Non-commutative stochastic distributions and applications to linear systems theory

Countable linear combinations of characters on commutative Banach algebras

An elegant but elementary result of Wolff from 1921, when interpreted in terms of Banach algebras, shows that it is possible to find a sequence of distinct characters $ϕ_n$ on the disc algebra and an $\ell_1$ sequence of complex numbers $λ_n$, not all zero, such that $\sum_{n=1}^\infty λ_n ϕ_n =0.$ We observe that, even for general commutative, unital Banach algebras, this is not possible if the closure of the countable set of characters has no perfect subsets.

Linear orthogonality preservers between function spaces associated with commutative JB$^*$-triples

It is known, by Gelfand theory, that every commutative JB$^*$-triple admits a representation as a space of continuous functions of the form $$C_0^{\mathbb{T}}(L) = \{ a\in C_0(L) : a(λt ) = λa(t), \ \forall λ\in \mathbb{T}, t\in L\},$$ where $L$ is a principal $\mathbb{T}$-bundle and $\mathbb{T}$ denotes the unit circle in $\mathbb{C}.$ We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB$^*$-triples. We show that each linear orthogonality preserver $T: C_{0}^{\mathbb{T}} (L_1)\to C_{0}^{\mathbb{T}} (L_2)$ decomposes in three …

可换极大不等式与保持态作用的可算群

Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups

本文研究了可算群在非可换$ L^1 $空间上保持态作用的极大不等式及遍历定理，并通过Neveu分解得到随机遍历定理。

Martin boundaries of representations of the Cuntz algebra

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of non-commutative operators; non-commutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these non-commutative algebras are the Cuntz algebra…

On Galois Correspondence and Non-Commutative Martingales

The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce non-abelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra $\Ma$ and a compact group $\Gr$ ac…

Commutative subalgebras of the algebra of smooth operators

We consider the Fréchet ${}^*$-algebra $L(s',s)$ of the so-called smooth operators, i.e. continuous linear operators from the dual $s'$ of the space $s$ of rapidly decreasing sequences into $s$. This algebra is a non-commutative analogue of the algebra $s$. We characterize all closed commutative ${}^*$-subalgebras of $L(s',s)$ which are at the same time isomorphic to closed ${}^*$-subalgebras of $s$ and we provide an example of a closed commutative ${}^*$-subalgebra of $L(s',s)$ which cannot be embedded into $s$.

从经典到量子：非交换概率中的多态性

From Classical to Quantum: Polymorphisms in Non-Commutative Probability

本文建立经典（标量值）与非交换（算子值）多态性之间的平行关系，重点应用于随机过程的条件分布，并借助正算子值测度（POVM）与量子态，阐明量子信息中的纠缠，揭示两种框架的相互作用。

Orthogonally additive polynomials on non-commutative $L^p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $τ$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(\mathcal{M},τ)$, with $0<p<\infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(\mathcal{M},τ)$ can be represented in the form $P(x)=Φ(x^m)$ $(x\in L^p(\mathcal{M},τ))$ for some continuous linear map $Φ\colon L^{p/m}(\mathcal{M},τ)\to X$.