Let LS(ℳ) be the algebra of locally measurable operators affiliated with a von Neumann algebra ℳ, equipped with an F-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between LS(ℳ) admits a canonical representation of the form Φ(x)=wJ(x), where w is a unitary element and J is a Jordan ^*-isomorphism, which extends classical results such as the Banach–Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type II_∞ and type III algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the F-norm and the pair (μ, D) of a probability measure and a dimension function, which fails for algebras with atomic centers.

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

Let E(ℳ,τ) and F(ℳ,τ) be two Calkin operator spaces affiliated with a semifinite von Neumann algebra ℳ equipped with a semifinite faithful normal trace τ. We show that if ℳ is atomless, τ is finite, and E(v,τ)⊈F(ℳ,τ), then every order-measure continuous and disjointness-preserving mapping T:E(ℳ,τ) F(ℳ,τ) is identical to the zero mapping, which establishes a noncommutative version of Abramovich's theorem. We also show that every positive isometry T from a normed ℳ-bimodule E(ℳ,τ) of τ-measurable operators into another F(ℳ,τ) preserves disjointness provided that the norm of F(ℳ,τ) is strictly monotone. As an application, we obtain the general form of T, which extends and unifies several results due to Abramovich, de Jager, Conradie, Veksler and Sukochev et al. .

The main objective of the present paper is to study the Grothendieck property of the sum and the intersection of two Banach function spaces over sigma-finite measure spaces. In particular, we show that for a reflexive symmetric function space E(0, infinity), the spaces (E boolean AND L-infinity)(0, infinity) and (E+L-infinity)(0, infinity) are Grothendieck spaces. As a consequence, we fully characterize those symmetric function spaces (L-p boolean AND L-q)(0, infinity) and (L-p + L-q)(0, infinity), 1 <= p, q <= infinity, possessing the Grothendieck property. We also show that the sum of a noncommutative L-p-space, 1 < p < infinity, and a noncommutative L-infinity-space has the Grothendieck property. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let ℳ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace τ. We establish Jensen's trace inequality in full generality and characterize its equality case, which answers two questions raised in [Kosaki2013] and [HaradaKosaki2008]. As an application, we derive noncommutative Lamperti-type inequalities and their equality conditions. Employing this result, we characterize linear isometries (not necessarily surjective) on a class of F-normed noncommutative Orlicz spaces, which provides a noncommutative Lamperti's theorem for linear isometries.

We solve Tingley's problem for Haagerup noncommutative L_p-spaces, where 1 .

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 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.

We fully describe the general form of a linear (or conjugate-linear) rank metric isometry on the Murray–von Neumann algebra associated with a II_1-factor. As an application, we establish Frobenius' theorem in the setting of II_1-factors, by showing that every determinant-preserving linear bijection between two II_1-factors is necessarily an isomorphism or an anti-isomorphism. This confirms the Harris–Kadison conjecture (1996).

We present two new compactness criteria in non-commutative quasi-Banach symmetric spaces associated to a finite von Neumann algebra, with focus on the non-commutative torus. The first result is novel, even in the commutative setting; while the second resembles the Kolmogorov-Riesz compactness theorem (see Theorems 4.1 and 5.7, respectively). The work contributes to understanding a conjecture of Brudnyi, adapted here for the non-commutative torus. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

This is a systematic study of isometries between noncommutative symmetric spaces. Let ℳ be a semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space ℋ equipped with a semifinite faithful normal trace τ. We show that for any noncommutative symmetric space corresponding to a symmetric function space E(0,∞) in the sense of Lindenstrauss–Tzafriri such that ·_E λ·_L_2, λ∈ℝ_+, any isometry on E(ℳ,τ) is of elementary form. This answers a long-standing open question raised in the 1980s in the non-separable setting [Math. Z. 1989], while the case of separable symmetric function spaces was treated in [Huang & Sukochev, JEMS, 2024]. As an application, we obtain a noncommutative Kalton–Randrianantoanina–Zaidenberg Theorem, providing a characterization of noncommutative L_p-spaces over finite von Neumann algebras and a necessary and sufficient condition for an operator on a noncommutative symmetric space to be an isometry. Having this at hand, we answer a question posed by Mityagin in 1970 [Uspehi Mat. Nauk] and its noncommutative counterpart by showing the any symmetric space E(ℳ,τ) L_p(ℳ,τ) over a noncommutative probability is not isometric to a symmetric space over a von Neumann algebra equipped with a semifinite infinite faithful normal trace. It is also shown that any noncommutative L_p-space, 1≤ p<∞, affiliated with an atomless semifinite von Neumann algebra has a unique symmetric structure up to isometries. This contributes to the resolution of an isometric version of Pełczyński's problem concerning the uniqueness of the symmetric structure in noncommutative symmetric spaces.

Let E(0, ∞) be a symmetric function space on (0, ∞) such that the set E(0, ∞) ∩L∞(0, ∞) is distinct from the set Lp(0, ∞)∩L∞(0, ∞), 1 ≤ p < 2, and let E( M) be the corresponding symmetric operator space associated with an atomless semifinite σ-finite von Neumann algebra M equipped with a semifinite infinite faithful normal trace τ. We show that there exists a noncommutative probability space (N,σ) such that E(0, ∞) embeds into L_p(N) if and only if there exists a noncommutative probability space (N̂,σ̂) such that E(M) embeds into L_p(N̂) . We also establish a discrete version of this result for symmetric sequence space ℓE. These extend and complement earlier results in [37,40,41,55].

We prove that the sequence space l(p,q) does not embed into L-p,L-q(M, tau) for any noncommutative probability space (M, tau) when 1 < p < infinity, 1 < q < infinity and p not equal q. Several applications to the isomorphic classification of noncommutative L-p,L-q-spaces are given, which generalize earlier results in [31,32,36, 43,61]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we present a version of the Schur inequality in the setting of Murray–von Neumann algebras, extending a result by Arveson and Kadison. We also describe the ring isomorphisms between * -subalgebras of two Murray–von Neumann algebras. A short proof of the commutator estimation theorem for Murray–von Neumann algebras is given as an easy application.

In this paper, we present a survey of recent developments concerning derivations with values in an order ideal of the ∗ * -algebra of all measurable operators affiliated with a semifinite von Neumann algebra.