We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.
In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued R-transforms and operator-valued moment series, under the compatibility. Also, we can see that there is a big gap between freeness and amalgamated freeness.
We show the dual spaces of conditional Hardy space and symmetric Hardy space of noncommutative martingales. We derive relationship between the symmetric Hardy space of noncommutative martingales and its conditioned version.
We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz-Hardy spaces and characterize their dual spaces.
These lecture notes were written during a mini-course on noncommutative Lp-spaces at the Basque Center of Applied Mathematics. It starts presenting the theory of weights and traces in von Neumann algebra, followed by the theory of noncommutative measure. After that, it presents a construction of these spaces in all detail and give some important results e.g. the noncommutative Radon-Nikodym theorem.
We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper.
We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
We prove that atomic decomposition for the Hardy spaces h_1 and H_1 is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales h_p and bmo form interpolation scales with respect to both complex and real interpolations.
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge bilaterally almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. In particular, these averages converge bilaterally almost uniformly in all noncommutative symmetric spaces with…
We prove that for any (trace-preserving) conditional expectation $\mathcal E$ on a noncommutative $L_p$ with $p>2$, $Id-\mathcal E$ is a contraction on the positive cone $L_p^+$.
Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of $n$-fold tensor products of Drury-Arveson spaces.
We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a $2$-co-lacunary subsequence. This extends the classical characterization, due to Räbiger.
This paper studies the $N$-tuple noncommutative Orlicz spaces $\bigoplus\limits_{j=1}^{n}L_{p,λ}^{(Φ_{j})}(\widetilde{\mathcal{M}},τ)$, where $L^{(Φ_{j})}(\widetilde{\mathcal{M}},τ)$ is noncommutative Orlicz spaces and $\widetilde{\mathcal{M}}$ is the $τ$-measurable operators. Based on the maximum principle, we give the Riesz-Thorin interpolation theorem on $\bigoplus\limits_{j=1}^{n}L_{p,λ}^{(Φ_{j})}(\widetilde{\mathcal{M}},τ)$ . As applications, the Clarkson inequality and some geometrical properties such as uniform convexity and unform smooth of noncommutative Orlicz space.
Let $\mathcal{M}$ be a $σ$-finite von Neumann algebra equipped with a normal faithful state $φ$, and let $Φ$ be a growth function. We consider Haagerup noncommutative Orlicz spaces $L^Φ(\M,φ)$ associated with $\M$ and $φ$, which are analogues of Haagerup $L^p$-spaces. We show that $L^Φ(\M,φ)$ is independent of $φ$ up to isometric isomorphism. We prove the Haagerup's reduction theorem and the duality theorem for this spaces. As application of these results, we extend some noncommutative martingale inequalities in the tracial case to the Haagerup noncommutative Orlicz space case.
Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the anal…
We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In clos…
In this paper, we will construct the graph free product of noncommutative probability space. This is the attempt to explain and observe the combinatorial-object-depending probabilistic structure.
We show that noncommutative $L_p$-spaces satisfy the axioms of the (nonunital) operator system with a dominating constant $2^{1 \over p}$. Therefore, noncommutative $L_p$-spaces can be embedded into $B(H)$ $2^{1 \over p}$-completely isomorphically and complete order isomorphically.
We extend the noncommutative L1-maximal ergodic inequality for semifinite von Neumann algebras established by Yeadon in 1977 to the framework of noncommutative L1-spaces associated with sigma-finite von Neumann algebras. Since the semifnite case of this result is one of the two essential parts in the proof of noncommutative maximal ergodic inequality for tracial Lp-spaces (1<p<infinity) by Junge-Xu in 2007, we hope our result will be helpful to establish a complete noncommutative maximal ergodic inequality for non-tracial Lp-spaces in the future.
For a noncommutative Orlicz space associated with a semifnite von Neumann algebra, a faithful normal semifnite trace and an Orlicz function satisfying $(δ_2,Δ_2)-$condition, an individual ergodic theorem is proved.
We prove a noncompact version of Haagerup and Rørdam's result about continuous paths of the rotation $C^*$-algebras. It gives a continuous Moyal deformation of Euclidean plane. Moveover, the construction is generalized to noncommutative Euclidean spaces of dimension $d\ge 2$. As a corollary, we obtain Lip$^{\frac12}$ continuous maps for the generators of noncommutative $d$-tori.
Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite commutative case which corresponds to a metric on a finite set, and also give some examples of computations in both commutative and noncommutative cases.
