Abstract
For an element a in a unital C *-algebraA, let V(a) and v(a) denote the numerical range and the numerical radius of a, respectively. First, we show that a is a unitary central element inAif and only if v(az) = v(z) for all normal elements z. A. We then use this result to show that if a and b are two elements in A with a b = 1, then v(azb) = v(z) for all normal elements z. A if and only if ba is a unitary central element in A and b =.baa * for some positive scalar.. If, however, the numerical radius is replaced by the numerical range, we show that such a condition is redundant and prove that V(azb) = V(z) for all normal elements z. A if and only if a is invertible such that a * a is a central element in A and b = a -1. Furthermore, we show that azb = z for all invertible elements z. A if and only if | a| is in the centre of A and | a|| b *| is a unitary element.