Improved Bounds for numerical radius and a-numerical radius in {C}^*-algebras

In this article, we derive several significant upper bounds for the numerical radius and $a$-numerical radius of an element in a ${C}^*$-algebra by improving inequalities for positive linear functionals. Our findings refine and generalize the existing inequalities. Furthermore, we introduce a new notion to derive improved upper bounds of the numerical radius for an element in a ${C}^*$-algebra using the Moore-Penrose inverse.