A Note on Values of Noncommutative Polynomials

Extensions of S-Lemma for Noncommutative Polynomials

We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix is positive semidefinite. Then we extend the S-lemma to three kinds of noncommutative polynomials: noncommutative polynomials whose coefficients are real numbers, matrix-valued noncommutative polynomials and hereditary polynomials.

Notes on noncommutative geometry

The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises is attached. Our notes are intended for the graduate students and faculty with interests in noncommutative geometry; they can be read by non-experts in the field.

关于通过非交换图研究稳定子形式体系的一则注记

A note on the stabilizer formalism via noncommutative graphs

本文利用紧群酉表示与有限维 Hilbert 空间算子构造非交换图，以此重新表述稳定子形式体系，并推广了非交换图反团存在性的判定结果。

Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions

We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial…

多项式的非交换Cauchy界和非交换Montel界

Noncommutative Cauchy Bound and Noncommutative Montel Bound for Roots of Polynomials

本文将复数多项式根的Cauchy界和Montel界推广到非交换多项式，利用系数范数给出算子根的上界。

Noncommutative Plurisubharmonic Polynomials Part II: Local Assumptions

We say that a symmetric noncommutative polynomial in the noncommutative free variables (x_1, x_2, ..., x_g) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if a noncommutative polynomial is noncommutative plurisubharmonic on a noncommutative open set, then the polynomial is actually noncommutative plurisubharmonic everywhere and has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are f…

关于非交换CW-谱的一个注记

A note on noncommutative CW-spectra

本文利用arXiv:2009.07539的机制，基于稳定模型范畴的著名结果，为非交换CW-谱范畴可建模为特定范畴M上的谱预层范畴这一结论提供了替代证明。

The A-polynomial from the noncommutative viewpoint

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.

On trace-convex noncommutative polynomials

To each real continuous function f there is an associated trace function on real symmetric matrices Tr f. The classical Klein lemma states that f is convex if and only if Tr f is convex. In this note we present an algebraic strengthening of this lemma for univariate polynomials f: Tr f is convex if and only if the noncommutative second directional derivative of f is a sum of hermitian squares and commutators in a free algebra. We also give a localized version of this result.

One noncommutative differential calculus coming from the inner derivation

We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain $C^*$-algebra, this calculus is closely related to the classical one when the algebra associates a deformation parameter.

关于非交换遍历定理的注记

Notes on noncommutative ergodic theorems

本文证明了半有限von Neumann代数中算子网在多种收敛下的完备性，并将逐点Cauchy性质扩展至完全对称空间，为非交换遍历理论提供了应用基础。

A Note on Quantum Odometers

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz Algebras.

Noncommutative Semialgebraic sets and Associated Lifting Problems

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen's discovery of the norm adjusting power of quasi-central approximate units. A projective C*-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show a noncommutative absolute retract results from the intersection of the approximate locus of a homogene…

A note on noncommutative unique ergodicity and weighted means

In this paper we study unique ergodicity of $C^*$-dynamical system $(\ga,T)$, consisting of a unital $C^*$-algebra $\ga$ and a Markov operator $T:\ga\mapsto\ga$, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that $(\ga,T)$ is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means {equation*} \frac{1}{p_1+...+p_n}\sum_{k=1}^{n}p_kT^kx {equation*} converge to $E_T(x)$ in $\ga$ for any $x\in\ga$, as $n\to\infty$, here $E_T$ is an projection of $\ga$ to the fixed point subspace of $T$. It is als…

A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere

The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.

Hölder Continuity of Cumulative Distribution Functions for Noncommutative Polynomials under Finite Free Fisher Information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of t…

Lectures on the classical moment problem and its noncommutative generalization

These notes contain a presentation of the noncommutative generalization of the classical moment problem introduced in [10] and [12]. They also contain a short summary of the classical moment problem in infinite dimension.

Noncommutative geometry as a functor

In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.

关于抽象Hardy空间插值中Pisier方法的注记

A note on Pisier's method in interpolation of abstract Hardy spaces

本文在更一般背景下提出并证明了Pisier方法的抽象版本，应用于非交换鞅变换研究。

正算子值非交换多项式是平方

Positive operator-valued noncommutative polynomials are squares

本文证明了正算子值非交换多项式可分解为单平方形式，并通过构造算子元和利用拓扑方法证明sos锥的闭合性。

非交换环面上的Schatten类：核条件

Schatten classes on noncommutative tori: Kernel conditions

本文研究了量子环面上积分算子的Schatten类隶属问题，通过引入非交换积分核准则并结合非交换Schwartz核定理，建立了判定有界算子Schatten类性质的具体测试方法。

在自由代数的张量积中正性是不可判定的

Positivity is undecidable in tensor products of free algebras

研究指出，在自由代数的张量积中，判断元素正性是coRE难问题，且非交换*多项式正性也具有类似复杂度。

行球上的非交换 Szegő 型定理

A Noncommutative Szegő-Type Theorem on the Row-Ball

本文借助非交换测度与 Schur 参数理论，建立 nc 测度和多 Toeplitz 核的对应，证明行球上的非交换 Szegő 型定理及正交多项式零点定理。

多项式遍历平均的非交换极大不等式

Noncommutative maximal inequalities for polynomial ergodic averages

本文证明了沿多项式序列的非交换极大遍历不等式，扩展了p的范围，通过Bourgain主弧策略及非交换Stein外插、组合方法和多线性Doob极大不等式等新工具实现。

全局迹正定非交换多项式与无界迹矩问题

Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem

本文首次建立了非交换多项式全局迹正定性的 Positivstellensatz，通过求解无界迹矩问题并结合半定优化，证明迹正定 nc 多项式可由 Hermitian 平方和与交换子显式逼近。