ABSTRACT

Let $\mathcal{A}$ be a C*-algebra with unit I and let $\mathcal{S}(\mathcal{A})$ be the state space of $\mathcal{A}$. Let $A\in \mathcal{A}$, an element $f\in \mathcal{S}(\mathcal{A})$ is said to be maximal for A if f(A*A) = ‖A‖². Denote the set of all maximal states for A by ${\mathcal{S}}_{max}(A)$ and define the algebraic maximal numerical range of A as follows $V_0(A):=\{f(A):f\in {\mathcal{S}}_{max}(A)\}.$ We say that A is orthogonal to I in the Birkhoff–James sense, written as A⊥_BJ I, whenever ‖A‖ ≤ ‖A − λ‖, for all complex numbers λ. In this paper, we give a characterization of A⊥_BJ I in terms of the algebraic maximal numerical range V₀(A). As applications, we give new numerical radius inequalities, generalize and improve earlier well-known results.

摘要

让$\mathcal{A}$是具有单位I的C *-代数并让$\mathcal{小号}(\mathcal{A})$是状态空间$\mathcal{A}$. 让$A\in \mathcal{A}$, 一个元素$F\in \mathcal{小号}(\mathcal{A})$如果f ( A * A ) = ‖ A ‖ ²，则被称为A的最大值。将A的所有最大状态的集合表示为${\mathcal{小号}}_{最大限度}(A)$并定义A的代数最大数值范围如下$V_0(A):=\{F(A):F\in {\mathcal{小号}}_{最大限度}(A)\}.$我们说A在 Birkhoff–James 意义上与I正交，写为A ⊥ _BJ I，只要 ‖ A ‖ ≤ ‖ A  −  λ ‖，对于所有复数λ。在本文中，我们根据代数最大数值范围V ₀ ( A )给出了A ⊥ _BJ I的表征。作为应用，我们给出了新的数值半径不等式，推广和改进了早期众所周知的结果。