arXiv subjects
Publications and source records attributed to Jinghao Huang.
We solve Tingley's problem for Haagerup noncommutative $L_p$-spaces, where $1<p\ne 2<\infty$, which gives an affirmative answer to Mori's problem \cite[Problem~6.3]{Mori}.
Let $\mathcal M$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. 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.
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 $(\mathcal M, \tau)$ possessing the Daugavet property as $L_1(\mathcal M,\tau)$ or $\mathcal M$ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of \cite{AKM12, KMMW13, AKM15} concerning the Daugavet property for real symmetric function spaces.
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 $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ admits a canonical representation of the form $\Phi(x)=wJ(x)$, 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 $\mathrm{II}_\infty$ and type $\mathrm{III}$ algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the $F$-norm and the pair $(\mu, D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.
Let $(\Omega,\mu)$ and $(\Lambda,\lambda)$ be complete atomless localizable semifinite measure spaces. Suppose that $E(\Omega,\mu)$ and $F(\Lambda,\lambda)$ are real rearrangement-invariant Banach function spaces with order-continuous norms, in the Banach-lattice sense, and that neither norm is proportional to the $L_2$-norm. Every surjective real-linear isometry $U:E(\Omega,\mu)\longrightarrow F(\Lambda,\lambda)$ has the form $Uf=w\Phi(f)$, where $w$ has full support and $\Phi$ is induced by a complete measure-class Boolean isomorphism. Both factors are uniquely determined by $U$. Let $(\mathcal{M},\tau)$ and $(\mathcal{N},\nu)$ be atomless semifinite von Neumann algebras, and let $E(\mathcal{M},\tau)$ and $F(\mathcal{N},\nu)$ be symmetric operator spaces satisfying the same assumptions on their norms. Every surjective real-linear isometry $V:E(\mathcal{M},\tau)_{\mathrm{sa}}\longrightarrow F(\mathcal{N},\nu)_{\mathrm{sa}}$ has the form $V(x)=hJ(x)$, where $J:\mathcal{M}\longrightarrow\mathcal{N}$ is a normal surjective Jordan $*$-isomorphism and $h\in LS(\mathcal{Z}(\mathcal{N}))_{\mathrm{sa}}$ is central with full support; again, the two factors are unique. We also identify the bounded skew-Hermitian operators on the real self-adjoint part and derive commutative and noncommutative isometric forms of Mityagin's question.
Let $\mathcal{M}$ be a factor equipped with a semi-finite faithful normal trace $\tau$. Let $E(0,\infty)$ be a symmetrically normed function space and $E(\mathcal{M},\tau)$ be the corresponding symmetrically normed operator space. Suppose that $a, b$ are $\tau$-measurable operators affiliated with $\mathcal{M}$. It is shown that the range of the multiplication operator $S_{a,b}: x\mapsto axb$ on $\mathcal{M}$ is contained in $E(\mathcal{M}, \tau)$ if and only if $\mu(a)\mu(b)$ belongs to $E(0, \infty)$, where $\mu(x)$ stands for the generalized singular value function of a $\tau$-measurable operators $x$ affiliated with $\mathcal{M}$. Moreover, we have $$ \|S_{a,b}\|_{\mathcal{M}\to E(\mathcal{M},\tau)}= \| \mu(a )\mu( b) \|_{E (0,\infty) }, $$ 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, $0<p<\infty$, is not monotone with respect to the logarithmic submajorisation.
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 $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. 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 K\"othe dual have the Kruglov property, the symmetric operator space $E(\mathcal{R})$ is isomorphic to some symmetric space $Z_E^2(\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. 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 $E(\mathcal{M},\tau)$ and $F(\mathcal{M},\tau)$ be two Calkin operator spaces affiliated with a semifinite von Neumann algebra $\mathcal{M}$ equipped with a semifinite faithful normal trace $\tau $. We show that if $\mathcal{M}$ is atomless, $\tau$ is finite, and $E(v,\tau)\not\subseteq F(\mathcal{M},\tau)$, then every order-measure continuous and disjointness-preserving mapping $T:E(\mathcal{M},\tau)\xrightarrow{\rm into} F(\mathcal{M},\tau)$ 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 $\mathcal{M}$-bimodule $E(\mathcal{M},\tau)$ of $\tau$-measurable operators into another $F(\mathcal{M},\tau)$ preserves disjointness provided that the norm of $F(\mathcal{M},\tau)$ 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. \cite{SV,HSZ20,Abra1991,vek,dC20}.
We give an affirmative answer to Tingley's problem for Schatten--von Neumann classes $C_p$ by a unified approach for all $0<p\ne 2<\infty$, extending results due to Fern\'andez-Polo et al.
The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal $\mathcal C_E$ of the algebra $\mathcal B(H)$ 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 $\mathcal C_E$ of $\mathcal B(H)$. Several applications are given to the study of $\mathcal C_E$-strictly singular operators, largest proper ideals in the algebra $\mathcal B(\mathcal C_E)$ of all bounded linear operators on $\mathcal C_E$ 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 $\ell_p$ and $L_p$, $1\le p <\infty $, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class $\mathcal C_p$, $1\le p<\infty $. For the crucial case, $p=1$, we establish that any operator $T\in\mathcal B(\mathcal C_1)$ is a commutator if and only if $T$ is not of the form $\lambda I+K$ for some $\lambda\neq 0$ and $\mathcal C_1$-strictly singular operator $K$.
This is a systematic study of isometries between noncommutative symmetric spaces. Let $\mathcal{M}$ 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 $\mathcal{H}$ equipped with a semifinite faithful normal trace $\tau$. We show that for any noncommutative symmetric space corresponding to a symmetric function space $E(0,\infty)$ in the sense of Lindenstrauss--Tzafriri such that $\left\|\cdot\right\|_E\ne \lambda \left\|\cdot\right\|_{L_2}$, $\lambda\in \mathbb{R}_+$, any isometry on $E(\mathcal{M},\tau)$ 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 $E(\mathcal{M},\tau)\ne L_p(\mathcal{M},\tau)$ 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, $1\le p<\infty$, 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 Pe\l czy\'nski's problem concerning the uniqueness of the symmetric structure in noncommutative symmetric spaces.
The problem of Motion Sickness (MS) among passengers significantly impacts the comfort and efficiency of In-Vehicle Infotainment Systems (IVIS) use. In this study, we innovatively designed SimPath, a visual design to effectively mitigate passengers' MS and boost their efficiency of using IVIS during driving. The study focuses on the problem of irregular motion conditions frequently encountered during actual driving. To validate the efficacy of this approach, two sets of real - vehicle experiments were carried out in real driving scenarios. The results demonstrate that this approach significantly reduces passenger's MS level to a certain extent. However, due to divided attention from visual content, it does not directly improve the IVIS efficiency. In conclusion, this study offers crucial insights for the design of a more intelligent and user friendly IVIS, based on the discussion of the principle, providing strong theoretical support and practical guidance for the development of future IVIS in autonomous vehicles.
With the advancement of drone technology, the volume of video data increases rapidly, creating an urgent need for efficient semantic retrieval. We are the first to systematically propose and study the drone video-text retrieval (DVTR) task. Drone videos feature overhead perspectives, strong structural homogeneity, and diverse semantic expressions of target combinations, which challenge existing cross-modal methods designed for ground-level views in effectively modeling their characteristics. Therefore, dedicated retrieval mechanisms tailored for drone scenarios are necessary. To address this issue, we propose a novel approach called Multi-Semantic Adaptive Mining (MSAM). MSAM introduces a multi-semantic adaptive learning mechanism, which incorporates dynamic changes between frames and extracts rich semantic information from specific scene regions, thereby enhancing the deep understanding and reasoning of drone video content. This method relies on fine-grained interactions between words and drone video frames, integrating an adaptive semantic construction module, a distribution-driven semantic learning term and a diversity semantic term to deepen the interaction between text and drone video modalities and improve the robustness of feature representation. To reduce the interference of complex backgrounds in drone videos, we introduce a cross-modal interactive feature fusion pooling mechanism that focuses on feature extraction and matching in target regions, minimizing noise effects. Extensive experiments on two self-constructed drone video-text datasets show that MSAM outperforms other existing methods in the drone video-text retrieval task. The source code and dataset will be made publicly available.
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).
Humans can discern scene-independent features of objects across various environments, allowing them to swiftly identify objects amidst changing factors such as lighting, perspective, size, and position and imagine the complete images of the same object in diverse settings. Existing object-centric learning methods only extract scene-dependent object-centric representations, lacking the ability to identify the same object across scenes as humans. Moreover, some existing methods discard the individual object generation capabilities to handle complex scenes. This paper introduces a novel object-centric learning method to empower AI systems with human-like capabilities to identify objects across scenes and generate diverse scenes containing specific objects by learning a set of global object-centric representations. To learn the global object-centric representations that encapsulate globally invariant attributes of objects (i.e., the complete appearance and shape), this paper designs a Disentangled Slot Attention module to convert the scene features into scene-dependent attributes (such as scale, position and orientation) and scene-independent representations (i.e., appearance and shape). Experimental results substantiate the efficacy of the proposed method, demonstrating remarkable proficiency in global object-centric representation learning, object identification, scene generation with specific objects and scene decomposition.
We prove that the sequence space $\ell_{p,q}$ does not embed into $L_{p,q}(\mathcal{M},τ)$ for any noncommutative probability space $(\mathcal{M},τ)$, $1< p<\infty $, $1\le q<\infty$, $p\ne q$. Several applications to the isomorphic classification of noncommutative $L_{p,q}$-spaces are given, which extend and complement several earlier results.
We fully characterize those von Neumann algebras having the ball-covering property. We also study the ball-covering property of noncommutative symmetric spaces. In particular, we provide a number of new examples of non-separable (commutative and noncommutative) Banach spaces having the ball-covering property.