Dilations and rigid factorisations on noncommutative L^p-spaces

Dilation theory yesterday and today

Paul Halmos' work in dilation theory began with a question and its answer: Which operators on a Hilbert space can be extended to normal operators on a larger Hilbert space? The answer is interesting and subtle. The idea of representing operator-theoretic structures in terms of conceptually simpler structures acting on larger Hilbert spaces has become a central one in the development of operator theory and, more generally, noncommutative analysis. The work continues today. In this article we summarize some of these diverse results and their history.

Noncommutative Riesz transforms -- a probabilistic approach

For $2\le p<\infty$ we show the lower estimates \[ \|A^{\frac 12}x\|_p \kl c(p)\max\{\pl \|Γ(x,x)^{1/2}\|_p,\pl \|Γ(x^*,x^*)^{1/2}\|_p\} \] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps on a von Neumann algebra with negative generator $T_t=e^{-tA}$, and gradient form \[ 2Γ(x,y)\lel Ax^*y+x^*Ay-A(x^*y)\pl .\] As additional hypothesis we assume that $Γ^2\gl 0$ and the existence of a Markov dilation for $(T_t)$. We give applications to quantum metric spaces and show the equivalence of semigroup Hardy norms and martingale Hardy norms derived from the Markov …

Surjective separating maps on noncommutative $L^p$-spaces

Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and separating (i.e., for any $x,y\in L^p({\mathcal M})$ such that $x^*y=xy^*=0$, we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions ${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$, ${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, such that $T=T_1+T_2…

Maximal ergodic inequalities for some positive operators on noncommutative $L_p$-spaces

In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong…

On ergodic theorems for free group actions on noncommutative spaces

We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres $s_{2n}$ of even radius. Here we study state preserving actions of free groups on a von Neumann algebra $A$ and the behaviour of $(s_{2n}(x))$ for $x$ in noncommutative spaces $L^p(A)$. For the Cesàro means $\frac{1}{n}\sum_{k=0}^{n-1} s_k$ and $p = +\infty$, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren'' theorem. To this end, we int…

Choquet型关系与Arveson超刚性猜想的态空间层面修正

Choquet-Type Relations and a State Space Level Amendment of Arveson's Hyperrigidity Conjecture

本文针对Arveson超刚性猜想，建立态空间方法，引入强扩张与积分细分关系，证明其性质并给出Clouâtre-Thompson定理的另一证明，还明确了两类关系在交换情形下的一致性。

Noncommutative de Leeuw theorems

Let H be a subgroup of some locally compact group G. Assume H is approximable by discrete subgroups and G admits neighborhood bases which are "almost-invariant" under conjugation by finite subsets of H. Let $m: G \to \mathbb{C}$ be a bounded continuous symbol giving rise to an Lp-bounded Fourier multiplier (not necessarily cb-bounded) on the group von Neumann algebra of G for some $1 \le p \le \infty$. Then, $m_{\mid_H}$ yields an Lp-bounded Fourier multiplier on the group von Neumann algebra of H provided the modular function $Δ_H$ coincides with $Δ_G$ over H. This is a noncommutative form of…

Factorization and dilation problems for completely positive maps on von Neumann algebras

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M_n(C) for all n\geq 3, as well as an example of a one-parameter semigroup (T(t))_{t\geq 0} of Markov maps on M_4(C) such that T(t) fails t…