论文标题
C* - 代数Schoenberg猜想
C*-algebraic Schoenberg Conjecture
论文作者
论文摘要
基于Schoenberg猜想\ textit {[Amer。数学。每月。，1986]}/malamud-pereira theorem \ textit {[j。数学。肛门。 Appl，2003]}，\ textIt {[trans。阿米尔。数学。 Soc。，2005]}我们制定了以下猜想，我们称之为C*-Algebraic Schoenberg猜想。\\\\\\\\\\\ textbf {C*-algebraic Schoenberg猜想：让$ \ Mathcal {a} $ Be a C*-algebra。令$ d \ in \ Mathbb {n} \ setMinus \ {1 \} $，$ p（z）=（z-a_1）（z-a_1）（z-a_2）\ cdots（z-a_d）$是$ \ \ m nathcal {a} $ a_1，a_1，a_1，a_2，a_2，a_d a_d a_d a_ d a y_d a a_d a a_ d a a ____________如果$ p'$可以写为$ p'（z）= d（z-b_1）（z-b_2）\ cdots（z-b_ {d-1}）$ on $ \ m varycal {a} $，带有$ b_1，b_1，b_2，\ dots，b_2，\ dots，b_ {d-1} \ sum_ {k = 1}^{d-1} b_kb_k^*\ leq \ frac {1} {d^2} \ left [\ sum_ {j = 1}^{d}^{d} a_j \ right] \ frac {d-2} {d} \ sum_ {j = 1}^{d} a_ja_j^*\ end {align*}和\ begin {align*} \ sum_ {k = 1} \ frac {1} {d^2} \ left [\ sum_ {j = 1}^{d} a_j \ right]^*\ left [\ sum_ {j = 1}^{d}^{d} a_j \ \ right] \ end {align*}}我们表明，C*-Algebraic Schoenberg的猜想在C*-algebras上持有2 C*-Algebraic多项式。
Based on Schoenberg conjecture \textit{[Amer. Math. Monthly., 1986]}/Malamud-Pereira theorem \textit{[J. Math. Anal. Appl, 2003]}, \textit{[Trans. Amer. Math. Soc., 2005]} we formulate the following conjecture which we call C*-algebraic Schoenberg Conjecture.\\ \textbf{ C*-algebraic Schoenberg Conjecture : Let $\mathcal{A}$ be a C*-algebra. Let $d\in \mathbb{N}\setminus\{1\}$, $P(z)= (z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1, a_2, \dots, a_d \in \mathcal{A} $. If $P'$ can be written as $P'(z)= d(z-b_1)(z-b_2)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1, b_2, \dots, b_{d-1} \in \mathcal{A} $, then \begin{align*} \sum_{k=1}^{d-1}b_kb_k^*\leq \frac{1}{d^2}\left[\sum_{j=1}^{d}a_j\right]\left[\sum_{j=1}^{d}a_j\right]^*+ \frac{d-2}{d}\sum_{j=1}^{d}a_ja_j^* \end{align*} and \begin{align*} \sum_{k=1}^{d-1}b_k^*b_k\leq \frac{1}{d^2}\left[\sum_{j=1}^{d}a_j\right]^*\left[\sum_{j=1}^{d}a_j\right]+ \frac{d-2}{d}\sum_{j=1}^{d}a_j^*a_j. \end{align*}} We show that C*-algebraic Schoenberg conjecture holds for degree 2 C*-algebraic polynomials over C*-algebras.
