Abstract
Let U be a unital C*-algebra with the unit 1 and S(U) be the set of all states on U. For a positive element a of U, let parallel to x parallel to(a), v(a)(x), and V-a(x) denote the a-operator semi-norm, the a-numerical radius, and the a-numerical range of x is an element of U, respectively. Let U-a = {x is an element of U : parallel to x parallel to(a) < infinity} and S-a(U) = {f/f(a) : f is an element of S(U), f (a) not equal 0}. Let also U-a be the set of all elements in U that admit a-adjoints. In this paper, we first characterize an element x is an element of U for which vertical bar f(ax)vertical bar = f (a) for every pure state f. Next, we prove that if a linear map f : aU(a) -> C satisfies f (a) = 1 and vertical bar f (ay)vertical bar <= parallel to y parallel to(a) for all y is an element of U-a, then there exists g is an element of S-a(U) such that f (ay) = g(ay) for all y is an element of U-a. We also show that v(a)(x) = parallel to x parallel to(a) if and only if there is f is an element of S-a (U) such that root f (x* ax) = parallel to x parallel to(a) and vertical bar f(ax)parallel to = v(a)(x). In addition, we prove that parallel to x parallel to(a) < infinity if and only if v(a)(x) < infinity. Finally, we show that V-a (x) = boolean AND is an element of C D(zeta, v(a) (x - zeta 1)).