Andreas Debrouwere, Lenny Neyt, Jasson Vindas
We provide explicit commutative sequence space representations for classical function and distribution spaces on the real half-line. This is done by evaluating at the Fourier transforms of the elements of an orthonormal wavelet basis.
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Let $\cal M$ be a semi-finite von Neumann algebra equipped with a distinguished faithful, normal, semi-finite trace $τ$. We introduce the notion of equi-integrability in non-commutative spaces and show that if a rearrangement invariant quasi-Banach function space $E$ on the positive semi-axis is $α$-convex with constant 1 and satisfies a non-trivial lower $q$-estimate with constant 1, then the corresponding non-commutative space of measurable operators $E({\cal M}, τ)$ has the following property: every bounded sequence in $E({\cal M}, τ)$ has a subsequence that splits into a $E$-equi-integrabl…
We adapt the "royal road" method used to simplify automatic analyticity theorems in noncommutative function theory to several complex variables. We show that certain families of functions must be real analytic if they have certain nice properties on one dimensional slices. Let $E \subset \mathbb{R}^d$ be open. A function $f:E \to \mathbb{R}$ is matrix monotone lite if $f(φ_1(t), \ldots, φ_d(t))$ is a matrix monotone function of $t$ whenever $t \in (0,1)$, the $φ_i$ are automorphisms of the upper half plane, and the tuple $(φ_1(t), \ldots, φ_d(t))$ maps $(0,1)$ into $E$. We use the "royal road"…
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.
The paper deal with the noncommutative Fréchet ${}^*$-algebra $\mathcal{L}(s',s)$ of the so-called smooth operators, i.e. linear and continuous operators acting from the space $s'$ of slowly increasing sequences to the Fréchet space $s$ of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative ${}^*$-subalgebras of this algebra and we show that every closed subspace of $s$ with basis is isomorphic (as a Fréchet space) to some closed commuta…
Endomorphisms of the measure algebra of commutative hypergroups are investigated. We focus on derivations and higher order derivations which are closely related to moment function sequences of higher rank. We describe the exact connection between those higher order derivations which are endomorphisms of the measure algebra if it is considered as a module over the ring of continuous functions.
In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(α_{n}))$ is downward half Cauchy whenever $(α_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(α_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $α_{m}-α_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down…
In this paper we establish a multivariable non-commutative generalization of Löwner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here characterizes several variable operator monotone functions, not assumed to be free analytic or even continuous, as free functions that admit free analytic continuation mapping the upper operator poly-halfspace into the upper operator halfspace over an arbitrary Hilbert space. We establ…
We investigate various notions of peaking behaviour for states on a $\mathrm{C}^*$-algebra, where the peaking occurs within an operator system. We pay particularly close attention to the existence of sequences of elements forming an approximation of the characteristic function of a point in the state space. We exploit such characteristic sequences to localize the $\mathrm{C}^*$-algebra at a given state, and use this localization procedure to verify a variation of Arveson's hyperrigidity conjecture for arbitrary operator systems.
Non-commutative Law of iterated logarithm
本文证明了鞅和独立随机变量序列的经典迭代对数律的最优非交换类比，通过改进的指数不等式导出非交换Stout型LIL，并由此得到非交换Hartman-Wintner型LIL。
Boehmians are quotients of sequences which are constructed by using a set of axioms. In particular, one of these axioms states that the set $S$ from which the {\it denominator} sequences are formed should be a commutative semigroup with respect to a binary operation. In this paper, we introduce a generalization of abstract Boehmian space, called $G$-Boehmian space, in which $S$ is not necessarily a commutative semigroup. Next, we provide an example of a $G$-Boehmian space and we discuss an extension of the Hartley transform on it. Finally, we compare the Hartley transform in this paper with th…
In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L_p-space, then it is either a script L_p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non--Hilbertian complemented operator subspaces of non commutative L_p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every $2<p< \infty$ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space the…
We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are Hölder on semifinite von Neumann algebras. The main tool is a new form of Hölder inequality for non commutative Lp spaces with weights.
Two different aspects of parabolic iteration in the complex upper half-plane are considered here. First, from a noncommutative probability perspective, a Berry-Esseen type estimate for the convergence speed of the monotone central limit theorem is proved. Secondly, if the underlying measure in this central limit process is singular to the Lebesgue measure on the real line, then the iteration is shown to be an infinite-measure preserving dynamical system that has a regularly varying return sequence of index 1/2.
We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator s…
The structure of non-commutative multiple correlation sequences and applications
该研究刻画了非交换二重多重相关序列的结构，分析其正性性质，推广Chu不等式得到量化形式，并应用于线性形式乘积的多重 recurrence 定理及二次方程的分划正则性结果，推广了Frantzikinakis与Host的定理。
It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plan…
We denote by $\Hp$ the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane $σ>1/2$ is an interpolating sequence for $\Hp$ if and only if it is an interpolating sequence for the Hardy space $H^2$ of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane $σ>1/2$.
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