arXiv subjects

Junsheng Fang

Publications and source records attributed to Junsheng Fang.

At least 19 recordsLinked to original sources

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

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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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

On Diximier's averaging theorem for operators in type ${\rm II}_1$ factors

Let $\M$ be a type ${\rm II_1}$ factor and let $\tau$ be the faithful normal tracial state on $\M$. In this paper, we prove that given finite elements $X_1,\cdots X_n \in \M$, there is a finite decomposition of the identity into $N \in \NNN$ mutually orthogonal nonzero projections $E_j\in\M$, $I=\sum_{j=1}^NE_j$, such that $E_jX_iE_j=\tau(X_i) E_j$ for all $j=1,\cdots,N$ and $i=1,\cdots,n$. Equivalently, there is a unitary operator $U \in \M$ such that $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jX_iU^j=\tau(X_i)I$ for $i=1,\cdots,n$. This result is a stronger version of Dixmier's averaging theorem for type ${\rm II}_1$ factors. As the first application, we show that all elements of trace zero in a type ${\rm II}_1$ factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [10]. As the second application, we prove that any self-adjoint element in a type ${\rm II}_1$ factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [15]. As the third application, we show that if $(\mathcal{M},\tau)$ is a finite factor, $X \in \mathcal{M}$, then there exists a normal operator $N \in \mathcal{M}$ and a nilpotent operator $K$ such that $X= N+ K$. This result answers affirmatively Question 1.1 in [9].

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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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arXiv2021

On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

Let $\mathcal{M}$ be a type ${\rm II_1}$ factor and let $τ$ be the faithful normal tracial state on $\mathcal{M}$. In this paper, we prove that given an $X \in \mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N \in \mathbb{N}$ mutually orthogonal nonzero projections $E_j\in\mathcal{M}$, $I=\sum_{j=1}^NE_j$, such that $E_jXE_j=τ(X) E_j$ for all $j=1,\cdots,N$. Equivalently, there is a unitary operator $U \in \mathcal{M}$ with $U^N=I$ and $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jXU^j=τ(X)I.$ As the first application, we prove that a positive operator $A\in \mathcal{M}$ can be written as a finite sum of projections in $\mathcal{M}$ if and only if $τ(A)\geq τ(R_A)$, where $R_A$ is the range projection of $A$. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if $X\in \mathcal{M}$, $X=X^*$ and $τ(X)=0$, then there exists a nilpotent element $Z \in \mathcal{M}$ such that $X$ is the real part of $Z$. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let $X_1,\cdots,X_n\in \mathcal{M}$. Then there exist unitary operators $U_1,\cdots,U_k\in\mathcal{M}$ such that $\frac{1}{k}\sum_{i=1}^kU_i^{-1}X_jU_i=τ(X_j)I,\quad \forall 1\leq j\leq n$. This result is a stronger version of Dixmier's averaging theorem for type ${\rm II}_1$ factors.

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arXiv2020

Strong sums of projections in type ${\rm II}$ factors

Let $M$ be a type ${\rm II}$ factor and let $τ$ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type ${\rm II}_\infty$ case and normalized by $τ(I)=1$ in the type ${\rm II}_1$ case. Given $A\in M^+$, we denote by $A_+=(A-I)χ_A(1,\|A\|]$ the excess part of $A$ and by $A_-=(I-A)χ_A(0,1)$ the defect part of $A$. V. Kaftal, P. Ng and S. Zhang provided necessary and sufficient conditions for a positive operator to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal) in type ${\rm I}$ and type ${\rm III}$ factors. For type ${\rm II}$ factors, V. Kaftal, P. Ng and S. Zhang proved that $τ(A_+)\geq τ(A_-)$ is a necessary condition for an operator $A\in M^+$ which can be written as the sum of a finite or infinite collection of projections and also sufficient if the operator is "diagonalizable". In this paper, we prove that if $A\in M^+$ and $τ(A_+)\geq τ(A_-)$, then $A$ can be written as the sum of a finite or infinite collection of projections. This result answers affirmatively a question raised by V. Kaftal, P. Ng and S. Zhang.

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arXiv2018

A note on relative amenable of finite von Neumann algebras

Let $M$ be a finite von Neumann algebra (resp. a type II$_{1}$ factor) and let $N\subset M$ be a II$_{1}$ factor (resp. $N\subset M$ have an atomic part). We prove that the inclusion $N\subset M$ is amenable implies the identity map on $M$ has an approximate factorization through $M_m(\mathbb{C})\otimes N $ via trace preserving normal unital completely positive maps, which is a generalization of a result of Haagerup. We also prove two permanence properties for amenable inclusions. One is weak Haagerup property, the other is weak exactness.

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arXiv2018

A note on characterizations of relative amenability on finite von Neumann algebras

In this paper, we give another two characterizations of relative amenability on finite von Neumann algebras, one of which can be thought of as an analogue of injective operator systems. As an application, we prove a stable property of relative amenable inclusions. We prove that under certain assumptions, the inclusion $N=\int_{X} \bigoplus N_{p} d μ\subset M=\int_{X} \bigoplus M_{p} d μ$ is amenable if and only if $N_p\subset M_p$ is amenable almost everywhere.

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arXiv2018

On irreducible operators in factor von Neumann algebras

Let $\mathcal M$ be a factor von Neumann algebra with separable predual and let $T\in \mathcal M$. We call $T$ an irreducible operator (relative to $\mathcal M$) if $W^*(T)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(T)'\cap \mathcal M={\mathbb C} I$. In this note, we show that the set of irreducible operators in $\mathcal M$ is a dense $G_δ$ subset of $\mathcal M$ in the operator norm. This is a natural generalization of a theorem of Halmos.

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arXiv2018

Multidimensional Free Poisson Limits on Free Stochastic Integral Algebras

In this paper, we prove four-moment theorems for multidimensional free Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that, under mild technical conditions, a bi-indexed sequence of free stochastic integrals in free Wigner algebra or free Poisson algebra converges to a free sequence of free Poisson random variables if and only if the moments with order not greater than four of the sequence converge to the corresponding moments of the limit sequence of random variables. Similar four-moment theorems hold when the limit sequence is not free, but has a multidimensional free Poisson distribution with parameters $λ>0$ and $α=\{α_i: 0\ne α_i\in \mathbb{R}, i=1, 2, \cdots\}$.

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arXiv2017

A note on the $C$-numerical radius and the $λ$-Aluthge transform in finite factors

We prove that for any two elements $A$, $B$ in a factor $M$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical radius $ω_{C}(\cdot)$ is a weakly unitarily invariant norm on finite factors and we also prove some inequalities on the $C$-numerical radius on finite factors. As an application, we show that for an invertible operator $T$ in a finite factor $M$, $f(\bigtriangleup_λ(T))$ is in the weak operator closure of the set $\{\sum_{i=1}^{n}z_{i}U_{i}f(T)U_{i}^{*}|n\in\mathbb{N},(U_{i})_{1\leq i\leq n}\in \mathscr{U}(M),\sum_{i=1}^{n}|z_{i}|\leq 1\}$, where $f$ is a polynomial, $\bigtriangleup_λ(T)$ is the $λ$-Aluthge transform of $T$ and $0\leqλ\leq 1$.

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arXiv2016

Mixing and weakly mixing abelian subalgebras of type II$_1$ factors

This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which are absolutely continuous but not Lebesgue. We also show that there exist uncountably many pairwise non--conjugate mixing masas in the free group factors each with Pukánszky invariant $\{1,\infty\}$.

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arXiv2014

On a class of operators in the hyperfinite ${\rm II}_1$ factor

Let $R$ be the hyperfinite ${\rm II}_1$ factor and let $u,v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2πiθ} uv$ for an irrational number $θ$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.

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arXiv2013

Upper triangular Toeplitz matrices and real parts of quasinilpotent operators

We show that every self--adjoint matrix B of trace 0 can be realized as B=T+T^* for a nilpotent matrix T of norm no greater than K times the norm of B, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self--adjoint n-by-n matrix of trace 0, then there is a unitary matrix V=XU_n, where X is an n-by-n permutation matrix and U_n is the n-by-n Fourier matrix, such that the upper triangular part, T, of the conjugate V^*DV of D has norm no greater than K times the norm of D. This matrix T is a strictly upper triangular Toeplitz matrix such that T+T^*=V^*DV. We apply this and related results to give partial answers to questions about real parts of quasinilpotent elements in finite von Neumann algebras.

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