "verifying" operators of conditional mathematical expectations {if such an operator leaves an order ideal in $L^1% invariant, then the ideal is an interpolation space}.

$C(X)$ 上的条件期望算子

Conditional expectation operators on $C(X)$

本文肯定回答了在$C(X)$空间中构造Kuo等人意义下条件期望算子的可能性，并参与$C(X)$空间积分的相关讨论。

Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.

The ascent and descent of weighted conditional expectation operators

In this paper we prove that the ascent of a weighted conditional expectation operator of the form of $M_wEM_u$ on $L^p$-spaces is always finite and is equal to 2. Also we get that under a weak condition the descent of $M_wEM_u$ is finite and is equal to 2 too. Moreover, we give some necessary and sufficient conditions for $M_wE M_u$ to be power bounded. In the sequel we apply some results in operator theory on ascent and descent to $M_wEM_u$. Finally we find that $T=M_wEM_u$ is Cesaro bounded if and only if $\widehat{T}$ is Cesaro bounded.

Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete…

Conditional expectations onto maximal abelian *-subalgebras

We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.

Conditional expectation on non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces : semifinite case

In this paper we investigate the conditional expectation on the non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.

Paranormal weighted conditional type operators

In this paper, some sub-classes of paranormal weighted conditional expectation type operators, such as *-paranormal, quasi-*-paranormal and (n; k)-quasi-*-paranormal weighted conditional expectation type opera- tors on $L^2(Σ)$ are investigated. Also, some applications about the spectrum, point spectrum, joint point spectrum, approximate point spectrum and joint approximate point spectrum of these classes are presented.

Hyperbolic valued random variables and conditional expectation

In this paper, we introduce the concept of hyperbolic valued random variables, their expectation and moments. We develop the hyperbolic analogue of Binomial and Poisson distributions. We study some of the properties of expectation on the basis of decomposition of a hyperbolic number into idempotent components. Finally we define conditional expectation of hyperbolic valued random variable and study some of its basic properties. Our random variable can take values which are zero divisors and this is the important part of this study.

On Fourier algebra of a hypergroup constructed from a conditional expectation on a locally compact group

We prove that the Fourier space of a hypergroup constructed from a conditional expectation on a locally compact group has a Banach algebra structure.

Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $C^*$-dynamical systems.

Conditionally expectations and martingales in noncommutative $L_p$-spaces associated with center-valued traces

In this paper we prove the existence of conditional expectations in the noncommutative $L_p(M,Φ)$ spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of the martingales in noncommutative $L_p(M,Φ)$ spaces.

具有 RNP 的 Banach 空间中的条件期望

Conditional Expectations in Banach spaces with RNP

本文研究具有 RNP 的 Banach 空间中多值函数的条件期望，利用提升证明其 Effros 可测条件期望存在并给出拟选择表示，且应用于 Pettis 可积多值函数鞅。

$H^p$ 空间 $(0<p<1)$ 上的压缩投影、条件期望与幂等系数乘子

Contractive projections, conditional expectations, and idempotent coefficient multipliers on $H^p$ spaces $(0<p<1)$

本文研究了 $0<p<1$ 时 Hardy 空间上的压缩投影、条件期望与幂等系数乘子，通过建立概率 $L^p$ 空间中压缩投影的延拓定理，完整刻画了 $d$ 维环面上此类乘子，补充了现有结果。

无附条件期望的 Z-吸收 C*-代数的自同态

Endomorphisms of Z-absorbing C*-algebras without conditional expectations

本文构造 Jiang-Su 代数 $\mathcal{Z}$ 上无附条件期望的自同态，回答 Kirchberg 的问题，并据此证明单位可分核 $\mathcal{Z}$-吸收 C*-代数在 $\mathcal{O}_2$ 中不可迁移。

On conditional expectations in L^p(mu;L^q(nu;X))

Let $(A,\mathscr{A},μ)$ and $(B,\mathscr{B},ν)$ be probability spaces, let $\mathscr{F}$ be a sub-$σ$-algebra of the product $σ$-algebra $\mathscr{A}\times\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\mathscr{F}$ defines a bounded linear operator from $L^p(μ;L^q(ν;X))$ onto $L^p_{\mathscr{F}}(μ;L^q(ν;X))$, the closed subspace in $L^p(μ;L^q(ν;X))$ of all functions having a strongly $\mathscr{F}$-measurable representative.

向量格无测度环境下条件遍历过程的Kakutani-Rokhlin分解

A Kakutani-Rokhlin decomposition for conditionally ergodic process in the measure-free setting of vector lattices

本文在向量格无测度框架下给出条件遍历系统的Kakutani-Rokhlin分解，并证明每个非周期保条件期望系统均可由周期系统逼近。

The linear conditional expectation in Hilbert space

The linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important rôle in approximate Bayesian inference, especially the Bayes linear approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert--Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the conditional mean embeddin…

使用紧致化算子的条件期望

Conditional expectation using compactification operators

本文提出一种基于核积分算子紧致化的算子理论方法，将去噪和流形学习等任务统一为再生核Hilbert空间中的线性逆问题以估计条件期望，并证明了数据驱动实现的收敛性及其实际应用效果。

关于有限指标条件期望

On conditional expectations of finite index

本文刻画了 C*-代数上有限指标条件期望的等价条件，证明其与完全正性条件等价，并推广了 Pimsner、Popa、Watatani 等人的结果，同时描述了 Jones 塔存在的情形及正规条件期望与 W*-型分解的交换性。

Smooth paths of conditional expectations

Let A be a von Neumann algebra with a finite trace $τ$, represented in $H=L^2(A,τ)$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval $I$. Let $E_t:A\to B_t$ be the unique $τ$-preserving conditional expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in A$ and $v \in H$, the map $$ I\ni t\mapsto E_t(a)v\in H $$ is continuously differentiable. This condition implies the existence of the derivative operator $$ dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v. $$ If this operator verifies the additional boundedness condition, $$ \int_J \|dE_t(a)\|_2^2 d t\le C_J…

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In clos…

Real positive maps and conditional expectations on operator algebras

Most of this article is an expanded version of our conference talk. It is essentially a survey, but some part, like most of the lengthy Section 5, is comprised of new results whose proofs are unpublished elsewhere. We begin by reviewing the theory of real positivity of operator algebras initiated by the author and Charles Read. Then we present several new general results (mostly joint work with Matthew Neal) about real positive maps. The key point is that real positivity is often the right replacement in a general algebra $A$ for positivity in C*-algebras. We then apply this to studying contra…

Strong completeness of a class of L2-type Riesz spaces

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T, in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional expectation operators, Positivity, 19 (2015), 647-657] as T-strong convergence and convergence in T- conditional probability, respectively. Generalized Lp spaces for the cases of p = 1; 2;1, were discussed in the setting of Riesz spaces as Lp(T) spaces in [C. C. A. Labuschagne, B. A. Watson, Discrete stochastic integration in Riesz spaces, Positivity, 14 (2010…

The Kac formula and Poincaré recurrence theorem in Riesz spaces

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in $L^1(Ω,{\mathcal A},μ)$, where $μ$ is a probability measure, new conditional versions…

Conditional probability in Renyi spaces

In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. This note introduces a general concept of conditional probability in Renyi spaces. Keywords: Measure theory; conditional probability space; conditional expectation