A-Roberts orthogonality in C*-algebras and its characterization via a-numerical ranges

‎Let \mathcal{A} be a unital C^{*} -algebra with unit ۱_{\mathcal{A}} and let a\in\mathcal{A} be a positive and invertible element‎. ‎Set \[ \mathcal{S}_a (\mathcal{A})=\{ \dfrac{f}{f(a)} \‎, : ‎\‎, ‎f \in \mathcal{S}(\mathcal{A})‎, ‎\‎, ‎f(a)\neq ۰\}‎, ‎\]where \mathcal{S}(\mathcal{A}) is the set of all states on \mathcal{A} ‎. ‎In this paper‎, ‎by using the concept of algebraic a-Davies-Wielandt shell of elements of \mathcal{A}‎, ‎we obtain a characterization of Roberts orthogonality with respect to the norm‎:‎\[ \|x\|_a = \sup_{\varphi \in \mathcal{S}_a(\mathcal{A})} \sqrt{\varphi(x^* ax)}\quad (x\in \mathcal{A}),\]‎ ‎in C^*-algebra \mathcal{A}‎, ‎so called‎, ‎a-Roberts orthogonality‎.‎More precisely‎, ‎for any a-isometry x\in\mathcal{A}‎, ‎we prove that x is a-Roberts orthogonal to ۱_{\mathcal{A}} if and only if algebraic a-numerical range of x is symmetric with respect to the origin.

کلیدواژه ها:

C*-algebra‎ ، ‎state space of C*-algebras‎ ، ‎a-Birkhoff-James orthogonality‎ ، ‎a-Roberts orthogonality‎ ، ‎algebraic a-numerical range