a-roberts orthogonality in c*-algebras and its characterization via a-numerical ranges

‎let $ mathcal{a} $ be a unital $ c^{*} $-algebra with unit $1_{mathcal{a}}$ and let $ ainmathcal{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 0}‎, ‎]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 (xin mathcal{a}),]‎ ‎in $c^*$-algebra $mathcal{a}$‎, ‎so called‎, ‎$a$-roberts orthogonality‎.‎more precisely‎, ‎for any $a$-isometry $xinmathcal{a}$‎, ‎we prove that $x$ is $a$-roberts orthogonal to $1_{mathcal{a}}$ if and only if algebraic $a$-numerical range of $x$ is symmetric with respect to the origin.