Symmetric Spaces Of Measurable Operators Research Articles

Higher derivatives of operator functions in ideals of von Neumann algebras

Let M be a von Neumann algebra and a be a self-adjoint operator affiliated with M. We define the notion of an “integral symmetrically normed ideal” of M and introduce a space OC[k](R)⊆Ck(R) of functions R→C such that the following holds: for any integral symmetrically normed ideal I of M and any f∈OC[k](R), the operator function Isa∋b↦f(a+b)−f(a)∈I is k-times continuously Fréchet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if f∈B˙11,∞(R)∩B˙1k,∞(R) and f′ is bounded, then f∈OC[k](R). Finally, we prove that all of the following ideals are integral symmetrically normed: M itself, separable symmetrically normed ideals, Schatten p-ideals, the ideal of compact operators, and – when M is semifinite – ideals induced by fully symmetric spaces of measurable operators.

Solving 3D fractional Schrödinger systems on the basis of Phragmén–Lindelöf methods

Our aim in this article is to study the existence and uniqueness of time periodic solutions to 3D fractional Schrödinger systems. The techniques in this work are based on Phragmén–Lindelöf method (Liess, Necessary conditions in Phragmén–Lindelöf type estimates and decomposition for non-divergence type elliptic equations and mixed boundary conditions. Math Nachr 290(8–9):1328–1346, 2017) and Matsaev’s theory (Sukochev, Ferleger, Matsaev’s theorem for symmetric spaces of measurable operators. Math Notes 56(5–6):1185–1189, 1995). Particularly, the fractional maximal effects and the potential nonlinear equation are taken into account which play a key role in the Green–Schrödinger growth properties, especially in the selection of the Schrödinger norm.

Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces

We provide asymmetric forms of Doob maximal inequality in the contexts of moments associated with convex functions and symmetric spaces of measurable operators. More precisely, let (M,τ) be a noncommutative measurable space equipped with a filtration (Mn)n≥1. Let (En)n≥1 be the conditional expectations corresponding to (Mn)n≥1. A sample result for the case of convex functions states that if Φ is an Orlicz function that is p-convex and q-concave for 1≤p≤q<2 and 1−p/2<θ<1, then for every x in the column Hardy space HΦc(M), there exist positive operators a,b, and a sequence of contractions (un)n≥1 in M such that for every n≥1,En(x)=aunb andτ(Φ(a1/(1−θ)))+τ(Φ(b1/θ))≲τ(Φ(Sc(x))). For the case of symmetric space, as an illustration, we obtain that if E is a separable symmetric Banach function space which is an interpolation of (Lp,Lq) for 1<p≤q<2, then for every z∈E(M), there exist positive operators α,β, two sequences of contractions (υn)n≥1 and (ηn)n≥1 in M such that for every n≥1,En(z)=αυn+ηnβ and‖α‖E(M)+‖β‖E(M)≲‖z‖E(M).Our approach is based on new algebraic atomic decompositions and improved Davis type decompositions which we develop for both the case of Hardy spaces associated with separable noncommutative symmetric spaces that are interpolation of the couple (Lp,Lq) for 1<p≤q<2 as well as the case of Hardy spaces associated with convex functions that are p-convex and q-concave for 1≤p≤q<2. Some other inequalities are also provided which are of independent interest.

Local Ergodic Theorems in Symmetric Spaces of Measurable Operators

Local mean and individual (with respect to almost uniform convergence in Egorov’s sense) ergodic theorems are established for actions of the semigroup $${\mathbb {R}}_+^d$$ in symmetric spaces of measurable operators associated with a semifinite von Neumann algebra.

Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Polya, Kothe duality, the spaces $L_p(\mathcal{M},\tau)$, $1\le p<\infty$, the identification between $C_E$ and $G(B(H), \rm{tr})$ for some symmetric function space $G$, the commutative case when $E$ is identified with $E(\mathcal{N}, \tau)$ for $\mathcal{N}$ isometric to $L_\infty$ with the standard integral trace, trace preserving $*$-isomorphisms between $E$ and a $*$-subalgebra of $E(\mathcal{M},\tau)$, and a general method of removing the assumption of non-atomicity of $\mathcal{M}$. The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, $k$-extreme points and $k$-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodým property and stability in the sense of Krivine-Maurey. We also state some open problems.

Erratum to: Banach envelopes in symmetric spaces of measurable operators

Erratum to: Banach envelopes in symmetric spaces of measurable operators

Banach envelopes in symmetric spaces of measurable operators

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (HC) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes \(\widehat{E}\) are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let \(\mathcal {M}\) be a non-atomic, semifinite von Neumann algebra with a faithful, normal, \(\sigma \)-finite trace \(\tau \) and E be as symmetric function space on \([0,\tau (\mathbf 1 ))\) or symmetric sequence space. We compute Banach envelope norms on \(E(\mathcal {M},\tau )\) and \(C_E\) for any quasi-normed symmetric space E. Then we show under assumption that \(E\in (HC)\) that the Banach envelope \(\widehat{E(\mathcal {M},\tau )}\) of \(E(\mathcal {M},\tau )\) is equal to \(\widehat{E}\left( \mathcal {M},\tau \right) \) isometrically. We also prove the analogous result for unitary matrix spaces \(C_E\).

Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators

We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of τ \tau -measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties of operators dominated in the sense of complete positivity by compact or by Dunford-Pettis operators.

K-Extreme Points in Symmetric Spaces of Measurable Operators

Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$ and $E$ be a strongly symmetric Banach function space on $[0,\tau(1))$. We show that an operator $x$ in the unit sphere of $E\left(\mathcal{M},\tau\right)$ is $k$-extreme, $k\in\mathbb N$, whenever its singular value function $\mu(x)$ is $k$-extreme and one of the following conditions hold (i) $\mu(\infty,x)=\lim_{t\to\infty}\mu(t,x)=0$ or (ii) $n(x)\mathcal{M} n(x^*)=0$ and $|x|\geq \mu(\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. The converse is true whenever $\mathcal{M}$ is non-atomic. The global $k$-rotundity property follows, that is if $\mathcal{M}$ is non-atomic then $E$ is $k$-rotund if and only if $E\left(\mathcal{M},\tau\right)$ is $k$-rotund. As a consequence of the noncommutive results we obtain that $f$ is a $k$-extreme point of the unit ball of the strongly symmetric function space $E$ if and only if its decreasing rearrangement $\mu(f)$ is $k$-extreme and $|f|\geq \mu(\infty,f)$. We conclude with the corollary on orbits $\Omega(g)$ and $\Omega'(g)$. We get that $f$ is a $k$-extreme point of the orbit $\Omega(g)$, $g\in L_1+L_{\infty}$, or $\Omega'(g)$, $g\in L_1[0,\alpha)$, $\alpha<\infty$, if and only if $\mu(f)=\mu(g)$ and $|f|\geq \mu(\infty,f)$. From this we obtain a characterization of $k$-extreme points in Marcinkiewicz spaces.

Complex uniform rotundity in symmetric spaces of measurable operators

Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a symmetric Banach function space on [0,τ(1)). We show that E is complex uniformly rotund if and only if E(M,τ)+ is complex uniformly rotund. Moreover, under the assumption that E is p-convex for some p>1, complex uniform rotundity of E implies complex uniform rotundity of E(M,τ). Therefore if E has non-trivial convexity, complex uniform convexity of E is equivalent with complex uniform convexity of E(M,τ). We obtain an analogous result for the unitary matrix space CE and a symmetric Banach sequence space E. From the above we conclude that E(M,τ)+ is complex uniformly rotund if and only if its norm ‖⋅‖E(M,τ) is uniformly monotone.

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators $${E(\mathcal{M},\tau)}$$ associated to a semifinite von Neumann algebra $${(\mathcal{M}, \tau)}$$ . We prove that x is a smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function $${f\in S_{E^{\times}}}$$ supporting μ(x), or s(x *) = 1. Under the assumption that the trace τ on $${\mathcal{M}}$$ is σ-finite, we show that x is strongly smooth point of the unit ball in $${E(\mathcal{M}, \tau)}$$ if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Frechet smoothness of the spaces E and $${E(\mathcal{M},\tau)}$$ .

Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space $E$, and of the unit ball of the space $E({\cal M},\tau)

Banach–Saks properties in symmetric spaces of measurable operators

Banach–Saks properties in symmetric spaces of measurable operators

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on ( 0 , ∞ ) . Let E ( M ) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E ( M ) into a Hilbert space H corresponds a positive norm one functional f ∈ E ( 2 ) ( M ) ∗ such that ∀ x ∈ E ( M ) ‖ T ( x ) ‖ 2 ⩽ K 2 ‖ T ‖ 2 f ( x ∗ x + x x ∗ ) , where E ( 2 ) denotes the 2-concavification of E and K is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for E ( M ) when E is either 2-concave or 2-convex and q-concave for some q < ∞ . We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.

Local Uniform Convexity and Kadec‐Klee Type Properties in K‐interpolation spaces II

We study local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces of Lorentz couples. We show that a wide class of Banach couples of (commutative and) non‐commutative Lorentz spaces possess the (so‐alled) (DGL)‐property originally introduced by Davis, Ghoussoub and Lindenstrauss in the context of renorming order continuous Banach latties. This property is used as a key tool to show that local uniform convexity and certain Kadec‐Klee type properties in non‐commutative symmetric spaces of measurable operators may be inferred from corresponding properties of the parameter space of the K‐interpolation method. Further applications are given to renorming properties of separable symmetric Banach function spaces and their non‐commutative counterparts.

Weak compactness criteria in symmetric spaces of measurable operators

The principal result of the paper reduces the study of certain weakly compact sets in Banach spaces of measurable operators to that of the corresponding sets of generalized singular value functions. In particular, under natural conditions, it is shown that the orbit of a relatively weakly compact subset of the Köthe dual of a symmetric space of measurable operators affiliated with some semi-finite von Neumann algebra is again relatively weakly compact.

The kadec-klee property in symmetric spaces of measurable operators

We show that ifE is a separable symmetric Banach function space on the positive half-line thenE has the Kadec-Klee property if and only if, for every semifinite von Neumann algebra (M, τ), the associated spaceE(M, τ) ofτ-measurable operators has the Kadec-Klee property.

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {xγ}γ∈G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.

Lifting of Kadec-Klee properties to symmetric spaces of measurable operators

We show that if E is a separable symmetric Banach function space on the positive half-line, then E has the Kadec-Klee property (respectively, uniform Kadec-Klee property) for local convergence in measure if and only if, for every semifinite von Neumann algebra (M, 'r), the associated space E(M, 'r) of 'r-measurable operators has the same property.

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.