A non-commutative F&M Riesz Theorem

Subdiagonal algebras with the Beurling type invariant subspaces

Let $\mathfrak A$ be a maximal subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$. If every right invariant subspace of $\mathfrak A$ in the non-commutative Hardy space $H^2$ is of Beurling type, then we say $\mathfrak A$ to be type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the non-commutative $H^1$ space. We show that the right analytic Toeplitz algebra on the non-commutative Hardy space $H^p$ associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.

Analytic functionals for the non-commutative disc algebra

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals must be given as integration against an absolutely continuous measure on the circle. We develop generalizations of this result to the multivariate non-commutative setting, upon reinterpreting the classical result. In one direction, we show that the GNS representation naturally associated to an analytic functional on the Cuntz algebra cannot have any singula…

Lebesgue decomposition of non-commutative measures

The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions are obtained as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be viewed as the unital norm-closed operator algebra of the shift operator on the Hardy Space, $H^2$ of the disk. Replacing square-summable Taylor series indexed by the non-negative integers, i.e. $H^2$ of the disk, with square-summable power series indexed b…

一类 Hilbert C*-模上的 Riesz-Fredholm 型定理

A Riesz-Fredholm type theorem on certain Hilbert C*-modules

本文研究满足性质 $\mathbb{[H]}$ 的 Hilbert C*-模上紧模算子 $C$ 与 $I-C$ 的 Riesz-Fredholm 理论，证明存在唯一幂次使 $L^r$ 为 EP 算子，并给出核与值域的直和分解及方程可解性框架。

Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely func…

Notes on ergodic theorems in non-commutative symmetric spaces

In this paper we establish individual ergodic theorem for positive kernels (or so called Danford Shwartz (DS+) operators acting on non commutative symmetric spaces.

Interpolation theorems for operators in non-commutative vector-valued symmetric spaces

We prove the version of interpolation theorem for non-commutative vector-valued fully symmetric spaces associated with fully symmetric Banach function spaces and a von Neumann algebra equipped with a faithful semifinite normal trace.

A note on the Voiculescu's theorem for commutative C$^*$-algebras in semifinite von Neumann algebras

In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.

A Non-commutative Wiener-Wintner Theorem

For a von Neumann algebra $\cal M$ with a faithful normal tracial state $τ$ and a positive ergodic homomorphism $α:\mathcal L^1(\mathcal M,τ)\to \mathcal L^1(\mathcal M,τ)$ such that $α$ does not increase the norm in $\mathcal M$ and $τ\circ α=τ$, we establish a non-commutative counterpart of the classical Wiener-Wintner Theorem.

Non-commutative ergodic Theorems for actions of the hyperbolic groups

The goal of this notice is to establish Not-commutative Point- wise Ergodic Theorems for actions of the Hyperbolic Groups. Similar non-commutative results were done by Bufetov, Khristoforov and Kli- menko, and later by Pollicott and Sharp. We were interested to expand short notice in Policott and Sharp's paper about non-commutative er- godic theorems.

A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes

In this paper, we consider the real rank zero $\mathrm{C}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.

A Proof of a Non-Commutative Central Limit Theorem by the Lindeberg Method

A Central Limit Theorem for non-commutative random variables is proved using the Lindeberg method. The theorem is a generalization of the Central Limit Theorem for free random variables proved by Voiculescu. The Central Limit Theorem in this paper relies on an assumption which is weaker than freeness.

Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B

The present material addresses several problems left open in the Trans. AMS paper " Non-crossing cumulants of type B" of P. Biane, F. Goodman and A. Nica. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the S-transform as well as proving analogue results to Central Limit Theorem and Poisson Limit Theorem.

非交换空间上的Hormander-Mikhlin型定理

Hormander-Mikhlin type theorem on non-commutative spaces

本文在非交换空间上引入傅里叶型形式化方法，推导出Hormander-Mikhlin Lp乘法定理的两个版本，并应用于非交换演算方程。

Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.

Rényi divergences as weighted non-commutative vector valued $L_p$-spaces

We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter $α= \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $α$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $α\to \{\frac{1}{2},1,\infty\…