论文标题
Hilbert C*-Modules和确切常数的非交换性$ \ ell_1- \ ell_2 $
The noncommutative $\ell_1-\ell_2$ inequality for Hilbert C*-modules and the exact constant
论文作者
论文摘要
令$ \ mathcal {a} $为Unital C*-ergebra。然后，希尔伯特C*-modules的理论告诉\ begin {align*} \ sum_ {i = 1}^{n}（a_ia_i^*） \ left（\ sum_ {i = 1}^{n} a_ia_i^*\ right）^\ frac {1} {2}，\ quad \ forall n \ in \ mathbb {n}，\ forall a_1，a_1，\ in a_n \ in \ in \ in \ in \ in \ mathcal}。通过修改Botelho-Andrade，Casazza，Cheng和Tran在2019年给出的\ End {Align*}，对于某些元组$ x =（a_1，\ dots，a_n）\ in \ nathcal {a} a}^n $，我们给出了一种积极的元素$ c_x $ c*-al c*-al c*-al平等 \ begin {Align*} \ sum_ {i = 1}^{n}（a_ia_i^*） \ end {align*}保持。我们为G. G. Kasparov的积分提供了申请。我们还得出了连续$ \ ell_1- \ ell_2 $不等式的确切常数的公式。
Let $\mathcal{A}$ be a unital C*-algebra. Then the theory of Hilbert C*-modules tells that \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\leq \sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}, \quad \forall n \in \mathbb{N}, \forall a_1, \dots, a_n \in \mathcal{A}. \end{align*} By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain tuple $x=(a_1, \dots, a_n) \in \mathcal{A}^n$, we give a method to compute a positive element $c_x$ in the C*-algebra $\mathcal{A}$ such that the equality \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=c_x \sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}. \end{align*} holds. We give an application for the integral of G. G. Kasparov. We also derive the formula for the exact constant for the continuous $\ell_1-\ell_2$ inequality.
