arXiv subjects

Chunlan Jiang

Publications and source records attributed to Chunlan Jiang.

34 records · Page 2Linked to original sources

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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arXiv2019

A classification of inductive limit $C^{*}$-algebras with ideal property

Let $A$ be an $AH$ algebra $A=\lim\limits_{n\to \infty}(A_{n}=\bigoplus\limits_{i=1}\limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m})$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth---that is, $sup_{n,i}dim(X_{n,i})<+\infty$. This result generalizes and unifies the classification of $AH$ algebras of real rank zero in [EG] and [DG] and the classification of simple $AH$ algebras in [G5] and [EGL1]. This completes one of two important possible generalizations of [EGL1] suggested in the introduction of [EGL1]. The invariants for the classification include the scaled ordered total $K$-group $(\underline{K}(A), \underline{K}(A)_{+},ΣA)$ (as already used in real rank zero case in [DG]), for each $[p]\inΣA$, the tracial state space $T(pAp)$ of cut down algebra $pAp$ with a certain compatibility, (which is used by [Stev] and [Ji-Jiang] for $AI$ algebras with the ideal property), and a new ingredient, the invariant $U(pAp)/\overline{DU(pAp)}$ with a certain compatibility condition, where $\overline{DU(pAp)}$ is the closure of commutator subgroup $DU(pAp)$ of the unitary group $U(pAp)$ of the cut down algebra $pAp$. In [GJL] a counterexample is presented to show that this new ingredient must be included in the invariant. The discovery of this new invariant is analogous to that of the order structure on the total K-theory when one advances from the classification of simple real rank zero $C^*$-algebras to that of non simple real rank zero $C^*$-algebras in [G2], [Ei], [DL] and [DG] (see Introduction below).

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arXiv2019

On the Decomposition Theorems for C*-algebras

Elliott dimension drop interval algebra is an important class among all $C^*$-algebras in the classification theory. Especially, they are building stones of $\mathcal{AHD}$ algebra and the latter contains all $AH$ algebras with the ideal property of no dimension growth. In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra. Our results are key steps in classifying all $AH$ algebras with the ideal property of no dimension growth.

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arXiv2019

On the inductive limit of direct sums of simple TAI algebras

An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},ϕ_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).

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arXiv2019

Hausdorffifized algebraic $K_1$ group and invariants for $C^*$-algebras with the ideal property

A $C^*$-algebra $A$ is said to have the ideal property if each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. $C^*$-algebras with the ideal property are generalization and unification of real rank zero $C^*$-algebras and unital simple $C^*$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see the introduction), consisting of scaled ordered total $K$-group $(\underline{K}(A), \underline{K}(A)^{+},ΣA)_Λ$ (used in the real rank zero case), the tracial state space $T(pAp)$ of cutting down algebra $pAp$ as part of Elliott invariant of $pAp$ (for each $[p]\inΣA$) with a certain compatibility, is the complete invariant for certain well behaved class of $C^*$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we will construct two non isomorphic $A\mathbb{T}$ algebras $A$ and $B$ with the ideal property such that $Inv^0(A)\cong Inv^0(B)$. The invariant to differentiate the two algebras is the Hausdorffifized algebraic $K_1$-groups $U(pAp)/\overline{DU(pAp)}$ (for each $[p]\inΣA$) with a certain compatibility condition. It will be proved in [GJL] that, adding this new ingredients, the invariant will become the complete invariant for $AH$ algebras (of no dimension growth) with the ideal property.

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arXiv2017

A Reduction theorem for $AH$ algebras with ideal property

Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{ϕ_{1,2}}A_{2}\xrightarrow{ϕ_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ with $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that $\sup_{n,i}dim(X_{n,i})<+\infty$. In this article, we prove that $A$ can be written as the inductive limit of $$B_{1}\longrightarrow B_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$ where $B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ are $\{pt\}$, $[0,1]$,$ S^{1}$,$ T_{II, k},$ $T_{III, k}$ and $S^{2}$ (all of them are connected simplicial complexes of dimension at most three), $s_{n}$ and $\{n,i\}$ are positive integers and $Q_{n,i}\in M_{\{n,i\}}(C(Y_{n,i}))$ are projections. This theorem unifies and generalizes the reduction theorem for real rank zero $AH$ algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple $AH$ algebras due to Gong (see [G4]).

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arXiv2017

Reduction to dimension two of local spectrum for $AH$ algebra with ideal property

A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.

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arXiv2014

Noncommutative Uncertainty Principles

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's $λ$-lattices, modular tensor categories etc.

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arXiv2014

Curvature and the Second fundamental form in classifying quasi-homogeneous holomorphic curves and operators in the Cowen-Douglas class

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen-Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen-Douglas class obtained recently by the last author and A. Korányi. Specifically, the complex geometric invariants used for our classification are the curvature and the second fundamental forms inherent in the definition of a quasi-homogeneous operator. We show that these operators are irreducible and determine when they are strongly irreducible. Applications include the equality of the topological and algebraic K-group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos on similarity for these operators.

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arXiv2014

Rigidity of the flag structure for a class of Cowen-Douglas operators

The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid, that is, the unitary equivalence class of the operator and the flag structure determine each other. A complete set of unitary invariants, which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class, are obtained.

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arXiv2014

Flag structure for operators in the Cowen-Douglas class

The explicit description of homogeneous operators and localization of a Hilbert module naturally leads to the definition of a class of Cowen-Douglas operators possessing a flag structure. These operators are irreducible. We show that the flag structure is rigid in the sense that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.

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arXiv2012

Direct integrals of strongly irreducible operators

Strongly irreducible operators can be considered as building blocks for bounded linear operators on complex separable Hilbert spaces. Many bounded linear operators can be written as direct sums of at most countably many strongly irreducible operators. In this paper, we show that a bounded linear operator A is similar to a direct integral of strongly irreducible operators if its commutant contains a bounded maximal abelian set of idempotents. We find that bounded linear operators which are similar to direct integrals of strongly irreducible operators form a dense subset of L(H) in the operator norm.

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arXiv2012

A similarity invariant of a class of n-normal operators in terms of K-theory

In this paper, we prove an analogue of the Jordan canonical form theorem for a class of $n$-normal operators on complex separable Hilbert spaces in terms of von Neumann's reduction theory. This is a continuation of our study of bounded linear operators, the commutants of which contain bounded maximal abelian set of idempotents. Furthermore, we give a complete similarity invariant for this class of operators by $K$-theory for Banach algebras.

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arXiv2012

On generalized universal irrational rotation algebras and the operator $u+v$

We introduce a class of generalized universal irrational rotation $C^*$-algebras $A_{θ,γ}=C^*(x,w)$ which is characterized by the relations $w^*w=ww^*=1$, $x^*x=γ(w)$, $xx^*=γ(e^{-2πiθ}w)$, and $xw=e^{-2πiθ}wx$, where $θ$ is an irrational number and $γ(z)\in C(\mathbb{T})$ is a positive function. We characterize tracial linear functionals, simplicity, and $K$-groups of $A_{θ,γ}$ in terms of zero points of $γ(z)$. We show that if $A_{θ,γ}$ is simple then $A_{θ,γ}$ is an $A{\mathbb T}$-algebra of real rank zero. We classify $A_{θ,γ}$ in terms of $θ$ and zero points of $γ(z)$. Let $A_θ=C^*(u,v)$ be the universal irrational rotation $C^*$-algebra with $vu=e^{2πiθ}uv$. Then $C^*(u+v)\cong A_{θ,|1+z|^2}$. As an application, we show that $C^*(u+v)$ is a proper simple $C^*$-subalgebra of $A_θ$ which has a unique trace, $K_1(C^*(u+v))\cong \mathbb{Z}$, and there is an order isomorphism of $K_0(C^*(u+v))$ onto $\mathbb{Z}+\mathbb{Z}θ$. {Moreover, $C^*(u+v)$ is a unital simple $A{\mathbb T}$-algebra of real rank zero.} We also calculate the spectrum and the Brown measure of $u+v$.

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