To any action of a locally compact group $G$ on a pair $(A,B)$ of von Neumann algebras is canonically associated a pair $(π\_A^α, π\_B^α)$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $π\_A^α(μ)$ and $π\_B^α(μ)$ for bounded measures $μ$ on $G$. We have a twofold aim. First to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about $(π\_A^α, π\_B^α)$. Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.
We introduce a class of unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norms with repect to $τ$ on a von Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracial weight $τ$. We prove a Beurling-Chen-Hadwin-Shen theorem for $H^\infty$-invariant spaces of $L^α(\mathcal{M},τ)$, where $α$ is a unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norm with respect to $τ$, and $H^\infty$ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the $H^\infty$-invariant subspaces of a noncommutati…
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 investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class Sq^n in Sp(lq^m) with optimal order m = n^2. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0,1] is min(p,p') (max(p,p')). Similar techniques are used to show that the operator space notions of B-convexity and K-convexity are equivalent.
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…
In the first part of this work we show how certain techniques from quantum information theory can be used in order to obtain very sharp embeddings between noncommutative $L_p$-spaces. Then, we use these estimates to study the classical capacity with restricted assisted entanglement of the quantum erasure channel and the quantum depolarizing channel. In particular, we exactly compute the capacity of the first one and we show that certain nonmultiplicative results hold for the second one.
We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p'…
We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<\infty$.
We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite dimensional matrices yield classical plurisubharmonic functions as the degree tends to infinity, and show that they are comparable to certain noncommutative versions of the Siciak extremal function. We prove estimates for Siciak functions associated to free products of distributions, and use the classical theory of plurisubharmonic functions in order to propose a n…
A mathematical notion of interaction is introduced for noncommutative dynamical systems, i.e., for one parameter groups of *-automorphisms of $\Cal B(H)$ endowed with a certain causal structure. With any interaction there is a well-defined "state of the past" and a well-defined "state of the future". We describe the construction of many interactions involving cocycle perturbations of the CAR/CCR flows and show that they are nontrivial. The proof of nontriviality is based on a new inequality, relating the eigenvalue lists of the "past" and "future" states to the norm of a linear functional on a…
In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $% α$ on a tracial von Neumann algebra $\left( \mathcal{M},τ\right) $ where $α$ is $\left\Vert \cdot \right\Vert _{1}$-dominating with respect to $τ$. In the paper, we first define a class of norms $% N_{Δ}\left( \mathcal{M},τ\right) $ on $\mathcal{M}$, called determinant, normalized, unitarily invariant continuous norms on $\mathcal{M}$. If $α\in N_{Δ}\left( \mathcal{M},τ\right) $, then there exists a faithful normal tracial state $ρ$ on $\mathcal{M}…
We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…
We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen's discovery of the norm adjusting power of quasi-central approximate units. A projective C*-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show a noncommutative absolute retract results from the intersection of the approximate locus of a homogene…
For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_γ)$ where $B_γ$ is a Stolz domain. Moreover, we introduce the `column square functions' $\norm{x}_{T,c,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1}|T^{k-1}(I-T)^α(x)|^2\Big)^{1/2}}_{L^p(M)}$ and the `row square functions' $\norm{x}_{T,r,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1} |\Big(T^{k-1}(I-T)^α(x)\Big)^*|^2\Big)^{1/2}}_{L^p(M)}$ for any $α>0$ and any $x\in L^p(M)$. Then, we provide an example of Ritt operator which admits a completely bounde…
Let $\mathfrak A$ be a type 1 subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $Φ$. We consider a Riesz type factorization theorem in noncommutative $H^p$ spaces associated with $\mathfrak A$. It is shown that if $1\leq r,p,q<\infty$ such that $\frac1r=\frac1p+\frac1q$, then for any $h\in H^r$, there exist $h_p\in H^p$ and $h_q\in H^q$ such that $h=h_ph_q$. Beurling type invariant subspace theorem for noncommutative $L^p(1< p<\infty)$ space is obtained. Furthermore, we show that a $σ$-weakly closed subalgebra contain…
We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a v…
In this paper we prove the existence of conditional expectations in the noncommutative $L_p(M,Φ)$ spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of the martingales in noncommutative $L_p(M,Φ)$ spaces.
Let $\mathcal{M}$ be a ($σ$-finite) von Neumann algebra associated with a normal faithful state $ϕ.$ We prove a complex interpolation result for a couple of two (quasi) Haagerup noncommutative $L_p$-spaces $L_{p_0} (\mathcal{M}, ϕ)$ and $L_{p_1} (\mathcal{M}, ϕ), 0< p_0 < p_1\leq \infty,$ which has further applications to the sandwiched $p$-Rényi divergence.
In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp-spaces, 1 < p < infinity, was established and, among other things, corresponding maximal ergodic inequalities and individual ergodic theorems were derived. In this article, we derive maximal ergodic inequalities in noncommutative Lp-spaces directly from [25] and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in partic…
We continue our study of outer elements of the noncommutative H^p spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in H^p actually satisfy the stronger condition that there exist a_n in A with h a_n in Ball(A) and h a_n \to 1 in p-norm.
We introduce notions of dimension and dynamical entropy for unital C*-algebras ``metrized'' by means of c-Lip-norms, which are complex-scalar versions of the Lip-norms constitutive of Rieffel's compact quantum metric spaces. Our examples involve UHF algebras and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical c-Lip-norm coincides with the topological entropy of its commutative analogue.
In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von Neumann algebra $\M$ with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight $\{a(\kb)\}_{\kb\in\bn^d}$ and every $x\in L_{p}(\M)$, ($p>1$) the averages A_{\Nb}(x)=\frac{1}{|\Nb|}\sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x). converge bilaterally almost uniformly in $L_{p}(\M)$.
In 2008, Blecher and Labuschagne extended Beurling's classical theorem to $H^\infty$-invariant subspaces of $L^p(\mathcal{M},τ)$ for a finite von Neumann algebra $\mathcal{M}$ with a finite, faithful, normal tracial state $τ$ when $1\le p\le \infty$. In this paper, using Arveson's non-commutative Hardy space $H^\infty$ in relation to a von Neumann algebra $\mathcal{M}$ with a semifinite, faithful, normal tracial weight $τ$, we prove a Beurling-Blecher-Labuschagne theorem for $H^\infty$-invariant spaces of $L^p(\mathcal{M},τ)$ when $0<p\leq\infty$. The proof of the main result relies on proofs …
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 almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x)\to 0$ as $t\to\infty$ for every $x\in E$, where $μ_t(x)$ is the non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmet…
In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace $\t$, and $\{α_ t\} $ a strongly continuous extension to $L^p(M,\t)$ of a semigroup of absolute contractions on $L^1 (M,τ)$. By means of a non-commutative Banach Principle we prove for a Besicovitch function b and $x\in L^p(M,\t)$, the averages \frac{1}{T}\int_0^Tb(t)\a_t(x)dt converge bilateral almost uniform in $L^p(M,\t)$ as $T\to 0$.
Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L1, where some recent results by the first-named author can be used. We find applications for noncommutative Khincthine/Rosenthal type inequalities and for noncommutative Lp embedding theory.
For a closed cocompact subgroup $Γ$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $ρ:\hat{K}\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(\hat{G}/Γ, ρ)$ of the homogeneous space $G/Γ$, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/Γ$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(\hat{G}/Γ, ρ)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metr…
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 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 $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More prec…
Let $Γ$ be a discrete subgroup of a unimodular locally compact group $G$. In Math. Ann. 388, 4251-4305 (2024), it was shown that the $L_p$ norm of a Fourier multiplier $m$ on $Γ$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m$. In the context where $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to: 1) The adjoint representation of the semisimple quotient $\mathfrak{s} = \mathfrak{g}/\mathfrak{r}$ of $\mathfrak{g}$ by the radical $…
Let M be a finite von Neumann algebra with a faithful trace $τ$. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of $τ$, $||x||_p=τ(|x|^p)^{1/p}$, $p\ge 1$. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d'_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined i…
This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},τ,G,σ)$, where $(\mathcal{M},τ)$ is a von Neumann algebra with a normal faithful finite trace and $(G,σ)$ is a connected amenable locally compact group with a well defined representation on $\mathcal{M}$, we try to find the largest noncommutative function spaces constructed from $\mathcal{M}$ on which the individual ergodic theorems hold. By using the Emerson-Greenleaf's structure theorem, we transfer the key question to proving the ergodic t…
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $ϕ$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $ϕ$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $ϕ$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: \[\|x\|_p^2\ge\bigl \|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad x\in L_p(\mathcal{M}),1<p\le2,\] which extends the celebrated Ball-Ca…
