Abstract
Let U be a unital C*-algebra and U' be its topological dual space. Let a be a positive element in U, and set f(a)(U) := {f is an element of U' : f >= 0, f(a) = 1 . The a-numerical range and a-numerical radius of any element x is an element of U are defined by
v(a)(x) := {f (ax) : f is an element of fa(U)},
and
v(a)(x) := sup (vertical bar z vertical bar : Z is an element of V-a(X)),
respectively. In this paper, we establish some permanence properties of the a-numerical range and a-numerical radius of elements in U. In particular, we investigate when the a-numerical range of an element of U is closed, and provide explicit formulas for the a-numerical radius of the so-called a-hermitian elements of U Furthermore, given a positive operator A on a complex Hilbert space H. we study and investigate the relationship between the algebraic and spatial A-numerical ranges of bounded linear operators on H.