Universal Non-Completely-Continuous Operators

Boundedness of completely additive measures with application to 2-local triple derivations

We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.

Scaled-free objects II

This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps completely bounded on a generation set to a completely bounded linear map of the matrix-normed space. Moreover, the universal matrix-normed algebra is used to prove the existence of a free product for matrix-normed algebras using algebraic methods.

Universally complete spaces of continuous functions

We characterise Tychonoff spaces X so that C(X) is universally σ-complete and universally complete, respectively.

A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics

In this note we prove a Birkhoff type transitivity theorem for continuous maps acting on non-separable completely metrizable spaces and we give some applications for dynamics of bounded linear operators acting on complex Fréchet spaces. Among them we show that any positive power and any unimodular multiple of a topologically transitive linear operator is topologically transitive, generalizing similar results of S.I. Ansari and F. León-Saavedra V. Müller for hypercyclic operators.

关于具有非连续权重采样的随机基展开中通用逼近的高概率性

On high probability of universal approximation in random basis expansions with non-continuous weight sampling

研究随机基展开的通用逼近性质，将其从连续权重采样和实值激活函数扩展到非连续权重分布及复值激活函数，证明此类随机基具有任意高概率的通用逼近性质。

Universal Continuous Calculus for Su*-Algebras

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e. "strictly" positive elements are invertible, and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for C*-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any rep…

Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"ıssé limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the concommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. We also prove that $\mathbb{NG}$ is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal TRO to the triple envelope is an isomorphism. We deduce from this fact that $\mathbb{NG}$ does not e…

Towards a Koopman theory for dynamical systems on completely regular spaces

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article we look at dynamics given by continuous semiflows on completely regular spaces which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato's equality). Finally, we demonst…

通过线性测量和标量非线性来逼近紧集上连续函数的通用性

Universal Approximation of Continuous Functionals on Compact Subsets via Linear Measurements and Scalar Nonlinearities

本文提出通过线性测量和标量非线性来逼近紧集上连续函数的通用性，为算子学习和成像提供了理论支持。

Representations of Group Algebras in Spaces of Completely Bounded Maps

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi…

完全可度量化群的自同态的混沌与Fréchet空间上的线性算子

Chaos for endomorphisms of completely metrizable groups and linear operators on Fréchet spaces

本文利用拓扑动力学技巧统一处理完全可度量化群连续自同态的三种混沌，通过半非正则点存在性进行刻画，并应用于Fréchet空间线性算子改进现有结果。

On the relation between completely bounded and $(1,cb)$-summing maps with applications to quantum XOR games

In this work we show that, given a linear map from a general operator space into the dual of a C$^*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(1,cb)$-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.

Completely positive multipliers of quantum groups

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group $\G$ (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of $\G$. It follows that there is an order bijection between the completely positive multipliers of $L^1(\G)$ and the positive functionals on the universal quantum group $C_0^u(\G)$. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing e…

Lifting fixed points of completely positive semigroups

Let $\{ϕ_s\}_{s\in S}$ be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra $N$. Assume there exists a semigroup $\{α_s\}_{s\in S}$ of weak*-continuous *-endomorphisms of some larger von Neumann algebra $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that $α_s(1-p)\le 1-p$ for every $s\in S$ and $ϕ_s(y)=pα_s(y)p$ for all $y\in N$. If $\inf_{s\in S}α_s(1-p)=0$ then we show that the map $E:M\to N$ defined by $E(x)=pxp$ for $x\in M$ induces a complete isometry between the fixed point spaces of $\{α_s\}_{s\in S}$ …

关于无点函数空间的泛完备化

On the universal completions of pointfree function spaces

本文通过两种方法构造完全正则标架上连续实函数 Riesz 空间的泛完备化，表征了泛完备标架，并提出了 Maeda-Ogasawara-Vulikh 表示定理的无点版本。