In this paper we study dominated orthogonally additive (in general nonlinear) operators on Köthe–Bochner spaces. We resolve the open problem stated in [12], showing that there exist Köthe–Bochner spaces E(X), F(Y) and an orthogonally additive operator T:E(X)→ F(Y) , such that T is f-locally dominated for each f∈ E(X) , but is not dominated. We also prove that there is a C-bounded orthogonally additive operator T:E(X)→ F(Y) which is not strongly C-bounded, resolving the open problem stated in [25].

We establish a noncommutative version of a result due to Linden-strauss and Tzafriri [Classical Banach spaces. I, Ergebnisse der Mathematikund ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 92,Springer-Verlag, Berlin-New York, 1977]. Precisely, every bounded sequence{x(i)}(infinity)(i=1)in a noncommutative quasi-Banach M-bimodule epsilon subset of L-p(M,tau)+M(here,Mstands for a semi finite von Neumann algebra),p>0, having order continuous quasi-norm 1) either satisfies that there exists a constant c>0 such that, for every choice{a(i)}(infinity)(i=1)of scalars,integral(1)(0)parallel to & sum;(i=1)r(i)(t)a(i)x(i)parallel to Edt >= c(n & sum;(i=1)|a(i)|(2))(1/2),n=1,2,(2) or has a subsequence which is equivalent to a sequence of disjoint ele-ments in epsilon. As a consequence, we answer a question by Randrianantoanina

Quantum information theory deals with the way information can be exchanged using the laws of quantum mechanics. The field is relatively young, but has seen an extaordinary rapid development over the past decade. The particular focus of this workshop is the analysis of quantum information theory and its connections to infinite dimensional systems and operator algebras. On the other hand the theory of noncommutative harmonic analysis has established itself further in a parallel development, with successful attempts at generalizing the theory of singular integrals and Fourier multipliers in the noncommutative realm. While there are close ties between the mathematics underpinning noncommutative harmonic analysis and quantum information theory, the two fields have so far remained largely disjoint, despite ample evidence that there could be a fruitful cross pollination between the two. In this workshop we have brought together some of the leading experts in both fields, as well as talented young mathematicians.

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.

Under mild assumptions, the main results of this paper characterize a (real or complex) symmetric operator space affiliated with a semi-finite atomless von Neumann algebra (ℳ, τ) possessing the Daugavet property as L_1(ℳ,τ) or ℳ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of concerning the Daugavet property for real symmetric function spaces.

The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class of) symmetric spaces over the hyperfinite II_1 factor ℛ and certain symmetric operator space over the hyperfinite II_∞ factor ℛ⊗̅ℒ(H)). Precisely, we show that for any symmetric function space E(0,1) (in the sense of Lindenstrauss and Tzafriri) such that both E(0,1) and its Köthe dual have the Kruglov property, the symmetric operator space E(ℛ) is isomorphic to some symmetric space Z_E^2(ℛ⊗̅ℒ(H)). This result establishes a noncommutative version of a well-known result due to Johnson, Maurey, Schechtman and Tzafriri, and answers the noncommutative version of a question due to Mityagin.

Let R-lambda,R-j be the jth Bessel-Riesz transform, where n >= 1, lambda > 0, and j = 1, ... , n + 1. In this article, we establish a Weyl-type asymptotic for [R-lambda,R-j, M-f], the commutator of R-lambda,R-j with the multiplication operator M-f, based on building a preliminary result that the endpoint weak Schatten norm of [R-lambda,R-j ,M-f] can be characterized via homogeneous Sobolev norm (W) over dot(1,n+1)(R-+(n+1)) of the symbol f. Specifically, the asymptotic coefficient is equivalent to ||f||((W) over dot1,n+1(R+n+1)). Our main strategy is to relate Bessel-Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.

Let (X, G) be a d-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator Delta(G), and let Pi(X) be the C*-algebra obtained by locally transferring the C*-algebra generated by multiplication operators and Riesz transforms on Rd. Denote by sym(X)the principal symbol mapping of Pi(X). For any S is an element of Pi(X), we prove that, in the framework of C*-algebra, lim(t ->infinity), t(1/p)mu(1 + Delta(G))(-d/2p)) = (2 pi (d)root d)(-1/p) ||sym(X) (S) || L-p (T*X,e(-qG) d lambda), 100 where 0 < p <00, e-96 is the canonical weight on X, and da is the Liouville measure on the cotangent bundle T*X.

Let ℳ be a factor equipped with a semi-finite faithful normal trace τ. Let E(0,∞) be a symmetrically normed function space and E(ℳ,τ) be the corresponding symmetrically normed operator space. Suppose that a, b are τ-measurable operators affiliated with ℳ. It is shown that the range of the multiplication operator S_a,b: x↦ axb on ℳ is contained in E(ℳ, τ) if and only if μ(a)μ(b) belongs to E(0, ∞), where μ(x) stands for the generalized singular value function of a τ-measurable operators x affiliated with ℳ. Moreover, we have S_a,b_ℳ→ E(ℳ,τ)= μ(a )μ( b) _E (0,∞) , which answers a question by Fialkow and Loebl (1984). We also consider the quasi-normed case, and show that the natural quasi-norm of weak L_p-space, 0<p<∞, is not monotone with respect to the logarithmic submajorisation.

We establish a version of the classical results concerning descriptions of isometries on C^*-algebras and noncommutative L_p-spaces due to Kadison (1951) and Yeadon (1981) in the setting of Haagerup–Schultz algebras. Precisely, we show that (not necessarily surjective) isometries on such algebras are necessarily implemented by partial isometries and trace-preserving Jordan ^*-monomorphisms.

Let a be a normal locally measurable operator affiliated with a semifinite von Neumann algebra & Mscr;. Let b be a locally measurable operator affiliated with & Mscr; commuting with a. Then for any locally measurable operator x such that [a, x] + b E (L1 + L infinity)(& Mscr;, tau), we have b-- [a, x] + b, where-- stands for the Hardy-Littlewood-P & oacute;lya submajorization. This extends several results in the existing literature. We also present two applications of the main result. Firstly, we show that the kernels of normal generalized inner derivations on fully symmetric spaces with order continuous norm whose K & ouml;the dual is contained in the ideal of tau-compact operators are orthogonally complemented. Secondly, we establish the regularity of every normal operator acting on a fully symmetric space with order continuous norm, where the norm is not proportional to the Hilbert space norm and the K & ouml;the dual consists of tau-compact operators. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal 𝒞_E of the algebra ℬ(H) of all bounded linear operators on the separable infinite-dimensional Hilbert space H. In this paper, we extend and strengthen Arazy's decomposition theorem to the setting of general bounded linear operators on a separable (quasi-Banach) operator ideal 𝒞_E of ℬ(H). Several applications are given to the study of 𝒞_E-strictly singular operators, largest proper ideals in the algebra ℬ(𝒞_E) of all bounded linear operators on 𝒞_E and complementably homogeneous Banach spaces among others. Our versions of decomposition theorems supply tools for a noncommutative generalization of deep commutator theorems for operators on ℓ_p and L_p, 1≤ p <∞, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class 𝒞_p, 1≤ p<∞. For the crucial case, p=1, we establish that any operator T∈ℬ(𝒞_1) is a commutator if and only if T is not of the form λI+K for some λ≠ 0 and 𝒞_1-strictly singular operator K.

Abstract We show that weak sequential convergence in a semifinite von Neumann algebra equipped with a faithful normal semifinite trace implies almost uniform convergence if and only if is of type with and that weak sequential convergence in a finite von Neumann algebra with implies (quasi‐)norm convergence in noncommutative spaces associated with for all .

We characterise the Schur bounded patterns of ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals 𝒞_p with 0<p<1. Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns.

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of τ-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of L_p-spaces for 1<p<∞. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.

The main objective of this paper is to investigate the distributional estimates for (i) commutators with Calderón–Zygmund integral operators; (ii) Marcinkiewicz multipliers; (iii) Littlewood–Paley square function, via semigroup {𝒞^α}_α >0 generated by Cesàro operator. In each of the cases (i)–(iii) we obtain new estimates of the distribution of elements in the range of the underlying operators in terms of the distribution function of the input function. Our method allows us to obtain optimal estimates shedding additional light at the results due to Pérez (J Funct Anal 128(1):163–185, 1995), Tao and Wright (Rev Mat Iberoam 17(3):521–558, 2001), Bakas et al. (Rev Mat Iberoam 2024), Bourgain (On the Behavior of the Constant in the Littlewood–Paley Inequality, Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., vol. 1376, pp. 202–208. Springer, Berlin, 1989). The main feature of the distributional form inequalities lies in its broad applicability across diverse problems in analysis, e.g. they allow obtaining estimates in wide range of symmetric quasi-Banach interpolation spaces between L_p and L_q ( 1

In this article we study the homotopical properties of linear groups of some Banach spaces. Our first main result asserts that for 1 < p q 0 We conclude with few open problems. p (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For a normal measurable operator a affiliated with a von Neumann factor M we show that if M is infinite, then there is lambda(0 )is an element of C so that for epsilon > 0 there are u(epsilon) = u(epsilon)(& lowast;), v(epsilon) is an element of U(M) with v(epsilon)|[a,u(epsilon) ]|v(epsilon)(& lowast;) (1-epsilon)(|a- lambda(0)1|+u(epsilon) |a- lambda(0)1|u(epsilon) ). If M is finite, then there is lambda(0)is an element of C and u, v is an element of U(M) root 3 so that v|[a, u]|v & lowast; 2 (|a- lambda(0)1|+ u|a-lambda(0)1|u & lowast;). These bounds are optimal for infinite factors, II1-factors and some In-factors. Furthermore, for finite factors applying kk1-norms to the inequality provides estimates on the norm of the inner derivation delta(a) : M -> L1(M, tau) associated to a. While by [3, Theorem 1.1] it is known for finite factors and self-adjoint a is an element of L1(M, tau) thatk||delta(a)|| ->(L1(,tau)) = 2 minz is an element of C ka- zk1, we present concrete examples of finite factors and normal operators a is an element of M for which this fails.

Let (xk)nk=1 be a sequence of positive elements in the noncommutative Lebesgue space Lp(M), and let (Ek)nk=1 be a sequence of conditional expectations with respect to an increasing subalgebras (Mn)k⩾1 of the finite von Neumann algebra M. We establish the following sharp noncommutative dual Doob inequalities: ∥∥∑nk=1xk∥∥Lp(M)⩽1p∥∥∑nk=1Ek(xk)∥∥Lp(M),p∈(0,1], and ∥∥∑nk=1Ek(xk)∥∥Lp(M)⩽p∥∥∑nk=1xk∥∥Lp(M),p∈[1,2]. As applications, we obtain several noncommutative martingale inequalities with better constants.

In the first part of the paper we describe the structure of the Boolean algebra FT of all fragments of a positive orthogonally additive operator T : E -> F, generalizing classical results of Aliprantis, Burkinshaw, de Pagter to the setting of orthogonally additive (in general nonlinear) operators on Banach lattices. In the second part of the article we present the nonlinear version of the well known Dodds-Fremlin's theorem. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.