Noncommutative Lp Research Articles

Noncommutative Multi-Parameter Subsequential Wiener–Wintner-Type Ergodic Theorem

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class Δ instead of the weight class case. In our subsequence class, every term of k̲∈Δ is one of the three kinds of nonzero density subsequences we consider. As key ingredients, we give the maximal ergodic inequalities of multi-parameter subsequential averages and obtain a noncommutative subsequential analogue of the Banach principle. Then, by combining the critical result of the uniform convergence for a dense subset of the noncommutative Lp(M) space and the noncommutative Orlicz space, we immediately obtain the main theorem.

Quantitative mean ergodic inequalities: Power bounded operators acting on one single noncommutative Lp space

In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative Lp space with 1<p<∞, which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems relies on noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calderón-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under consideration from operator theory, and a noncommutative version of the classical transference method due to Coifman and Weiss.

Beurling type representation for certain invariant subspaces of maximal subdiagonal algebras

Let A be a maximal subdiagonal algebra in a σ-finite von Neumann algebra M with respect to a faithful normal conditional expectation Φ. We consider certain invariant subspaces of A in the Haagerup noncommutative Lp space Lp(M) (1≤p≤∞). Suppose that φ is a faithful normal state on M with φ∘Φ=φ and {σtφ:t∈R} is the modular automorphism group of M associated with φ, we give a Beurling type representation for those right invariant subspaces of A in Lp(M) which are also invariant under {σtφ:t∈R}. Furthermore, we give an algebraic characterization for a subdiagonal algebra to be type 1 and also give a type decomposition for a central projection in M associated with the center of D.

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

AbstractIn this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.

An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces

The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical “T(1)∈BMO” assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical T(1) criterion still guarantees their L2-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by R-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative Lp spaces for all 1<p<∞, without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of Hong et al. for p≠2.

Fractional powers on noncommutative Lp for p < 1

We prove that the homogeneous functional calculus associated to x↦|x|θ or x↦sgn(x)|x|θ for 0<θ<1 is θ-Hölder on selfadjoint elements of noncommutative Lp-spaces for 0<p⩽∞ with values in Lp/θ. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.

Conditional expectations and martingales in noncommutative Lp-spaces associated with center-valued traces

In this paper we prove the existence of conditional expectations in the noncommutative Lp(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 weighted averages of martingales in noncommutative Lp(M,Φ)-spaces.

Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

In this paper, we study the noncommutative Orlicz space \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\), which generalizes the concept of noncommutative Lp space, where \(\mathcal {M}\) is a von Neumann algebra, and φ is an Orlicz function. As a modular space, the space \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\) possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace \(E_{\varphi }(\tilde {\mathcal {M}},\tau )=\overline {\mathcal {M}\bigcap L_{\varphi }(\tilde {\mathcal {M}},\tau )}\) in \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\), which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function φ satisfies the Δ2-condition, then \(L_{\varphi }(\tilde {\mathcal {M}},\tau )\) is uniformly monotone, and convergence in the norm topology and measure topology coincide on the unit sphere. Hence, \(E_{\varphi }(\tilde {\mathcal {M}},\tau )=L_{\varphi }(\tilde {\mathcal {M}},\tau )\) if φ satisfies the Δ2-condition.

Smooth Fourier multipliers in group algebras via Sobolev dimension

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative Lp spaces. This general principle —exploited in Euclidean harmonic analysis during the last 40 years— is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more flexibility to measure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredients yields new Lp estimates for smooth Fourier multipliers in group algebras.

An inequality in noncommutative Lp-spaces

We prove that for any (trace-preserving) conditional expectation E on a noncommutative Lp with p>2, Id−E is a contraction on the positive cone Lp+.

Non-commutative Nash inequalities

A set of functional inequalities—called Nash inequalities—are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative 𝕃p spaces, where their relationship to Poincaré and log-Sobolev inequalities is fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.

Complemented subspaces of spaces of multilinear forms and tensor products, II. Noncommutative [formula omitted] spaces

We study tensor products of the Schatten classes Sp, and their duals. It is shown that if p1−1+⋯+pk−1+q−1=1, then the space of multilinear forms B(Sp1,…,Spk;C) contains a complemented subspace isometric to Sq. We construct explicit embeddings of Srn into Spn⊗ˆSqn for r−1=p−1+q−1 whose range is complemented by a “natural” norm-one projection. As a byproduct we compute the nuclear norm of some multiplication operators: if r−1+1=p−1+q−1, with p,q≥1, then, given an n-by-n matrix ϕ, the nuclear norm of the multiplication operator h∈Spn↦h⋅ϕ∈Sqn is n times the norm of ϕ in Srˆn, where rˆ=max⁡(1,r). A number of results on general noncommutative Lp spaces are also included.

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let \(\mathcal{M}\) be a σ-finite von Neumann algebra and let \(\mathfrak{A} \subseteq \mathcal{M}\) be a maximal subdiagonal algebra with respect to a faithful normal conditional expectation Φ. Based on the Haagerup’s noncommutative Lp space \(L^p \left( \mathcal{M} \right)\) associated with \(\mathcal{M}\), we consider Toeplitz operators and the Hilbert transform associated with \(\mathfrak{A}\). We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space \(H^2 \left( \mathcal{M} \right)\) is just the right analytic Toeplitz algebra. Furthermore, the Hilbert transform on noncommutative \(L^p \left( \mathcal{M} \right)\) is shown to be bounded for 1 < p < ∞. As an application, we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative \(H^1 \left( \mathcal{M} \right)\) as a concrete space of operators.

Automatic continuity of orthogonality preservers on a non-commutative [formula omitted] space

Elements a and b of a non-commutative Lp(τ) space associated to a von Neumann algebra N, equipped with a normal semifinite faithful trace τ, are called orthogonal if l(a)l(b)=r(a)r(b)=0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from Lp(N,τ) to a normed space Y is said to be orthogonality-to-p-orthogonality preserving if ‖T(a)+T(b)‖p=‖a‖p+‖b‖p whenever a and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(N,τ) (1⩽p<∞, p≠2) to a Banach space X is automatically continuous, whenever N is a separably acting von Neumann algebra. If N is a semifinite factor not of type I2, we establish that every orthogonality-to-p-orthogonality preserving linear mapping T:Lp(N,τ)→X is continuous, and invertible whenever T≠0. Furthermore, there exists a positive constant C(p) (1⩽p<∞, p≠2) so that ‖T‖‖T−1‖⩽C(p)2, for every non-zero orthogonality-to-p-orthogonality preserving linear mapping T:Lp(N,τ)→X. For p=1, this inequality holds with C(p)=1 – that is, T is a multiple of an isometry.

Tensor Products of Noncommutative Lp-Spaces

We consider the notion of tensor product of noncommutative Lp spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative Lp spaces associated with σ-finite von Neumann algebras.

Automatic continuity of orthogonality or disjointness preserving bijections

Elements a and b of a non-commutative Lp(M,τ) space associated to a von Neumann algebra, M, equipped with a normal semi-finite faithful trace τ, are called orthogonal if l(a)l(b)=r(a)r(b)=0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from Lp(M,τ) to a normed space X is said to be orthogonality-to-p-orthogonality preserving if ∥T(a)+T(b)∥p=∥a∥p+∥b∥p whenever a and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(M,τ) to a Banach space is automatically continuous if 1≤p<∞, and M is either an abelian von Neumann algebra or a discrete von Neumann algebras. Furthermore, any complete p-additive norm on such Lp(M,τ) is equivalent to the canonical norm.

A Noncommutative Version of H p and Characterizations of Subdiagonal Algebras

Let \({\mathcal M}\) be a σ-finite von Neumann algebra and \({\mathfrak A}\) a maximal subdiagonal algebra of \({\mathcal M}\) with respect to a faithful normal conditional expectation \({\Phi}\) . Based on Haagerup’s noncommutative Lp space \({L^p(\mathcal M)}\) associated with \({\mathcal M}\) , we give a noncommutative version of Hp space relative to \({\mathfrak A}\) . If h0 is the image of a faithful normal state \({\varphi}\) in \({L^1(\mathcal M)}\) such that \({\varphi\circ \Phi=\varphi}\) , then it is shown that the closure of \({\{\mathfrak Ah_0^{\frac1p}\}}\) in \({L^p(\mathcal M)}\) for 1 ≤ p < ∞ is independent of the choice of the state preserving \({\Phi}\) . Moreover, several characterizations for a subalgebra of the von Neumann algebra \({\mathcal M}\) to be a maximal subdiagonal algebra are given.

Noncommutative Lp-spaces without the completely bounded approximation property

For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for r large enough depending on p. We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have the Approximation Property of Haagerup and Kraus. This provides examples of exact C^*-algebras without the operator space approximation property.

Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

Abstract We show that the multiplier algebra of the Fourier algebra on a locally compact group G can be isometrically represented on a direct sum on non-commutative Lp spaces associated with the right von Neumann algebra of G. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative Lp spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative Lp spaces, say . It is shown that is isometric to L1(G), generalising the abelian situation.

The General Form of γ-Family of Quantum Relative Entropies

We use the Falcone–Takesaki non-commutative flow of weights and the resulting theory of non-commutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and classical relative entropies of Zhu–Rohwer, and belong to an intersection of families of Petz relative entropies with Bregman relative entropies. For the purpose of this task, we generalise the notion of Bregman entropy to the infinite-dimensional non-commutative case using the Legendre–Fenchel duality. In addition, we use the Falcone–Takesaki duality to extend the duality between coarse-grainings and Markov maps to the infinite-dimensional non-commutative case. Following the recent result of Amari for the Zhu–Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre–Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.