arXiv subjects

Chunlan Jiang

Publications and source records attributed to Chunlan Jiang.

At least 19 recordsLinked to original sources

arXiv2026

Kaplansky's second test problem in operator algebras

Kaplansky's second test problem on similarity asks: if $T$ and $S$ are elements in a unital Banach algebra $\mathcal{B}$ and $T\oplus T$ is similar to $S\oplus S$ in $\mathbb{M}_2(\mathcal{B})$, is $T$ similar to $S$ in $\mathcal{B}$? We answer this problem affirmatively if $T$ is an operator with property $(J)$ in a type $\mathrm{I}_n$ von Neumann algebra $\mathcal{M}$, i.e., $\{T\}'\cap\mathcal{M}$ contains a bounded maximal abelian family of idempotents. Moreover, the condition of property $(J)$ can be removed for $1\leqslant n\leqslant 3$. A similar result is proved if $T$ is an element in a unital Banach algebra $\mathcal{B}$ with essentially finite-dimensional commutant, i.e., the relative commutant of $T$ in $\mathcal{B}$ is finite-dimensional modulo its Jacobson radical. Finally, we point out that one of our main results can be applied to the implementation of local unitary (LU) equivalence of quantum states.

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arXiv2026

Intertwining Properties for Bimodule Quantum Markov Semigroups

In this paper, we study the Bakry-\'{E}mery estimates for GNS- and KMS-symmetric semigroups in terms of the Fourier multiplier of the gradient form and the iterated gradient form in the framework of quantum Fourier analysis. We also systematically investigate the intertwining properties for bimodule GNS- and KMS-symmetric quantum Markov semigroups and compare with the Bakry-\'{E}mery estimates. A number of examples of GNS- and KMS-symmetric semigroups satisfying these intertwining properties are presented.

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arXiv2026

Completely Bounded Representations Into Von Neumann Algebras And Connes Embedding Problem

In this paper, we prove that if $\mathcal{A}$ is a unital separable $C^*$-algebra, $\mathcal{M}$ is a von Neumann algebra which has the Kirchberg's quotient weak expectation property (QWEP), and $\phi:\, \mathcal{A}\rightarrow \mathcal{M}$ is a unital completely bounded representation, then there is an invertible operator $S\in \mathcal{M}$ such that $S\phi(\cdot) S^{-1}$ is a $\ast$-representation. On the other hand, Gilles Pisier proved the following result: a unital $C^*$-algebra $\mathcal{A}$ is nuclear if and only if for every unital completely bounded representation $\phi$ of $\mathcal{A}$ into an arbitrary von Neumann algebra $\mathcal{M}$ there is an invertible operator $S\in \mathcal{M}$ such that $S\phi(\cdot) S^{-1}$ is a $\ast$-representation. This implies that there exist von Neumann algebras which are not QWEP. Eberhard Kirchberg showed that every von Neumann algebra has QWEP if and only if every tracial von Neumann algebra embeds into the ultrapower $\mathcal{R}^w$ of the hyperfinite type ${\rm II}_1$ factor $\mathcal{R}$. This provides a negative answer to the Connes Embedding Problem. This paper relies on previous work of Gilles Pisier and Florin Pop.

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arXiv2025

Bimodule KMS Symmetric Quantum Markov Semigroups and Gradient Flows

The bimodule KMS symmetry of a bimodule quantum Markov semigroup extends the classical KMS symmetry of a quantum Markov semigroup. Compared with (bimodule) GNS symmetry, the (bimodule) KMS symmetry retains significantly more of the underlying noncommutativity. In this paper, we study bimodule KMS symmetric quantum Markov semigroups and introduce directional matrices for such semigroups, which reduce to diagonal matrices in the GNS symmetric setting. Using these directional matrices, we establish a corresponding gradient-flow structure. As a consequence, we obtain both a modified logarithmic Sobolev inequality and a Talagrand inequality for bimodule KMS symmetric quantum Markov semigroups.

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arXiv2025

On Klee's problem of convex bodies in Banach spaces

It is well known that every convex body in a finite dimensional normed space can be uniformly approximated by strictly convex and smooth convex bodies. However, in the case of infinite dimensions, little progress has been made since Klee asked how it is in the case of infinite dimensions in 1959. In this paper, we show that for an infinite dimensional Banach space $X$, (1) every convex body can be uniformly approximated by strictly convex bodies if and only if $X$ admits an equivalent strictly convex norm; (2) every convex body can be uniformly approximated by G\^{a}teaux smooth convex bodies if the dual $X^*$ of $X$ admits an equivalent strictly convex dual norm; in particular, (3) if $X$ is either separable, or reflexive, then every convex body in $X$ can be uniformly approximated by strictly convex and smooth convex bodies. They are done by showing that some correspondences among the sets of all convex bodies endowed with the Hausdorff metric, all continuous coercive Minkowski functionals and Fenchel's transform defined on all quadratic homogenous continuous convex functions equipped with the metric induced by the sup-norm of all bounded continuous functions defined on the closed unit ball $B_X$ are actually locally Lipschitz isomorphisms.

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arXiv2025

A local quantization principle for inclusions of tracial von Neumann algebras

We study the local quantization principle (after Sorin Popa~\cite{popa 94} and \cite{popa 95}) of inclusions of tracial von Neumann algebras. Let $(\mathcal{M},\tau)$ be a type ${\rm II}_1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\rm II}_1$ von Neumann subalgebra. Let $x_1,\ldots, x_m \in \mathcal{M}$ and $ \epsilon> 0$. Then there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\left\|\sum_{i=1}^n p_{i}\left(x_j-E_{\mathcal{N}'\cap \mathcal{M}}(x_j)\right)p_{i}\right\|_{2}<\epsilon,\quad 1\leq j\leq m.\] In particular, if $\mathcal{N}\subseteq \mathcal{M}$ is an inclusion of type $\rm II_{1}$ factors with $[\mathcal{M}:\mathcal{N}]=2$, then for any $x_{1},\ldots, x_{m}\in \mathcal{M}$, there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\sum_{i=1}^n p_ix_jp_i=\tau(x_j)1, \quad 1\leq j\leq m.\] Equivalently, there exists a unitary operator $u\in \mathcal{N}$ such that \[\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=\tau(x_j)1, \quad 1\leq j\leq m.\]

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arXiv2025

The invariant subspace problem and Rosenblum operators I

Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\|T^n\|$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\|T^n+\lambda T^{-n}\|$, where $n\in \mathbb{N}$ and $\lambda$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators.

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arXiv2025

Density of irreducible operators in the trace-class norm

In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in $B(H)$ with respect to the trace-class norm topology? Precisely, for each operator $T $ in $B(H)$ and every $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$? For $p>1$, to prove the $\Vert \cdot \Vert_p$-norm density of irreducible operators in $B(H)$, a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case $p=1$, where by $\Vert \cdot \Vert_p$-norm we denote the Schatten $p$-norm. In the current paper, for a large family of operators in $B(H)$, we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type $\mathrm{II}_1$ von Neumann algebras.

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arXiv2025

The Jordan decomposition and Kaplansky's second test problem for Hermitian holomorphic vector bundles

In 1954, I. Kaplansky proposed three test problems for deciding the strength of structural understanding of a class of mathematical objects in his treatise "Infinite abelian groups", which can be formulated for very general mathematical systems. In this paper, we focus on Kaplansky's second test problem in a context of complex geometry. Let $H^2_{\beta}$ be a weighted Hardy space. The Cowen-Douglas operator theory tells us that each $h\in\textrm{Hol}(\overline{\mathbb{D}})$ induces a Hermitian holomorphic vector bundle on $H^2_{\beta}$, denoted by $E_{h(S_\beta)}(\Omega)$, where $\Omega$ is a domain. We show that the vector bundle $E_{h(S_\beta)}$ is a push-forwards Hermitian holomorphic vector bundle and study the similarity deformation problems. Our main theorem is that if $H^2_{\beta}$ is a weighted Hardy space of polynomial growth, then for any $f\in \textrm{Hol}(\overline{\mathbb{D}})$, there exists a unique positive integer $m$ and an function $h\in\textrm{Hol}(\overline{\mathbb{D}})$ inducing an indecomposable vector bundle $E_{h(S_{\beta})}$, such that $E_{f(S_\beta)}$ is similar to $\bigoplus_1^m E_{h(S_\beta)}$, where $h$ is unique in the sense of analytic automorphism group action. That could be seemed as a Jordan decomposition theorem for the push-forwards Hermitian holomorphic vector bundles. Furthermore, we give the similarity classification of those push-forwards Hermitian holomorphic vector bundles induced by analytic functions, and give an affirmative answer to Kaplansky's second test problem for those objects. We also give an affirmative answer to the geometric version and generalized version of a problem proposed by R. Douglas in 2007, and obtain the $K_0$-group of the commutant algebra of a multiplication operator on a weighted Hardy space of polynomial growth. In addition, we give an example to show the setting of polynomial growth condition is necessary.

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arXiv2024

Phase Group Categories of Bimodule Quantum Channels

In this paper, we study the quantum channel on a von Neuamnn algebra $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely an $\mathcal{N}$-$\mathcal{N}$-bimodule unital completely positive map. By introducing the relative irreducibility of a bimodule quantum channel, we show that its eigenvalues with modulus 1 form a finite cyclic group, called its phase group. Moreover, the corresponding eigenspaces are invertible $\mathcal{N}$-$\mathcal{N}$-bimodules, which encode a categorification of the phase group. When $\mathcal{N}\subset \mathcal{M}$ is a finite-index irreducible subfactor of type II$_1$, we prove that any bimodule quantum channel is relatively irreducible for the intermediate subfactor of its fixed points. In addition, we can reformulate and prove these results intrinsically in subfactor planar algebras without referring to the subfactor using the methods of quantum Fourier analysis.

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arXiv2023

A new version of the Gelfand-Hille theorem

Let $\mathcal{X}$ be a complex Banach space and $A\in\mathcal{L}(\mathcal{X})$ with $\sigma(A)=\{1\}$. We prove that for a vector $x\in \mathcal{X}$, if $\|(A^{k}+A^{-k})x\|=O(k^N)$ as $k \rightarrow +\infty$ for some positive integer $N$, then $(A-\mathbf{I})^{N+1}x=0$ when $N$ is even and $(A-\mathbf{I})^{N+2}x=0$ when $N$ is odd. This could be seemed as a new version of the Gelfand-Hille theorem. As a corollary, we also obtain that for a quasinilpotent operator $Q\in\mathcal{L}(\mathcal{X})$ and a vector $x\in\mathcal{X}$, if $\|\cos(kQ)x\|=O(k^N)$ as $k \rightarrow +\infty$ for some positive integer $N$, then $Q^{N+1}x=0$ when $N$ is even and $Q^{N+2}x=0$ when $N$ is odd.

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arXiv2023

Composition operators on weighted Hardy spaces of polynomial growth

In the present paper, we study the composition operators acting on weighted Hardy spaces of polynomial growth, which are concerned with norms, spectra and (semi-)Fredholmness. Firstly, we estimate the norms of the composition operators with symbols of disk automorphisms. Secondly, we discuss the spectra of the composition operators with symbols of disk automorphisms. In particular, it is proven of that the spectrum of a composition operator with symbol of any parabolic disk automorphism is always the unit circle. Thirdly, we consider the Fredholmness of the composition operator $C_φ$ with symbol $φ$ which is an analytic self-map on the closed unit disk. We prove that $C_φ$ acting on a weighted Hardy space of polynomial growth has closed range (semi-Fredholmness) if and only if $φ$ is a finite Blaschke product. Furthermore, it is obtained that $C_φ$ is Fredholm if and only if $φ$ is a disk automorphism.

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arXiv2022

Analytic automorphism group and similar representation of analytic functions

In geometry group theory, one of the milestones is M. Gromov's polynomial growth theorem: Finitely generated groups have polynomial growth if and only if they are virtually nilpotent. Inspired by M. Gromov's work, we introduce the growth types of weighted Hardy spaces. In this paper, we focus on the weighted Hardy spaces of polynomial growth, which cover the classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and much broader. Our main results are as follows. $(1)$ We obtain the boundedness of the composition operators with symbols of analytic automorphisms of unit open disk acting on weighted Hardy spaces of polynomial growth, which implies the multiplication operator $M_z$ is similar to $M_{\varphi}$ for any analytic automorphism $\varphi$ on the unit open disk. Moreover, we obtain the boundedness of composition operators induced by analytic functions on the unit closed disk on weighted Hardy spaces of polynomial growth. $(2)$ For any Blaschke product $B$ of order $m$, $M_B$ is similar to $\bigoplus_{1}^m M_z$, which is an affirmative answer to a generalized version of a question proposed by R. Douglas in 2007. $(3)$ We also give counterexamples to show that the composition operators with symbols of analytic automorphisms of unit open disk acting on a weighted Hardy space of intermediate growth could be unbounded, which indicates the necessity of the setting of polynomial growth condition. Then, the collection of weighted Hardy spaces of polynomial growth is almost the largest class such that Douglas's question has an affirmative answer. $(4)$ Finally, we give the Jordan representation theorem and similarity classification for the analytic functions on the unit closed disk as multiplication operators on a weighted Hardy space of polynomial growth.

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arXiv2022

On the third problem of Halmos on Banach spaces

Assume that $X$ is a complex separable infinite dimensional Banach space and $\mathcal{B}(X)$ denotes the Banach algebra of all bounded linear operators from $X$ to itself. In 1970, P.R. Halmos raised ten open problems in Hilbert spaces. The third one is the following: If an intransitive operator $T$ has an inverse, is its inverse also intransitive? This question is closely related to the invariant subspace problem. Ever since Enflo's celebrated counterexample on $\ell_1$ answered the invariant subspace problem in negative, the Banach space setting of the third question of Halmos has become more interesting. In this paper, we give an affirmative answer to this problem under certain spectral conditions. As an application, we show that for an invertible operator $T$ with Dunford's Property ($C$), if $T^{-1}$ is intransitive and there exists a connected component $Ω$ of $intσ(T^{-1})^\land$ which is off the origin such that $Ω\capρ_F(T^{-1})\neq \emptyset$, then $T$ is also intransitive. In the end of the paper, we show that a sufficient and necessary condition for that there exists a bounded linear operator without non-trivial invariant subspaces on the infinite dimensional space $L_1(Ω,\sum,μ)$ (resp., $C(K)$, the space of bounded continuous functions on a complete metric space $K$) is that $(Ω,\sum,μ)$ is $σ$-finite (resp., $K$ is compact).

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arXiv2021

Cowen-Douglas operators and the third of Halmos' ten problems

Let $T$ be a bounded linear operator on a complex separable infinite dimensional Hilbert space $\mathcal{H}$. $T$ is called intransitive if it leaves invariant spaces other than 0 or the whole space $\mathcal{H}$; otherwise it is transitive. In 1970, P. R. Halmos raised ten open problems on operator theory. In the past more than 50 years, nine of Halmos' ten problems were answered, but only the third one has made little progress. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? In this paper, we establish a set of theoretical systems with the help of Cowen-Douglas operators and spectral analysis. We give an affirmative answer to this problem under certain spectral conditions, which make essential progress in the research of Halmos' third problem. As the first application, we show that for an invertible hyponormal operator $T$, if $T^{-1}$ is intransitive and int$σ(T^{-1})^{\land}$ is not connected, then $T$ is also intransitive. As the second application, we show that if $T^{-1}$ has a proper strictly cyclic invariant subspace and there exists a bounded open set $Ω$ which is a connected component of $ρ(T^{-1})$ such that $Ω\cap \mathcal{U}_0=\emptyset$, where $\mathcal{U}_0$ is the connected component of $int(σ(T^{-1})^\land)$ containing zero point, then $T$ is intransitive.

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arXiv2020

Geometric Similarity invariants of Cowen-Douglas Operators

In 1978, M. J. Cowen and R.G. Douglas introduce a class of operators (known as Cowen-Douglas class of operators) and associates a Hermitian holomorphic vector bundle to such an operator in a very influential paper. They give a complete set of unitary invariants in terms of involving the curvature and its covariant partial derivatives. At the same time they ask: can one use geometric ideas to characterize completely the similarity invariants of Cowen-Douglas operators? We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely characterize the similarity invariants for a norm dense class of Cowen-Douglas operators.

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arXiv2020

Quantum Fourier Analysis

{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^{p}$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outline several open problems.

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arXiv2019

Similarity Invariants of Essentially normal Cowen-Douglas Operators and Chern Polynomials

In this paper, we systematically study a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and prove a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

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